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Introduction to Path Integrals in Field Theory U. Mosel1 Institut fuer Theoretische Physik, Universitaet Giessen D-35392 Giessen, Germany SS 02 July 6, 2002

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http://theorie.physik.uni-giessen.de

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This manuscript on path integrals is based on lectures I have given at the University of Giessen. If you find any conceptual or typographical errors I would like to learn about them. In this case please send an e-mail to [email protected]

Contents I

Non-Relativistic Quantum Theory

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1 PATH INTEGRAL IN QUANTUM THEORY 1.1 Propagator of the Schrödinger Equation . . . . 1.2 Propagator as Path Integral . . . . . . . . . . . 1.3 Quadratic Hamiltonians . . . . . . . . . . . . . 1.3.1 Cartesian metric . . . . . . . . . . . . . 1.3.2 Non-Cartesian metric . . . . . . . . . . . 1.4 Classical Interpretation . . . . . . . . . . . . . .

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7 7 10 13 14 15 17

2 PERTURBATION THEORY 20 2.1 Free propagator . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Perturbative Expansion . . . . . . . . . . . . . . . . . . . . . . 22 2.3 Application to Scattering . . . . . . . . . . . . . . . . . . . . . 27 3 GENERATING FUNCTIONALS 3.1 Groundstate-to-Groundstate Transitions . . . . . . . . . . . . 3.1.1 Generating functional. . . . . . . . . . . . . . . . . . . 3.2 Functional Derivatives of Transition Amplitudes . . . . . . . . . . . . . . . . . . . . .

32 33 37

II

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Relativistic Quantum Field Theory

4 CLASSICAL RELATIVISTIC FIELDS 4.1 Equations of Motion . . . . . . . . . . . . . 4.1.1 Examples . . . . . . . . . . . . . . . 4.2 Symmetries and Conservation Laws . . . . . 4.2.1 Geometrical Space–Time Symmetries 4.2.2 Internal Symmetries . . . . . . . . .

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44 44 46 50 51 53

CONTENTS

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5 PATH INTEGRALS FOR SCALAR FIELDS 56 5.1 Generating Functional for Fields . . . . . . . . . . . . . . . . . 57 5.1.1 Euclidean Representation . . . . . . . . . . . . . . . . 59 6 EVALUATION OF PATH INTEGRALS 6.1 Free Scalar Fields . . . . . . . . . . . . . 6.1.1 Generating functional . . . . . . . 6.1.2 Feynman propagator . . . . . . . 6.1.3 Gaussian Integration . . . . . . . 6.2 Stationary Phase Approximation . . . . 6.3 Numerical Evaluation of Path Integrals . 6.3.1 Imaginary time method . . . . . 6.3.2 Real time formalism . . . . . . .

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62 62 62 64 68 71 74 74 75

7 S-MATRIX AND GREEN’S FUNCTIONS 7.1 Scattering Matrix . . . . . . . . . . . . . . . 7.2 Reduction Theorem . . . . . . . . . . . . . . 7.2.1 Canonical field quantization . . . . . 7.2.2 Derivation of the reduction theorem .

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78 78 80 80 82

8 GREEN’S FUNCTIONS 8.1 n-point Green’s Functions . . . . 8.1.1 Momentum representation 8.1.2 Operator Representations 8.2 Free Scalar Fields . . . . . . . . . 8.2.1 Wick’s theorem . . . . . . 8.2.2 Feynman rules . . . . . . . 8.3 Interacting Scalar Fields . . . . . 8.3.1 Perturbative expansion . .

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87 87 88 89 91 91 92 94 96

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99 99 101 102 103 103 104 107 109 109 110

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9 PERTURBATIVE φ4 THEORY 9.1 Perturbative Expansion of the Generating Function 9.1.1 Feynman rules . . . . . . . . . . . . . . . . . 9.1.2 Vacuum contributions . . . . . . . . . . . . 9.2 Two-Point Function . . . . . . . . . . . . . . . . . . 9.2.1 Terms up to O(g 0 ) . . . . . . . . . . . . . . 9.2.2 Terms up to O(g) . . . . . . . . . . . . . . . 9.2.3 Terms up to O(g 2 ) . . . . . . . . . . . . . . 9.3 Four-Point Function . . . . . . . . . . . . . . . . . 9.3.1 Terms up to O(g) . . . . . . . . . . . . . . . 9.3.2 Terms up to O(g 2 ) . . . . . . . . . . . . . .

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CONTENTS

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10 DIVERGENCES IN n-POINT FUNCTIONS 10.1 Power Counting . . . . . . . . . . . . . . . . . 10.2 Dimensional Regularization of φ4 Theory . . . . . . . . . . . . . . . . . . . 10.2.1 Two-point function . . . . . . . . . . . 10.2.2 Four-point function . . . . . . . . . . . 10.3 Renormalization . . . . . . . . . . . . . . . . . 10.3.1 Renormalization of φ4 Theory . . . . .

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116 117 118 122 125

11 GREEN’S FUNCTIONS FOR FERMIONS 11.1 Grassmann Algebra . . . . . . . . . . . . . . 11.1.1 Derivatives . . . . . . . . . . . . . . 11.1.2 Integration . . . . . . . . . . . . . . 11.2 Green’s Functions for Fermions . . . . . . . 11.2.1 Generating Functional for fermions . 11.2.2 Green’s Functions . . . . . . . . . . .

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128 128 130 132 138 138 141

12 INTERACTING FIELDS 12.1 Feynman Rules . . . . . . . . . 12.1.1 Fermion Loops . . . . . 12.2 Wick’s Theorem . . . . . . . . . 12.3 Removal of Degrees of Freedom: Yukawa Theory . . . . . . . . . 12.3.1 Perturbative Expansion

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113 . . . . . . . . . 114

144 . . . . . . . . . . . . . . . . . 144 . . . . . . . . . . . . . . . . . 145 . . . . . . . . . . . . . . . . . 148 . . . . . . . . . . . . . . . . . 150 . . . . . . . . . . . . . . . . . 152

13 PATH INTEGRALS FOR GAUGE FIELDS 13.1 Gauge invariance in Abelian theories . . . . . 13.2 Non-abelian gauge fields . . . . . . . . . . . . 13.3 Path integrals . . . . . . . . . . . . . . . . . . 13.3.1 Gauge Fixing . . . . . . . . . . . . . . 13.4 Feynman Rules . . . . . . . . . . . . . . . . . 13.4.1 Faddeev-Popov Determinant . . . . . . 13.4.2 Ghost fields . . . . . . . . . . . . . . . 13.4.3 Feynman Rules . . . . . . . . . . . . .

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157 . 158 . 162 . 164 . 169 . 171 . 171 . 175 . 176

14 EXAMPLES FOR GAUGE FIELD THEORIES 14.1 Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . 14.2 Quantum Chromodynamics . . . . . . . . . . . . . . . . . . 14.3 Electroweak Interactions . . . . . . . . . . . . . . . . . . . .

184 . 184 . 184 . 186

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CONTENTS

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A Units and Metric 189 A.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 A.2 Metric and Notation . . . . . . . . . . . . . . . . . . . . . . . 190 B Functionals B.1 Definition . . . . . . . . . B.2 Functional Integration . . B.2.1 Gaussian integrals B.3 Functional Derivatives . .

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192 . 192 . 193 . 193 . 196

C RENORMALIZATION INTEGRALS

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D GRASSMANN INTEGRATION FORMULA

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E BIBLIOGRAPHY

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Part I Non-Relativistic Quantum Theory

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Chapter 1 PATH INTEGRAL IN QUANTUM THEORY In this starting chapter we introduce the concepts of propagators and path integrals in the framework of nonrelativistic quantum theory. In all these considerations, and the following chapters on nonrelativistic quantum theory, we work with one coordinate only, but all the results can be easily generalized to the case of d dimensions.

1.1

Propagator of the Schr¨ odinger Equation

We start by considering a nonrelativistic particle in a one-dimensional potential V (x). The Schrödinger equation reads Hψ(x, t) = −

∂ψ(x, t) h ¯ 2 ∂ 2 ψ(x, t) + V (x)ψ(x, t) = i¯ h . 2m ∂x2 ∂t

(1.1)

This equation allows us to calculate the wavefunction ψ(x, t) at a later time, if we know ψ(x, t0 ) at the earlier time t0 < t. For further calculations we rewrite this equation into the following form

∂ i¯ h − H ψ(x, t) = 0 . ∂t

(1.2)

Next, we consider the function K (x, t; xi , ti ) which is defined as solution of the equation

∂ i¯ h − H K (x, t; xi , ti ) = i¯ hδ(x − xi )δ(t − ti ) . ∂t 7

(1.3)

CHAPTER 1. PATH INTEGRAL IN QUANTUM THEORY

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K is the “Green’s function” of the Schrödinger equation (K is also often called “propagator”) with the initial condition K(x, ti + 0; xi , ti ) = δ(x − xi ) .

(1.4)

The solution of the Schrödinger equation (1.2) can be written as

ψ(x, t) =

K (x, t; xi , ti ) ψ(xi , ti ) dxi

(1.5)

for t > ti (Huygen’s principle). Relation (1.5) can be proven by inserting the lhs into the Schrödinger equation

∂ i¯ h −H K (x, t; xi , ti ) ψ(xi , t) dxi ∂t

= i¯ h

δ (t − ti ) δ (x − xi ) ψ (xi , t) dxi

= i¯ hδ (t − ti ) ψ(x, t) = 0

for t > ti .

(1.6)

Thus the ψ defined by (1.5) is indeed a solution of the Schrödinger equation for all times t > ti . K (x, t; xi , ti ) is the probability amplitude for a transition from xi , at time ti , to the position x, at the later time t. The restriction to later times preserves causality. We can find an explicit form for the propagator, if the solutions of the stationary Schrödinger equation, ϕn (x), and the corresponding eigenvalues, En , are known. Since the ϕn form a complete system, K can certainly be expanded in this basis (for t ≥ ti ) K (x, t; xi , ti ) =

an ϕn (x)e− h¯ En t Θ (t − ti ) . i

(1.7)

n

Here the stepfunction Θ(t) = 0 for t < 0 and Θ(t) = 1 for t ≥ 0 takes explicitly into account that we propagate the wavefunction only forward in time. The expansion coefficients obviously depend on xi , ti an = an (xi , ti ) .

(1.8)

Because of the initial condition K(x, ti + 0; xi , ti ) = δ (x − xi ) we have δ (x − xi ) =

i

an (xi , ti )ϕn (x)e− h¯ En ti .

(1.9)

n

The lhs is time-independent; thus we must have i

an (xi , ti ) = an (xi )e+ h¯ En ti ,

(1.10)

CHAPTER 1. PATH INTEGRAL IN QUANTUM THEORY and consequently δ (x − xi ) =

an (xi ) ϕn (x) .

9

(1.11)

n

This can be fulfilled by

an (xi ) = ϕ∗n (xi )

(1.12)

(closure relation). Thus we have a representation of K (x, t; xi , ti ) in terms of the eigenfunctions and eigenvalues of the underlying Hamiltonian K (x, t; xi , ti ) = Θ(t − ti )

ϕ∗n (xi ) ϕn (x)e− h¯ En (t−ti ) . i

(1.13)

n

It is easy to show that this propagator fulfills (1.3). In Dirac’s bra and ket notation this result can also be written as K (x, t; xi , ti ) =

ϕ∗n (xi ) ϕn (x)e− h¯ En (t−ti ) Θ (t − ti ) i

n

=

n|xi e− h¯ En (t−ti ) x|n Θ (t − ti ) i

n

=

ˆ

ˆ

n|e+ h¯ Hti |xi x|e− h¯ Ht |n Θ (t − ti ) i

i

n ˆ

= x|e− h¯ H(t−ti ) |xi ≡ x|Uˆ (t, ti ) |xi Θ (t − ti ) .(1.14) i

Thus the propagator is nothing else than the time development operator i ˆ Uˆ (t, ti ) = e− h¯ H(t−ti )

(1.15)

for t > ti in the x representation. It is also often written as ˆ

K (x, t; xi , ti ) = x|e− h¯ H(t−ti ) |xi Θ (t − ti ) ≡ xt|xi ti . i

(1.16)

The notation here is that of the Heisenberg representation of quantum mechanics. In this representation the physical state vectors are time-independent and the operators themselves carry all the time-dependence whereas this is just the opposite for the Schrödinger representation. For example, for the position operator xˆ in the Schrödinger representation we obtain the timedependent operator i ˆ i ˆ (1.17) xˆH (t) = e h¯ Ht xê− h¯ Ht and xˆH (t)|xt = x|xt with

i

ˆ

|xt = e h¯ Ht |x .

(1.18) (1.19)

The state |xt is thus the eigenstate of the operator xˆH (t) with eigenvalue x and not the state that evolves with time out of |x; this explains the sign of the frequency in the exponent.


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t

t

t1

ti x xi

x

Figure 1.1: Possible paths from xi to x, corresponding to (1.22).

1.2

Propagator as Path Integral

We start by dividing the time-interval between ti and t by inserting the time t1 . The wavefunction is first propagated until t1 and then, in a second step, until t

ψ (x1 , t1 ) = ψ(x, t) =

K (x1 , t1 ; xi , ti ) ψ (xi , ti ) dxi

(1.20)

K(x, t; x1 , t1 )ψ (x1 , t1 ) dx1 .

Taking these two equations together we get

ψ(x, t) =

K (x, t; x1 , t1 ) K (x1 , t1 ; xi , ti ) ψ(xi , ti ) dxi dx1 .

(1.21)

Comparing this result with (1.5) yields

K (x, t; xi , ti ) =

K (x, t; x1 , t1 ) K (x1 , t1 ; xi , ti ) dx1 .

(1.22)

We can thus view the transition from (xi , ti ) to (x, t) as the result of a transition first from (x, t) to all possible intermediate points (x1 , t1 ), which is then followed by a transition from these intermediate points to the endpoint. We could also say that the integration in (1.22) is performed over all possible paths between the points (xi , ti ) and (x, t), which consist of two straight line segments with a bend at t1 . This is illustrated in Fig. 1.1.


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We now subdivide the time interval further into (n + 1) equal parts of length Δt = η. We then have in direct generalization of the previous result

K (x, t; xi , ti ) =

...

dx1 dx2 . . . dxn

(1.23)

× [K (x, t; xn , tn ) K (xn , tn ; xn−1 , tn−1 ) . . . K (x1 , t1 ; xi , ti )] . The integrals run here over all possible paths between (xi , ti ) and (x, t) which consist of (n + 1) segments with boundaries that are determined by the time steps ti , t1 , . . . , tn , t. We now calculate the propagator for a small time interval Δt = η from tj to tj+1 . For this propagation we have according to (1.16) ˆ

K (xj+1 , tj+1 ; xj , tj ) = xj+1 |e− h¯ Hη |xj (1.24) i ˆ ∼ = xj+1 |1 − Hη|x j h ¯ i ˆ j = δ (xj+1 − xj ) − ηxj+1 |H|x h ¯ 1 i p(xj+1 −xj ) iη ˆ j = e h¯ dp − xj+1 |H|x 2π¯ h h ¯ i

with the representation for the δ-function δ(x − x ) =

1 ik(x−x ) e dk . 2π

(1.25)

ˆ is given by We now assume that H ˆ = Tˆ(ˆ H p) + Vˆ (ˆ x) .

(1.26)

Here Tˆ, pˆ, Vˆ , xˆ are all operators; we assume that T (ˆ p) and V (ˆ x) are Taylorexpandable. In this case, where the p- and x-dependences separate, we can also bring the last term in (1.24) into an integral form. We have ˆ j = xj+1 |Tˆ + Vˆ |xj . xj+1 |H|x

(1.27)

First, we consider the first summand xj+1 |Tˆ|xj =

=

=

dp dpxj+1 |p p |Tˆ(ˆ p)|pp|xj dp dpxj+1 |p δ(p − p)T (p)p|xj dpxj+1 |pT (p)p|xj .

(1.28)


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With the normalized momentum eigenfunctions x|p = √

i 1 e h¯ px 2π¯ h

(1.29)

we thus obtain i 1 xj+1 |Tˆ(ˆ p)|xj = T (p) e h¯ p(xj+1 −xj ) dp . 2π¯ h

(1.30)

While there is an operator pˆ on the lhs of this equation there are only numbers p on its rhs. For the potential part an analogous transformation can be performed xj+1 |Vˆ |xj = V (xj ) δ (xj+1 − xj ) 1 i p(xj+1 −xj ) e h¯ dp V (xj ) = 2π¯ h

(1.31)

Again, on the lhs Vˆ is an operator, while the rhs of this equation contains no operators. In summary, we have for the propagator over a time-segment η i 1 dp e h¯ p(xj+1 −xj ) 2π¯ h i i iη 1 1 − dp T (p)e h¯ p(xj+1 −xj ) + dp e h¯ p(xj+1 −xj ) V (xj ) h ¯ 2π¯ h 2π¯ h i 1 iη = dp e h¯ p(xj+1 −xj ) 1 − H (p, xj ) 2π¯ h h ¯ 1 i −→ dpj exp (1.32) [pj (xj+1 − xj ) − ηH (pj , xj )] . η→0 2π¯ h h ¯

K (xj+1 , tj+1 ; xj , tj ) =

Here H = T + V is a function of the numbers x and p and no longer an operator! In the last step we have renamed the integration variable into pj to indicate that it may be viewed as the momentum of a classical particle moving from xj to xj+1 between times tj and tj+1 . We now insert (1.32) into (1.23) and obtain K (x, t; xi , ti ) =

lim

n→∞

n k=1

dxk

n dpl l=0

2π¯ h

⎧ n ⎨i

exp ⎩

h ¯ j=0

(1.33) ⎫ ⎬

[pj (xj+1 − xj ) − ηH (pj , xj )]⎭ .

(with x0 = xi and xn+1 = x). The asymmetry in the range of the products over x- and p-integrations comes about because with n intermediate steps between xi and x there are n + 1 intervals and corresponding momenta.


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The integrand here is, for finite n, a complex function of all the coordinates x1 , x2 , . . . , xn and the momenta p1 , p2 , . . . , pn . In the limit n → ∞ it depends on the whole trajectory x(t), p(t). Here we note that p is not the momentum canonically conjugate to the coordinate x, but instead just an integration variable. In the limit n → ∞ we obtain for the exponent n

[pj (xj+1 − xj ) − ηH (pj , xj )]

j=0

= −→

xj+1 − xj − H (pj , xj ) η pj η j=0 n

t

n→∞

dt [p(t )x(t ˙ ) − H(p(t ), x(t ))] .

(1.34)

ti

With this result we rewrite (1.33) in an abbreviated, symbolic form

K (x, t; xi , ti ) =

Dx

Dp e

i h ¯

t

dt [p(t )x(t ˙ )−H(p(t ),x(t ))]

ti

,

(1.35)

h). The integrals here are where Dx stands for dxk and Dp for dpl /(2π¯ limits of n-dimensional integrals over x and p for n → ∞, they are integrals over all functions x(t) and p(t) and are defined by (1.33). Equation (1.35) represents an important result. It allows to calculate the propagator and thus the solution of the Schrödinger equation in terms of a path integral over classical functions.

1.3

Quadratic Hamiltonians

Even though the propagator (1.35) looks like a path integral over an exponential function of the action, this is in general not the case, because ˙ p) px˙ − H(p, x) = L(x, x,

(1.36)

is not equal to the classical Lagrange function since p is not the canonical momentum as already stressed above. Therefore, in general one cannot express the path integral (1.35) in terms of the action. Such a simplification, however, is possible for a special p-dependence of the Hamiltonian. If H depends at most quadratically on p, then the path integration over the momentum p can be performed and the action appears in the exponent. This will be discussed in the next 2 sections.


1.3.1

14

Cartesian metric

In the last section we have made the special ansatz H = T (p) + V (x) in which the momenta and coordinates are separated. For the special case, in which H depends only quadratically on p with constant coefficient, e.g. p2 + V (x) , 2m

H=

(1.37)

we can further simplify the path-integral (1.33) K (x, t; xi , ti ) =

lim n→∞

n

dxk

k=1

⎧ ⎨i

n

n dpl

× exp ⎩ η h ¯ j=0

l=0

(1.38)

2π¯ h

⎫

⎬ p2j xj+1 − xj pj − − V (xj ) ⎭ . η 2m

Using the integral relation +∞ −∞

e

−ap2 +bp+c

dp =

π b2 +c e 4a a

(1.39)

for Gaussian integrals, discussed in more detail in App. B.2.1, we obtain by performing the p-integration

K =

lim n→∞ ×

m i2π¯ hη

n+1 2

(1.40) ⎧ ⎨i

n

n

⎡

m ⎣ dxk exp ⎩ η h ¯ j=0 2 k=1

xj+1 − xj η

2

⎤⎫ ⎬ − V (xj )⎦⎭ .

Thus in this special case (H = p2 /2m + V ) the propagator K is given (again in abbreviated notation) by

K (x, t; xi , ti ) = N

Dx e

i h ¯

t

=N

with the Lagrangian L(x, x) ˙ = and the action

t

S[x(t)] =

L(x,x)dt ˙

ti

i

Dx e h¯ S[x(t)] ,

m 2 x˙ − V (x) 2

L(x(t ), x(t ˙ )) dt .

(1.41)

(1.42) (1.43)

ti

N represents the factor in front of the integral in (1.40). There is a problem with N : the factor N is complex and becomes infinite for n → ∞, η →


15

0. We will see, however, later, for example in Sect. 2.1, that the whole pathintegral leads to a well-defined expression. In addition, this problem will be bypassed in later developments where we show that physically relevant is only a normalized propagator in which N has been removed. The propagator K has thus been reduced to a one-dimensional pathintegral, which is only possible for Hamiltonians which are quadratic in p. This is a quite important result that we will use throughout all the following sections. Eq. (1.41) shows that the propagator is given by the phase exp ¯hi S[x(t)] summed over all possible trajectories x(t) with fixed starting and end points. This is reminiscent of the partition function in statistical mechanics which is obtained by summing the Boltzmann factor exp (−En /T ) over all possible states of the system. At this point we should realize that in going from (1.38) to (1.40) we have integrated an oscillatory integrand (eif (p) ) over an infinite interval. This was only possible by a mathematical trick: in applying the Gaussian integration formula (B.18) we have in effect used the quantity iη in (1.38) as if it were real. In other words: we have analytically continued expression (1.38) into the complex plane by setting the time interval η → −iη with η real and positive. Then (1.38) becomes a well-behaved Gaussian integral. After performing the integration we have then gone back to the original η. This analytical continuation is in general possible only if no singularities are encountered while going to the real variable η . Also, one has to worry about phase ambiguities connected with the appearance of the squareroot of a complex number.

1.3.2

Non-Cartesian metric

The momentum integration is even possible for Hamiltonians of a much more general form. As an example we consider H=

1 f1 (x)p2 + f2 (x)p + f3 (x) . 2m

(1.44)

Here a problem arises because the canonical quantization of such a Hamiltonian is ambiguous. This is so because the classical coordinates and momenta commute, so that H can be brought into various forms that are classically all equal, but differ after quantization because the operators pˆ and xˆ do not commute. Even though the time development operator (1.24) could still be evaluated for any of these various forms, these would in general not lead to a path integral of the form (1.35). We, therefore, turn the question around and ask:


16

Given a path integral

Dx

Dp e

i h ¯

t

dt (px−H(p,x)) ˙

ti

(1.45)

with the classical Hamiltonian (1.52), can this still be identified as a propagator and, if so, for which Hamiltonian? The answer to this question is given here without proof1 :

Dx

Dp e

i h ¯

t

dt [px−H(p,x)] ˙

ti

i

ˆ

= x |e− h¯ (t−ti )HW |x .

(1.46)

ˆ W is the “Weyl-ordered” Hamiltonian Here H ˆW = H

1 2 pˆ f1 (x) + 2ˆ pf1 (x)ˆ p + f1 (x)ˆ p2 8m 1 + [ˆ p] + f3 (x) . pf2 (x) + f2 (x)ˆ 2

(1.47)

In this form the momentum- and coordinate-dependent terms are symmetrized. Note that the path integral in (1.46) is well determined because all quantities on the rhs of this equation are classical, commuting quantities. The lhs of (1.46) can be simplified by performing the integration over the momenta in (1.33) K =

lim n→∞

n

dxk

k=1

⎧ n ⎨i

× exp ⎩

h ¯ j=0

n

dpl l=0

(1.48)

2π¯ h

⎫ ⎬

p2j pj (xj+1 − xj ) − η f1 (xj ) + f2 (xj )pj + f3 (xj ) 2m

⎭

.

Using again the Gaussian integral formula (1.39) this gives

K =

lim n→∞

m i2π¯ hη

n+1 n 2

⎧ ⎪ n ⎨i

⎛

m × exp ⎪ η⎜ ⎝ ¯ j=0 2 ⎩h 1

dxk

k=1

!

n

l=0

(xj+1 −xj ) η

The proof can be found in [LEE], p. 475

1

(1.49)

f1 (xl )

"2

− f2 (xj )

f1 (xj )

⎞⎫ ⎪ ⎬ − ηf3 (xj )⎟ ⎠ . ⎪ ⎭


17

The exponent (in round brackets) is !

"2

xj+1 −xj − f2 (xj ) m η (. . .) = − f3 (xj ) 2 f1 (xj ) 2 ˙ 2 (x) + f22 (x) −→ m x˙ − 2xf − f3 (x) . η→0 2 f1 (x)

(1.50)

This last expression is just the Lagrangian L=

m g1 (x)x˙ 2 + g2 (x)x˙ + g3 (x) 2

(1.51)

(a kinetic term of this form appears, for example, when rewriting the kinetic energy of a particle from cartesian into polar coordinates). For this L the corresponding classical Hamiltonian is given by H = px˙ − L = (mg1 x˙ + g2 ) x˙ −

m g1 x˙ 2 − g2 x˙ − g3 2 2

m m p − g2 = − g3 g1 x˙ 2 − g3 = g1 2 2 mg1 1 1 2 g2 1 g22 = p − p+ − g3 . 2m g1 mg1 2m g1

(1.52)

With

1 g2 g22 , f2 = − , f3 = − g3 g1 mg1 2mg1 this is just the Hamiltonian (1.44) that we started out with. We thus have for the complete propagator in this case f1 =

K (x, t; xi , ti )

=

lim

n→∞

m i2π¯ hη

n+1 n 2

k=1

dxk

n

l=0

⎧ ⎨i

(1.53)

(1.54)

n

⎫ ⎬

g1 (xl ) exp ⎩ η L(xj , x˙ j )⎭ . h ¯ j=1

Thus, in this case the path integral is changed. The square root of a function that determines the metric of the system appears in the integrand. For Hamiltonians even more general than (1.44) an additional potential term appears in the Lagrangian [Grosche/Steiner].

1.4

Classical Interpretation

The simple form (1.41) for the path integral allows a very physical interpretation of the classical limit to quantum mechanics. For the classical path the


18

variation of the action is, according to Hamilton’s principle, equal to zero, i.e. the action is stationary2 t

δS =

L (xcl + δx, x˙ + δ x˙ cl ) dt −

ti

t

L (xcl , x˙ cl ) dt = 0 .

(1.55)

ti

This implies that all the paths close to the classical path give about equal contributions to the path integral. For each path (C1 ) somewhat more removed from the classical one there will also be another one, (C2 ), whose h. Then we have action differs from that on C1 just by π¯ i

i

i

e h¯ S(C1 ) = e h¯ S(C2 )+iπ = −e h¯ S(C2 ) ,

(1.56)

so that the contributions from these two paths cancel each other. Sizeable contributions to the path integral thus come from paths close to the classical one. Quantum mechanics then describes the fluctuations of the action in a narrow range around the classical path. This observation forms the basis for a semiclassical approximation. This can be formulated by expanding the action functional S[x(t), x(t)] ˙ in terms of fluctuations δx around the classical path xcl (t). This gives

1 δ2L δ2L δ2L 2 S[x, x] ˙ = Scl + (δx) + 2 ˙ 2 + ..... δx δ x˙ + 2 (δ x) 2 2 δx δxδ x˙ δ x˙ 2 (1.57) ≡ Scl + δ S + . . . . Here all the derivatives have to be taken at the classical path. Because S is stationary at the classical path, there is no first derivative in this equation. The propagator (1.41) now becomes

K (x, t; xi , ti ) = N

Dx e

i S h ¯

= Ne

i S h ¯ cl

&

'

1i 2 Dx exp δ S + .... . 2h ¯

(1.58)

Note that this result (without higher order terms) is exact for Lagrangians that depend at most quadratically on x. The second factor gives the effects of quantum mechanical fluctuations around the classical path. An interesting observation on the character of these fluctuations can be made. The main contribution to the integrand in (1.40) for η → 0 comes from exponents 2 η m xj+1 − xj

1, (1.59) h ¯ 2 η 2

For an explanation of functionals and their derivatives see App. B


19

i.e. from velocities vj ≈ 2¯ h/(ηm) which diverge with η → 0. This implies √ that the average displacement d within a time-step η is proportional to η, so that d2 ∼ η (not d2 = v 2 dt2 = v 2 η 2 !), just as for a random walk. The main contribution to the path integral, and therefore to the quantum mechanical fluctuation around the classical trajectory, thus comes from paths that are continous, but have no finite derivative.

