COHERENT STATES Some Applications in Quantum

Göteborg ITP 93-10 November 1993

COHERENT STATES Some Applications in Quantum Field Theory and Particle Physics Bo-Sture Skagerstam Institute of Theoretical Physics Chalmers University of Technology and University of Göteborg S-412 96 Göteborg, Sweden and University of Kalmar Box 905, S-391 29 Kalmar, Sweden

To appear in the Proceedings of the International Symposium on Coherent States: Past, Present, and Future, Oak Ridge, June 14th - 17th, 1993. Eds. D. H. Feng, J. R. Klauder and M. R. Strayer (World Scientific, 1994). Abstract

We review some applications of the concept of coherent states in theoretical physics with special focus on quantum field theory and the physics of elementary particles.

COHERENT STATES SOME APPLICATIONS IN QUANTUM FIELD THEORY AND PARTICLE PHYSICS BO-STURE SKAGERSTAM1 Institute of Theoretical Physics, Chalmers University of Technology and University of Göteborg, S-412 96 Göteborg, Sweden and University of Kalmar, Box 905, S-391 29 Kalmar, Sweden Abstract

We review some applications of the concept of coherent states in theoretical physics with special focus on quantum field theory and the physics of elementary particles.

1. Introduction “Truth and clarity are complementary.” N. Bohr In this presentation of a what I find an extremely fascinating subject, we will discuss some applications of the concept(s) of coherent states in quantum field theory with special focus on the physics of the elementary particles. It is not only because of limited time and space at our disposal that we have to limit our choice of subjects to be discussed. In fact, it is our view that since its advent the space of applications of coherent states has expanded to such an extent that it is now virtually impossible to have a rather complete overview of the field. There are, however, many excellent text-books [1, 2, 3], recent reviews [4] and other expositions of the subject [5] to which we will refer to for details and/or other aspects of the subject under consideration. Already at this point we apologise to the many authors whose work we have no time to review and/or we may have overlooked and to those who feel that their work should have been referred to. I will not touch much 1

Email address: [email protected]. Research supported by the Swedish National Research Council under contract no. 8244-311 and the Knut and Alice Wallenberg Foundation.

upon the historical aspects of the subjects to be discussed - one may find discussions on such aspects elsewhere (see e.g. the recent discussion by Steiner [6]). We will not try to make a general enough definition of the concept of coherent states (for such an attempt see e.g. the introduction of Ref.[5]). We rather intend to illustrate in some rather elementary and explicit examples the use of the concept of coherent states and, when appropriate, stress the physics involved. One basic issue, which one may raise in any elementary presentation of quantum mechanics, is the transition to the domain of classical physics, a topic which has been discussed at great detail during this wonderful symposium. Quantum mechanics we believe is, after all, the fundamental framework for the description of all known natural physical phenomena. Still we are, however, very often puzzled about the role of concepts from the domain of classical physics within this framework. The interpretation of the theoretical framework of quantum mechanics is, of course, directly connected to the “classical picture” of physical phenomena. We often talk about quantization of the classical observables in particular so with regard to classical dynamical systems in the Hamiltonian formulation as has so beautifully been discussed by Dirac (see e.g. Ref.[7]) and others. I believe that the concept of coherent states is very useful in trying to orient the inquiring mind in this jungle of conceptually difficult issues when connecting classical pictures of physical phenomena with the fundamental notion of quantum-mechanical probability-amplitudes and probabilities. As is well-known, coherent states appears in a very natural way when considering the classical limit of quantum electrodynamics (QED). In the conventional and extremely successful application of perturbative quantum field theory in this context, the number-operator Fock-space representation is the natural Hilbert space in realising the canonical commutation relation of the electro-magnetic field even though this may lead to mathematical difficulties when interactions are taken into account. Over the years we have, however, in practice learned how to deal with some of these mathematical difficulties. In presenting the theory of the second-quantized electro-magnetic field in an elementary

course in quantum mechanics, it is tempting to exhibit the apparent “paradox” of Erhenfest “theorem” in quantum mechanics and the existence of the classical Maxwell’s equations - any average of the electromagnetic field-strengths in the physically natural number-operator basis is zero and hence these averages will not obey the classical equations of motion. The solution of this apparent paradox is, as is by now a wellknown fact, that the classical fields in Maxwell’s equations corresponds to quantum states with an arbitrary number of photons. In classical physics, we may neglect the quantum structure of the charged sources. Let j(x, t) be such a classical current, like the classical current in a coil, and A(x, t) the second-quantized radiation field (in e.g. the radiation gauge). The interaction Hamiltonian HI (t) then has the form HI (t) = −

Z

d3 x j(x, t) · A(x, t) ,

(1.1)

and the quantum states in the interaction picture, |tiI , obey the Schrödinger equation, i.e. d i |tiI = HI (t)|tiI . (1.2) dt For reasons of simplicity, we will consider only one specific mode of the electromagnetic field described in terms of a canonical creation operator (a∗) and an annihilation operator (a). The general case then easily follows by considering a system of such independent systems. It is therefore sufficient to consider the following interaction Hamiltonian: f (t) HI (t) = − √ (a exp[−iωt] + a∗ exp[iωt]) , (1.3) 2ω where the real-valued function f (t) describes the classical current. The “free” part H0 of the total Hamiltonian in natural units then is H0 = ω(a∗ a + 1/2) .