Chapter 2 PERTURBATION THEORY In this chapter we discuss first how to calculate the propagator of a free particle and derive its analytic form. In most cases, however, with a potential included the exact propagator cannot be calculated. Thus one has to resort to perturbation theory which will also be developed in this chapter.

2.1

Free propagator

We start with the free propagator K0 , given by

i

Dx e h¯ S0

K0 = N

=

lim

n→∞

m 2πi¯ hη

⎡

n+1 +∞

n 2 −∞

n i m dxk exp ⎣ η h ¯ j=0 2 k=1

xj+1 − xj η

2 ⎤ ⎦ . (2.1)

This path-integral can be performed exactly. With (B.19) we obtain for the free propagator

K0 =

lim

n→∞

m 2πi¯ hη

n+1 ⎛ 2

⎝

(n + 1)

⎞1 2

in π n

i m × exp (x − xi )2 n + 1 2¯ hη

m 2¯ hη

n ⎠

(2.2)

,

since xn+1 = x, x0 = xi . With (n + 1)η = t − ti this becomes (

K0 (Δx, Δt) =

2 i m(x−xi ) m h ¯ 2(t−ti ) = e 2π¯ hi (t − ti )

20

i mΔx2 m e h¯ 2Δt 2π¯ hiΔt

(2.3)

CHAPTER 2. PERTURBATION THEORY

21

with Δx = x−xi , Δt = t−ti . This is the propagator of a free particle for Δt ≥ 0; for Δt < 0 it has to be supplemented by the condition K = 0. Because of Galilei invariance and time homogeneity the free particle propagator depends only on the space- and time-distances. The last step, from (2.2) to (2.3), shows nicely how in the limit n → ∞ the infinite normalization factor combines with another equally ill defined factor from the path integral into a well defined product in (2.2). Since a free particle has conserved momentum, it is advantageous to transform K0 into the momentum representation 1 − i pΔx e h¯ K0 (Δx, Δt) dΔx 2π¯ h m − i pΔx i m Δx2 1 e h¯ e h¯ 2Δt dΔx . = √ hiΔt 2π¯ h 2π¯

K0 (p, Δt) = √

(2.4)

We can now use again the integral relation (B.18) in the form +∞

e

−aΔx2 +bΔx

dΔx =

−∞

π b2 e 4a a

(2.5)

(with a = −im/(2¯ hΔt) and b = −ip/¯ h) to write

(

1 m π2¯ hΔt − i p2 Δt e h¯ 2m K0 (p, Δt) = 2π¯ h iΔt −im 2 i p 1 e− h¯ 2m Δt , = √ 2π¯ h

(2.6)

so that we obtain 1 i pΔx K0 (x, t; xi , ti ) = √ e h¯ K0 (p, Δt) dp 2π¯ h p2 Δt 1 h¯i pΔx− 2m e dp Θ(Δt) , = 2π¯ h

(2.7)

where in the last line the causality condition Θ(Δt) has been written explicitly; it takes care of the boundary condition K = 0 for Δt < 0. Eq. (2.7) is just the Fourier representation of the propagator (2.3). The step function can be rewritten using the relation +∞ 1 eiωΔt Θ(Δt) = dω 2πi ω − iε −∞

(ε > 0) ,

(2.8)


22

which follows directly from the residue theorem: for Δt > 0 the integral can be closed in the upper half of the complex ω plane; Cauchy’s integral theorem then gives 2πi in the limit ε → 0 so that Θ(t) = 1. If Δt < 0, on the other hand, then the loop integration can only be closed in the lower half-space thus missing the pole at ω = +iε). Multiplying (2.6) with (2.8) gives

K0 (x, t; xi , ti ) =

1 (2π)2 h ¯i

+∞

dp dω

e

i h ¯

!

−∞

We now substitute E=

pΔx−

p2 −¯ hω 2m

" Δt

ω − iε

.

p2 −h ¯ω 2m

(2.9)

(2.10)

and obtain K0 (x, t; xi , ti ) =

i 1 i¯ h (pΔx−EΔt) h ¯ dp dE e 2 p (2π¯ h)2 E − 2m + iε

(2.11)

(the “−” sign coming from the substitution is cancelled by another sign obtained by inverting the integration boundaries). Since we will need these expressions later on in three space dimensions we give them here in a straightforward generalization of (2.7) and (2.11) i p2 −t) p ·( x − x )− (t 1 2m K0 (x , t ; x, t) = e h¯ d3p Θ(t − t) , (2π¯ h)3

(2.12)

and K0 (x, t; xi , ti ) =

i 1 3 i¯ h ( p·Δ x−EΔt) h ¯ . d p dE e 2 p (2π¯ h)4 E − 2m + iε

(2.13)

The integrand on the right-hand side of (2.11) is the free propagator in the energy-momentum representation. Since p and E are independent variables in the integral, we see that propagation also takes place at energies E = p2 /2m. The classical dispersion relation does show up as a pole in the propagator.

2.2

Perturbative Expansion

We now assume that the unperturbed particle moves freely and that the perturbing interaction is given by V (x, t). We furthermore assume that H


23

has the special form H = p2 /2m + V (x, t), so that the propagator is given by i (2.14) K (x, t; xi , ti ) = N Dx e h¯ S with

t

t

S=

L(x, x) ˙ dt = ti

ti

m 2 x˙ − V (x, t) dt . 2

(2.15)

Since the integrand here is a classical function, we have

e

i h ¯

i S h ¯

=e

t ti

m 2 x˙ 2

i dt − h ¯

e

t

V (x,t) dt

ti

.

(2.16)

The second factor can now be expanded in powers of the potential, thus yielding a perturbative expansion of the action i h ¯

e

t

V (x,t) dt

ti

⎛ t

⎞2

1 1 ⎝ i ∼ V (x, t) dt − V (x, t) dt⎠ + . . . . =1− 2 h ¯ 2! h ¯ t t t

i

(2.17)

i

When we substitute this expansion into the expression (2.14) we obtain

K=N

) i

Dx e h¯ S0

⎡ ⎢

× ⎣1 −

i h ¯

t

⎤ ⎛ t ⎞2 , 1 1 ⎝ ⎥ V (x, t) dt − V (x, t) dt⎠ + . . .⎦ 2

2! h ¯

ti

ti

= K0 + K1 + K2 + . . . ,

(2.18)

i.e. a sum ordered in powers of the interaction. First order propagator. We next determine the first-order propagator K1 . According to (2.18) it is given by i i = − N Dx e h¯ S0 V (x, t) dt h ¯ tf

K1

ti

= − ×

m i lim h ¯ n→∞ 2π¯ hiη +∞

n

−∞ i=1

dxi

(2.19)

n k=1

n+1 2

⎛

V (xk , tk ) η exp ⎝i

n m

2¯ hη

j=0

⎞

(xj+1 − xj )2 ⎠ .


24

Here the time-integral has been written as a sum. In a next step we now split the sum in the exponent into two pieces, one running from j = 0 to j = k − 1 and the other from j = k to j = n and separate the corresponding integrals. This gives

K1

n i = − n→∞ η dxk V (xk , tk ) lim h ¯ k=1 ⎛

⎜

k

2 ×⎜ ⎝N

dx1 dx2 · · · dxk−1 e

⎛ n−k+1 ×⎜ ⎝N 2

dxk+1 · · · dxn e

with N≡

(2.20)

m i 2¯ hη

m i 2¯ hη

-

k−1 j=0

n j=k

(xj+1 −xj )

(xj+1 −xj )2

⎞ 2

⎟ ⎟ ⎠

⎞ ⎟ ⎠

m . 2π¯ hiη

(2.21)

The term in the first bracket is nothing else than the propagator from ti to tk (K0 (xk , tk ; xi , ti )) , and that in the second bracket is that from tk to t (K0 (x, t; xk , tk )). Thus we have K1 (xf , tf ; xi , ti ) = +∞ tf i dx dt K0 (xf tf ; x, t) V (x, t)K0 (x, t; xi , ti ) . − h ¯ −∞

(2.22)

ti

The time integral over the interval from ti to tf can be extended to ∞ by noting that K0 (x, t; xi , ti ) = 0 K0 (xf , tf ; x, t) = 0

for t < ti for tf < t .

(2.23)

This gives K1 (xf , tf ; xi , ti ) = +∞ +∞ i − dt dx K0 (xf , tf ; x, t) V (x, t)K0 (x, t; xi , ti ) . h ¯ −∞

−∞

(2.24)


25

xf tf

x2 t2 =

+

+ x1 t1

+ ... x1 t1

xi ti Figure 2.1: Born series for the propagator Higher order propagators. Similarly, the higher order terms in the perturbation expansion (2.18) can be obtained. This yields finally K (xf , tf ; xi , ti ) =

i K0 (xf , tf ) − K0 (xf , tf ; x1 , t1 ) V (x1 , t1 ) K0 (x1 , t1 ; xi , ti ) dx1 dt1 h ¯ ! 1 − 2 K0 (xf , tf ; x1 , t1 ) V (x1 , t1 ) K0 (x1 , t1 ; x2 , t2 ) h ¯ " × V (x2 , t2 ) K0 (x2 , t2 ; xi , ti ) dx1 dt1 dx2 dt2 + ··· .

(2.25)

Note that the time-integrations in (2.25) are effectively time-ordered because of the implicit Θ (tf − ti ) function in each of the propagators, leading to tf > t1 > t2 > · · · > ti . This fact explains why, for example, the factor 1/(2!) in the second order term of (2.18) no longer appears in (2.25) 1 2!

2

dtV (x, t)

1 = V (x, t)V (x, t ) dt dt 2!

1 = [V (x, t)V (x, t )Θ (t − t) + V (x, t)V (x, t )Θ(t − t )] dt dt 2! =

V (x, t )Θ (t − t) V (x, t) dt dt .

(2.26)

The last line follows from a simple change of variables; the Θ function in it is absorbed into the propagator in (2.25). A similar argument holds for all


26

the higher-order terms in the interaction. In each case the prefactor (1/n!) stemming from the expansion of the exponential in (2.17) is cancelled by the time ordering inherent in the propagators. Equation (2.25) is the Born series for the propagator; it can be represented graphically as shown in Fig. 2.1. A time axis runs here from bottom to top. The straight lines denote the free propagation of the particles (K0 ) and the dots stand for the interaction vertices (−iV /¯ h) (the circles mark the interaction range) and a space and time integration appears at each vertex. Bethe-Salpeter equation. The expansion (2.25) can be formally summed. This can be seen as follows (in obvious shortterm notation) K = K0 + K 0 U K 0 + K 0 U K 0 U K 0 + · · · = K0 + K0 U (K0 + K0 U K0 + · · ·)

(2.27) with U =

− ¯hi V

.

The expression in parentheses is just again K, so that we obtain the BetheSalpeter equation (2.28) K = K0 + K0 U K . The Bethe-Salpeter equation is an integral equation for the full interacting propagator K as can be seen most easily from its space-time representation K (xf , tf ; xi , ti ) = K0 (xf , tf ; xi , ti ) (2.29) i − K0 (xf , tf ; x, t) V (x, t)K (x, t; xi , ti ) dx dt . h ¯ We can also represent the Bethe-Salpeter equation in a diagrammatic way. If we denote the so-called “dressed propagator” K, that includes all the effects of the interactions, by a double line = K (x2 , t2 ; x1 , t1 ) ,

(2.30)

then this equation can be graphically represented as shown in Fig. 2.2. Each graph in Fig. 2.2 finds a one-to-one correspondence in (2.28): the single lines represent the free propagator, the double line the dressed propagator and the dot stands for the interaction U . (2.28) shows that the factors in the second term of this equation have to be written from left to right against the time-arrow in Fig. 2.2. The Bethe-Salpeter equation can also be written in an equivalent form for the interacting wavefunction

Ψ (xf , tf ) =

K (xf , tf ; xi , ti ) Ψ (xi , ti ) dxi


=

27

+

Figure 2.2: Bethe-Salpeter equation (2.28). The arrows indicate the time direction.

=

K0 (xf , tf ; xi , ti ) Ψ (xi , ti ) dxi

i K0 (xf , tf ; x, t) V (x, t)K (x, t; xi , ti ) Ψ (xi , ti ) dxi dx dt ¯ h = K0 (xf , tf ; xi , ti ) Ψ (xi , ti ) dxi (2.31) −

i − K0 (xf , tf ; x, t) V (x, t)Ψ(x, t) dx dt . h ¯ This constitutes an integral equation for the unknown wavefunction Ψ(x, t).

2.3

Application to Scattering

Let us now apply the results of the last section to a scattering process. In this case the particle is free at t = −∞, then undergoes the scattering interaction and then, at t = +∞, is free again. We treat this problem as usual by adiabatically switching on and off the interaction V (x, t). The initial condition (for t → −∞) for the wavefunction then is (2.32) Ψin (x, t) = N ei(ki ·x−ωi t) . We choose here √ a box normalization with periodic boundary conditions so that N = 1/ V . The scattering state that evolves from this incoming state is denoted by Ψ(+) (x, t). The superscript (+) indicates that the state evolves forward in time, starting from Ψin at t = −∞; it thus fulfills the boundary condition (2.33) Ψ(+) (x, t → −∞) = Ψin (x, t) . In a scattering experiment one looks at t → +∞ for a free scattered particle


28

with definite momentum; the corresponding final state is denoted by

Ψout (x, t) = N ei(kf ·x−ωf t) .

(2.34)

The probability amplitude for the presence of Ψout in the scattered state Ψ(+) is given by

Sf i =

Ψ∗out (xf , tf ) Ψ(+) (xf , tf ) d3xf

for

tf → ∞ .

(2.35)

This is just the transition amplitude from the initial state i to the final state f (S-matrix). Expansion (2.29) yields the Bethe-Salpeter equation for the scattering wavefunction

Ψ(+) (xf , tf ) = −

K0 (xf , tf ; xi , ti ) Ψin (xi , ti ) d3xi

(2.36)

i K0 (xf , tf ; x, t) V (x, t)K(x, t; xi , ti )Ψin (xi , ti ) d3xi d3x dt . h ¯

Inserting this into (2.35) gives for the S-matrix

Sf i =

Ψ∗out (xf , tf ) K0 (xf , tf ; xi , ti ) Ψin (xi , ti ) d3xi d3xf

−

i h ¯

(2.37)

Ψ∗out (xf , tf ) K0 (xf , tf ; x, t) V (x, t)

× K(x, t; xi , ti )Ψin (xi , ti ) d3xi d3x dt d3xf . Since Ψin is a plane wave, we know that

φ (xf , tf ) =

K0 (xf , tf ; xi , ti ) Ψin (xi , ti ) d3xi

(2.38)

is also a plane wave state, since K0 is the free propagator so that no interaction takes place. φ is actually the same wavefunction as Ψin , only taken at a later time and space point

φ (xf , tf ) = N ei(ki ·xf −ωi tf ) .

(2.39)

This means that the first integral in (2.37) can be easily evaluated

Ψ∗out (xf , tf ) φ (xf , tf ) d3xf = δ 3 kf − ki

.

(2.40)

We thus have i 3 Sf i = δ kf − ki − d xf d3x d3xi dt (2.41) h ¯ ×[Ψ∗out (xf , tf ) K0 (xf , tf ; x, t) V (x, t)K(x, t; xi , ti )Ψin (xi , ti ) ] . 3


29 xf tf

xt

xi ti Figure 2.3: First order scattering diagram, corresponding to the second term in (2.41). Time runs from left to right. The amplitude for a scattering process is given by the second term. If K on the rhs of (2.41) is now represented by the Born series expansion of the Bethe-Salpeter equation (2.27) this amplitude can be graphically represented as shown in Fig. 2.3. This diagram can be translated into the amplitude just given by writing for each straight-line piece xt t1

x2 t2

= K0 (x2 , t2 ; x1 , t1 )

(2.42)

and for each interaction vertex xt

i = − V (x, t) + integration over x, t . h ¯

(2.43)

The rules are completed by multiplying Ψin and Ψ∗out at the corresponding sides of the diagram and then integrating over the spatial variables of these wavefunctions and over all intermediate times. The ordering of factors is such that in writing the various factors from left to right one goes against the flow of time in the figure. These rules are illustrated for a second-order scattering process in Fig. 2.4. In this figure the time runs from ti to tf from left to right. The corresponding amplitude is given by (2)

A

=

&

d xi d xf d x1 d x2 dt1 dt2 Ψ∗out (xf , tf ) K0 (xf , tf ; x2 , t2 ) 3

3

3

3

'

i i − V (x2 , t2 ) K0 (x2 , t2 ; x1 , t1 ) − V (x1 , t1 ) K0 (x1 , t1 ; xi , ti ) Ψin (xi , ti ) . h ¯ h ¯ (2.44)


30 xf tf

x1 t1

x2 t2 xi ti Figure 2.4: Second order scattering diagram, corresponding to (2.44). We now evaluate the first order scattering amplitude (cf. (2.41)) A

i ! ∗ = − Ψout (xf , tf ) K0 (xf , tf ; x, t) V (x, t)K0 (x, t; xi , ti ) h ¯ " × Ψin (xi , ti ) d3x dt d3xi d3xf (2.45)

(1)

somewhat further by using expression (2.6) for the free propagator (extended to three dimensions) p2 (t −t) 1 h¯i p·(x −x)− 2m K0 (x , t ; x, t) = e d3p Θ(t − t) . 3 (2π¯ h)

(2.46)

Inserting this into the expression for A(1) gives A(1) = −

e

i −h ¯

i 1 1 3 d x dt d3xi d3xf d3p d3q 6 h ¯ V (2π¯ h)

(pf ·xf −Ef tf ) e

i h ¯

2

p p ·( xf − x)− 2m (tf −t)

V (x, t)e

(2.47) i h ¯

2

q q·( x− xi )− 2m (t−ti )

e

i ( pi · xi −Ei ti ) h ¯

,

where the time-ordering tf > t > ti is understood and we have abbreviated Ei = p2i /(2m) and Ef = p2f /(2m). We integrate first over xi and xf . This yields

d3xi −→

(2π¯ h)3 δ 3 (pi − q)

d3xf −→ (2π¯ h)3 δ 3 (pf − p) .

Performing next the integrals over p and q gives for the amplitude (1)

A

i 3 =− d x dt h ¯V

(2.48)


× e

i +h E t ¯ f f

e

i h ¯

p2

31

f − pf · x− 2m (tf −t)

V (x, t)e

i h ¯

p2

i (t−t ) p i · x− 2m i

i i h¯i (−pf ·x+Ef t) = − e V (x, t)e h¯ (+pi ·x−Ei t) d3x dt . h ¯V

e

i −h Et ¯ i i

(2.49)

The time-integration is performed next. For that we assume that V (x, t) acts only during a finite, but long time-interval ([−T, T ]), in which it is timeindependent; at the boundaries of the interval it is adiabatically, i.e. without any significant energy-transfer, turned on and off. We then obtain

dt V (x, t)e (Ef −Ei )t dt = i h ¯

T

T →∞

V (x)ei(ωf −ωi )t dt −→ V (x) 2πδ (ωf − ωi ) ,

−T

(2.50) with ωf − ωi = (Ef − Ei )/¯ h. Thus A(1) becomes A(1) = −

i i 2πδ (ωf − ωi ) e h¯ (pi −pf )·x V (x) d3x . h ¯V

This is the well-known lowest-order Born-approximation result.

(2.51)

Chapter 3 GENERATING FUNCTIONALS In this section we consider the transition amplitude in the presence of an external “source” J(t), so that the Hamiltonian reads ¯ J(t)x . HJ (x, p) = H(x, p) − h

(3.1)

A classical example is that of a harmonic oscillator with externally driven equilibrium position x0 (t). Its Hamiltonian is p2 1 + k(x − x0 (t))2 2m 2 = H − kx0 (t)x + O(x20 ) .

H x0 =

(3.2)

With h ¯ J(t) = kx0 (t) and for small amplitudes x0 we just have the form of (3.1). It is evident that the states of this system will change as time develops, because of its changing equilibrium position. Suppose now that the system was in its groundstate at t → −∞ and that x0 (t) acts only for a limited time period. We could then calculate the transition probability for the system to be still in its groundstate at t → +∞ by using the techniques developed in sections 2.2 and 2.3. There we saw (cf. (2.25) and (2.44)) that this probability is determined by matrixelements of time-ordered products of the interaction, i.e. of the operator xˆ in the present case. In this chapter we will show that these matrixelements can all be generated once only the functional dependence of the probability for the system to remain in its groundstate on an external source is known.

32

CHAPTER 3. GENERATING FUNCTIONALS

3.1

33

Groundstate-to-Groundstate Transitions

Generalizing the special example of the introduction we assume that an arbitrary physical system is at first (at ti ) stationary and then changes under the influence of an external source J(t)x of finite duration. After the source has been turned off, the groundstate of the system is still the same (up to a phase), but the system may be excited. The propagator for this system, is quite generally, given by

xf , tf |xi , ti J =

Dx

Dp e

i h ¯

tf

[px−H(x,p)+¯ ˙ hJ(t)x] dt

ti

.

(3.3)

The value of this propagator depends obviously on the source function J(t); it is a functional of the source J(t). In the following paragraphs we will now discuss this functional dependence for tf → +∞ and ti → −∞. We will also show that it determines the vacuum expectation values of time-ordered xˆ operators. We start by calculating the propagator (3.3) of a system under the influence of a source, i.e. for a system described by (3.1). We first assume that the source is nonzero only for a limited time between −T and +T J(t) = 0

for |t| > T .

(3.4)

We can then write for the propagator (ti < −T , tf > T ) xf tf |xi ti J =

dx dx xf tf |x T x T |x −T J x −T |xi ti .

(3.5)

Note that the two outer propagators are taken with the sourceless Hamiltonian (J = 0) because of condition (3.4). They are given by, e.g., ˆ

x −T |xi ti = x|e− h¯ H(−T −ti ) |xi i

=

ϕn (x)ϕ∗n (xi ) e− h¯ En (−T −ti ) . i

(3.6)

n

Similarly we get ˆ

xf tf |x T = xf |e− h¯ H(tf −T ) |x i

=

ϕn (xf )ϕ∗n (x )e− h¯ En (tf −T ) . i

(3.7)

n

ˆ without a source; we Here the ϕn are eigenstates of the Hamiltonian H ˆ is bounded from below with eigenvalues assume that the spectrum of H En ≥ E0 . The dependence on ti and tf is now isolated.


34

Taking now the limits ti → −∞ and tf → +∞ is not straightforward because both times appear as arguments of oscillatory functions. These functions oscillate the more rapidly the higher the eigenvalues En are. We can thus expect that the dominant contribution will come from the lowest eigenvalue E0 . Wick Rotation. This can indeed be shown by by a mathematical trick, the so-called Wick rotation. In this method one looks at the propagators (3.6) and (3.7) as functions of ti and tf , respectively, and continues these variables analytically from the real axis into the complex plane. One then performs the limits and after that rotates back to real times. Mathematically this is achieved by replacing the physical Minkowski time t by a complex time τ (3.8) t −→ τ = te−iε . The direction of the rotation is mandated by the requirement that there are no singularities of the integrand encountered in the rotation of the timeaxis. This is the case for 0≤ε 0 , except for n = 0. Thus we get e− h¯ E0 τi x −T |xi τi = ϕ0 (x)ϕ∗0 (xi ) e h¯ E0 T . i

lim

τi →−∞(cos ε−i sin ε)

i

(3.12)

Analogously we also obtain e+ h¯ E0 τf xf τf |x T = ϕ0 (xf )ϕ∗0 (x ) e h¯ E0 T . i

lim

τf →+∞(cos ε−i sin ε)

i

(3.13)

Both expressions, (3.12) and (3.13), are the analytical continuations of the corresponding limits on the real time axis from ε = 0 to ε > 0. Since the right hand sides of these equations do not depend on ε they can be continued back to the real time axis (ε = 0) without any change. By means of the Wick rotation we have thus been able to make the expression (3.12) and (3.13) convergent; in this process we have found that only the groundstate contributes to the transition probability. We now insert these expressions into (3.5) and get lim

=

e h¯ E0 (τf −τi ) xf τf |xi τi J i

τi →−∞(cos ε−i sin ε) τf →∞(cos ε−i sin ε)

i ϕ0 (xf )ϕ∗0 (xi ) e h¯ E0 2T

(3.14)

dx dx ϕ∗0 (x ) x T |x −T J ϕ0 (x)

The integral in the last line can be rewritten, using ϕ0 (x) = x|0. This gives

dx dx

ϕ∗0 (x ) x T |x

−T J ϕ0 (x) =

dx dx 0|x x T |x −T J x|0

= 0T |0 −T J

(3.15)

so that we obtain for (3.14) e h¯ E0 (τf −τi ) xf τf |xi τi J i

lim

τi →−∞(cos ε−i sin ε) τf →∞(cos ε−i sin ε)

= ϕ0 (xf )ϕ∗0 (xi ) e h¯ E0 2T 0T |0 −T J . i

(3.16)

We obtain by Wick-rotating back e h¯ E0 (tf −ti ) e− h¯ E0 2T 0T |0 −T J = t lim xf tf |xi ti J . i →−∞ ϕ∗0 (xi )ϕ0 (xf ) t →+∞ i

i

(3.17)

f

With (in the limit ti → −∞, tf → +∞) xf tf |xi ti J=0 = ϕ∗0 (xi )ϕ0 (xf )e− h¯ E0 (tf −ti ) i

(3.18)


36

(3.17) becomes 0T |0 −T J = t lim →−∞

i tf →+∞

xf tf |xi ti J − i E0 2T e h¯ . xf tf |xi ti 0

(3.19)

This immediately gives for the free (J = 0) transition amplitude 0T |0 −T 0 = e− h¯ E0 2T , i

(3.20)

as it should. Equation (3.17) implies that the groundstate-to-groundstate transition amplitude is – up to a factor – given by a path integral from arbitrary xi to arbitrary xf and thus does not depend on these quantities, if only the corresponding times are taken to infinity. Gs-to-gs transition rate. The vacuum transition rates 0T |0 −T J deserve some explanation. Formally, they are given by 0T |0 −T J = 0|e− h¯ (H−¯hJx)2T |0 . i

(3.21)

The vacuum state |0 is assumed to be unique, if there is no source J present, and normalized to 1. It is the vacuum of the theory before and after the i action of the source. On the other hand, e− h¯ (H−¯hJx)2T |0 is the state that the vacuum at t = −T has evolved into at t = +T under the influence of the external source J(t). If this external perturbation has acted adiabatically (very gently turned on and off again) the vacuum at t > T differs from that of the source-free theory only by a phase. The matrixelement (3.21) is then the probability amplitude to find the original vacuum in the time-evolved vacuum state. The absolute value of this probability amplitude is surely 1, so that the two states can differ only by a J-dependent real phase i

0T |0 −T J = e h¯ (S[J]−E0 2T )

(3.22)

where we have taken out the free propagation contribution. This phase is the quantity determined by (3.17). To conclude these considerations we note that instead of using the Wickrotation to make the transition rates well behaved for very large times we could also have added a small negative imaginary term −iεEn to all eigenvalues En . This would have given a damping factor to the oscillating exponentials that becomes larger with n and thus would have led to a suppression of all higher excitations. This becomes apparent by looking at expressions (3.6) and (3.7). At the end of the calculation the limit ε → 0 would have to be performed.


3.1.1

37

Generating functional.

This groundstate-to-groundstate transition rate is a functional of the source J(t) which we denote by xf tf |xi ti J − i E0 2T e h¯ . xf tf |xi ti J=0

W [J] = 0 +∞|0 −∞J = t lim →−∞

i tf →+∞ T →∞

(3.23)

W [J] is called a generating functional for reasons that will become clear in the next section. In order to get rid of the phase that is produced already by a source-free propagation (exp − ¯hi E0 2T ) we define now a normalized generating functional Z[J] =

xf tf |xi ti J 0 +∞|0 −∞J W [J] = t lim = i →−∞ xf tf |xi ti J=0 W [0] 0 +∞|0 −∞J=0 t →+∞

(3.24)

f

with Z[0] = 1. The functional Z[J] describes the processes relative to the unperturbed (J = 0) time-development. The numerator in (3.24) is a transition amplitude and can therefore be written as a path integral xf + ∞|xi − ∞J =

Dx

Dp e

i h ¯

+∞

[px−H(p,x)+¯ ˙ hJ(t)x] dt

−∞

.

(3.25)

If H is quadratic in p and of the form H = p2 /(2m)+V (x), the propagator can be rewritten as (see Sect. 1.3) xf + ∞|xi − ∞J = N

Dx e

i h ¯

+∞

[L(x,x)+¯ ˙ hJ(t)x] dt

−∞

.

(3.26)

In the normalized functional Z[J] the (infinite) factor N cancels out because it is independent of J Z[J] =

W [J] 0 +∞|0 −∞J = W [0] 0 +∞|0 −∞J=0

=

Dx e

i h ¯

+∞

[L(x,x)+¯ ˙ hJ(t)x] dt

−∞

Dx e

i h ¯

+∞ −∞

[L(x,x)] ˙ dt

(3.27)


3.2

38

Functional Derivatives of Transition Amplitudes

In this section we will show – by using the methods outlined in App. B – that the groundstate expectation value of a time-ordered product of interaction operators – in this present case of the operators xˆ(t) – can be obtained as functional derivatives of the functional W [J] with respect to J. We start with the definition of a path integral as a limit of a finite dimensional integral (see (1.33)) xf tf |xi ti J =

n

lim

n→∞

dxk

n dpl

k=1

l=0

⎛

(3.28)

2π¯ h

⎞

n i [pj (xj+1 − xj ) − ηH (pj , xj ) + h ¯ Jj xj ]⎠ . × exp ⎝ h ¯ j=0

and calculate its functional derivative with respect to J. In order to become familiar with functional derivatives we do this in quite some detail. Using the definition (B.26) for the functional derivative we get (with the abbreviation F (xj , pj ) = pj (xj+1 − xj ) − ηH(pj , x¯j )) δxf tf |xi ti J δJ(t1 ) )

n n 1 dpl = lim lim dxk ε→0 ε n→∞ h k=1 l=0 2π¯ ⎡

(3.29)

⎛

⎞

n i × ⎣exp ⎝ {F (xj , pj ) + h ¯ xj [Jj + εδ(tj − t1 )]}⎠ h ¯ j=0

⎞⎤⎫

⎛

n ⎬ i − exp ⎝ (F (xj , pj ) + h ¯ xj Jj )⎠⎦ ⎭ h ¯ j=0

=

lim

n→∞

n

dxk

k=1

⎛

l dpl

⎞

n i (F (xj , pj ) + h ¯ xj Jj )⎠ , ix1 exp ⎝ 2π¯ h h ¯ j=0 l=0

which can be written as δxf tf |xi ti J δJ(t1 )

= i

Dx

Dp x(t1 )e

i h ¯

tf ti


.

(3.30)

We now want to relate this derivative of a classical functional to quantum mechanical expressions and thus understand its physical significance and


39

meaning. In order to do so, we go back to the definition of the propagator. There we had (in (1.23)) xf , tf |xi ti

(3.31)

dx1 . . . dxn xf tf |xn , tn xn tn |xn−1 tn−1 . . . x1 t1 |xi ti .

=

Now, in (3.30) for J = 0, we have one factor more on the righthand side of this equation

dx1 . . . dxn xf tf |xn tn . . . x1 t1 |xi ti x1

=

dx1 . . . dxn xf tf |xn tn . . . x1 t1 |ˆ x(t1 )|xi ti ;

(3.32)

the last step is possible because |x1 is an eigenstate of the xˆ operator with eigenvalue x1 (cf. (1.18)). The last integral is obviously equal to xf tf |ˆ x(t1 )|xi ti ,

(3.33)

i.e. to a matrixelement of the position operator. We thus have . δxf tf |xi ti J .. . δJ(t1 ) .

= i

Dx

Dp x(t1 )e

i h ¯

tf

[px−H(x,p)] ˙ dt

ti

J=0

= ixf tf |ˆ x(t1 )|xi ti .

(3.34)

For the case that H is separable in x and p and quadratic in p, this relation reads . δxf tf |xi ti J .. . δJ(t1 ) .

= iN

Dx x(t1 )e

i h ¯

tf

L(x,x) ˙ dt

ti

J=0

= ixf tf |ˆ x(t1 )|xi ti .

(3.35)

The functional derivative of the propagator with respect to the source thus gives the transition matrix element of the coordinate x. The higher order functional derivatives yield δ n xf tf |xi ti J δJ(t1 )δJ(t2 ) . . . δJ(tn ) n

= (i)

Dx

Dp x(t1 )x(t2 ) . . . x(tn ) e

(3.36) i h ¯

tf ti


.


40

One might guess that .

. δ n xf tf |xi ti J . . = in xf tf |ˆ x(t1 )ˆ x(t2 ) . . . xˆ(tn )|xi ti , δJ(t1 )δJ(t2 ) . . . δJ(tn ) .J=0

(3.37)

but this equation is not quite correct. We see this by considering explicitly the second derivative. We can proceed there exactly in the same way as for the first. We have for the rhs of (3.36) in the case of the second derivative . δ xf tf |xi ti J .. . δJ(tα )δJ(tβ ) .

2

= i2

J=0

= i2

Dx

Dp x(tα )x(tβ )e

i h ¯

tf

[px−H(x,p)] ˙ dt

ti

dxi . . . dxn xf tf |xn tn · · · xl tl |ˆ x(tα )|xl−1 tl−1

x(tβ )|xk−1 tk−1 · · · x1 t1 |xi ti . · · · xk tk |ˆ

(3.38)

Here we have assumed that tα > tβ , since each of the infinitesimal Green’s functions propagates only forward in time. In this case (3.38) is indeed equal to x(tα )ˆ x(tβ )|xi ti . (3.39) i2 xf tf |ˆ However, if tα < tβ , then these two times appear in a different ordering in the rhs of (3.38) and thus of the matrix element. The two cases can be combined by introducing the time-ordering operator T )

x(t2 )] = T [ˆ x(t1 )ˆ

xˆ(t1 )ˆ x(t2 ) xˆ(t2 )ˆ x(t1 )

t1 > t2 t2 > t1 .

(3.40)

With the time-ordering operator we have . δ 2 xf tf |xi ti J .. . δJ(t1 )δJ(t2 ) .

2

= i

Dx

Dp x(t1 )x(t2 )e

i h ¯

tf

[px−H(x,p)] ˙ dt

ti

J=0

= i2 xf tf |T [ˆ x(t1 )ˆ x(t2 )] |xi ti .

(3.41)

The same reasoning leads to the following result for higher-order derivatives n 1

i

=

.

. δ n xf tf |xi ti J . . δJ(t1 )J(t2 ) . . . δJ(tn ) .J=0

Dx

Dp x(t1 )x(t2 ) . . . x(tn ) e

i h ¯

tf

[px−H(x,p)] ˙ dt

ti

= xf tf |T [ˆ x(t1 )ˆ x(t2 ) . . . xˆ(tn )] |xi ti .

(3.42)


41

This is the generalization of (3.41). We thus have n

1 i

n

1 i

=

=

.

. δn Z[J]... δJ(t1 )δJ(t2 ) . . . δJ(tn ) J=0

Dx Dp x(t1 )x(t2 ) . . . x(tn ) e

=

.

δn xf tf |xi ti J .. . δJ(t1 )δJ(t2 ) . . . δJ(tn ) xf tf |xi ti J=0 .J=0

Dx Dp e

i h ¯

tf

i h ¯

tf

[px−H(x,p)] ˙ dt

ti

[px−H(x,p)] ˙ dt

ti

x(t1 )ˆ x(t2 ) . . . xˆ(tn )] |xi ti xf tf |T [ˆ , xf tf |xi ti

(3.43)

where the limit ti → −∞, tf → +∞ is understood and all the times t1 , . . . , tn lie in between these limits. If the Hamiltonian is quadratic in p and separates in p and x, then we have x(t1 )ˆ x(t2 ) . . . xˆ(tn )] |xi ti xf tf |T [ˆ = xf tf |xi ti

i

Dx x(t1 )x(t2 ) . . . x(tn ) e h¯ S[x(t)]

i

Dx e h¯ S[x(t)]

,

(3.44) where S[x(t)] is the action that depends functionally on the trajectory x(t). We now rewrite this equation. The numerator of the lhs becomes (in the limit ti → −∞, tf → +∞, indicated by the arrow) xf tf |T [. . .]|xi ti J=0 −→ xf tf |00|T [. . .]|00|xi ti J=0 = 0|T [. . .]|0 ϕ0 (xf )e− h¯ E0 tf ϕ∗0 (xi )e+ h¯ E0 ti , i

i

(3.45)

while the denominator can be written as (c.f. (3.18)) ˆ

ˆ

xf tf |xi ti = xf |e− h¯ Htf e+ h¯ Hti |xi −→ ϕ0 (xf )e− h¯ E0 tf ϕ∗0 (xi )e+ h¯ E0 ti i

i

i

i

(3.46)

where we have used in both cases (1.19), inserted a complete set of states and – through the limit of infinite times – projected out the groundstate. The gs wavefunctions and the time-dependent exponentials cancel out so that we obtain (for quadratic Hamiltonians)

x(t2 ) . . . xˆ(tn )] |0 = 0|T [ˆ x(t1 )ˆ

i

Dx x(t1 )x(t2 ) . . . x(tn ) e h¯ S[x(t)]

i

Dx e h¯ S[x(t)] . n . 1 δn = Z[J]... i δJ(t1 )δJ(t2 ) . . . δJ(tn ) J=0 (3.47)


42

+∞ with S[x(t)] = −∞ L(x(t), x(t)) ˙ dt. This is a very important result. It shows that the groundstate expectation value of a time-ordered product of position operators, the so-called correlation function, can be obtained as a functional derivative of the functional Z[J] ˆ as can be seen defined in (3.24). Note that |0 is the groundstate of H, from (3.45). Thus the groundstate appearing on the lhs of (3.47) and the propagator Z[J] are linked together: if Z[J] contains, for example, only a free Hamiltonian, then |0 is the groundstate of a free theory. If, on the other hand, Z[J] contains interactions, then |0 is the groundstate of the full interacting theory. Since all the correlation functions can be generated from Z[J] this functional is called a generating functional. For the remainder of this book we will be concerned with these generating functionals. Coming back to our example of the driven harmonic oscillator, discussed at the start of this section, we see that the time-ordered vacuum expectation values of xˆ are just the matrix elements that would appear in a timedependent perturbation theory treatment of the groundstate of this system. Thus, if all the correlation functions are known, then the perturbation series expansion is also known.

Part II Relativistic Quantum Field Theory

43

Chapter 4 CLASSICAL RELATIVISTIC FIELDS In this chapter a few essential facts of classical relativistic field theory are summarized. It will first be shown how to derive the equations of motion of a field theory, for example the Maxwell equations of electrodynamics, from a Lagrangian. Second, the connection between symmetries of the Lagrangian and conservation laws will be discussed1 .

4.1

Equations of Motion

The equations of motion of classical mechanics can be obtained from a Lagrange function by using Hamilton’s principle that the action for a given mechanical system is stationary for the physical space–time development of the system. The equations of motion for fields that determine their space–time dependence can be obtained in an analogous way by identifying the field amplitudes at a coordinate x with the dynamical variables (coordinates) of the theory. Let the functions that describe the fields be denoted by Φα (x) with xμ = (t, x) ,

(4.1)

where α labels the various fields appearing in a theory. The Lagrangian L of the system is expressed in terms of a Lagrange density L, as follows:

L=

L(Φα , ∂μ Φα ) d3 x

1

(4.2)

The units, the metric and the notation used in this and the following chapters is explained in Appendix A.

44

CHAPTER 4. CLASSICAL RELATIVISTIC FIELDS

45

where the spatial integration is performed over the volume of the system. The action S is then defined as usual by t1

L dt =

S= t0

L(Φα , ∂μ Φα ) d4 x

(4.3)

Ω

with the Lorentz-invariant four-dimensional volume element d4 x = d3 x dt. The, in general finite, space–time volume of the system is denoted by Ω. As pointed out before the fields Φα (x) play the same role as the generalized coordinates qi in classical mechanics; the analogy here is such that the fields Φα correspond to the coordinates q and the points x and α to the classical indices i. The corresponding velocities are given in a direct analogy by the time derivatives of Φα : ∂t Φα . Lorentz covariance then requires that also the derivatives with respect to the first three coordinates appear; this explains the presence of the four-gradients ∂μ Φα in (4.2). In order to derive the field equations from the action S by Hamilton’s principle, we now vary the fields and their derivatives Φα → Φα = Φα + δΦα ∂μ Φα → (∂μ Φα ) = ∂μ Φα + δ(∂μ Φα ) .

(4.4)

This yields δL = L(Φα , (∂μ Φα ) ) − L(Φα , ∂μ Φα ) ∂L ∂L δΦα + δ(∂μ Φα ) = ∂Φα ∂(∂μ Φα ) ∂L ∂L = δΦα + ∂μ (δΦα ) . ∂Φα ∂(∂μ Φα )

(4.5)

According to the Einstein convention a summation over μ is implicitly contained in this expression. In going from the second to the third line differentiation and variation could be commuted because both are linear operations. The equations of motion are now obtained from the variational principle

δS = Ω

∂L ∂L ∂L − ∂μ δΦα + ∂μ δΦα ∂Φα ∂(∂μ Φα ) ∂(∂μ Φα )

= 0

d4 x (4.6)

for arbitrary variations δΦα under the constraint that δΦα (t0 ) = δΦα (t1 ) = 0 ,


46

where t0 and t1 are the time-like boundaries of the four-volume Ω. The last term in (4.6) can be converted into a surface integral by using Gauss’s law; for fields which are localized in space this surface integral vanishes if the surface is moved out to infinity. Since the variations δΦα are arbitrary the condition δS = 0 leads to the equations of motion ∂ ∂xμ

∂L ∂L − =0. ∂(∂μ Φα ) ∂Φα

(4.7)

The relativistic equivalence principle demands that these equations have the same form in every inertial frame of reference, i.e. that they are Lorentz covariant. This is only possible if L is a Lorentz scalar, i.e. if it has the same functional dependence on the fields and their derivatives in each reference frame. In a further analogy to classical mechanics, the canonical field momentum is defined as ∂L ∂L = . (4.8) Πα = ˙ ∂(∂0 Φα ) ∂ Φα From L and Πα the Hamiltonian H is obtained as

H=

Hd x= 3

(Πα Φ˙ α − L) d3 x .

(4.9)

The Hamiltonian H represents the energy of the field configuration.

4.1.1

Examples

The following sections contain examples of classical field theories and their formulation within the Lagrangian formalism just introduced. We start out with the probably best-known case of classical electrodynamics, then generalize it to a treatment of massive vector fields and then move on to a discussion of classical Klein–Gordon and Dirac fields that will play a major role in the later chapters of this book.

Electrodynamics The best-known classical field theory is probably that of electrodynamics, in which the Maxwell equations are the equations of motion. The two homogeneous Maxwell equations allow us to rewrite the fields in terms of a four-potential , (4.10) Aμ = (A0 , A)


47

defined via = ∇ ×A , B = −∇A 0 − ∂A . E (4.11) ∂t Note that the 2 homogeneous Maxwell equations are now automatically fulfilled. The two inhomogeneous Maxwell equations2 ·E = ∇ ×B − ∂E = ∇ ∂t with external density ρ and current j can be ∂μ F μν =

ρ, j

(4.12)

rewritten as

∂F μν = jν ∂xμ

,

(4.13)

with the four-current j ν = (ρ, j)

(4.14)

and the antisymmetric field tensor F μν =

∂Aν ∂Aμ − = ∂ μ Aν − ∂ ν Aμ . ∂xμ ∂xν

(4.15)

The tensor Fμν is the dyadic product of two fourvectors and thus a Lorentztensor. Equation (4.13) is the equation of motion for the field tensor or the four-vector field Aμ . It is easy to show that (4.13) can be obtained from the Lagrangian 1 L = − Fμν F μν − j ν Aν 4

(4.16)

by using (4.7); the fields Aν here play the role of the fields Φα in (4.7). We have ∂L = −j ν , ∂Aν 1 ∂L (4.17) = − 2(+F μν − F νμ ) = −F μν . ∂(∂μ Aν ) 4 The last step is possible because F is an antisymmetric tensor. The equation of motion is therefore ∂ ∂xμ 2

∂L ∂F μν ∂L − =− + jν = 0 , μ ∂(∂μ Aν ) ∂Aν ∂x

Here the Heaviside units are used with c = 1 and 0 = μ0 = 1.

(4.18)


48

in agreement with (4.13). It is now easy to interpret the two terms in L (4.16): the first one gives the Lagrangian for the free electromagnetic field, whereas the second describes the interaction of the field with charges and currents. The two homogeneous Maxwell equations can also be expressed in terms of the field tensor by first introducing the dual field tensor 1 F˜ μν = μνρσ Fρσ ; 2

(4.19)

here μνρσ is the Levi–Civita antisymmetric tensor which assumes the values +1 or −1 according to whether (μνρσ) is an even or odd permutation of (0,1,2,3), and 0 otherwise. In terms of F˜ the homogeneous Maxwell equations read (4.20) ∂μ F˜ μν = 0 . Eqs. (4.13) and (4.20) represent the Maxwell equations in a manifestly covariant form. The Lagrangian (4.16) is obviously Lorentz-invariant since it consists of invariant contractions of two Lorentz-tensors (the first term) and two Lorentz-vectors (the second term). It is also invariant under a gauge transformation (4.21) Aμ −→ Aμ = Aμ + ∂ μ φ because F itself is gauge-invariant by construction and the contribution of the interaction term to the action is gauge-invariant for an external conserved current. The same then holds for the equation of motion (4.18). Symmetry (Lorentz-invariance), gauge-invariance and simplicity (there are no higher order terms in (4.16)) thus determine the Lagrangian of electrodynamics. The gauge freedom can be used to impose constraints on the four components of the vector field Aμ . In addition, for free fields this gauge freedom can be used, for example, to set the 0th component of the four-potential equal to zero. Thus, a free electromagnetic field has only two degrees of freedom left.

Massive Vector Fields Vector fields in which – in contrast to the electromagnetic field – the field quanta are massive are described by the so-called Proca equation: ∂μ F μν + m2 Aν = j ν .

(4.22)

Operating on this equation with the four-divergence ∂ν gives, because F is antisymmetric, (4.23) m2 ∂ν Aν = ∂ν j ν .


49

For m = 0 and a conserved current, this reduces the equation of motion (4.22) to 2 + m2 Aν = j ν ; ∂ν Aν = 0 . (4.24) Thus for massive vector fields the freedom to make gauge transformations on the vector field is lost. The condition of vanishing four-divergence of the field reduces the degrees of freedom of the field from 4 to 3. The space-like components represent the physical degrees of freedom. The Lagrangian that leads to (4.22) is given by 1 1 L = − F 2 + m2 A2 − j · A . 4 2

(4.25)

Klein–Gordon Fields A particularly simple example is provided by the so-called Klein–Gordon field φ that obeys the equation of motion

∂μ ∂ μ + m 2 φ = 2 + m 2 φ = 0 ;

(4.26)

such a field describes scalar particles, i.e. particles without intrinsic spin. The Lagrangian leading to (4.26) is given by L= since we have

and

1 (∂μ φ) (∂ μ φ) − m2 φ2 2

(4.27)

∂L = ∂ μφ ∂ (∂μ φ)

(4.28)

∂L = −m2 φ . ∂φ

(4.29)

It is essential to note here that the Lagrangian density (4.27) that leads to (4.26) is not unique; unique is only the action S = L d4x. For localized fields for which the surface contributions vanish we can perform a partial integration of the kinetic term in (4.26)

so that we obtain

d4x (∂μ φ) (∂ μ φ) = −

d4x φ2φ

(4.30)

1 (4.31) L = − φ 2 + m2 φ . 2 The Lagrangians (4.31) and (4.27) are equivalent. Since both give the same action they also lead to the same equation of motion (4.26).


50

An interesting case occurs if we consider two independent real scalar fields, φ1 and φ2 , with the same mass m. The total Lagrangian is then simply given by a sum over the Lagrangians describing the individual fields, i.e. L=

1 1 (∂μ φ1 ) (∂ μ φ1 ) − m2 φ21 + (∂μ φ2 ) (∂ μ φ2 ) − m2 φ22 . 2 2

(4.32)

On the other hand, we can also construct two complex fields from the two real fields φ1 and φ2 , namely 1 φ = √ (φ1 + iφ2 ) 2

(4.33)

and its complex conjugate. In terms of these the Lagrangian (4.32) can be rewritten to L = (∂μ φ)∗ (∂ μ φ) − m2 φ∗ φ =

− φ∗ 2 + m2 φ .

(4.34)

Dirac Fields A particularly simple example is provided by the Dirac field Ψ for which the equation of motion is just the Dirac equation (iγ μ ∂μ − m) Ψ = 0 ,

(4.35)

where the γμ are the usual (4 × 4) matrices of Dirac theory. Ψ itself is a (4 × 1) matrix of four independent fields, a so-called spinor. The corresponding Lagrangian is given by ¯ (iγ μ ∂μ − m) Ψ . L=Ψ

(4.36)

This can be seen by identifying the fields Φα in (4.7) with the four components ¯ the equation ¯ = Ψ† γ0 . Since L does not depend on ∂μ Ψ of the Dirac spinor Ψ of motion is simply given by ∂L μ ¯ = (iγ ∂μ − m) Ψ = 0 . ∂Ψ

4.2

(4.37)

Symmetries and Conservation Laws

As in classical mechanics there is also in field theory a conservation law associated with each continuous symmetry of L. The theorem which describes


51

the connection between the invariance of the Lagrangian under a continuous symmetry transformation and the related conserved current is known as Noether’s theorem. In the following, this will be illustrated for different types of symmetries which then lead to the well-known conservation laws. The common expression in the arguments to follow is the change of the Lagrangian density under a change of the fields and their derivatives (see (4.4)). According to (4.5) and the Lagrange equations of motion (4.7) this change is given by ∂L (4.38) δΦα . δL = ∂μ ∂(∂μ Φα )

4.2.1

Geometrical Space–Time Symmetries

In this section we investigate the consequences of translations in four-dimensional space–time, i.e. infinitesimal transformations of the form xν → xν = xν + ν

,

(4.39)

where ν is a constant infinitesimal shift of the coordinate xν . Under such transformations the change of L is given by δL = ν

∂L = ν ∂ ν L , ∂xν

(4.40)

since L is a scalar. If now L is required to be form-invariant under translations, it does not explicitly depend on xν . In this case, δL is also given by (4.38). The changes of the fields Φα appearing there are for the space–time translation considered here given by ∂Φα = ν ∂ ν Φα . (4.41) δΦα = ν ∂xν Inserting (4.41) into (4.38) yields

δL = ν ∂μ

∂L ∂ ν Φα ∂(∂μ Φα )

.

(4.42)

=0 ,

(4.43)

Equating (4.42) and (4.40) finally gives

∂μ

∂L ∂ ν Φα − L g μν ∂(∂μ Φα )

since the ν are arbitrary. By defining the tensor T μν as T μν ≡

∂L ∂ ν Φα − L g μν ∂(∂μ Φα )

(4.44)


52

this equation reads ∂μ T μν = 0 .

(4.45)

Relation (4.45) has the form of a continuity equation. Spatial integration over a finite volume yields ⎛

⎞

/ d ⎝ 0μ ∂T iμ 3 (μ) · n dS . S T (x)d3 x⎠ = − d x = − dt ∂xi V

V

(4.46)

S

(μ) Here n is a unit vector vertical on the surface S pointing outwards and S is a three-vector: (μ) = (T 1μ , T 2μ , T 3μ ) . S (4.47) The surface integral on the rhs of (4.46) is taken over the surface S of volume V. For localized fields it can be made to vanish by extending the volume towards infinity. It is then evident that the quantities

Pμ =

T 0μ d3 x

(4.48)

are conserved. These are the components of the four-momentum of the field, as can be verified for the zeroth component,

P

0

=

T

=

00

3

d x=

∂L ∂ 0 Φα − L d3 x ∂(∂0 Φα )

(Πα Φ˙ α − L) d3 x = H

,

(4.49)

according to (4.8) and (4.9). The spatial components of the field momentum are ∂L k 0k 3 P = T d x= (4.50) ∂ k Φα d3 x . ∂(∂0 Φα ) T μν , as defined in (4.44), has no specific symmetry properties. It can, however, always be made symmetric in its Lorentz-indices because (4.45) does not define the tensor T uniquely. We can always add a term of the form ∂λ Dλμν , where Dλμν is a tensor antisymmetric in the indices λ and μ, such that T becomes symmetric.3 3

In classical mechanics the form invariance of the Lagrangian under rotations leads to the conservation of angular momentum. Analogously, in a relativistic field theory the form invariance of L under four-dimensional space–time rotations (Lorentz covariance) leads to the conservation of a quantity that is identified with the angular momentum of the field. To obtain the same form for the angular momentum as in classical mechanics it is essential that Tμν is symmetric.


53

Comparing (4.46) with (4.50) and assuming T to be symmetric we see that (μ) in (4.47) describe the momentum the normal components of the vectors S flow through the surface S of the volume V and thus determine the “radiation pressure” of the field.4 These properties allow us to identify Tμν as the energymomentum tensor of the field. For the Lagrangian (4.16) of electrodynamics Tμν is just the well-known Maxwell’s stress tensor. As already mentioned at the beginning of this chapter these conservation laws are special cases of Noether’s theorem, which can be summarized for the general case as follows: Each continuous symmetry transformation that leaves the Lagrangian invariant is associated with a conserved current. The spatial integral over this current’s zeroth component yields a conserved charge.

4.2.2

Internal Symmetries

Relativistic field theories may contain conservation laws that are not consequences of space–time symmetries of the Lagrangian, but instead are connected with symmetries in the internal degrees of freedom such as, e.g., isospin or charge. We therefore now allow for a mixture of the different fields under the transformation (4.51) Φα (x) → Φα (x) = e−iεqαβ Φβ , where is an infinitesimal parameter and the qαβ are fixed c-numbers. We then have (4.52) δΦα (x) = Φα (x) − Φα (x) = −iεqαβ Φβ (x) . The change of the Lagrangian is given by (4.38)

δL = ∂μ

∂L δΦα ∂(∂μ Φα )

.

(4.53)

If L is invariant under this variation δΦα , then we have

δL = ∂μ

∂L δΦα = 0 . ∂(∂μ Φα )

(4.54)

Equation (4.54) is in the form of a continuity equation for the “current” j μ (x) = 4

1 ∂L δΦα . ∂(∂μ Φα ) ε

(4.55)

More precisely, S k(μ) denotes the flux of the μth component of the field momentum in the direction xk .


54

Inserting the field variations δΦα yields for the current j μ (x) = −i

∂L qαβ Φβ . ∂(∂μ Φα )

(4.56)

Equations (4.54) and (4.56) imply that the “charge”

Q=

j (x) d x = −i 0

3

∂L qαβ Φβ d3 x ∂(∂0 Φα )

(4.57)

of the system is conserved. The physical nature of these “charges” and “currents” has to remain open. It depends on the specific form of the symmetry transformation (4.63) and can be determined only by coupling the system to external fields.

Example: Quantum Electrodynamics To illustrate this conservation law, the theory of electromagnetic interactions is used as an example. However, in contrast to the considerations in Sect. 4.1.1 we now consider a coupled system of a fermion field Ψ(x) and the electromagnetic field Aμ (x) to determine the physical meaning of the conserved current. Together with a quantization procedure this theory is called Quantum Electrodynamics (QED). The Lagrangian is given by 1 ¯ [iγ μ (∂μ + ieAμ ) − m] Ψ . L = − Fμν F μν + Ψ 4

(4.58)

L contains a part that describes the free electromagnetic field (first term). The second term describes the fermion Lagrangian; it is obtained from the free particle Lagrangian of (4.36) by replacing the derivative ∂μ by the covariant derivative (4.59) Dμ = ∂μ + ieAμ (minimal coupling). Here e is the electron’s charge (e = −|e|). The Lagrangian (4.58) is obviously invariant under a variation of the fermion fields of the form Ψ → Ψ = e−ie Ψ .

(4.60)

Comparison with (4.51) gives qαβ = e δαβ so that the conserved “current” given by (4.56) is: ¯ μΨ . (4.61) jμ (x) = e Ψγ Note that this conserved current is exactly the quantity that couples to the electromagnetic field in (4.58). This property allows one to identify the current (4.61) as the electromagnetic current of the electron fields.


55

Example: Scalar Electrodynamics The Lagrangian for the case of a complex scalar field interacting with an electromagnetic field is given by 1 L = − Fμν F μν + (Dμ φ)∗ (Dμ φ) − m2 φ∗ φ . 4

(4.62)

This Lagrangian is simply the sum of the free electromagnetic Lagrangian (4.16) and the Lagrangian for a complex scalar field (4.34), where in the latter again the derivative ∂μ has been replaced – through minimal substitution – by the covariant derivative Dμ (4.59). The Lagrangian (4.62) is obviously invariant under the phase transformations φ(x) −→ e−iεe φ(x) φ∗ (x) −→ e+iεe φ∗ (x)

(4.63)

The conserved current connected with this invariance can be obtained from the definition (4.56)

j

μ

∂L ∂L = −i eφ + (−e)φ∗ ∗ ∂ (∂μ φ) ∂ (∂μ φ ) ∗ μ μ = ie (φ D φ − φ (D φ)∗ ) .

(4.64)

The conserved charge is then given by

Q=

3

0

d x j (x) = ie

∗

d3 x φ∗ D0 φ − φ D0 φ

.

(4.65)

It is remarkable that now the electromagnetic field Aμ appears in the conserved current (through the covariant derivative Dμ = ∂ μ + ieAμ ). Again the conserved current provides the coupling to the electromagnetic field. If the scalar field is real then it is invariant under the transformation (4.63) only for e = 0. Eq. (4.65) then shows that in this case the conserved charge Q = 0.

Chapter 5 PATH INTEGRALS FOR SCALAR FIELDS In this chapter we apply the methods developed in chapter 3 to the case of scalar fields. There we showed that the vacuum expectation values of timeordered products of xˆ operators could be obtained as functional derivatives of a generating functional. All these results can be taken over into field theory remembering that fields play the role of the coordinates of the theory and the spatial locations x correspond to the indices of the classical coordinates. This implies that we can obtain the vacuum expectation values of time-ordered field operators by performing the derivatives on an appropriate functional. To discuss this functional is the main purpose of this chapter. It is easy to see that these vacuum expectation values of time-ordered products of field operators play an important role in quantum field theory. Each field operator creates or annihilates particles and a time-ordered product of field operators can thus describe the probability amplitude for a physical process in which particles are created and annihilated. The quantitative information about such a process is contained in the S matrix which can be obtained from the vacuum expectation values of time-ordered field operators by means of the so-called reduction theorem which we will derive in a later chapter. The rest of this manuscript will therefore be concerned with calculating these expectation values and with developing perturbative methods for their determination when an exact calculation is not possible.

56

CHAPTER 5. PATH INTEGRALS FOR SCALAR FIELDS

5.1

57

Generating Functional for Fields

We assume that the system is described by a Lagrangian of the form L (φ, ∂μ φ) =

1 μ ∂ φ∂μ φ − m2 φ2 − V (φ) . 2

(5.1)

In order to obtain the functional W [J] for fields we note that the fields play the role of the coordinates of the theory and that sums over the different coordinates have to be replaced by integrals over the space-time coordinates. In order to define more stringently what is actually meant by a path integral for fields we write it down here in detail for a free field Lagrangian (V = 0). In order to do so we bring it into a form as close as possible to the classical definition (1.38). We first Fourier-expand the field 1 qk (t)eik·x . φ(x, t) = √ V k For a real field φ we have

∗ q− k = qk .

(5.2)

(5.3)

Inserting this expansion into the free Lagrangian gives1

1 μ L = Ld x = ∂ φ∂μ φ − m2 φ2 d3x 2 " 1 ! = qk qk k · k − m2 + q˙k q˙k ei(k+k )·x d3x . 2V kk 3

(5.4)

Integration over d3x gives V δk,−k so that we have 1 q˙k q˙−k − (k 2 + m2 )qk q−k 2 k 1 = (q˙k q˙k∗ − ωk qk qk∗ ) 2 k

L =

(5.5)

with ωk = k 2 + m2 . Using (5.3), writing qk = Xk + iYk , and grouping then the coordinates Xk and −iYk into a new vector xk gives L= 1

1 2 x˙ k − ωk2 x2k . 2 k

For ease of notation we drop the vector arrows in the indices

(5.6)


58

In the general spirit of field theory we introduce a source density J which we also expand 1 jk (t)eik·x . (5.7) J(x, t) = √ V k With this expansion we obtain for the source term

d3x J(x)φ(x) =

jk (t)q−k (t) =

k

Jk (t)xk (t)

(5.8)

k

where we have grouped jk and −ijk into a new vector Jk . The Lagrangian (5.6) has the structure of a Lagrangian with quadratic momentum dependence and constant coefficient, discussed in Sect. 1.3.1; also the souce term reads formally just the same as for the nonrelativistic systems treated in chapter 3. We can thus again integrate the momentum dependence out and obtain for the vacuum-to-vacuum transition amplitude (cf. (3.23)) W [J] =

lim 0tf |0ti J

tf →+∞

ti →−∞

1 = lim η→0 2π¯ hiη

n+1 n 2

with Ll = L (xl , x˙ l ) =

iη

dxkj e

n -

(Ll +Jk xkl )

l=0

,

(5.9)

k j=1

1 2 x˙ kl − ωk2 x2kl , 2 k

(5.10)

where the first index (k) denotes the coordinate and the second (j) the time interval. If we now identify the integration measure as

1 Dφ = lim η→0 2π¯ hiη

n+1 2

n

dxkj ,

(5.11)

k j=1

we can write the generating functional also as

" 1! W [J] = Dφ exp i ∂μ φ∂ μ φ − m2 − iε φ2 + Jφ d4x 2 & ' ε 2 4 = Dφ exp i L (φ, ∂μ φ) + Jφ + i φ d x . (5.12) 2

Here the volume element d4x is given by the Lorentz-invariant expression d4x = d3x dt .

(5.13)

The term iεφ2 /2 with positive ε has been introduced in an ad hoc manner to ensure the convergence of W when taking the fields to infinity with the


59

understanding that ultimately ε has to be taken to 0 (cf. the discussion at the end of Sect. (3.1)). The second line of (5.12) also gives the generating functional for an interacting theory with the interaction V included in L. This can be easily proven by including an additional interaction in the Lagrangian (5.4) and Fourier-expanding it. The generating functional for a scalar field theory is thus given by

W [J] =

Dφ e−i

d4x[ 12 φ(2+m2 −iε)φ+V (φ)−Jφ]

(5.14)

where the Lagrangian in the form (4.31) has been used. In analogy to (3.24) we define a normalized functional Z[J] =

W [J] . W [0]

(5.15)

Again the infinite normalization factor inherent in W [J] cancels out in this definition. Since W [J] involves an exponential it will often be convenient in the following discussions to introduce its logarithm iS[J] which is itself a functional of J W [J] ≡ eiS[J] =⇒ S[J] = −i ln W [J] = −i ln Z[J] − i ln W [0] .

5.1.1

(5.16)

Euclidean Representation

In the preceding section we have ensured convergence of the generating functional by introducing the ε-dependent term in the energies. In this subsection we go back to the alternative method that relies on the Wick rotation that we introduced in Sect. 3.21 and discuss the Euclidean representation of the generating functional. This discussion also illustrates in some more detail the remarks on integrating oscillatory functions made at the end of Sect. 1.3.1. The real Euclidean space is obtained from Minkowski space by rotating the real axis in the x0 plane by δ = −π/2 into the negative imaginary axis (Wick rotation). We denote a space-time point in Euclidean space by xE ; it is related to the usual space-time point x in Minkowski space by xE = (x, x4 )

with x4 = ix0 = it .

(5.17)

Under the Wick rotation t → −it and x4 thus becomes real. With this definition we can extend the usual Minkowski-space definitions of volume element and space-time distance to Euclidean space d4 xE ≡ d3 x dx4 = d3 x idt = id4 x dx2E =

3 j=1

dx2j + x24 = −dx2 .

(5.18)


60

The d’Alembert operator is then given by 2=

4 2 ∂2 ∂2 2 =− ∂ −∇ 2=− − ∇ ≡ −2E . 2 ∂t2 ∂x24 a=1 ∂xa

(5.19)

With these transformations the generating functional for a free scalar field in Minkowski space (5.14) becomes in its Euclidean representation

WE0 [J] =

Dφ e−

d4xE { 12 [(∂E φ)2 +m2 φ2 ]−Jφ}

,

(5.20)

with (∂E φ)2 = − (∂φ)2 . Because x4 is now real, (∂E φ)2 is always positive and the exponent is negative definite; the integral thus converges and is welldefined even without adding in the ε-dependent term. Since the exponent is furthermore quadratic in the fields, WE0 [J] can be evaluated by using the techniques for Gaussian integrals that are explained in App. B. Physical results are then obtained by rotating backwards after all integrations have been performed. Remembering that in field theory the fields play the role of the coordinates of a Lagrangian theory we can now directly generalize some of the results of Chapt. 3 to field theory. In particular, we have that WE [J] of eq. (5.20) is the transition amplitude from the vacuum state of the free theory at t → −∞ to that at t → +∞ under the influence of the external source J (cf. (3.27) so that the normalized transition amplitude ZE is given by ZE0 [J] =

WE0 [J] 0 + ∞|0 − ∞J , = 0 WE [0] 0 + ∞|0 − ∞0

(5.21)

where |0 is the vacuum state of the free theory. Eq. (5.20) shows that the normalized transition amplitude can be understood as an integration of the source action exp (+ d4xE Jφ) with the weights wE0 (φ) = e− 2 1

over all fields φ

ZE0 [J] =

d4xE [(∂E φ)2 +m2 φ2 ]

(5.22)

4 Dφ wE0 (φ)e d xE Jφ Dφ wE0 (φ)

.

(5.23)

Eq. (3.47) then shows that for a function of the fields O(φ)

Dφ wE0 (φ) j Oj (φ(xj )) ˆ j ))]|0 . = 0|T [ Oj (φ(x Dφ wE0 (φ) j

The field operators on the rhs here are those of free fields.

(5.24)


61

In the interacting case all these relations still hold if wE0 is replaced by a weight function for the interacting theory and the vacuum state is now that of the interacting theory which we will denote by |˜0. To obtain the weight function of the interacting theory we use a perturbative treatment of the interaction V . By writing in analogy to (5.22) wE (φ) = e− = e−

d4xE { 12 [(∂E φ)2 +m2 φ2 ]+V (φ)}

1

V (φ) d4xE − 2

e

[(∂E φ)2 +m2 φ2 ] d4xE ≡ e−

(5.25) V (φ) d4xE

0 (φ) wE

we get with (5.24)

˜0|T [

ˆ ˜0 = Oj (φ)]|

j

=

Dφ wE (φ)

j

Dφ wE0 (φ)e−

0|T [

Oj (φ) V (φ) d4xE

Dφ wE0 (φ)e−

=

ˆ − j Oj (φ0 )e

0|T [e−

V

j

(φ) d4x

Oj (φ) E

V (φˆ0 ) d4xE

V (φˆ0 ) d4xE

]|0

]|0

.

(5.26)

Here |0 on the rhs is the vacuum state of the non-interacting free theory (V = 0) and all the field operators on the rhs are free field operators φˆ0 at all times if they were free at t → −∞. This can be seen from the path integral which contains the free weight wE0 connected with free propagation. In the following chapters it is often useful to think in terms of this probabilistic interpretation even when we work in Minkowski metric. This is possible by relating Euclidean and Minkowski space by analytic continuation.

Chapter 6 EVALUATION OF PATH INTEGRALS Only a limited class of path integrals can be evaluated analytically [GroscheSteiner] so that one is forced to use either numerical or perturbative methods. In this section we first calculate the generating functional for free scalar fields, both by a direct reduction method and, to gain familiarity with this technique, with the the methods of Gaussian integration developed in App. B.2.1. We then show how more general types of path integrals can approximately be reduced to the Gaussian form. In this reduction we find a systematic method for a semiclassical expansion in terms of the Planck constant h ¯ . Finally, we briefly discuss methods for the numerical evaluation of path integrals.

6.1

Free Scalar Fields

The generating functional for a free scalar field theory plays a special role in the theory of path integrals. This is so because of two reasons: first, a free-field theory is the simplest possible field theory and, second, as we have just seen in the last section the effects of the interaction term V (φ) can be described with the help of perturbation theory in which the functional of the full, interacting theory is expanded around that of the free theory.

6.1.1

Generating functional

The representation of the generating functional of a free field theory (V = 0 in (5.14)) as a path integral is still quite cumbersome for practical applications. For these it would be very desirable if we could factorize out the functional dependence on J in form of a normal integral; the remaining path integral 62

CHAPTER 6. EVALUATION OF PATH INTEGRALS

63

would then disappear in the normalization. In the following we will therefore separate the path integral (5.9) into 2 factors, one depending on J and the other one being an integral over φ, i.e. just a number. For this purpose we start with the generating functional

W [J] =

Dφ e−i

d4x[ 12 φ(2+m2 −iε)φ−Jφ]

(6.1)

where the Lagrangian in the form (4.31) has been used. The field φ here is an integration variable; it does not fulfill a Klein-Gordon equation! We now introduce a field φ0 that does just that !

"

2 + (m2 − iε) φ0 (x) = J(x) .

(6.2)

J thus plays the role of a source to φ0 . We now take this field φ0 as a reference field and expand φ around it, setting φ = φ0 + φ . We thus obtain for the integrand in the exponent in (6.1) 1 (φ0 + φ ) 2 (φ0 + φ ) + (φ0 + φ ) m2 − iε (φ0 + φ ) − J (φ0 + φ ) 2 " 1 ! = φ 2 + m2 − iε φ 2 " " 1 ! 1 ! + φ0 2 + m2 − iε φ0 + φ0 2 + m2 − iε φ 2 2 " 1 ! 2 + φ 2 + m − iε φ0 − Jφ0 − Jφ . (6.3) 2

The third and the fourth terms on the rhs give the same contribution when integrated over. The second term gives, according to (6.2), 12 Jφ0 . Collecting all terms we therefore have for the action in (6.1)

" 1 ! S[φ, J] = − d x φ 2 + m2 − iε φ − Jφ 2 " 1 ! 1 = − d4x φ 2 + m2 − iε φ + Jφ0 2! 2 " 0 2 + φ 2 + m − iε φ0 − Jφ0 − Jφ . 4

(6.4)

In the last line of this equation we can again apply (6.2) to obtain " 0 1 4 1 ! d x φ 2 + m2 − iε φ − Jφ0 S[φ, J] = − 2

(6.5)

We are now very close to our aim to factorize out the J dependence. To do so we solve (6.2) by writing φ0 (x) = −

DF (x − y)J(y) d4y ,

(6.6)


64

where DF , the so-called Feynman propagator, fulfills the equation !

2 + m2 − iε

"

DF (x) = −δ 4 (x)

(6.7)

in complete analogy to the nonrelativistic propagator K in chapter 1. Using

1 δ (x) = 2π

4

4

we obtain DF (x − y) =

d4k e−ikx

1 e−ik(x−y) d4k . (2π)4 k 2 − m2 + iε

(6.8)

(6.9)

Substituting (6.6) into the action (6.5) gives

S[φ, J] = =

! " 1 4 − d x φ 2 + m2 − iε φ + J(x) DF (x − y)J(y) d4y 2 " 0 1 4 1 ! − d x φ 2 + m2 − iε φ 2 1 − J(x)DF (x − y)J(y) d4x d4y . (6.10) 2

The exponential of the last term no longer depends on φ and can, therefore, be pulled out of the pathintegral (6.1). The pathintegral involving the exponential of the first term appears also in the denominator of the normalized generating function (5.15) and thus drops out. We thus obtain now for the normalized generating functional Z0 [J] =

i W [J] 4 4 = e− 2 J(x)DF (x−y)J(y) d x d y . W [0]

(6.11)

This is the vacuum-to-vacuum transition amplitude for a free scalar field theory. Note that it no longer involves a path integral.

6.1.2

Feynman propagator

The imaginary part of the mass, originally introduced to achieve convergence for the path integrals, appears here now in the denominator of the propagator DF 1 e−ikx DF (x) = (6.12) d4k . (2π)4 k 2 − m2 + iε It determines the position of the poles in DF , which are, in the k0 -integration, at (6.13) k02 = k 2 + m2 − iε


65

Im k0

–ω k+iδ Re k0 +ω k–iδ

Figure 6.1: Location of the poles in the Feynman propagator. or

k0 = ± k 2 + m2 ∓ iδ .

(6.14)

The poles are therefore located as indicated in Fig. 6.1. The location of the poles, originally introduced only in an ad-hoc way to achieve convergence of the path integrals, determines now the properties of the propagator of the free Klein-Gordon equation, the Feynman propagator. This can be seen by rewriting the Feynman propagator DF in the following form 1 4 e−ikx d k (6.15) (2π)4 k 2 − m2 + iε e−ikx 1 3 = d k dk 0 (2π)4 k02 − k 2 − m2 + iε 1 1 1 3 −ikx 1 = d k dk0 e − (2π)4 2ωk k0 − ωk + iδ k0 + ωk − iδ

DF (x) =

with ωk2 = k 2 + m2 . We now first perform the integration over k0 . Since the exponential contains a factor e−ik0 t , the path can be completed in the upper half plane for negative times and in the lower half for t > 0. Cauchy’s theorem then gives 2πi × the sum of the residues of the enclosed poles i 3 eik·x +iωk t −iωk t d k −Θ(−t)e − Θ(t)e DF (x) = (2π)3 2ωk

=

i 3 eik·x +iωk t −iωk t − d k Θ(−t)e + Θ(t)e (2π)3 2ωk

(6.16)


66

(in the second term here an extra “−” sign appears because of the negative direction of the contour integral). It can now be shown that DF propagates free fields with negative frequencies backwards in time and those with positive frequencies forwards. To demonstrate this we write i 3 eik·x +iωk x0 −iωk x0 d k Θ(−x )e + Θ(x )e DF (x) = − 0 0 (2π)3 2ωk i 1 ikx = − Θ(−x0 ) d3k e 3 (2π) 2ωk i 3 1 −ikx − Θ(x0 ) dk e . (6.17) (2π)3 2ωk

Here we have changed k → −k in the first integral, which does not change its value under this substitution. The integrands are products of orthogonal, normalized solutions of the Klein-Gordon equation (±) ϕk (x) =

1 (2π)3 2ωk

e∓ikx

(6.18)

which fulfill the normalization and orthogonality conditions appropriate for a scalar field (cf. (4.65) without an A field)

i

d3x ϕk (x)ϕ˙ k (x) − ϕ˙ k (x)ϕk (x) = ±δ 3 (k − k) . (±)∗

and i

(±)

(±)∗

(±)∗

(∓)

(±)

(±)∗

(∓)

(6.19)

d3x ϕk (x)ϕ˙ k (x) − ϕ˙ k (x)ϕk (x) = 0 .

(6.20)

We can thus write 1 3 1 1 √ dk √ eikx iDF (x) = Θ(−x0 ) 3 (2π) 2ωk 2ωk 1 3 1 1 √ + Θ(x0 ) dk √ e−ikx 3 (2π) 2ωk 2ωk

= Θ(−x0 )

d

(−)∗ (−) k ϕk (0)ϕk (x)

3

+ Θ(x0 )

(6.21)

(+)∗

d3k ϕk

(+)

(0)ϕk (x) .

Comparison with (1.13) shows that negative-frequency solutions are propagated backwards in time, and positive-frequency solutions forward. This particular behavior is a consequence of the location of the poles relative to the integration path. This location is fixed by the sign of the ε term which in turn was needed to achieve convergence for the generating functional.


67

Euclidean representation. The Feynman propagator can also be given in a Euclidean representation. We can define a corresponding Euclidean momentum space by requiring that k 0 x0 = k4E xE4 ; this condition ensures that a plane wave propagating forward in time does so both in Euclidean and in Minkowski space. We thus get with k4 = −ik0 . (6.22) kE = (k, k4 ) The eigenvalues of 2E become k 2 = k02 − k 2 = −(k42 + k 2 ) = −kE2 < 0. This gives for the momentum space volume element and the energy-momentum distance in Euclidean space d4 kE ≡ d3 k dk4 = −d3 k idk0 = −id4 k dkE2

=

3

dkj2 + dk42 = −dk 2 .

(6.23)

j=1

Combining both of these definitions yields for the typical exponent of a plane wave kx = kμ xμ = k0 x0 − k · x = ik4 (−i)x4 − k · x = k4 x4 − k · x = kE xE .

(6.24)

Note that this is not equal to the Euclidean scalar product. However, in Fourier transforms, where this expression often appears, we always have an integration over d3 k and can thus change k → −k; thus in these Fourier integrals – and only there – we can replace kx by kE xE . The Feynman propagator can now be rewritten. For that purpose we choose a different path for the integration over k0 after we have Wick-rotated the time axis. Instead of integrating along the real energy (k0 ) axis we integrate along the imaginary energy axis. We then close the integration path in the right half of the k0 plane for t > 0 and in the left half for t < 0. Since in this way the same poles are included as on the original path the value of the integral does not change. This fact makes this rotation in the k0 -space different from the one in the t-space where the imaginary time-axis has to be rotated back to the real one after the calculations in order to obtain physical results. The Euclidean propagator then reads i e−ikE xE 4 d kE (2π)4 kE2 + m2 i e−ikE xE = − d4kE . (6.25) 4 2 2 2 (2π) k + k4 + m Because k4 is real, the integral no longer contains any poles on its integration path and is therefore well defined. DF (x) =

−


6.1.3

68

Gaussian Integration

In Sect. 1.3 we have already used a Gaussian integral relation to integrate out the p-dependence of the path integral. In many cases the generating functions appearing in field theory are of a form that contains the fields and their derivatives only in quadratic form so that again a Gaussian method can be used. In order to gain familiarity with this technique we derive in this section again the generating functional for the free scalar field theory. We thus apply the Gaussian integration formulas of Sect. B.2.1 to the generating functional (6.1)

Dφ e

W [J] =

− 2i

φ(2+m2 −iε)φ d4x i

e

Jφ d4 x

.

(6.26)

In order to make the matrix structure of the exponent more visible we use two-fold partial integration and write

W [J] =

= →

Dφ e− 2 i

Dφ e

− 2i

φ(x)δ 4 (x−y)(2y +m2 −iε)φ(y) d4x d4y i

e

φ(x)[(2y +m2 −iε)δ 4 (x−y)]φ(y) d4x d4y i

e

Jφ d4 x

Jφ d4 x

1 E 2 4 4 Dφ e− { 2 φ(xE )[(−2y +m )δ(xE −yE )]φ(yE ) d yE −Jφ} d xE . (6.27)

In the last step we have gone over to the Euclidean representation of the generating functional (cf. (5.20)), using 2E = −2 and δ 4 (xE −yE ) = −iδ 4 (x− y) (cf. Sect. 5.1.1). The integrals appearing here are now of Gaussian type and can thus be integrated by using the expressions developed in the last section. We first identify the matrices A and B in the Gaussian integration formula (B.18) as 2 4 AE (x, y) = −(2E y − m )δ (xE − yE )

B(x) = −J T (x) .

(6.28)

AE is real with positive eigenvalues (kE2 +m2 > 0) as required by the derivation in section B.2.1. It is also symmetric as can be seen by writing the d’Alembert operator in a discretized form, e.g. .

. 1 d2 . . φ(x) = lim [φ(xi + h) − 2φ(xi ) + φ(xi − h)] 2 . h→0 h2 dx xi

= Aij φ(xj ) with φ(xi + h) = φ(xi+1 ) etc. and Aij = δi,j−1 − 2δi,j + δi,j+1 .

(6.29)


69

Thus the necessary conditions for the application of (B.13) are fulfilled. Applying now (B.18) gives !

"− 1

W E [J] = det(−2E + m2 )δ(xE − yE )

2

1

e2

−1

J(xE )(AE (xE ,yE ))

J(yE ) d4 xE d4 yE

. (6.30) The Wick rotation back to real times is easily performed by the transformation (5.19) 2E → −2 − iε; it reintroduces the term +iε to guarantee the proper treatment of the poles. In this case A becomes AE → A = (2 + m2 − iε)iδ(x − y) .

(6.31)

and the volume element d4xE d4yE → −d4 xd4 y

(6.32)

so that we finally obtain1 W [J] = √

−1 1 2 4 4 4 1 e− 2 J(x)[i(2y +m +iε)δ (x−y)] J(y) d x d y . det A

(6.33)

The determinant of a matrix A is in general given by the product of its eigenvalues αi . Therefore we have

ln(det A) = ln

αi =

i

ln αi = tr ln A .

(6.34)

i

In the present case, though, A contains an operator and the notation det A deserves some explanation. The matrix is given by A(x, y) =

2 + m2 iδ(x − y) = 2 + m2 i

1 4 ik(x−y) dke (2π)4

i 4 2 2 d k −k + m eik(x−y) (2π)4 1 1 4 = i d k d4k eik x −k 2 + m2 δ 4 (k − k ) e−iky . 4 4 (2π) (2π)

=

Thus the momentum representation of the operator A is given by

A(k, k ) = −k 2 + m2 δ 4 (k − k ) .

(6.35)

We now evaluate the trace of the logarithm of this matrix where – in accordance with (6.34) – the logarithm of a matrix is explained by taking the 1

Note that formally we could have obtained this also by using (B.18) with A = i(2y + m − iε)δ 4 (x − y), B = −iJ, C = 0. 2


70

logarithm of each of the diagonal elements, i.e. the eigenvalues, after the matrix has been diagonalized. This yields in the present case

d4k 4 4 tr ln A(x, y) = d xd y d k δ (x − y)ei(k x−ky) ln(−k 2 + m2 )δ 4 (k − k ) 4 (2π) 4 dk d4x ln(−k 2 + m2 ) (6.36) = 4 (2π) 4

4

Now we use that the inverse operator appearing in (6.33) is just the Feynman propagator. This can be seen by deriving the equation of motion for the inverse operator δ (x − y) =

A(x, z)A−1 (z, y) d4z

4

=

!

= i

"

i 2z + m2 − iε δ 4 (x − z) A−1 (z, y) d4z

δ 4 (x − z) 2z + m2 − iε A−1 (z, y) d4z

= i 2x + m2 − iε A−1 (x, y) .

(6.37)

This is just the defining equation for −DF (6.7), so that we have A−1 = iDF .

(6.38)

The propagator can thus be obtained as the inverse of the operator between the two fields in the Lagrangian for the free field (4.31). For the normalized generating functional we thus obtain Z0 [J] =

i W [J] 4 4 = e− 2 J(x)DF (x−y)J(y) d x d y . W [0]

(6.39)

Equation (6.39) is the result derived earlier in section 5.1 (cf. (6.1.1)). The propagator that appears here is just given by the inverse of the Klein-Gordon Operator −1 DF (x − y) = − 2 + m2 δ(x − y) , (6.40) i.e. of that operator that appears between the two fields in the Lagrangian for a free Klein-Gordon field 1 L = − φ(2 + m2 )φ . 2

(6.41)


6.2

71

Stationary Phase Approximation

If the path integral in question is not that over a Gaussian, it can be approximately brought into a Gaussian form by using the so-called stationary phase or saddle point method. In this method one first looks for the stationary point of the exponent in the path integral. As explained earlier this will give a major contribution to the path integral. The remaining contributions are approximated by expanding the exponent around the stationary point. We illustrate this method here for the case of a scalar field with selfinteractions. The Lagrangian is given by " 1 ! L = − φ 2 + m2 φ − V (φ) 2

(6.42)

and the action S[φ, J] =

d4x (L + Jφ)

(6.43)

is a functional of the field φ and the source J. We next determine the stationary point by looking for the zero of the functional derivative .

δS[φ, J] .. ! . = − 2 + m2 φ0 (x) − V (φ0 (x)) + J(x) = 0 ; . δφ(x) φ0

(6.44)

this is the classical equation of motion corresponding to the action S[φ, J]. The stationary field is just the classical field; the corresponding classical action is & ' 1 4 2 (6.45) S[φ0 , J] = − d x φ0 2 + m φ0 + V (φ0 ) − Jφ0 . 2 We now expand S[φ, J] around this stationary field (cf. (B.35),(B.37)) S[φ, J] = S[φ0 , J] (6.46) . . 1 δ2S . . [φ(x1 ) − φ0 (x1 )] [φ(x2 ) − φ0 (x2 )] + · · · . d4x1 d4x2 + 2 δφ(x1 )δφ(x2 ) .φ0 The second functional derivative appearing here can be obtained by varying the first derivative (6.44). We thus get from the definition (B.26) .

.

. " . δ ! δ2S . . 2 . = − . (2 + m )φ + V (φ) − J 1. δφ(x2 )δφ(x1 ) .φ0 δφ(x2 ) φ0

(6.47)

Using now (B.27) we get .

. ! " δ2S . . = − 2 + m2 + V (φ0 ) δ 4 (x2 − x1 ) 1 δφ(x2 )δφ(x1 ) .φ0

(6.48)


72

which is an operator. The index 1 here means that the corresponding expressions are to be taken at the point x1 . The action (6.45) is calculated at the fixed classical field φ0 . It can, therefore, be taken out of the path integral so that we finally obtain

W [J] =

i

Dφ e h¯

i

= e h¯ S[φ0 ,¯hJ]

d4x (L+Jφ)

Dφ exp −

1

× [φ(x1 ) − φ0 (x1 )]

1!

(6.49) i 2¯ h

d4x1 d4x2 "

0

2 + m + V (φ0 ) δ (x2 − x1 ) [φ(x2 ) − φ0 (x2 )]} 2

4

1

+ ... .

(6.50)

In order to facilitate the following discussion we have put the unit of action, h ¯ , explicitly into this expression by setting i → i/¯ h and J → h ¯ J. The path integral remaining here is now in a Gaussian form. It can be evaluated after a Wick rotation, just as in the developments leading to (6.33). After a “coordinate transformation” φ → φ = φ − φ0 and after scaling the √ ¯ φ we get for the generating functional fields by φ → h 1

!

"0− 1

W [J] = e h¯ S[φ0 ,¯hJ] det i 2 + m2 + V (φ0 ) δ 4 (x2 − x1 ) i

2

.

(6.51)

We now perform a normalization with respect to the free case (6.33), i.e. to 1

!

"0− 1

det i 2 + m2 δ 4 (x2 − x1 )

W0 [0] =

2

= {A(x1 , x2 )}− 2 1

(6.52)

with A from (6.31); the index 0 on W denotes V = 0. This gives for the normalized generating functional ˜ [J] = W [J] = e h¯i S[φ0 ,¯hJ] W W0 [0]

× det

&

(6.53)

−1

' − 1

d z A (x2 , z) (A(z, x1 ) + iV (φ0 (x1 ))) δ (z − x1 ) 4

4

2

With A−1 = iDF (6.38) we get 1

!

"0− 1

˜ [J] = e h¯i S[φ0 ,¯hJ] det δ 4 (x1 − x2 ) − DF (x2 − x1 )V (φ0 (x1 )) W

2

(6.54)

Our aim is now to write the inverse root of the determinant as a correction term to the classical action. For this purpose we use (6.34) − 12

{det[. . .]}

1 − tr ln[. . .] =e 2 .

(6.55)

.


73

The matrix is given by !

"

x1 |1 − DF V (φ0 )|x2 = δ 4 (x1 − x2 ) − DF (x2 − x1 )V (φ0 (x1 )) .

(6.56)

The trace of its logarithm is then given by

tr ln[. . .] = We can now write

d4x ln [1 − DF (0)V (φ0 (x))] .

(6.57)

˜ [J] = e h¯i S[φ0 ,J] W

(6.58)

with S[φ0 , J] =

−

&

d4x

'

1 φ0 2 + m2 φ0 + V (φ0 ) + h ¯ d4x Jφ0 2

i 4 + h h2 ) . ¯ d x ln [1 − DF (0)V (φ0 (x))] + O(¯ 2

(6.59)

The first line is just the classical action S[φ0 , J]. The two terms can be summed with a resulting action S[φ0 , J] = −

&

1 d x φ0 2 + m2 φ0 + Veff (φ0 ) + h ¯ Jφ0 2

'

4

(6.60)

with the effective potential i Veff (φ(x)) = V (φ(x)) − h ¯ ln [1 − DF (0)V (φ0 (x))] . 2

(6.61)

Expression (6.59) shows that the saddle point approximation amounts to an expansion of the action in powers of h ¯ . This is in accordance with the discussion in Sect. 1.4 that quantum mechanics describes the fluctuations of the action around the classical path. The potential Veff incorporates the effects of these fluctuations into a classical potential. Eq. (6.59) suggests a perturbative treatment through an expansion of the logarithm (ln(1 + x) = x − x2 /2 + x3 /3 − . . .) in terms of DF V , i.e. the strength of the potential. The trace corresponds to an integration over x such that the initial and final space-time points in the individual terms in the expansion are identical, i.e. to a closed loop integration. With higher orders in the expansion of the logarithm more and more vertices appear, but they are always located on this one closed loop. This loop expansion approximates the quantum mechanical behavior, whereas the perturbation treatment takes interactions into account. If we take φ0 as a constant field, i.e. if we neglect its space-time dependence through the d’Alembert operator in (6.44), then the operator 1 +


74

DF V (φ0 ) becomes local in momentum space. In this case we can evaluate the trace of its logarithm by integrating over the eigenvalues x1 | ln [1 − DF V (φ0 )] | 1 d4k −ik(x2 −x1 ) e ln 1 − 2 V (φ0 ) . = (2π)4 k − m2 + iε

(6.62)

This equation writes the original matrix in x-space as a result of a unitary transformation of a diagonal matrix in k-space. The logarithm is taken of this diagonalized matrix which is then transformed back to x-space.

6.3

Numerical Evaluation of Path Integrals

An alternative method for the evaluation of path integrals is that of direct numerical computation; with rapidly increasing computer power this method becomes more and more important nowadays.

6.3.1

Imaginary time method

The generating functional is in general given by ˆ

W [J] = 0|e−i(H+J)(tf −ti ) |0

(6.63)

for ti → −∞ and tf → +∞, as we have seen in chapter 3. After a Wick rotation this becomes ˆ

W [J] = lim 0|e−β(H+J) |0 ,

(6.64)

β→∞

where β denotes the real Euclidean time. It is immediately obvious that (6.64) also equals the groundstate expectation value of the statistical operator of quantum statistics if we identify β with the inverse temperature, i.e. β = 1/T . Inserting the explicit definition (5.9) and performing the Wick rotation gives

n+1 W [J] = lim n→0 2π¯ hβ

n+1 n 2

with Hl = Hl (xl , x˙ l ) =

dxkj e

β − n+1

n l=0

(Hl +Jk xkl )

,

(6.65)

k j=1

" 1 ! 2 x˙ kl + ωk2 x¯2kl + V (xkl ) , 2 k

(6.66)


75

so that W [J] can be written as

β n + 1

W [J] = lim dxkj ρ(xkj )e− n+1 l V (xkl ) n→0 2π¯ hβ k j

with

β

ρ(x) = e− n+1

-

1 (x˙ 2kl +ωk2 x2kl )+Jk xkl l 2

[

].

(6.67)

(6.68)

The function ρ(x) is of Gaussian shape and can, therefore, analytically be normalized into ρ(x) P (x) = . (6.69) dx ρ(x) The multiple integrals appearing here can be evaluated by a Monte-Carlo technique which samples the integrand at a large number of points, where each ‘point’ really corresponds to a full path x(t). Given a certain point x = x1 , . . . , xn one randomly chooses a new point x = x1 , . . . , xn , often by just changing one single coordinate. One then evaluates r=

P (x ) . P (x)

(6.70)

If r is larger than 1, the new point is accepted. If r < 1, on the other hand, then a random number ρ between 0 and 1 is picked. If ρ < r, then the new point is also accepted, otherwise it is rejected. This method is repeated until a large enough number of points is sampled. In this way the most important regions in x space are sampled, thus generating finally M accepted points xm . The integral is then approximated by W [J] =

M -n β 1 e− n+1 l=0 V ((xkl )m ) . M m=1

(6.71)

The sampling algorithm just described is known after its inventor as the Metropolis algorithm; it plays an important role in numerical evaluations of statistical physics expressions. In the present case of evaluating the generating functional for gs to gs transitions one has to choose a fixed value of β, i.e. the Euclidean time. The calculations then have to be performed for several values of β with a subsequent extrapolation to β → ∞ (or T → 0).

6.3.2

Real time formalism

The major difficulty in evaluating a path integral numerically stems from the oscillatory character of the integrand. The discretized form of W [J] (5.9) can


76

– in an obvious abbreviation – be written as the limit of a multidimensional integral (6.72) W = dx eiS(x) , where x is a n-dimensional vector. Importance sampling such as the one just discussed in the last section cannot directly be used because there is no positive probability weight function in the integrand. A weight function can, however, be inserted by a mathematical trick. Using (B.18) in the form

we can write

W =

dx0 det(A/2π)e− 2 (x−x0 ) 1

T A(x−x ) 0

dxdx0 det(A/2π)e− 2 (x−x0 ) 1

=1

T A(x−x ) 0

(6.73)

eiS(x) .

(6.74)

The Gaussian factor under the integral ensures that values of x close to x0 will contribute the most to the integral. The function S(x) can, therefore be expanded around x0 1 S(x) = S(x0 ) + S1 (x)(x − x0 ) + (x − x0 )T S2 (x0 )(x − x0 ) + . . . 2 .

with

dS .. . S1 (x0 ) = dx .x0

(6.75)

.

d2 S .. . S2 (x0 ) = . dx2 .x0

and

(6.76)

After inserting this expansion we can perform the x-integration and obtain from (B.18)

W =

dx0 e

iS(x0 )

det(A) det[A − iS2 (x0 )]

1 2

T

e− 2 S1 (x0 )[A−iS2 (x0 )] 1

−1

S1 (x0 )

. (6.77)

If we now use that (6.73) is still approximately valid even if A is a function of x0 , we can choose A = A(x0 ) = iS2 (x0 ) + c−1 1

(6.78)

with c > 0 and obtain

W =

!

"− 1

dx0 eiS(x0 ) det(1 − iS2 (x0 )A−1 )

The function

c

T

2

c

T

e− 2 S1 (x0 )S1 (x0 ) .

P (x0 ) = e− 2 S1 (x0 )S1 (x0 )

(6.79)

(6.80)


77

thus provides a probability distribution for sampling the remaining integrand and the expression can be evaluated with the Monte-Carlo method discussed in the last section. We obtain, therefore, M 1 1 W = eiS(xi ) [det(1 + icS2 (xi )]− 2 , M i=1

(6.81)

where the M points xi are taken randomly from the distribution P (x). For large values of c the points with S1 ≈ 0 will contribute the most to W . This is just the stationarity condition discussed in section 6.2 which corresponds to the classical solution. Since only a few points close to this configuration will contribute significantly, a good sampling with good statistics can be obtained. On the other hand, for small c the probability distribution becomes broad and the statistics correspondingly worse; however, quantum mechanical effects beyond the one-loop approximation now start to contribute. Thus, by choosing the proper value of c quantum mechanics can be switched on.

Chapter 7 S-MATRIX AND GREEN’S FUNCTIONS The ultimate aim of all our fieldtheoretical developments in this book is to calculate reaction and transition rates for processes involving elementary particles. In this chapter, we therefore, now derive a connection between these transition rates and expectation values of field operators, the so-called reduction theorem.

7.1

Scattering Matrix

The typical initial state of a scattering experiment is that of widely separated on-shell particles at t → −∞. On-shell here means that these particles fulfill the free energy-momentum dispersion relation. The groundstate of the system is the state of lowest-energy, i.e. the state with no particles present. At large times t → +∞ the final state is again that of free, non-interacting on-shell particles. The vacuum state of the theory is unique and is therefore the same as the initial vacuum state. The transition rate of any quantum process, be it a scattering process m + n → m + n is determined by the so-called S-matrix. We define the Smatrix as the probability amplitude for a process that leads from an ingoing state |α, in to an outgoing state |β, out . The particles are assumed to move freely in these asymptotic states outside the range of the interaction; both of these states can therefore be characterized by giving the momenta of all participating particles and possibly other quantum numbers as well all of which are denoted by α and β. The S-matrix is thus given by Sβα = β, out|α, in . 78

(7.1)

CHAPTER 7. S-MATRIX AND GREEN’S FUNCTIONS

79

We can then also introduce an operator Sˆ that transforms in-states (bras) into out-states (bras) β, out| = β, in|Sˆ (7.2) so that we have ˆ in . Sβα = β, in|S|α,

(7.3)

Sˆ is a unitary operator. This can be seen by taking the hermitean conjugate of (7.2) and writing β, in|SˆSˆ† |α, in = β, out|α, out = δα,β .

(7.4)

SˆSˆ† = 1.

(7.5)

Thus we have

The field operators transform in the standard way under the unitary transformation S (7.6) φout = Sˆ† φîn Sˆ . In Sect. 2.3 we have expressed the matrix element Sβα in terms of wavefunctions. There we showed that S (see (2.35)) could be written as

Sβα =

Ψ∗β (x, t → +∞)Ψ(+) x, t → +∞) d3x . α (

(7.7)

Here Ψ(+) was a wavefunction that fulfilled an “in” boundary condition, i.e. it evolved forward in time, starting from an incoming free plane wave at t → −∞. Ψβ , on the other hand, was a free plane wave for t → +∞. We now generalize these considerations to fields and introduce the free asymptotic fields φin and φout φin = φout =

lim φ(x, t)

t→−∞

lim φ(x, t) .

(7.8)

t→+∞

Using now the time-development operator U on φin gives for the S- matrix Sβα = φout |U (+∞, −∞)|φin

(7.9)

with the time-development operator (1.1). With U (+∞, −∞) = U (+∞, t0 )U (t0 , −∞) we can write the S-matrix also as (−)

(+)

Sβα = φout |U (+∞, t0 )U (t0 , −∞)|φin = φout (t0 )|φin (t0 ) .

(7.10)


80

This form corresponds to (7.7). As is obvious from its definition the S-matrix determines all the transition rates possible within the field theory. For example, a cross-section for a 1 + 1 → 1 + 1 collision is simply given by the absolute square of S, multiplied with the available phase-space of the outgoing particles and normalized to the incoming current. The Reduction Theorem to be derived in the following section provides a link between the S-matrix and expectation values of timeordered products of field operators that can be calculated as derivatives of generating functionals.

7.2

Reduction Theorem

Since the asymptotic states appearing in the S matrix are those of free, onshell particles we can describe them as non-interacting quantum excitations of the vacuum of the theory with free dispersion relations. We, therefore, first review the basic properties of free field creation and annihilation operators in the next subsection.

7.2.1

Canonical field quantization

The field operators are obtained by quantizing the free asymptotic fields φin and φout . This is done in the usual way by imposing commutator relations for the fields and their momenta. We impose the canonical commutator relations of quantum mechanics for coordinates and corresponding canonical momenta. Remembering that in field theory the fields play the role of the classical coordinates we thus impose1 [Π(x, t), φ(x , t)] = − iδ 3 (x − x ) " ˙ x , t) = 0 Π(x, t), φ(

!

[φ(x, t), φ(x , t)] = 0 .

(7.11)

We now employ the normal mode expansion of the fields (see the discussion at the start of section 5.1) and momenta 1 1 √ φ(x, t) = √ ak (t) eik·x + a†k (t) e−ik·x 2ωk V k

i ωk √ Π(x, t) = − √ ak (t) eik·x − a†k (t) e−ik·x 2ωk V k 1

(7.12)

From here on all the fields φ in this section are operators; only for ease of notation we do not write the ’operator-hats’ explicitly


81

with ωk = k 2 + m2 . Since the fields φ and the momenta Π are now operators, the ’expansion coefficients’ a and a† are operators as well. The inverse Fourier transformation is given by 1 = √ eik·x (ωk φ(x, t) + iΠ(x, t)) d3x 2ωk V 1 † √ eik·x (ωk φ(x, t) − iΠ(x, t)) d3x ak (t) = 2ωk V

a†k (t)

(7.13)

Using the commutator relations (7.11) we obtain also the commutator relations for the operators ak and a†k ! !

"

ak (t), a†k (t)

= δk,k

ak (t), ak (t)

=

"

!

"

a†k (t), a†k (t) = 0 .

(7.14)

The asymptotic in and out fields are free fields so that the time-dependence of their operators a and a† is harmonic ak (t) = ak (0)e−iωk t a†k (t) = a†k (0)e+iωk t

(7.15)

as can be obtained from the Heisenberg equation of motion. The asymptotic in fields φin are then given by 1 1 √ φin (x, t) = √ ak (0) e−ikx + a†k (0) e+ikx ; 2ωk V k

(7.16)

with ikx = i(ωk t − k · x). The fields φout can be represented in the same way. ˙ the free field annihilation and Remembering that for free fields Π = Φ, creation operators for the in and out states are given in terms of the fields and momenta as −i 3 −ikx d xe (∂t φ(x, t) + iωk φ(x, t)) 2ωk V V i ak (0) = √ d3x e+ikx (∂t φ(x, t) − iωk φ(x, t)) 2ωk V V

a†k (0) = √

(7.17)

with kx = ωk t − k · x. The operators (7.17) are the same as the ones used in the well known algebraic treatment of the harmonic oscillators for the normal field modes. The a† are the creation and the a the annihilation operators for free field quanta and the vacuum (groundstate) of the free field theory is given by a|0 = 0.


7.2.2

82

Derivation of the reduction theorem

In a realistic physics situation the scattering or decay processes that we aim to describe involve interactions between the particles. Simulating the situation in a scattering experiment we, therefore, now assume that the interactions between the particles are adiabatically being switched on and off; adiabatically here means without energy transfer. The asymptotic “in” and “out” states are thus free states which can be described by the free field operators (7.17). The operators a† (t) and a(t) in (7.13) have a harmonic time-dependence (7.15) only for times t → ±∞. The free field operators (7.17) acting on the vacuum of the full, interacting theory create and annihilate field quanta only at times t → ±∞ while they do so at all times when acting on the vacuum state of the noninteracting theory. They can thus be used to describe the asymptotic states. We can thus write for the S matrix element (7.1) Sβα = β, out|α, in = β, out|a†in (k)|α − k, in ,

(7.18)

where we have assumed that the in-state |α, in contained a free particle with three-momentum k ; |α−k, in is then that in state in which just this particle is missing. We can further write this as β, out|α, in = β, out|a†out (k)|α − k, in + β, out|a†in (k) − a†out (k)|α − k, in .

(7.19)

In the first term β, out|a†out (k) = β − k, out| (= 0, if β, out| does not contain a particle with momentum k). We now rewrite the in and out creation operator into a more compact form ↔ † (7.20) ain,out (k) = −i d3x fk (x) ∂ t φin,out (x, t) with

↔

f (t) ∂ t φ(x, t) = f and

∂φ ∂f − φ ∂t ∂t

(7.21)

1 fk (x) = √ e−ikx . 2ωk V

With this expression we can get the S-matrix element into the form β, out |α, in = β − k, out|α − k, in − iβ, out|

d3x fk (x)

(7.22) ↔ ∂t

[φin (x) − φout (x)] |α − k, in .


83

For only 2 particles in the in and out states the first term on the rhs represents a single particle transition matrixelement. In this case it can obviously only contribute if both particles do not change their energy and momentum, i.e. if |out and |in are identical. It is then just the forward scattering amplitude. The rhs of (7.22) is time-independent. We can see this by calculating the time derivative of its integrand

↔

∂t f (x) ∂ t φin (x) = f ∂t2 φin − ∂t2 f φin .

(7.23)

Here f (x) = e−ikx and φin both solve the free Klein-Gordon equation with the same mass. We therefore have

∂t f (x)

↔ ∂t

2 φin − (∇ 2 f )φin . φin (x) = f ∇

(7.24)

The integral over this expression vanishes after twofold partial integration of one of the terms. The same holds, of course, for the term involving φout in (7.22). The rhs of (7.22) is thus indeed time-independent. We can, therefore, take it at any time and in particular also at t → ±∞. Then we can replace the in and out fields at these times by the limits of the field φ(x). This gives for the S-matrix element β, out|α, in = β − k, out|α − k, in

↔

d3x fk (x) ∂ t φ(x)|α − k, in

lim iβ, out|

+

t→+∞

−

↔

d3x fk (x) ∂ t φ(x)|α − k, in .

lim iβ, out|

t→−∞

(7.25)

We now write this expression in a covariant form by using

lim − lim

t→+∞

=

=

↔

! " ↔ ∂ dx f (x) ∂ t φ = d4x f ∂t2 φ − ∂t2 f φ ∂t 3

dt −∞

d3x f (x) ∂ t φ

t→−∞

+∞

!

2 − m2 )f d4x f ∂t2 − (∇

"

φ.

(7.26)

Note that here φ is an interacting field, since we integrate now over all times. Thus, in contrast to (7.23) φ does not solve the free Klein-Gordon equation and, consequently, this integral does not vanish. Twofold partial integration in the second term on the rhs allows us now to roll the Laplace operator from f over to φ. This gives

4

dx

f ∂t2

φ−

(∂t2 f )φ

=

d4x f (x)(2 + m2 ) φ(x) .

(7.27)


84

We thus have β, out|α, in = β − k, out|α − k, in

(7.28)

d4x fk (x)(2 + m2 )β, out|φ(x)|α − k, in .

+i

In this expression we have removed one particle from the in state. We now continue by removing one particle with the momentum k from the out state by going through exactly the same steps as before. Disregarding the first term in (7.28), that contributes only to forward scattering, we get β, out|φ(x)|α − k, in = β − k , out|aout (k )φ(x)|α − k, in = β − k , out|φ(x)ain (k )|α − k, in (7.29) + β − k , out|aout (k )φ(x) − φ(x)ain (k )|α − k, in . We next replace the annihilation operators by the corresponding field operators as in (7.20) (here we have to take the hermitian conjugate operator) and obtain β, out|φ(x)|α − k, in = β − k , out|φ(x)|α − k − k , in

+i

↔

(7.30)

d3x β − k , out| fk∗ (x ) ∂ t [φout (x )φ(x) − φ(x)φin (x )] |α − k, in .

Taking now the limits t → ±∞ gives, as above, for this expression β − k , out|φ(x)|α − k − k , in ↔ 4 ∂ ∗ + iβ − k , out| d x fk (x ) ∂ t T [φ(x )φ(x)] |α − k, in ∂t = . . . + iβ − k , out|

d4x fk∗ (x )∂t2 T [φ(x )φ(x)]

− ∂t2 fk∗ (x ) T [φ(x )φ(x)] |α − k, in .

(7.31)

We now use again (7.27) and obtain

= · · · + i β − k , out|

d4x fk∗ (x )(2 + m2 )T [φ(x )φ(x)] |α − k, in .

Combining this result with (7.28) finally gives (neglecting the forward scattering amplitude) β, out|α, in = i2

d4x d4x fk∗ (x )fk (x)

(7.32)

×(2 + m2 )(2 + m2 )β − k , out|T [φ(x )φ(x)] |α − k, in .


85

This reduction can obviously be continued on both sides until we have (with n particles with momenta k in the out state and m particles with momenta k in the in state) Sβα = βn k , out|αm k, in = im+n

m

d4xi

n

i=1

(2j

j=1

(7.33)

d4xj fk∗j (xj )fki (xi ) ×

+ m )(2i + m )0|T [φ(x1 )φ(x2 ) . . . φ(xn )φ(x1 )φ(x2 ) . . . φ(xm )] |0 . 2

2

This is the so-called Reduction Theorem that enables us to express the Smatrix in terms of the (n + m)-point Green’s function, somestimes also called correlation function G(x1 , x2 , . . . , xn , x1 , x2 , . . . , xm ) = 0|T [φ(x1 )φ(x2 ) . . . φ(xn )φ(x1 )φ(x2 ) . . . φ(xm )] |0 .

(7.34)

Note that in the reduction theorem (7.33) the information about the interaction of the particles is contained in the (interacting) field operators φ. Also the vacuum appearing here is that of the full, interacting theory. The n+m-point function appearing there is, therefore, also that of the interacting theory! The physical process described by the reduction theorem is that of m onshell particles in the initial state at the asymptotic space-time coordinates x1 , . . . , xm and n on-shell particles at the space-time coordinates x1 , . . . , xn with an interaction region in between these sets of coordinates. As we will see later, the Klein-Gordon operators 2 + m2 when acting on G just remove the propagators from the interaction region out to the asymptotic points, creating so-called vertex functions Γ(x1 , . . . , xn , x1 , . . . , xm ). The reduction theorem (7.33) then gives the transition rate Sβα as the Fourier transform of this vertex function. The Fourier transform in (7.33) contains factors of the form exp(ik x ) for the outgoing particles and exp(−ikx) for the incoming ones. This could be symmetrized by changing all outgoing momenta k → −k . This gives Sβα = βn − k , out|αm k, in m+n

= i

m

i=1

4

d xi

n

j=1

d4xj

1 1 e−i(kj xj +ki xi ) 2ωkj V 2ωki V

×Γ(x1 , x2 , . . . , xn , x1 , x2 , . . . , xm )

.

(7.35)

The connection between the S-matrix and the Green’s function as expressed by the reduction theorem has been derived here only for scalar fields,


86

but it is valid in general. The only formal difference is that the Klein-Gordon operator 2 + m2 has to be replaced by the corresponding free-field operator. It is also important to note that the method of canonical quantization used here to derive the reduction theorem has been used only for the asymptotic states. Thus, even for fields where this method runs into difficulties when interactions are present, like, e.g., the gauge fields to be treated in chapter 13, the reduction theorem holds in the form given above. In the remainder of this book we will be concerned with calculating the correlation functions by using path integral methods. Once these correlation functions are known the reduction theorem allows us to calculate any reaction rate or decay probability.

Chapter 8 GREEN’S FUNCTIONS In chapter 7 we have found that all the S-matrix elements can be calculated once the correlation, or Green’s, functions are known. In this chapter we discuss how these functions can be obtained as functional derivatives of the generating functionals of the theory.

8.1

n-point Green’s Functions

In chapter 7 we have seen that the correlation function, i.e. the vacuum expectation value of time-ordered field operators, !

"

ˆ 1 )φ(x ˆ 2 ) . . . φ(x ˆ n ) |0 . G(x1 , x2 , . . . , xn ) = 0|T φ(x

(8.1)

determines the transition rate for all physical processes. Remembering that in field theory the field operators play the role of the coordinates in classical quantum theory we can now directly use the results obtained in Chapt. 3 and Sect. 5.1.1 and write using (3.47) !

"

ˆ 2 ) . . . φ(x ˆ n ) |0 ˆ 1 )φ(x G(x1 , x2 , . . . , xn ) = 0|T φ(x

=

with S[φ] =

+∞ −∞

Dφ φ(x1 )φ(x2 ) . . . φ(xn )eiS[φ]

Dφ eiS[φ]

(8.2)

L(φ, ∂μ φ)d4x. The vacuum here is that of the full, interacting

Hamiltonian. The latter expression can also be obtained as a functional derivative of the generating functional of the theory (cf. (3.47)) so that we can also equiv87

CHAPTER 8. GREEN’S FUNCTIONS

88

alently define the n-point Green’s function by G(x1 , x2 , . . . , xn ) =

n 1

i

.

. δ n Z[J] . . . δJ(x1 )δJ(x2 ) . . . δJ(xn ) .J=0

(8.3)

We note that G(x1 , . . . , xn ) is a symmetric function of its arguments. Therefore, according to (B.37) the following relation holds Z[J] =

1 n

n!

dx1 . . . dxn in G(x1 , x2 , . . . , xn )J(x1 )J(x2 ) . . . J(xn ) . (8.4)

Connected Green’s Functions. Guided by (3.22) we define a functional S[J] by the relation (8.5) Z[J] = eiS[J] and introduce the so-called connected Green’s functions Gc in terms of S[J] defined by the relation n−1

Gc (x1 , . . . , xn ) =

1 i

.

. δnS . . . δJ(x1 ) . . . δJ(xn ) .J=0

(8.6)

The name of this correlation function and its physics content will become clear later in this section.

8.1.1

Momentum representation

Very often it is advantageous to work in momentum space because the external lines of Feynman graphs represent free particles with good momentum. In general the transformation of the Green’s function into the momentumrepresentation is given by

e−i(p1 x1 +p2 x2 +...+pn xn ) G(x1 , x2 , . . . , xn ) d4x1 d4x2 . . . d4xn = (2π)4 δ 4 (p1 + p2 + . . . + pn ) G(p1 , p2 , . . . , pn )

(8.7)

The δ-function here reflects the momentum conservation due to translational invariance. This can be seen by performing pairwise transformations of two space-time points to their cm. point and their relative coordinate. If we then assume that G depends only on the latter, the integral over the c.m. coordinate can be performed and yields the δ-function. As in our discussion around (7.35) we take all the momenta as pointing into the vertex (see Fig. 8.1).

CHAPTER 8. GREEN’S FUNCTIONS p1

89 p4 p5

p2

p6 p7

p3

Figure 8.1: Momentum representation of the n-point function. Note that all momenta are pointing into the shaded interaction region.

8.1.2

Operator Representations

Operator representation of the generating functional. For completeness, we now derive an alternative expression for the generating functional Z[J]. We start by defining the operator functional ˆ = T ei Z[J]

ˆ J(x)φ(x) d4x

(8.8)

ˆ where φˆ is an operator! If we form the functional derivatives of Z[J] we get, in analogy to (3.43), n

1 i

! " δ n Zˆ ˆ n )Z[J] ˆ 1 ) . . . φ(x ˆ , = T φ(x δJ(x1 ) . . . δJ(xn )

(8.9)

ˆ = 1, so that, because of Z[0] n

1 i

.

. ! " ˆ δ n 0|Z[J]|0 . ˆ 1 ) . . . φ(x ˆ n ) |0 . φ(x = 0|T δJ(x1 ) . . . δJ(xn ) .J=0 n

=

1 i

.

. δnZ . . . (8.10) δJ(x1 ) . . . δJ(xn ) .J=0

ˆ Thus all the functional derivatives of 0|Z[J]|0 agree with those of Z[J] at J = 0. According to (8.3) and (8.4) the two expressions therefore have to be equal ˆ Z[J] = 0|Z[J]|0 . (8.11) Functional form of the scattering operator. After having seen that the Green’s functions can be obtained as functional derivatives of a generating functional in this section we show that the scattering operator Sˆ can also be


90

written in a functional form as δ i φîn (x)(2+m2 ) δJ(x) d4 x ˆ S = :e : Z[J]|J=0 (8.12) ∞ k k i δ = d4x1 . . . d4xk : φîn (x1 ) . . . φîn (xk ) : Z[J] |J=0 . δJ(x1 ) . . . δJ(xk ) k=0 k! Here the : : symbol denotes the so-called normal ordered product of field operators. This normal-ordered product is defined in such a way that all operators in it are reordered so that all the annihilation operators are moved to the right. This reordering takes place without a sign change for boson fields and with a sign-change for each pairwise exchange for fermion fields. δ in (8.12) acts only on Z[J]. The operator (2 + m2 ) δJ(x) The matrix elements of the operator (8.12) indeed agree with (7.33). This can be seen by considering again a matrixelement with n particles in the out states and m particles in the in state. In the expansion of the exponential in (8.12) only that term can contribute that contains exactly m + n powers of the fields. We thus have ˆ n, out|S|m, in = im+n

m

i=1

4

d xi

m+n

d4xj

(8.13)

j=m+1

1 n| : φin (x1 ) . . . φin (xm+n ) : |m (m + n)! × (21 + m2 )(22 + m2 ) . . . (2m+n + m2 ) im+n G(x1 , x2 , . . . , xm+n ) .

×

Here the in field operator φin (x) can simply be replaced by operators of the free field (7.8). The normal product reorders the expansion (8.13) such that all the annihilation operators are on the right. Since each field contains two independent sums over positive and negative energy eigenstates, respectively, we have in total 2m+n operator products; of these only the term with n creation operators and m annihilation operators can contribute. This gives with the expansion (7.12) n| : φin (x1 ) . . . φin (xm+n ) : |m n n+m

(m + n)! n| = fk∗k (xk )a†k fkl (xl )al |m m! n! k=1 l=n+1 n n+m

(m + n)!

∗ fkk (xk ) fkl (xl ) . = m! n! k=1 l=n+1

(8.14)

The degeneracy factor in front of the matrix element follows from the binomial expansion of the individual terms in the normal mode expansion (fk ak + fk∗ a†k )m+n . The integration in (8.13) over the xi and xj just gives an extra degeneracy factor m!n!. Taking this result together with (8.13) is the same as (7.33).


8.2

91

Free Scalar Fields

We consider first the case of free fields. In this case the generating functional can be given analytically (6.11) Z0 [J] = e− 2 i

J(x)DF (x−y)J(y) d4x d4y

.

(8.15)

The first functional derivative vanishes at J = 0 because the integral (8.15) is Gaussian. For the second functional derivative we obtain i δ 2 Z0 [J] 4 4 = −iDF (x1 − x2 ) e− 2 J(x)DF (x−y)J(y) d x d y δJ(x1 )J(x2 )

2

+ (−i)

d4x d4y DF (x1 − x)DF (x2 − y)J(x)J(y) e− 2 i

so that we have

(8.16) J(x)DF (x−y)J(y) d4x d4y

,

.

δ 2 Z0 [J] .. . = iDF (x1 − x2 ) . G(x1 , x2 ) = − δJ(x1 )δJ(x2 ) .J=0

(8.17)

The two-point function is thus just the Feynman propagator. It is therefore also a solution of (cf. (6.7)) !

2 + m2 − iε

8.2.1

"

G(x1 , x2 ) = −iδ 4 (x)

(8.18)

Wick’s theorem

The higher order derivatives can be most easily obtained by expanding (8.15) ∞ 1

n i J(x)DF (x − y)J(y) dx dy (8.19) 2 n=0 n! ∞ i n 1 = 1+ − dx1 . . . dx2n D12 D34 . . . D2n−1 2n J1 J2 . . . J2n 2 n=1 n!

Z0 [J] =

−

with the shorthand notation Dij = DF (xi − xj ), Jk = J(xk ). Noting that Z always contains even powers of J, it is immediately evident that all n-point functions with odd n vanish because an odd functional derivative of an even function always vanishes at J = 0. Taking now the 2k-th functional derivative of Z0 and using (8.3) and (B.36) we obtain 2k

1 δ 2k Z0 | i δJ1 . . . δJ2k J=0 (i)k = k Dp1 p2 . . . Dp2k−1 p2k 2 k! P

G(x1 , x2 , . . . , x2k ) =

(8.20)


92

where the sum runs over all permutations (p1 , p2 , . . . , p2k ) of the numbers (1, 2, . . . , 2k). The factor in front of the sum removes the doublecounting because of the symmetry Dp2 p1 = Dp1 p2 (2k ) and because of the random order of factors under the sum (k!). Equation (8.20) states that the n-point function of a system of free bosons can be written as a properly normalized and symmetrized product of twopoint functions. This is the so-called Wick’s theorem. As an example we consider the case n = 4. We then have G(x1 , x2 , x3 , x4 ) = −

1 Dp p Dp p . 8 P ∈S4 1 2 3 4

(8.21)

Among the 24 terms in the sum, 12 are pairwise equal because the product of the two propagators commutes. Furthermore, the propagator Dp1 p2 and Dp2 p1 are pairwise equal. Thus, there are only 24 : 2 : 2 : 2 = 3 essentially distinct terms in the sum; the factor 1/8 just takes care of all the others. We thus have G(x1 , x2 , x3 , x4 ) =

− DF (x1 − x2 )DF (x3 − x4 ) − DF (x1 − x3 )DF (x2 − x4 ) − DF (x1 − x4 )DF (x2 − x3 ) .

(8.22)

The first few n-point Green’s functions are therefore – according to (8.17), (8.20) and (8.22) – given by G(x1 ) G(x1 , x2 ) G(x1 , x2 , x3 ) G(x1 , x2 , x3 , x4 )

= = = = =

etc.

0 0|T [φ(x1 )φ(x2 )] |0 = iDF (x1 − x2 ) 0 0|T [φ(x1 )φ(x2 )φ(x3 )φ(x4 )] |0 − DF (x1 − x2 )DF (x3 − x4 ) − DF (x1 − x3 )DF (x2 − x4 ) − DF (x1 − x4 )DF (x2 − x3 )

(8.23)

(8.24)

Here |0 is the vacuum state of the free Hamiltonian because it was obtained as a functional derivative of the non-interacting functional Z0 (8.19). All n-point functions with odd n vanish.

8.2.2

Feynman rules

As in the classical case (cf. Sect. 2.3) we can again represent these results in a graphical way. The Feynman rules, that establish the connection between


93

the algebraic and the graphical representation, are for the case of free fields still rather trivial. They are given by 1) each Feynman propagator is represented by a line: = iDF (x − y) . x y 2) each source is represented by a cross:

= iJ(x) .

x

3) There is an integration over all space-time coordinates of the currents 4) Each diagram has a factor that takes its symmetry into account. For example, if there is an integration over the endpoints x and y, these could be exchanged without changing the result; the symmetry factor is, correspondingly, 1/2. With rule 1) we get, for example, for the fourpoint function (8.22), i.e. the two-particle Green’s function 1 2 G(x1 , x2 , x3 , x4 ) =

1 3

2 4

+

1

2

+

(8.25)

3 4 3 4 Each line connecting the two points x and y denotes the free propagator and gives a factor iDF (x − y). We now set (8.26) Z0 [J] = eiS0 [J] with

i 4 4 d x d y J(x)DF (x − y)J(y) iS0 [J] = − 2 By using rules 2), 3) and 4) we find immediately iS0 = Now we expand Z0 [J] = eiS0 [J] = e i = 1+ − J(x)DF (x − y)J(y) d4x d4y 2 2 i 1 − J(x)DF (x − y)J(y) d4x d4y + · · · + 2! 2

(8.27)

(8.28)

CHAPTER 8. GREEN’S FUNCTIONS = 1 + iS0 [J] + = 1+

94

1 1 (iS0 [J])2 + (iS0 [J])3 + . . . 2! 3! 1 1 + + + ... . 2! 3!

(8.29)

We now call all graphs that hang together connected graphs and all the others unconnected graphs. In our simple case here S0 [J] is represented by only one connected graph (8.28), whereas Z0 [J] – through the power expansion (8.29) of the exponential function – generates unconnected graphs as well. In the simple case of a free field discussed here there is only one connected graph (see (8.25)). The connected Green’s function can be obtained from its definition (8.6) as Gc (x1 , x2 ) =

1 δ 2 S0 = iDF (x1 − x2 ) . | i δJ(x1 )δJ(x2 ) J=0

(8.30)

All higher functional derivatives of S0 [J] vanish. Gc is in this free case thus just given by the Feynman propagator.

8.3

Interacting Scalar Fields

In this section we consider Lagrangians of the form L = L0 − V (φ) ,

(8.31)

where L0 is the free scalar Lagrangian (5.1) and V represents a selfinteraction of the field. For such a Lagrangian the generating functional for the n-point functions can no longer be given in closed form. In order to obtain the n-point function one has to resort to perturbative methods. The n-point function then follows from its definition (8.2)

G(x1 , x2 , . . . , xn ) =

Dφ φ(x1 )φ(x2 ) . . . φ(xn ) eiS[φ]

Dφ e

iS[φ]

(8.32)

with the exponential in the generating functional of the form eiS[φ] = ei

d4x (L0 −V +i 2ε φ2 )

.

(8.33)

The action exponential can be Taylor-expanded in the interaction strength eiS[φ] =

∞ 1 N =0 N !

−i

d4x V

N

eiS0 [φ]

(8.34)


95

with the free action

S0 [φ] =

ε d x L0 + i φ2 2

4

.

(8.35)

Inserting this into (8.32) gives the n-point function of the interacting theory in terms of powers of the interaction and the free-field action

G(x1 , x2 , . . . , xn ) =

Dφ φ(x1 )φ(x2 ) . . . φ(xn ) eiS[φ] Dφ e

Dφ φ(x1 )φ(x2 ) . . . φ(xn )

=

Dφ

∞ 1

∞ 1 N =0 N !

−i

N =0 N !

(8.36)

iS[φ]

−i

d4x V

N

N

4

d xV

eiS0 [φ]

eiS0 [φ]

By using (8.2) and the developments in Sects. 3.2 and 5.1.1 we can rewrite this equation also in terms of normalized vacuum expectation values. The last line of (8.36) involves the free action S0 and thus free propagation. Therefore, if the field operators are those of free in fields at asymptotic times they remain so even during propagation. Also the vacuum appearing in the quantum mechanical vacuum expectation value is then that of the non-interacting theory so that we get !

"

ˆ 2 ) . . . φ(x ˆ n ) |˜0 ˆ 1 )φ(x G(x1 , x2 , . . . , xn ) = ˜0|T φ(x

0|T φîn (x1 ) . . . φîn (xn ) = 0|T

∞ 1 N =0 N !

∞ 1 N =0

N!

−i

−i

Vˆ d4x

Vˆ d x

N

4

N

|0 .

(8.37)

|0

˜ denotes here the vacuum state of the full, interacting As in Sect. 5.1.1 |0 Hamiltonian, whereas |0 is that of the free Hamiltonian. With (8.37) we have achieved a remarkable result: (8.37) expresses the expectation value of the time-ordered product of field operators in the vacuum state of the full, interacting theory by a perturbative expansion over free field expectation values. The latter can be calculated as path integrals over products of classical fields and powers of the interaction (8.36). This enables us to calculate G, and ultimately also the scattering matrix S (Chapt. 7), perturbatively up to any desired order in the interaction.


8.3.1

96

Perturbative expansion

Eq. (8.32) allows us to calculate the perturbative expansion of the full Green’s function up to any desired order in V . Alternatively, these higher order terms can also be obtained as functional derivatives of the free generating functional Z0 [J] which is known. According to our general considerations the functional for the Lagrangian (8.31) is given by Z[J] = Z0 = Z0

Dφ ei

Dφ e−i

d4x (L0 −V (φ)+Jφ+i 2ε φ2 )

d4x V (φ) i

(8.38)

d4x (L0 +Jφ+i 2ε φ2 )

e

.

Here Z0 is just the inverse of the path integral for J = 0 Z0−1 =

Dφ ei

d4x (L0 −V (φ)+i 2ε φ2 )

.

(8.39)

We now use the relation 4 4 ε 2 ε 2 1 δ ei d x (L0 +Jφ+i 2 φ ) = φ(y) ei d x (L0 +Jφ+i 2 φ ) . i δJ(y)

(8.40)

This relation, read from right to left, will also be true for any function V (φ), as can be seen by expanding V into a series in powers of φ. We thus have also i

V [φ(y)] e

d4x (L0 +Jφ+i 2ε φ2 )

4 ε 2 1 δ ei d x (L0 +Jφ+i 2 φ ) =V i δJ(y)

(8.41)

and consequently, after exponentiation, also e−i

d4y V [φ(y)] i −i

= e

e

d4y V ( 1i

d4x (L0 +Jφ+i 2ε φ2 ) δ δJ(y)

) ei

d4x

L0 +Jφ+i 2ε φ2

(

(8.42) ).

This relation allows us to take the V -dependent factor in (8.38) out of the path-integral Z[J] = Z0 e

−i

δ d4y V ( 1i δJ(y) )

Dφ e

i

d4x (L0 +Jφ+i 2ε φ2 )

.

(8.43)

The last factor in (8.43) has been expressed in terms of the free two-particle propagator introduced in the last section. We thus have Z[J] = Z0 e−i

δ d4z V ( 1i δJ(z) ) − 2i

4

4

J(x)DF (x−y)J(y) d x d y e 4 δ 1 = Z0 e−i d x V ( i δJ(x) ) eiS0 [J] 4 1 δ = Z0 e−i d x V ( i δJ(x) ) Z0 [J] .

(8.44)


97

Expanding the exponential that contains the interaction V then gives the perturbative expansion for Z[J] Z[J] = Z0

∞ 1 N =0

−i

N!

4

d xV

1 δ i δJ(x)

N

Z0 [J] .

(8.45)

Since we will later on be mostly interested in connected graphs we do not need Z[J] directly but instead its logarithm. We, therefore, now expand the functional iS[J] = ln Z[J] in powers of the interaction V . We start by inserting a factor 1 = exp (+iS0 ) exp (−iS0 ) between Z0 and the exponential in (8.44) and taking the logarithm

ln Z[J] = ln Z0 + ln 1 · e−i

= ln Z0 + iS0 + ln e

d4x V ( 1i

δ δJ

−iS0 −i

e

&

) eiS0 [J]

d4x V iS0

e

−iS0 [J]

= ln Z0 + iS0 [J] + ln 1 + e

e

−i

δ d4x V ( 1i δJ )

−1 e

= iS[J] .

' iS0 [J]

(8.46)

A perturbation theoretical treatment is now based on a Taylor expansion of the logarithm. For that purpose we abbreviate

ε[J] = e−iS0 [J] e−i

d4x V

− 1 eiS0 [J]

(8.47)

and obtain iS[J] = ln Z[J] = ln Z0 + iS0 [J] + ln(1 + ε[J]) 1 2 3 = ln Z0 + iS0 [J] + ε[J] − ε [J] + O(ε ) . 2

(8.48)

Equation (8.48) represents an expansion of S in powers of the (for V → 0) small quantity ε. In order to obtain a perturbative expansion in terms of the potential V we now rearrange the expansion (8.48). First we expand ε[J] of (8.47) in terms of the strength of the interaction. This gives ) −iS0 [J]

−i

ε[J] = e

+

1 −i 2!

4

d xV

d4x V

1 δ i δJ

1 δ i δJ

(8.49) ⎫ ⎬

2

+ · · ·⎭ e+iS0 [J] .

We now insert this expression into (8.48) and obtain

1 iS[J] = ln Z0 + iS0 [J] + ε[J] − ε2 [J] + . . . 2


98

−iS0 [J]

= ln Z0 + iS0 [J] + e

1 + e−iS0 [J] −i d4x V 2! )

−i

1 −iS0 [J] − e −i d4x V 2

4

d xV

1 δ i δJ

2

1 δ i δJ

eiS0 [J]

eiS0 [J]

1 δ i δJ

= ln Z0 + iS0 [J] + iS1 [J] + iS2 [J] −

,2

e

iS0 [J]

+ O(V 3 )

1 (iS1 [J])2 + O(V 3 ) , 2 (8.50)

with i J(x)DF (x − y)J(y) d4x d4y 2 1 δ −iS0 [J] 4 iS1 [J] = e (−i) d x V e+iS0 [J] i δJ

iS0 [J] = −

1 −iS0 [J] iS2 [J] = −i d4x V e 2!

1 δ i δJ

2

e+iS0 [J] .

(8.51)

Equation (8.50) represents a perturbative expansion of S in powers of V . Note that the term of second order in the interaction V receives contributions both from the linear and the quadratic term in the original expansion (8.48) in ε. The term ∼ S1 obviously just contains an iteration of the linear term.

Chapter 9 PERTURBATIVE φ4 THEORY In this chapter we apply the formalism developed in the preceding chapter to the so-called φ4 theory whose Lagrangian is given by L = L0 − V (φ) = L0 −

g 4 φ . 4!

(9.1)

Here g is a coupling constant. This φ4 theory is a prototype of a field theory with selfinteractions. It serves as a didactical example which exhibits all phenomena of more complex field theories.

9.1

Perturbative Expansion of the Generating Function

We start with the generating functional for connected Green’s functions (8.50) iS[J] = ln Z0 + iS0 [J] + iS1 [J] + iS2 [J] −

1 (iS1 [J])2 + O(V 3 ) . 2

(9.2)

Inserting the interaction of φ4 theory (9.1) into (8.51) we obtain i 4 4 iS0 [J] = − d z d y J(z)DF (z − y)J(y) , 2 ig −iS0 [J] 4 δ4 iS1 [J] = − e d x 4 eiS0 [J] 4! δJ (x) and 1 −ig 2 −iS0 [J] 4 4 δ4 δ4 iS2 [J] = e d xd y 4 eiS0 [J] . 2! 4! δJ (x) δJ 4 (y)

99

(9.3)

CHAPTER 9. PERTURBATIVE φ4 THEORY

100

For notational convenience in the following we now introduce the Si defined by ig iS1 [J] = − S˜1 [J] 4! −ig 2 ˜ S2 [J] . iS2 [J] = 4!

(9.4)

The generating function for the connected Green‘s functions in φ4 theory reads then iS[J] = ln Z0 + iS0 [J] +

−ig ˜ −ig S1 [J] + 4! 4!

2

2

1 −ig ˜ S1 [J] S˜2 [J] − 2 4!

+ ... .

(9.5) Since each of the Si [J] contains functional derivatives of the known S0 [J] we can now evaluate the functional derivatives of S[J] and obtain all the Green’s functions. First, we calculate the terms linear in g

δ4 e+iS0 [J] 4 δJ (x) i δ4 4 4 = e−iS0 [J] d4x 4 e− 2 J(z)DF (z−y)J(y) d z d y . δJ (x)

S˜1 [J] = e−iS0 [J]

d4x

(9.6)

The fourth functional derivative is most easily obtained by going to a discrete representation. Noting that ∂ 4 − i Ji Dij Jj e 2 = [−3Dkk Dkk + 6iDkk (DJ)k (DJ)k ∂Jk4

(9.7)

+ (DJ)k (DJ)k (DJ)k (DJ)k ] e− 2 Ji Dij Jj i

we obtain (k =x) ˆ S˜1 [J] = −3

+ 6i

+

DF (x − x)DF (x − x) d4x DF (y − x)DF (x − x)DF (x − z)J(y)J(z) d4x d4y d4z

[DF (x − y)DF (x − z)DF (x − v)DF (x − w) "

× J(y)J(z)J(v)J(w) d4x d4y d4v d4w d4z .

(9.8)

The first term has no sources and will, therefore, not contribute to any Green’s function, the second term is quadratic in J and thus contributes to the two-point function and the last term here has the structure of a point interaction of four fields generated by independent sources at y, z, v, and w and thus contributes only to the four-point function.


9.1.1

101

Feynman rules

We can again represent these results in a graphical form by using the rules given in section 8.2.2. These were iDF (x − y) =

1) propagator: 2) source:

iJ(x) =

x

y

x

3) Integration over the space-time coordinates of the sources 4) Symmetry factor for each diagram We supplement these now for the interacting theory by the additional rules −ig = 4!

5) Each interaction is represented by a dot:

6) Integration

d4x for each loop.

If we represent the interacting connected functional iS[J] by a double line (9.9) iS[J] = we can draw the graphs for iS[J] = ln Z0 + iS0 [J] +

−ig ˜ S1 [J] + O(g 2 ) 4!

(9.10)

as = ln Z0 + ⎛

+⎜ ⎝

w x

+

+ y

x

z

y

x

z

⎞

(9.11)

⎟ 2 ⎠ + O(g )

v

The first graph on the right-hand side is again the zeroth order term (9.3), whereas the graphs in the parentheses represent all the terms of first order in the interaction (9.8). The first one of these describes a process without any external lines; this is a vacuum process that takes place regardless if physical particles are present or not. It constitutes a background to all physical processes. The second graph with the single loop describes a mass change due to the selfinteraction that we will discuss in the next section. The third graph, finally, describes a true interaction process.


102

The symmetry factors in these graphs are the factors in front of the integrals in (9.8); they can be obtained as follows. The first graph carries the factor 1/2 as explained in section 8.2.2. To construct the vacuum graph we pick one of the 4 legs of the interaction vertex and then connect with any of the other three free legs; there are always 2 pairwise equal loops. Thus in total we get a weight of 4 × 3/(2 × 2) = 3. For the second term in parentheses in (9.11) we have four legs of the vertex to connect with the external line to y; this gives 4 possibilities. The external line to z can then still be connected with 3 remaining vertex legs. Since there is an exchange symmetry between y and z we get an additional factor 1/2 (as in S0 ), so that the weight of this vertex becomes 6. The last graph, finally, is obtained by joining one of the four legs of the vertex to one of the external points, say z. This generates 4 possibilities. Next we join any one of the 3 remaining free legs of the vertex to the external point y; there are obviously 3 ways to do this. The remaining 2 legs can be joined in 2 different ways with the two external points v and w. Thus, there are in total 4! = 24 possibilities. However, since all the external points v,w,y and z can be exchanged without changing any of the physics (v,w,y and z are integration variables), there are also 4! identical terms so that the last graph in the parentheses in Fig. 9.11 carries the weight 1. These weights (symmetry factors) have to be multiplied for each graph to the analytical expression obtained by following the rules given above for the translation of the pictorial representation into an analytical one. Indeed, using the symmetry factors just given and following the Feynman rules for the graphs (9.11) gives the expression (8.50) (together with (9.4) and (9.8)).

9.1.2

Vacuum contributions

We now consider the normalization term ln Z0 . Since we are working with normalized generating functionals, Z0 is given by W [0]−1 Z0−1

=

Dφ e

= e−i

i

d4x(L0 −V +i 2ε φ2 )

δ d4xV ( 1i δJ(x) ) − 2i

e

(9.12)

J(x)DF (x−y)J(y)d4x d4y |

J=0

.

We can now treat this expression in exactly the same way as we just did for S[J]; the only change being that we have to take the final result at J = 0. This gives (see (9.10)) ln Z0 = −iS[0] = −iS0 [0] −

(−ig) ˜ S1 [0] + O(g 2 ) . 4!

(9.13)


103

In the graphical representation this reads, using S0 [0] = 0 and S1 [0] = −3DF2 (0) d4x (cf. (9.8)), ln N = −

(9.14)

The normalization constant, or – in other words – the denominator of the generating functional, thus contains just the vacuum graph. Inserting (9.14) into (9.11) then removes the vacuum contribution giving, finally, for Z[J] the graphical representation up to terms of first order in the interaction =

+

+

(9.15)

Although we have shown here only for first-order coupling that the denominator in Z[J] (see (5.15),(5.16)) just removes the vacuum contributions, this is a general result that holds to all orders of perturbation theory.

9.2

Two-Point Function

The connected n-point function can now be obtained from its definition in (8.6) . n−1 . 1 δnS . . . (9.16) Gc (x1 , . . . , xn ) = i δJ(x1 ) . . . δJ(xn ) .J=0 In this section we work out the connected two-point function in the lowest orders of the coupling constant.

Terms up to O(g 0 )

9.2.1

There is only one connected Green’s function in the free case which is just given by the Feynman propagator (8.30). 9.2.1.1


We now evaluate the lowest order (in g) two-point function in momentum space (cf. (8.7)). Taking the Fourier-transform of the two-point function (8.30) gives

e−i(p1 x1 +p2 x2 ) Gc (x1 , x2 ) d4x1 d4x2

=

e−i(p1 x1 +p2 x2 )

i (2π)4

(9.17)

d4q

e−iq(x1 −x2 ) d4x1 d4x2 . 2 2 q − m + iε


104

−p

p

Figure 9.1: Two-point function of a scalar theory. We first perform the integrations over x1 and x2 and obtain, according to the definition (8.1.1), for the rhs (2π)4 δ 4 (p1 + p2 )Gc (p1 , p2 ) = (2π)4 δ 4 (p1 + p2 )

p21

i . − m2 + iε

(9.18)

From this equation we can read off the momentum representation of the propagator. The momenta p1 and p2 are incoming momenta that point towards a vertex. Thus, the momentum representation of the free propagator is given by i G0 (p, p = −p) = 2 , (9.19) p − m2 + iε pictured in Fig. 9.1. Note that here the second momentum appears with a negative sign. This is due to our notation to take all momenta as incoming (see Fig. 8.1).

9.2.2

Terms up to O(g)

Up to terms linear in the coupling strength we obtain from (9.5) .

. δ2 S . . Gc (x1 , x2 ) = −i δJ(x1 )δJ(x2 ) .J=0

(9.20) .

. δ 2 S0 (−ig) δ 2 S1 . . = −i + O(g 2 ) . − δJ(x1 )δJ(x2 ) 4! δJ(x1 )δJ(x2 ) .J=0

With S0 [J] from (9.3) and S1 [J] from (9.8) this gives for the two-point function Gc (x1 , x2 ) = iDF (x1 − x2 ) ig − − 12i d4xDF (x − x)DF (x − x1 )DF (x − x2 ) + O(g 2 ) 4! g 4 d xDF (x2 − x)DF (x − x)DF (x − x1 ) = iDF (x1 − x2 ) − 2 + O(g 2 ) . (9.21)


105

This is the connected propagator, up to terms of O(g), of the interacting theory. We can represent this equation in the following graphical form, where denotes the “dressed” propagator x1

x2

=

x1

x2

+

(9.22)

x1 x2 with the rules developed above. Since the n-point functions involve derivatives with respect to the source current, taken at zero source, the external lines of all Feynman graphs do not contain crosses, that depict sources, anymore. They are instead given by free propagators. The weight factors for these diagrams have been explained at the end of section 8.2.2. Since we deal here with Green’s functions with definite, fixed external points, the exchange symmetry factors must not be divided out here. Thus, the first diagram on the rhs in (9.22) carries the weight 1 and the second, the so-called tadpole diagram, the weight 12. 9.2.2.1


In Sect. 8.1.1 we have introduced the momentum representation of the Green’s function and in Sect. 9.2.1.1 we have already evaluated it for the free case. Here we now determine it for the φ4 theory up to terms of order O(g). Taking the Fourier-transform of the two-point function (9.21) gives

e−i(p1 x1 +p2 x2 ) Gc (x1 , x2 ) d4x1 d4x2

=

e−i(p1 x1 +p2 x2 ) ⎡

i (2π)4

(9.23)

d4q

e−iq(x1 −x2 ) d4x1 d4x2 2 2 q − m + iε

g ⎣ −i(p1 x1 +p2 x2 ) 4 1 − e dx 2 (2π)4 ×

3

e−iq2 (x2 −x) e−iq1 (x−x1 ) d q1 d q2 d q3 2 d4x1 d4x2 . (q1 − m2 + iε)(q22 + m2 − iε)(q32 − m2 + iε) 4

4

4

As in Sect. 9.2.1.1 we first perform the integrations over x1 , x2 and x and obtain for the rhs i (2π)4 δ 4 (p1 + p2 )Gc (p1 , p2 ) = (2π)4 δ 4 (p1 + p2 ) 2 p1 − m2 + iε g 1 1 − (2π)4 δ 4 (p1 + p2 ) 2 2 2 2 p1 − m + iε p2 − m2 + iε d4q 1 × . (9.24) 4 2 (2π) q − m2 + iε


106

This equation is used to read off the momentum representation of the propagator (see (8.7)). The momenta p1 and p2 are incoming momenta that point towards a vertex. Thus, the momentum representation of (9.21) is given by i (9.25) − m2 + iε −ig d4q i i i +S 2 2 4 2 2 2 p − m + iε 4! (2π) q − m + iε p − m2 + iε

Gc (p, p = −p) =

p2

where the symmetry factor is S = 12. Equation (9.25) gives the momentum representation of the two-point function up to terms of order g. The first term on the rhs gives the free propagator (9.19) already obtained in Sect. 9.2.1.1, whereas the second term gives the contribution of the interaction to this two-point function. Momentum space Feynman rules. The Feynman rules for (9.25) are now i . 1) each line gives a factor 2 q − m2 + iε 2) each vertex gives a factor

−ig . 4!

3) there is four-momentum conservation for the sum of all momenta flowing into a vertex.

4) each internal line gives an integration

d4q . (2π)4

5) to each diagram a weight factor has to be multiplied as explained above. Selfenergy. With the abbreviation Σ=

g d4 q i 4 2 2 (2π) q − m2 + iε

(9.26)

and the free two-point function G0 from (9.19 we can write the two-point function (9.25) as

Σ Σ ΣG0 Gc (p, −p) = G0 + G0 G0 = G0 1 + G0 ≈ G0 1 − i i i i 1 = 2 2 p − m + iε 1 − Σ p2 −m1 2 +iε i . = 2 2 p − m − Σ + iε

−1

(9.27)


107

h

a)

b)

c)

Figure 9.2: Feynman graphs for the two-point function up to O(g 2 ). Here we have consistently kept terms up to O(g). The quantity Σ appears like an additional mass term in the final result. It is, therefore, called a selfenergy and the second graph on the rhs in (9.22) is called a selfenergy insertion. The appearance of this selfenergy is a first indication that the mass m appearing in the Lagrangian is the mass of the particle only in a classical theory. In quantum theory it gets changed by the interactions.

9.2.3

Terms up to O(g 2 )

As noted at the end of Sect. 8.3.1 there are two distinct contributions to the second order term, one being a genuine term of second order in V and the other just being an iteration of the first order term. With the help of the Feynman rules we can now construct the corresponding Feynman graphs up to terms of O(g 2 ). For the two-point function these are given in Fig. 9.2. Graph (a) in Fig. 9.2 obviously just represents an iteration of the first order tadpole graph in (9.22). Its contribution to the two-point function is given by

i −ig d4q i (9.28) Ga (p, −p) = Sa 2 2 4 2 p − m + iε 4! (2π) q − m2 + iε i −ig d4q i i × 2 , 2 4 2 2 2 p − m + iε 4! (2π) q − m + iε p − m2 + iε Sa is the symmetry factor; it is simply given by the product of the corresponding factors for the one-loop graphs, Sa = 12 · 12 = 144. It is evident from Fig. 9.2a, as well as from its algebraic representation in (9.28), that the graph can be cut into two parts, each representing a first


108

order process. This reflects the appearance of the last term in (8.50) that is simply the square of the first order term. Such a graph that falls apart into 2 unconnected parts, if one internal line is cut, is called one-particlereducible; otherwise it is one-particle-irreducible (1PI). The reducible graph here is generated by the square of the first order term ∼ S˜12 in (9.5). In order to facilitate the following discussions we introduce now the vertex function Γ(p1 , p2 , . . . , pn ), sometimes also called connected proper vertex function, which describes only 1PI graphs and in which the propagators for the external lines are missing. The n-point vertex function is, therefore, given by (9.29) Γ(p1 , p2 , . . . , pn ) −1 −1 −1 = G (p1 , −p1 )G (p2 , −p2 ) . . . G (pn , −pn )Gc (p1 , p2 , . . . , pn ) . The free 1PI 2-point function is defined by Γ(p, −p) = p2 − m2 .

(9.30)

Note that the product of inverse two-point Green’s functions and the n-body function is just the combination that appears in the reduction theorem (cf. (7.35)). The 1PI part of the Green’s function for the graph 9.2a reads Γa (p, −p) = G−1 (p, −p)G−1 (−p, p)Ga (p, −p) −ig d4q i = Sa 4 2 4! (2π) q − m2 + iε

(9.31)

with Sa = 144. We then get for the graph in Fig. 9.2b, which is one-particle irreducible,

−ig 2 d4q d4u i i(2π)4 δ 4 (q − u) 4! (2π)4 (2π)4 q 2 − m2 + iε u2 − m2 + iε 4 dr i (9.32) × 4 2 (2π) r − m2 + iε

Γb (p, −p) = Sb

with Sb = 12 · 12 = 144. For the graph in Fig. 9.2c (also 1PI) we finally obtain

−ig 4!

2

d4q d4r d4s δ(p − (q + r + s)) (2π)4 (2π)4 (2π)4 i i i , (9.33) × 2 2 2 2 2 q − m + iε r − m + iε s − m2 + iε

Γc (p, −p) = Sc

with Sc = 4 · 4! = 96.

CHAPTER 9. PERTURBATIVE φ4 THEORY 3

109

4

x

1

2

Figure 9.3: Feynman graph for the four-point function.

9.3

Four-Point Function

It is easy to see that the three-point function vanishes for the model considered here since the third functional derivative (see (9.16)) of the action (8.50) at J = 0 vanishes.

9.3.1

Terms up to O(g)

The four-point function up to terms of O(g) is given by 3

1 i

Gc (x1 , x2 , x3 , x4 ) =

.

. δ4 S . . δJ(x1 ) . . . δJ(x4 ) .J=0 .

(9.34) .

. . δ 4 S0 (−ig) δ 4 S1 . . . . = i + δJ(x1 ) . . . δJ(x4 ) .J=0 4! δJ(x1 ) . . . δJ(x4 ) .J=0

+ O(g 2 ) where S is given by (9.5) and Fig. 9.2. Because Gc involves the fourth derivative with respect to the source and S0 depends on J only quadratically (see (9.3)), only the last term of S1 in (9.8) can contribute to the Green’s function. Thus we get Gc (x1 , . . . , x4 ) = −ig

d4xDF (x − x1 )DF (x − x2 )DF (x − x3 )DF (x − x4 ) .

(9.35) In momentum space this is simply the product of the four propagators (9.19) times the factor −ig. The corresponding Feynman graph is given in Fig. 9.3. It carries the symmetry factor 4!, corresponding to a symmetry under permutation of all external legs. Unconnected graphs. From the connected graphs calculated so far, we could reconstruct also the unconnected graphs. Up to terms linear in the


110

coupling constant g we get in symbolic notation Z[J] = e

= 1+

+

= 1+

1 ( 2!

+

1 + 2!

)2 + . . .

+

2

+

+

+O(g 2 ) .

(9.36)

The four-point function is generated by diagrams with four external legs (each external leg corresponds to a factor J in the generating functional), because G is given by a fourth functional derivative. Therefore, only the last diagram in the second line of (9.36) and the square of the first term in the last line can contribute to the four-point function in this order. We thus have for the four-point function up to terms linear in the coupling G(x1 , x2 , x3 , x4 ) =

+

+

(9.37)

In both of the first two diagrams the two particles just move by each other, without interaction. These unconnected graphs thus do not contribute to any interaction processes.

9.3.2

Terms up to O(g 2 )

. In order to become more familiar with Feynman graphs, we now construct the connected four-point function up to terms of order g 2 in a graphical way. This four-point function is shown in Fig. 9.4. The first line in Fig. 9.4 gives the four-point function just constructed, with a symmetry factor S1 = 4!. The four diagrams on the second line are just the basic vertices with self-energy insertions on each of the external legs; these insertions carry the extra symmetry factor S2 = 12, as we have seen in section 9.1.1. The last three diagrams in the third line are of a new topological structure. They represent modifications of the basic interaction vertex through the insertion of internal lines. Each one of these graphs has the same external lines. We thus have an overall factor for all graphs S1

4

2 k=1 pk

i , − m2

(9.38)


3

4

3

4

1

2

111

=

1

2 3

4

+

3

4

+ 1

2

3

4

4

+ 1

2

3 +

3

3

4

1

2

+ 1

2 3

4

q1

4 + O(g 3 )

+

+ q2 1 1

2

2

1

2

Figure 9.4: Four-point function of φ4 theory up to terms ∼ g 2 . with the external symmetry factor S1 = 4! so that the full four-point function is given by Gc (p1 , p2 , p3 , p4 ) = S1

4

2 k=1 pk

i (G1 + G2 + G3 ) , − m2

(9.39)

where Gi denotes the contribution from the i-th line in Fig. 9.4 without the external line symmetry factor. The first basic vertex then just gives the contribution G1 =

−ig . 4!

(9.40)

The graphs of the second line have one loop in addition. We thus have G2 = (

4 i i −ig 2 d4q ) S2 2 4! (2π)4 q 2 − m2 l=1 pl − m2

(9.41)


112

for their contribution, with S2 = 12. The internal momenta pl here are those between the loop and the four-point vertex. Since the loop carries no momentum away they are the same as the corresponding incoming momenta. The three graphs of the third line, finally, have two internal lines, which are, however, related by energy - and momentum conservation at the incoming vertex. They give i i −ig 2 d4q1 d4q2 ) S3 G3 = ( 2 2 4 4 2 4! (2π) (2π) q1 − m q2 − m2 × (2π)4

δ 4 (q1 + q2 − (pk + pl ))

(9.42)

kl

where the last sum runs only over the pairs of numbers (1,2), (1,3) and (1,4), i.e. the external legs at one of the vertices in each of the three graphs. The δfunction appears because the net momentum running into the dressed vertex has to be zero (see (8.7)). The symmetry factor S3 can, for example for the middle graph of Fig. 9.4, be obtained as follows. The external leg 1 can be connected with the left vertex in 4 different ways; the same holds for the leg 2 with the right vertex. Once these connections (4 × 4 possibilities) have been done, each of the 3 free legs of the left vertex can be connected with the external leg 3; the same holds for the connections of the right vertex to the external point 4 (3 × 3 possibilities). Finally, each of the remaining 2 legs of the left vertex can be connected to the right vertex; for the remaining leg there is then no freedom left (2 possibilities). Thus, the symmetry factor for the last 3 graphs in Fig. 9.4 is (4 · 4)(3 · 3)2 (4!)2 4! = = . (9.43) S3 = S1 2S1 2

Chapter 10 DIVERGENCES IN n-POINT FUNCTIONS Many of the expressions obtained in the preceding sections for two- and fourpoint functions are actually ill-defined because they diverge, as we will show in this section. We start with a rather general discussion of divergences in φ4 theory and then evaluate explicitly the two- and four-point functions. To illustrate the divergence of the Green’s functions obtained we consider, as an example, the two-point function Gc (p, −p) =

i − m2 + iε i g i + −i iD . (0) F 2 p2 − m2 + iε p2 − m2 + iε

p2

(10.1)

The loop contribution between the two Feynman propagators in the second term on the rhs is given by g g d4q i Σ = iDF (0) = . 4 2 2 2 (2π) q − m2 + iε

(10.2)

The integral here diverges: after integrating over q0 we obtain integrals of the form (cf. section 6.1.2)

d3q √ 2 . q + m2

(10.3)

By introducing an upper bound Λ for the integral over |q| and then taking Λ → ∞ we see that the integral diverges as Λ2 ; this is called a quadratic divergence. Because this divergence happens here for large q one also speaks of a quadratic ultraviolet divergence. 113

CHAPTER 10. DIVERGENCES IN N -POINT FUNCTIONS

114

Another way to see the degree of divergence is the so-called “powercounting”: There are 4 powers of q in the integration measure, but only two powers of q in the denominator of (10.2); the degree of divergence is then given by the net power (2) of q.

10.1

Power Counting

The power-counting just illustrated for the case of the tadpole diagram can be generalized to any Feynman graphs with an arbitrary number of loops. In order to see this we consider a theory with an interaction ∼ φp in n dimensions. Since each loop contributes according to the Feynman rules an integral dnq to the total expression and since each internal propagator gives a power q −2 , we have for the degree of divergence D in a diagram with L loops and I internal lines D = nL − 2I . (10.4) Note that here each loop has also at least 1 internal line. For example, the tadpole diagram has L = 1 and I = 1 , giving D = 2 in four dimensions. D > 0 clearly diverges, D = 0 corresponds to a logarithmic divergence, and D < 0 seems to be convergent. If a graph has V interaction vertices, then the total number of lines in φp theory is pV , since each vertex has p legs. These legs can be either external or internal lines. If they are internal, they count twice because each internal line originates and disappears at a vertex. Thus we have pV = E + 2I ,

(10.5)

where E is the number of external lines. In addition, the number of loops, L, is related to the number of vertices, V , by L=I −V +1 .

(10.6)

Combining equations (10.5) and (10.6) with (10.4) allows us to eliminate L and I to obtain

n(p − 2) p −p V − −1 E . D =n+ 2 2

(10.7)

The degree of divergence of a connected graph in this φp theory in n dimensions thus depends on the number of external legs and, in general, also on the number of vertices. In the perturbative treatment of field theory derived in Chap. 8.3 we have seen that the order of perturbation theory directly equals the number of


115

D = 4 · 1 − 2 · 1 = +2

D =4·1−2·2=0

D = 4 · 0 − 2 · 1 = −2

D = 4 · 1 − 2 · 3 = −2

Figure 10.1: Examples for graphs with different degree of divergence. On the right equation (10.4) is illustrated for n = 4. vertices, V , in a Feynman diagram. Thus, the degree of divergence becomes larger and larger with increasing order of a perturbative treatment, if the factor of V in (10.7) is positive. On the other hand, D is independent of this order if that factor is zero and D becomes even smaller when going to higher orders of perturbation theory if the factor is negative. Thus the perturbative treatment leads to a finite number of divergent terms if and only if n(p − 2) −p≤0 . 2

(10.8)

If this factor is zero, then the total number of divergent diagrams in a perturbative expansion can still be infinite, but in each order of perturbation theory only the same finite number of divergent diagrams appears. In this case the theory is called renormalizable. If the factor is negative, then even the total number of divergent terms is finite; in this case the theory is called superrenormalizable. In both cases one can add a finite number of so-called counter terms to the Lagrangian that just remove these divergences.


116

In order to become familiar with this power counting for φ4 theory in four dimensions we give three examples in Fig. 10.1. According to our earlier considerations we expect that D ≥ 0 diverges. However, this does not guarantee that graphs with D < 0 actually converge. This is illustrated by the lowest example, in Fig. 10.1, that has L = 1, I = 3, and thus D = 4 · 1 − 2 · 3 = −2, but diverges, because of the loop on the externals legs. Only when each possible subgraph has also D < 0, then the whole expression converges (Weinberg’s Theorem). Note that in the physical case n = 4 and p = 4 D does not depend on V , i.e. it is independent of the order of perturbation theory. φ4 theory is thus renormalizable.

10.2

Dimensional Regularization of φ4 Theory

Regularization serves to make all divergent expressions convergent, at the expense of introducing a parameter that has no physical meaning and ultimately has to be removed. If the integrals can be performed and evaluated as a function of this parameter then the infinite contributions can be separated from the finite ones. We show this procedure by evaluating now the divergent integrals by a modern technique called dimensional regularization. This technique starts by considering the theory in n dimensions. We first consider the dimensions in the Lagrangian L=

" g 1! μ ∂ φ∂μ φ − m2 φ2 − φ4 . 2 4!

Since the action S=

L dnx

(10.9)

(10.10)

is a dimensionless quantity (in units in which h ¯ = 1), L must have the −n dimension ( is a length). The kinetic energy term in L thus has also dimension −n and – since [∂μ ] = −1 – we get [φ] = 1−n/2 . Any potential term of the form gφp must have the same dimension as the Lagrangian, i.e. n −n ; we thus get [g] = −n /p(1−n/2) = −(n+p(1− 2 )) . The mass term (p = 2) thus has [m2 ] = −2 in n dimensions, as it should. Note that the dimension of g is just the factor of the number of vertices, V , in (10.7). Thus, the dimension of the coupling constant and the renormalizability of a theory are closely connected: if the dimension of g is that of a positive or zeroth power of mass, then the theory is superrenormalizable or renormalizable, respectively.


117

These considerations show that for p = 4 [g] = n−4 = mass4−n ;

(10.11)

i.e. in four dimensions the coupling constant of a φ4 theory is dimensionless and the theory is, therefore, renormalizable; in fact, only theories with p ≤ 4 are renormalizable. Therefore, according to the discussion in the preceding section the φ4 theory contains only a finite number of divergent vertices. In n = 4 dimensions, however, this is no longer true. If we want to keep the coupling constant dimensionless in order to ensure renormalizability also in n dimensions we have to modify the φ4 term in the Lagrangian (8.51) such that an additional factor with the dimension of (mass)4−n absorbs the dimension and g becomes dimensionless L = L0 −

g 4−n 4 μ φ . 4!

(10.12)

Note that μ here is an arbitrary mass.

10.2.1

Two-point function

The two-point function is completely determined once we know the selfenergy. We start by calculating this quantity in lowest order in the coupling by evaluating explicitly the contribution of the tadpole diagram. To do so we first go to n dimensions so that (10.2) becomes dnq i g . Σ = μ4−n 2 (2π)n q 2 − m2 + iε

(10.13)

This integral can be obtained analytically by going into a space of n-dimensional polar coordinates (see Appendix. C). The result is

n g μ4−n n−2 n m π2Γ 1− Σ= n 2 (2π) 2

.

(10.14)

The divergence of this expression is now manifest, since the Γ-function has poles at 0 and the negative integers, and thus also for n = 4. We now expand Γ around this pole. For that purpose we write

n ε = Γ −1 + Γ 1− 2 2

(10.15)

with ε = 4 − n and expand in powers of ε (cf. (C.4))

Γ −1 +

ε 2 = − − 1 + γ + O(ε) , 2 ε

(10.16)


118

where γ is the Euler-Mascheroni constant (γ ≈ 0.577..). We thus get for n close to 4

2 g με 2−ε 4−ε 2 m π − − 1 + γ + O(ε) Σ = 2 (2π)4−ε ε m2 = g 32π 2

4πμ2 m2

ε 2

We now use

2 − − 1 + γ + O(ε) ε

(10.17)

.

ε→0

xε = eε ln x −→ 1 + ε ln x

(10.18)

and obtain

gm2 ε 2 4πμ2 1 + − − 1 + γ + O(ε) ln Σ −→ 2 2 32π 2 m ε 4πμ2 gm2 2 = − − 1 + γ − ln + O(ε) 32π 2 ε m2 gm2 1 gm2 4πμ2 = − 1 − γ + ln + O(ε) . − 16π 2 ε 32π 2 m2 ε→0

(10.19)

We have thus split Σ into two parts. The first one is clearly diverging as n approaches 4 (ε → 0). The second term is finite and depends on the arbitrary mass μ that was originally introduced only to keep the coupling constant free of dimension. The appearance of the arbitrary mass μ in the finite part is related to the arbitrariness in separating an overall infinite expression into a sum of an infinite and a finite contribution. Note that Σ is independent of p. In next higher order the same is true for the selfenergy contribution of Fig. 9.2b on page 107 whereas that of Fig. 9.2c depends quadratically on p [Ramond].

10.2.2

Four-point function

In this section we will now evaluate the four-point function. By looking at Fig. 9.4 on page 111 we expect that the four graphs in the second line just contribute again to the selfenergy. On the other hand, we expect that the three diagrams in the lowest line can all graphically be contracted such that they contain only one interaction point with possibly modified interaction strength. As an example, we now evaluate the middle graph in the last line of Fig. 9.4. According to the Feynman rules its contribution to the 1PI vertex is


119

(see (9.42))

−ig 4−n ΔΓ(p1 , p2 , p3 , p4 ) = μ 4!

2

d4q i i S3 S1 4 2 2 (2π) q − m (p − q)2 − m2 (10.20) with p = p1 + p3 . The symmetry factor is S1 S3 = 12 · 4! (9.42). The two denominators in the integrand can be combined into one by a mathematical trick due to Feynman that is very often used for the evaluation of such expressions. This trick starts from the elementary integral relation b a

1 1 1 dx b−a = − |ba = − + = . 2 x x b a ab

(10.21)

By substituting now x = az + b(1 − z) we obtain

b a

dx dz = (a − b) 2 . 2 x [az + b(1 − z)] 1

(10.22)

0

(10.23)

Combining (10.21) and (10.23) gives 1 dz = 2 . ab [az + b(1 − z)] 0 1

(10.24)

We now apply this to the integrand in (10.20) and obtain 1 1 1 dz = 2 2 2 2 2 2 q − m (p − q) − m {(q − m )z + [(p − q)2 − m2 ] (1 − z)}2 0 1

= 0

dz . [q 2 − 2pq(1 − z) + p2 (1 − z) − m2 ]2 (10.25)

The first three terms in the denominator can be combined by substituting q = q − p(1 − z) .

(10.26)

This gives for expression in the denominator q 2 − 2pq(1 − z) + p2 (1 − z) − m2 = [q − p(1 − z)]2 − m2 − p2 (1 − z)2 + p2 (1 − z) = q 2 − m2 − p2 z(z − 1) .

(10.27)


120

The 1PI vertex now reads in n dimensions 1 1 2 2 4−n dnq 1 ΔΓ(p1 , p2 , p3 , p4 ) = g (μ ) dz 2 n 2 (2π) [q − m2 − sz(z − 1)]2 0 (10.28) with s = p2 = (p1 + p3 )2 . We now interchange the order of integration and evaluate the integral over q first with the help of (C.18) in Appendix C

dnq 1 (2π)n [q 2 − m2 − sz(1 − z)]2

" n−4 n Γ 2 − n i ! 2 2 = m + sz(1 − z) 2 π 2 n (2π) Γ(2)

.

(10.29)

The four-point function thus reads

n

1 2 2 4−n π 2 n Γ 2− ΔΓ(p1 , p2 , p3 , p4 ) = g (μ ) i n 2 (2π) 2 ×

1

!

" n−4

dz m2 + sz(1 − z)

2

.

(10.30)

0

We now introduce again ε = 4 − n. This gives − ε 1 2 1 2 ε 1 ε m + sz(1 − z) 2 ΔΓ(p1 , p2 , p3 , p4 ) = g μ i Γ dz . (10.31) 2 16π 2 2 4πμ2 0

Now the four-point function is in a form that allows to take the limit ε → 0. Using again xε = eε ln x → 1 + ε ln x (10.32) and (C.2)

Γ

ε 2 = − γ + O(ε) 2 ε

(10.33)

gives &

ΔΓ(p1 , p2 , p3 , p4 ) =

ig 2 με 2 − γ + +O(ε) 32π 2 ε ⎡

'

ε m2 + sz(1 − z) × ⎣1 − ln 2 4πμ2 1

0

⎡

⎤

dz ⎦

ig 2 με 1 ig 2 με ⎣ m2 + sz(1 − z) = γ + ln − 16π 2 ε 32π 2 4πμ2 1

⎤

dz ⎦

0

+ O(ε)

with

s = (p1 + p3 )2 .

(10.34)


121

Here the four-point function has been separated into a divergent part and a convergent term (for ε → 0). The integral appearing is a function of p, m, and μ; the latter dependence remains even when ε → 0. So far, we have only calculated the middle graph in the last line of Fig. 9.4 on page 111. It is evident, however, that the result can be directly taken over also to the other 2 graphs in the last line by taking for p the appropriate total momentum at one of the vertices. We, therefore, introduce now the three Lorentz-invariant Mandelstam variables s = (p1 + p3 )2 = p2 t = (p1 + p2 )2 u = (p1 + p4 )2 ,

(10.35)

with the property s + t + u = m21 + m22 + m23 + m24 . These variables represent the total squared four-momentum at the vertex that involves p1 in each of the graphs in the last line of Fig. 9.4. We thus get for the sum of all three diagrams in the last line of Fig. 9.4, the vertex correction diagrams, Γv (p1 , p2 , p3 , p4 ) =

3ig 2 με 1 (10.36) 16π 2 ε ig 2 με [3γ + F (s, m, μ) + F (t, m, μ) + F (u, m, μ)] . − 32π 2

Here F (s, m, μ) denotes the integral in (10.34). Of the diagrams in Fig. 9.4 only the first and the three last ones are 1PI graphs. The four diagrams in the second line are all 1P reducible; they just differ by the propagators on the external legs that are left out when we consider the 1PI four-point function. The complete 1PI four-point function is, therefore, given by Γ(p1 , p2 , p3 , p4 ) = −igμε + Γv (p1 , p2 , p3 , p4 ) 3ig 2 με 1 = −igμε + 16π 2 ε 2 ε ig μ − [3γ + F (s, m, μ) + F (t, m, μ) + F (u, m, μ)] 32π 2 3g 1 = − igμε 1 − (10.37) 16π 2 ε g [3γ + F (s, m, μ) + F (t, m, μ) + F (u, m, μ)] . + 32π 2 Equation (10.37) gives the effective, “dressed” interaction vertex. By comparison with the free 1PI four-point function, its forms suggests to absorb all


122

effects of the loop graphs contained in the curly brackets into a new effective coupling constant g −→ g {1 + δg(s, t, u, m, ε, μ)} .

(10.38)

This effective coupling constant depends on s, t, u, m and μ and, as the selfenergy, separates into a divergent and a finite term. It contains all the effects of the loops.

10.3

Renormalization

In the preceding two subsections we have seen that both the 1PI two-point and the 1PI four-point functions are (for ε = 0) regular functions of ε. The in four dimensions diverging quantities have thus been regularized. For n → 4 both have divergent and finite contributions from higher order diagrams. In this section we now show how to handle these divergences by the renormalization technique. Any renormalization procedure requires first a regularization, either by a cut-off for the upper bounds of diverging integrals or by going to n = 4 dimensions, to be followed by a procedure in which the dependence on these artifacts is removed. Since the separation of a divergent quantity into a finite and an infinite contribution is arbitrary there are various so-called renormalization schemes. They all have in common that they either add terms to the Lagrangian or scale the fields and coupling constants such that the original form of the Lagrangian is maintained. The most obvious scheme is the so-called minimal subtraction scheme that removes just the pole contributions in the dimensional regularization, i.e. the terms that go like powers of 1/ε. This scheme is straighforward and well-suited for the dimensional regularization, but it leads to expressions in which the parameters m (mass) and g (coupling) have no direct relation to measurable quantities. Here we discuss another scheme, in which we require that the parameters of the Lagrangian assume their physical, measured values, i.e. m is the physical mass and g the physical coupling constant. We start by looking at the structure of the most general two-point function. When we go to terms that depend on g 2 and higher orders of the interactions we always encounter 1P reducible graphs, such as the one in Fig. 9.2a. The general structure of the (reducible) two-point function is of the form Σ Gc (p, −p) = G0 (p, −p) + G0 (p, −p) G0 (p, −p) i Σ Σ + G0 (p, −p) G0 (p, −p) G0 (p, −p) + . . . , (10.39) i i


123

where Σ is the so-called proper self-energy. Note that (10.39) is an expansion in terms of irreducible diagrams. Equation (10.39) can be summed and gives

Σ Σ Σ G0 (p, −p) + G0 (p, −p) G0 (p, −p) + . . . i i i −1 Σ = G0 (p, −p) 1 − G0 (p, −p) i Σ −1 −1 = G0 (p, −p) − i i = 2 . (10.40) p − m2 − Σ + iε

Gc (p, −p) = G0 (p, −p) 1 +

The self-energy Σ is in general a function of the momentum p of the particle. We can therefore expand Σ(p2 ) around the on-shell point p2 = m2 , where m is the physical, observable mass Σ(p2 ) = Σ(m2 ) + (p2 − m2 )Σ1 + Σ2 (p2 ) . Here Σ1 =

∂Σ |2 2 ∂p2 p =m

Σ2 (m2 ) = 0 .

and

(10.41)

(10.42)

In (10.41) only the first two terms of the expansion of Σ have been written out explicitly; Σ2 (p2 ) denotes the whole remainder of the expansion. Σ, being the selfenergy insertion of a two point function, is quadratically divergent. Consequently, Σ1 as the first derivative of Σ with respect to p2 is logarithmically divergent and Σ2 as a second derivative is convergent since taking the derivate always adds one more power of q 2 in the denominator of the Feynman propagators. Inserting this expansion into (10.40) gives for the propagator Gc (p, −p) =

i

− − − − m2 )Σ1 − Σ2 (p2 ) + iε i 1 . (10.43) = 2 )+Σ (p2 ) Σ(m 2 2 2 1 − Σ1 p − m − + iε 1−Σ1 p2

m2

Σ(m2 )

(p2

The pole of this propagator should be at the physical mass and its residuum should be i. However, this is in general not the case, if we start with the observable, physical mass in the Lagrangian, because the selfinteractions contribute to the self-energy.


124

Counter terms. In order to get the pole to the correct, physical location we therefore have to change the mass in the Lagrangian by adding a so-called counterterm to it 1 1 with Z= . (10.44) Lcm = − δm2 Zφ2 2 1 − Σ1 Z is usually called “field renormalization constant” for reasons that will become obvious a little later (see (10.56)). Note that this counterterm has exactly the same form as the mass term in the original Lagrangian. It will thus also add a term −Zδm2 in the denominator of the dressed propagator (10.43). We determine the unknown δm2 by the requirement δm2 + Σ(m2 ) = 0

(10.45)

so that the new term just cancels the selfenergy contribution at the on-shell point. With the counterterm added, the propagator then becomes 1 i G(p, −p) = . (10.46) 2 2 1 − Σ1 p − m − ZΣ2 (p2 ) + iε ˜ 2 (m2 ) = 0 by definition, this propagator has the correct pole at the Since Σ physical mass m. Its residue, however, is – instead of being simply i – i = iZ . (10.47) 1 − Σ1 This deficiency can be cured by adding another, additional counterterm 1 Lcφ = (Z − 1) ∂μ φ∂ μ φ − m2 φ2 (10.48) 2 to the Lagrangian. Then the propagator becomes i G(p, −p) = Z 2 2 2 p − m − ZΣ2 (p ) + (Z − 1)(p2 − m2 ) + iε i = 2 . (10.49) 2 p − m − Σ2 (p2 ) + iε This propagator has the pole at the correct, physical mass m (because of Σ2 (m2 ) = 0) and the correct residue i. By adding the given counterterms we have thus removed the divergent quantities Σ(m2 ) and Σ1 from the propagator. The one remaining quantity Σ2 (p2 ) involves a second derivative of the selfenergy with respect to p2 and is convergent; it vanishes at the on-shell point. Note that the counterterms all have the structure of terms already present in the original Lagrangian. If this is the case, in general a theory is called renormalizable.


125

Renormalization of φ4 Theory

10.3.1

We now specify all of these general considerations to the example of φ4 theory. As we have discussed in Sect. 10.2 the φ4 theory is renormalizable, i.e. only a finite number of elementary vertices diverges. These are just the terms we have calculated in the last two subsections, namely the selfenergy contribution (10.19) and the vertex function (10.37). For the selfenergy we have, up to one-loop diagrams, (cf. (10.19)),

gm2 1 gm2 4πμ2 1 − γ + ln − Σ(p ) = − 16π 2 ε 32π 2 m2 2

+ O(ε) ,

(10.50)

i.e. Σ1 = Σ2 = 0 and Z = 1. Equation (10.45) reduces to δm2 = −Σ(0) = −Σ =⇒ Z = 1 .

(10.51)

Thus in this one-loop approximation, there is no field renormalization, but already in the two-loop approximation we would get Σ1 = 0 because the graph Fig. 9.2c is p-dependent. To be general we, therefore, keep the factor Z in the following expressions. So far we have not taken the change of the coupling constant due to higher order loop diagrams into account. In (10.37) we have indeed already seen that also the interaction vertex gets modified due to higher order loop corrections. The structure there was such that the coupling constant g was replaced by an effective coupling constant g(1 + δg(s, t, u, m, ε, μ)). If we again want to have the physical, observable coupling constant in our Lagrangian we must get rid of the modification by a proper counter term. We, therefore, define a vertex renormalization constant Zg by Zg = (1 + δg(s, t, u, m, ε, μ))−1 |r ,

(10.52)

where r denotes an in principle arbitrary renormalization point. Since we want to relate the coupling to a measurable quantity we use the so-called symmetric point 1 2 4 pα · pβ = m (10.53) δij − 3 3 for i, j = 1, . . . , 4. At this point we have s = t = u = −4m2 /3. We can then introduce the additional counter term Lcv = −

gμε (Zg − 1) φ4 . 4!

(10.54)

In this way we can ensure that the coupling constant g in the Lagrangian has its physical, observable value at r.


126

The Lagrangian with all the counterterms now reads L =

!

"

(∂μ φ)2 − m2 φ2 −

gμε 4!

! " 1 1 gμε (Zg − 1) φ4 + (Z − 1) ∂ μ φ∂μ φ − m2 φ2 − δm2 Zφ2 − 2 2 4! " Z! 1 gμε = (∂μ φ)2 − m2 φ2 − δm2 Zφ2 − (10.55) Zg φ4 . 2 2 4!

This Lagrangian leads to the correct physical behaviour for the 2-point Green’s function with the physical mass m and the proper residue. The fields φ, in terms of which the propagator is defined, are thus the physical fields, which include already the effects of selfinteractions. By introducing the “bare field” φ0 and the “bare mass” m0 by √ φ0 = Zφ 2 (10.56) m0 = m2 + δm2 and a “bare coupling constant” by g0 = gμε

Zg , Z2

(10.57)

then the whole Lagrangian can be expressed in terms of bare quantities only L=

" 1! g0 (∂μ φ0 )2 − m20 φ20 − φ40 . 2 4!

(10.58)

This bare Lagrangian has the same form as the original one, because all the counter terms had the same form as terms already appearing in the original Lagrangian. As a consequence it leads to finite physical quantities in all orders of perturbation theory. If this is the case, then the theory is well defined, i.e. it is said to be renormalizable. The bare Lagrangian is really considered to be the ‘true’ Lagrangian of the theory because it leads only to finite physical quantities. In the preceding considerations we have chosen 2 arbitrary renormalization points; we have required the propagator to have a pole at p2 = m2 , where m is the physical mass (cf. (10.49)), and for the vertex we have chosen the symmetric point r (s = t = u = −4m2 /3) (cf. (10.52)). Choosing other renormalization points will lead to other values for the mass and the coupling. This arbitrariness in the renormalization point reflects the fact that the divergent expressions for the selfenergy and the vertex all separate into a divergent and a finite term. Any scheme, that removes the infinite terms, will lead to finite expressions for the physical observables, independent of


127

what it does with the finite contributions. It will also lead to the same npoint function, and – with the help of the reduction theorem – to the same observable transition rates. In the dimensional renormalization discussed here all the counter terms and renormalization factors also depend on the arbitrary mass μ, even after ε → 0 and after the infinite terms have been removed. This μ was originally introduced to keep the coupling constant g dimensionless also for n = 4. It is obvious from our considerations above that – for specified renormalization points – choosing different values for μ will lead to different values for the wavefunction renormalization Z and the coupling constant g. This arbitrariness, however, is nothing else than the arbitrariness we have already encountered in the choice of the renormalization point. The physics must be independent of μ. This observation is the starting point for the development of the renormalization group method. It is essential to realize that even in a massless theory, which contains no dimensional scale parameters, μ introduces a scale that determines the momentum dependence of the coupling constant. Thus, at the quantum level, a bare Lagrangian is not enough to specify a theory, but a renormalization scheme must be added that introduces necessarily a scale into the theory.

Chapter 11 GREEN’S FUNCTIONS FOR FERMIONS For simplicity of notation we have so far in this book discussed only the path integrals and generating functions for scalar fields. All this formalism can be easily generalized also to vector fields which obey the Proca equation (4.24) and thus fulfill component by component the Klein-Gordon equation. For fermion fields, however, there is a problem. The main idea in using path integrals is to express quantum mechanical transition amplitudes by integrals over classical fields; the values of these fields at the discrete coordinatesites were taken to be commuting numbers. Such a formalism can, however, not “know” about the Pauli principle. For example, with the formalism developed so far, a fermion could be propagated to a point in configuration space which is already occupied. In nature, however, this propagation is Pauli-forbidden. For the description of fermions it is, therefore, necessary to extend the theory developed so far such that the Pauli principle is taken into account. This can be achieved by using an anticommuting algebra for the classical fields, the so-called Grassmann algebra.

11.1

Grassmann Algebra

In this section we outline the basic mathematical properties of the Grassman algebra as far as we will need them in the later developments. We define the n generators i , . . . , n of an n-dimensional Grassmann algebra by the anticommutation relations {i , j } ≡ i j + j i = 0 . 128

(11.1)

CHAPTER 11. GREEN’S FUNCTIONS FOR FERMIONS

129

Let us now consider series expansions in these variables. Since 21 = 22 = . . . = 2n = 0 ,

(11.2)

because of (11.1), any series in i must have the form φ() = φ0 +

φ1 (i)i +

i

+

φ2 (i, j)i j

(11.3)

i 0)

(B.6)

to this case. From the definition of the path integral it is obvious that we have to consider products of such integrals

exp −

n 1

2 k=1

ak x2k

(2π)n dx1 dx2 . . . dxn = n √ . ak k=1

(B.7)

We next assume that the n numbers ak are all positive and form the elements of a diagonal matrix A. We thus have n

ak

(B.8)

xk Akk xk = xT Ax ,

(B.9)

det(A) =

k=1

and

n k=1

ak x2k =

n k=1

APPENDIX B. FUNCTIONALS

194

where x is a column vector ⎛ ⎜ ⎜ x=⎜ ⎜ ⎝

⎞

x1 x2 .. .

⎟ ⎟ ⎟ . ⎟ ⎠

xn Thus (B.7) becomes

n

e

− 12 xT Ax

(2π) 2

n

dx=

.

(B.10)

det(A) So far, we have derived this equation only for a diagonal matrix A. It is, however, valid for a more general class of matrices. This can be seen by noting that for each real, symmetric matrix B there exists a real, orthogonal matrix O that diagonalizes B to A OT BO = A

(O real, orthogonal)

(B.11)

B = OAOT

(B real, symmetric) .

(B.12)

or, equivalently

We thus get

e

− 21 yT By

n

dy

=

e

− 12 yT OAOT y n

(2π) 2

=

e− 2 x 1

n

dy=

T Ax

dnx

n

(2π) 2

=

det(A)

,

(B.13)

det(B)

where we have substituted y = Ox and have used the fact that the Jacobian of an orthogonal transformation is 1 (because det(O) = 1). The last step is possible because the determinant of a matrix is invariant under an orthogonal transformation. Equation (B.10) is thus valid also for general symmetric matrices with positive eigenvalues; it can also be shown to hold for complex matrices with positive real parts. This result can also be extended to more general quadratic forms in the exponent. For a one-dimensional integral of such type we have +∞

e−ap

−∞

2 +bp+c

dp =

π b2 +c e 4a , a

(B.14)


195

We now assume an n-dimensional integral over such a form where the exponential is given by 1 T T (B.15) e−F (x) = e−( 2 x Ax+B x+C ) where A is a real symmetric matrix, B is a vector and C a constant. We bring F (x) into a quadratic form by writing 1 F (x) = (x − x0 )T A(x − x0 ) + F (x0 ) 2 where x0 is given by

(B.16)

x0 = −A−1 B

(B.17)

and F0 = F (x0 ) = C − 12 BT A−1 B. Setting now y = x − x0 gives

e

−( 12 xT Ax+BT x+C )

e− 2 y 1

n

dx =

T Ay−F 0

dny

n

(2π) 2

=

e2B 1

T A−1 B−C

.

(B.18)

det(A) Similar to the Gaussian integral (B.7) the following integral relation, which can be proven by induction from n to n + 1, holds also +∞

1

!

dx1 . . . dxn exp iλ (x1 − a)2 + (x2 − x1 )2 + . . . + (b − xn )2

−∞

=

in π n (n + 1)λn

1 2

iλ (b − a)2 exp n+1

"0

.

(B.19)

Complex integrals. We can generalize these formulas now to complex integration by noting that (B.6) can be squared and then be written as π −ax2 −ay2 2 2 dx e dy = e−a(x +y ) dx dy . = e a

(B.20)

Introducing now the complex variable z = x + iy gives π 1 −az∗ z ∗ e dz dz . = a 2i

(B.21)

This can be generalized as before to many coordinates (by replacing orthogonal matrices by unitary ones). We obtain

e

−z† Az

n ∗

n

dz dz=

∗

e−zi Aij zj dnz ∗ dnz =

(2πi)n , det(A)

(B.22)


196

where A is a Hermitian matrix with positive eigenvalues. Another convenient, often used relation in this context is ln det A = tr ln A ,

(B.23)

most easily proven for diagonal matrices. In this relation ln A is defined by its power series expansion ln A = ln(1 + A − 1) = A − 1 −

(A − 1)2 (A − 1)3 + − ... . 2 3!

(B.24)

With the help of this relation we have

B.3

e−z

† Az

dnz ∗ dnz = (2πi)n e−tr

ln A

.

(B.25)

Functional Derivatives

Suppose that F [f ] is a functional of the function f (x). The functional derivative of F is then defined by F [f (x) + εδ(x − y)] − F [f (x)] δF [f (x)] = lim . ε→0 δf (y) ε

(B.26)

For example, let us take F [f ] = 2f (x) ,

(B.27)

where 2 is the d’Alembert operator. We get from the definition (B.26) δF 1 = lim [2 (f (x) + εδ(x − y)) − 2f (x)] = 2δ(x − y) ε→0 δf (y) ε Another example is

(B.28)

+∞

F [f ] =

f (x)dx .

(B.29)

−∞

According to the definition just given we have δF 1 = lim ε→0 ε δf (y)

=

[f (x) + εδ(x − y)] dx −

f (x) dx

(B.30)

δ(x − y) dx = 1

A second example is given by F [f ] = ei

f (x)x dx

(B.31)


197

with δF 1 i [f (x)+εδ(x−y)]x dx e − ei f (x)x dx = lim ε→0 ε δf (y) 1 = lim ei f (x)xdx eiεy − 1 = iy ei f (x)x dx . ε→0 ε

(B.32)

A general formula that we will need quite often involves functionals F [J] of the form

F [J] =

dx1 . . .

dxn f (x1 , . . . , xn ) J(x1 )J(x2 ) . . . J(xn )

(B.33)

with f symmetric in all variables. Then the functional derivative with respect to J has the form δF [J] = n dx1 dx2 . . . dxn−1 f (x1 , x2 , . . . , xn−1 , x) δJ(x) × J(x1 )J(x2 ) . . . J(xn−1 ) .

(B.34)

In our later considerations the functional may also be defined by a power series expansion φ[J] =

∞ 1 n=1

n!

dx1 . . . dxn φn (x1 , . . . , xn )J(x1 )J(x2 ) . . . J(xn ) .

(B.35)

Then the functional derivative is given by .

. δ k φ[J] 1 . . = φk (yp1 , yp2 , . . . , ypk ) , δJ(y1 )δJ(y2 ) . . . δJ(yk ) .J=0 k! p

(B.36)

where the sum runs over all permutations p1 , . . . , pk of the indices 1, . . . , k. If we assume that φk is a symmetric function under exchange of any of the coordinates x1 , . . . , xk , then the functional derivative is given by .

. δ k φ[J] . . = φk (y1 , . . . , yk ) , δJ(y1 )δJ(y2 ) . . . δJ(yk ) .J=0

just as in a normal Taylor series.

(B.37)

Appendix C RENORMALIZATION INTEGRALS In Chapt. 10 we encountered divergent integrals in the calculation of higherorder two- and fourpoint functions. To evaluate these integrals analytically in n dimensional Minkowski space is the purpose of this appendix. We start with a discussion of the Gamma function whose properties play a role in the evaluation of the integral and its expansion into n ≈ 4 dimensions. Properties of the Gamma function. The Gamma function is defined by an integral representation Γ(x) =

∞

e−t tx−1 dt ;

(C.1)

0

it is single-valued and analytic everywhere except at the points z = n = 0, −1, −2, . . . where it has a simple pole with residue (−1)n /n!. It can thus be expanded around z = 0 Γ(z) =

1 − γ + O(z) , z

(C.2)

where γ is known as the Euler-Mascheroni constant (γ ≈ 0.577 . . .). Since the Gamma function also obeys the relation Γ(z + 1) = zΓ(z) = z!

(C.3)

we obtain

1 1 Γ(z) ≈ −(1 + z) −γ =− −1+γ . Γ(−1 + z) = z−1 z z 198

(C.4)

APPENDIX C. RENORMALIZATION INTEGRALS

199

It obeys the relation Γ(z) = (z − 1)Γ(z − 1) ≈ (z − 1)

−1 , z+1

(C.5)

where the second part of this equation is correct close to the pole at z = −1. Evaluation of integrals with powers of propagators in n dimensions. The typical integral to be evaluated reads

Il =

1 1 dq dq dq0 2 l = l . 2 2 (q − m + iε) (q0 − q 2 − m2 + iε) n

(C.6)

with one timelike (q 0 ) and n−1 spacelike (q k ) coordinates; the vector symbol denotes a (n − 1)-dimensional vector q = (q 1 , q 2 , . . . , q n−1 ). l is an integer parameter and m2 a real, positive number (mass squared). We first consider the integration over q0 . The situation here is exactly as in Sect. 6.1.2. The integrand has its only poles in the second and fourth quadrant of the complex q0 plane at q0 = ± [(q 2 + m2 ) − iδ] (cf. Fig. 5.1). Since the integral behaves as 1/(q0 )2l−1 for large q0 the integration along the q0 -axis can be closed in the lower half of the complex q0 plane without changing the integral’s value. According to Cauchy’s theorem this integration contour can now be changed into one that runs along the imaginary q0 axis and closes the contour in the right half of the complex q0 plane. Because the contour still encloses the same pole (the one in the fourth quadrant of the complex plane) the value of the integral does not change. This then gives the equality +∞ −∞

dq0

+i∞

1 (q02

−

q2

−

m2

l

+ iε)

=

−i∞

dq0

1 (q02

−

q2

l

− m2 + iε)

(C.7)

since in both cases the contribution of the half-circle that closes the contour vanishes. In the integral on the rhs the integration runs along the imaginary axis in the complex q0 plane. On that axis q0 is purely imaginary; the integrand has thus been analytically continued from the originally purely real q0 to a complex one. Using the transformations (6.22), (6.23) qn = −iq0

dn qE ≡ dqn dq = −idn q ,

(C.8)

with real qn we obtain the integral

Il = (−)l i

+∞

dq

−∞

dqn

1 (qE2 +

l m2 )

= (−)l i

dn q E

1 l

(qE2 + m2 )

(C.9)

APPENDIX C. RENORMALIZATION INTEGRALS with

n i 2

qE2 =

q

200

= q 2 + qn2 .

(C.10)

i=1

(C.8) is just the Wick rotation discussed in Sect. 5.1.1. We now introduce “polar coordinates” by defining an n − 1 dimensional solid angle element dΩn by the relation dnqE = qEn−1 dqE dΩn .

(C.11)

and get for the integral1 ∞

l

Il = (−) i

q n−1 dq

dΩn 0

1 . (q 2 + m2 )l

(C.12)

The integral over the solid angle can be analytically performed and yields

dΩn =

2π n/2 Γ

.

(C.13)

n 2

This can be seen by considering the n-th power of the Gaussian integral √ n

π

=

∞

=

0

dΩn

=

dx e

−x2

∞ 0

x

n

dx1 dx2 . . . dxn e−

=

n−1 −x2

e

dx =

1 n dΩn Γ 2 2

-n k=1

x2k

n−2 1 ∞ 2 dΩn d(x2 ) x2 2 e−x 2 0

.

(C.14)

Here the integral representation (C.1) of the Gamma function has been used in the last step. This gives for the integral ∞ n 2π 2

Il = (−) i n Γ 2 l

q n−1 dq

0

1 . (q 2 + m2 )l

(C.15)

The remaining integral can be evaluated with the help of Euler’s β function ∞ Γ(x)Γ(y) = 2 dt t2x−1 (1 + t2 )−x−y , B(x, y) = (C.16) Γ(x + y) 0

1

In order to simplify the notation we are no longer denoting the Euclidean vectors by the subscript E.

APPENDIX C. RENORMALIZATION INTEGRALS

201

(see Abramowitz (6.2.1)); we obtain

Γ 1 n n 1 q n−1 dq = mn−2l B( , l − ) = mn−2l l 2 2 2 2 (q 2 + m2 )

n 2

Γ l−

n 2

. (C.17)

Γ(l)

Thus the complete integral is now given by n

Γ 2π 2 1 Il = (−)l i n mn−2l 2 Γ 2

n 2

Γ l− Γ(l)

n 2

n 2

= (−)l iπ m

Γ l− n−2l Γ(l)

n 2

. (C.18)

In this expression the dependence of the integral on the dimension n is explicit; it can analytically be continued to the physical case n = 4 where it has a pole.

Appendix D GRASSMANN INTEGRATION FORMULA A more general proof than the specific examples given in section (11.1.2.2) uses the fact that any antisymmetric (n × n) dimensional matrix M can be brought into a block-diagonal form through an orthogonal transformation M −→ M = OT M O ⎛

0 M1 ⎜ 0 ⎜ −M1

⎜ ⎜ ⎜ ⎜ ⎜ M =⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

0 0

0 0

··· ···

M2 · · ·

0

0

0

0 .. .

0 .. .

−M2 .. .

0 .. .

0 0

0 0

0 0

0 0

···

(D.1) 0 0 .. . .. . .. .

··· ··· 0 · · · −Mn

0 0 .. . .. . .. .

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ Mn ⎠

(D.2)

0

for n even (for odd n the matrix M has one more row and one more column with all zeros, so that its determinant vanishes). The form of M shows clearly that (D.3) det(M ) = det(M ) = M12 M22 · · · Mn2 . We thus get

I(n) =

=

−η T M η

dη1 . . . dηn e

=

T M

d1 . . . dn J e−

dη1 . . . dηn e−η

T (OM O T )η

(D.4)

with = OT η and J being the Jacobian. In our special case here, O is orthogonal and therefore, [det(O)]−1 = J = 1. 202

APPENDIX D. GRASSMANN INTEGRATION FORMULA

203

We can now continue with the evaluation of I(n) in (D.4) and get

n n 1 d1 . . . dn (−) 2 n T M 2 , ! 2

I(n) =

(D.5)

because only that term in the expansion of the exponental can contribute in a n-dimensional integral that has exactly n factors of i . In detail, we have n (−) 2

I(n) = n 2

!

d1 . . . dn α Mαβ β

n 2

.

(D.6)

Since M has the special, block-diagonal form given above (D.2), we have = 0, except for α = 2α , β = α − 1 or α = 2α − 1, β = α + 1. Again, Mαβ there can be no two equal s under the integral, if this integral is to be nonzero. We thus have n (−) 2

I(n) =

n 2

d1 . . . dn 2α M2α ,2α −1 2α −1

!

n 2 2 (−)n

=

n 2

!

− 2α −1 M2α −1,2α 2α

d1 . . . dn 2α −1 M2α −1,2α 2α

n 2

n 2

,

(D.7)

since M is antisymmetric. Here the index α still runs from 1 to n/2; n is even so that the phase disappears in the following. We now perform the exponentiation n 22

I(n) = n 2

!

d1 . . . dn

α

2

...

β

34

n sums

ν

2α −1 2α · · · 2ν −1 2ν

5

× M2α −1,2α M2β −1,2β · · · M2ν −1,2ν . M2α −1,2α

(D.8)

· · · M2ν −1,2ν

just gives det(M ). Since The product of the factors the i appear always pairwise, they can be commuted to normal ordering (n · · · 1 ) without sign-change; the sums then give (n/2)! times the same result. Thus I(n) = 2

n 2 n

= 22

d1 . . . dn n · · · 1 det(M ) .

det(M ) (D.9)

APPENDIX D. GRASSMANN INTEGRATION FORMULA

204

The last step is possible because we have for each blockmatrix Mβα separately det(Mβα ) = − (M2α−1,2α ) (+M2α,2α−1 ) = + (M2α−1,2α )2 , and for det(M )

det(M ) =

det(Mβα ).

(D.10)

(D.11)

α

Using (D.3) we thus get the desired result

T M

d1 . . . dn e−

n

= 22

det(M ) .

(D.12)

Equation (D.12) corresponds to (B.10) for the boson case. Note that here – in contrast to the bosonic case – the determinant appears in the numerator!

Appendix E BIBLIOGRAPHY We list here a number of textbooks that provide for more information on the topics treated in this manuscript. Introductory texts 1. J.J. Sakurai, Modern Quantum Mechanics, Addison-Wesley, Redwood City, 1985; contains a very nice introduction into the non-relativistic path integral formalism. 2. U. Mosel, Fields, Symmetries, and Quarks, 2nd revised and enlarged edition: Springer, Heidelberg, 1999; contains a short introduction into fundamentals of field theories and gauge field theories. 3. T.D. Lee, Particle Physics and Introduction to Field Theory, Harwood, Chur, 1981; very physical treatment of field theory and high-energy phenomenology, somewhat annoying typesetting. 4. L.H. Ryder, Quantum Field Theory, Cambridge University Press, 1985; very didactical presentation of modern field theories. 5. C. Itzykson and J-B. Zuber, Quantum Field Theory, McGraw-Hill, New York, 1985; nearly comprehensive book on field theory, didactically not very well done. 6. M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, Reading, 1995; also didactically very good, rather comprehensive book on quantum field theory. The new standard book on this topic!

205

APPENDIX E. BIBLIOGRAPHY

206

Specialized Literature 1. Ta-Pei Cheng and Ling-Fong Li, Gauge Theory of elementary particle physics, Clarendon Press, Oxford, 1984; quite comprehensive book, well-structured, easy to read, but not always complete in its arguments and derivations. 2. P. Ramond, Field Theory, a modern Primer, Addison-Wesley, Redwood City, 1990; very nice introduction to field theories, also modern aspects, builds entirely on path integral formalism. 3. P.H. Frampton, Gauge Field Theories, Benjamin-Cummings, Menlo Park, 1987; bad typesetting, not easy to read. 4. S. Pomorski, Gauge Field Theories, Cambridge University Press, 1987; not very didactical, but quite deep. 5. D. Bailin and A. Love, Introduction to Gauge Field Theory, 2nd ed., Hilger, Bristol, 1994; very comprehensive treatment of gauge field theories based on path integral methods from the start, didactical, contains modern aspects of field theory beyond Standard Model. 6. C. Grosche and F. Steiner, Handbook of Feynman Path Integrals, Springer, Berlin, 1998; compact presentation of theory of path integrals and their history. Unique in its description of evaluation techniques and its tables of analytically calculable path integrals. Complete list of references. 7. H. Kleinert, Path Integrals in Quantum Mechanics, Statistics and Polymer Physics, World Scientific, Singapore 1995. Application of path integrals to wide range of physics problems. 8. G. Roepstorff, Path Integral Approach to Quantum Physics, Springer, Heidelberg, 1996

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