(1.4)

In terms of canonical “momentum” (p) and “position” (x) field-quadrature degrees of freedom defined by s ω 1 a = x + i√ p , 2 2ω s ω 1 a∗ = x − i√ p , (1.5) 2 2ω

we therefore see that we are formally considering a damped harmonic oscillator. The solution to Eq.(1.2) is easily found. We can write |tiI = exp[iφ(t)] exp[iA(t)]|t0 iI where A(t) = −

Z t t0

dt0 HI (t0 ) ,

,

(1.6) (1.7)

and where the c-number phase φ(t) is given by iZt 0 φ(t) = dt [A(t0), HI (t0 )] . 2 t0

(1.8)

We now define the unitary operator U (z) = exp[za∗ − z ∗ a]

.

(1.9)

Canonical coherent states |z; φ0i, depending on the (complex) parameter z and the fiducial normalized state number-operator eigenstate |φ0i, are defined by |z; φ0i = U (z)|φ0i , (1.10)

such that

Z d2 z d2 z 1= |zihz| = |z; φ0ihz; φ0 | . (1.11) π π Here the canonical coherent-state |zi corresponds to the choice φ0 = |0i, i.e. the Fock vacuum state. We then see that, up to a phase, the solution Eq.(1.6) is a canonical coherent-state. It can be verified that the expectation value of the second-quantized electromagnetic field in the state |tiI obeys the classical Maxwell equations of motion for any fiducial Fock-space state |t0 iI = |φ0i. Therefore the corresponding complex, and in general time-dependent, parameters z constitute an explicit mapping between classical phase-space dynamical variables and a pure quantum-mechanical state. In more general terms, quantummechanical models can actually be constructed which demonstrates that by the process of phase-decoherence one is naturally lead to such a correspondence between points in classical phase-space and coherent states (see e.g. Ref.[9]). Z

0.08

n =25

0.06 0.04

P (n)

0.02 0.00 0

10

20

30

40

50

Number of Photons n Figure 1: The photon number distribution P (n) for a coherent state with a mean number of photons n ¯ = 25 (solid curve) and a one-photon displaced state with n ¯ = 26 (dashed curve).

If the fiducial state |φ0i is a number operator eigenstate |mi, where m is an integer, the corresponding coherent-state |z; mi have recently been discussed in detail in the literature and is referred to as a semicoherent state [10, 11] or a displaced number-operator state [12]. We will now argue that a classical current can be used to amplify the information contained in the pure fiducial vector |φ0i. For a given initial fiducial Fock-state vector |mi, it is a trivial exercise to calculate the probability P (n) to find n photons in the final state, i.e. (see e.g. Ref.[8]) P (n) = lim |hn|tiI |2 , (1.12) t→∞

which then depends on the Fourier transform of f (t). In Figure 1, the solid curve gives P (n) for |φ0i = |0i, where we, for the purpose of illustration, have chosen the Fourier transform of f (t) such that the mean value of the Poisson number-distribution of photons is 25. This distribution P (n) then characterize a classical state of the radiation field. The dashed curve in Figure 1 corresponds to |φ0i = |1i, and we observe the characteristic oscillations. It may be a slight surprise that the

minor change of the initial state by one photon completely change the final distribution P (n) of photons. If |φ0i = |mi one finds in the same way that the P (n)-distribution will have m zeros. If we sum the distribution P (n) over the initial-state quantum number m we, of course, obtain unity as a consequence of the unitarity of the time-evolution. Unitarity is actually the simple quantum-mechanical reason why oscillations in P (n) must be present. We also observe that two canonical coherent states |tiI are orthogonal if the initial-state fiducal vectors are orthogonal. It is in the sense of oscillations in P (n), as described above, that a classical current can amplify a quantum-mechanical pure state |φ0i to a coherent-state with a large number of coherent photons. This effect is, of course, due to the boson character of photons. It has, furthermore, been shown that one-photon states localized in space and time can be generated in the laboratory [13]. It would be interesting if such a state could be amplified by means of a classical source in resonance with the typical frequency of the photon. It has been argued by Knight et al. [12] that an imperfect photon-detection by allowing for dissipation of field-energy does not necessarily destroy the appearance of the oscillations in the probability distribution P (n) of photons in the displaced number-operator eigenstates. It would, of course, be an interesting and striking verification of quantum coherence if the oscillations in P (n)-distribution could be observed experimentally. 2. Quantum Field Theory “It is robust.” A. S. Wightman In QED and in the soft-photon limit, we consider a classical current jµ (x) as a function of the space-time co-ordinate x = (x, t) describing the initial and final state of electrically charged particles in a scattering process, i.e. Z jµ (x) = d4 kjµ (k) , (2.1)

where the Fourier transform jµ (k) is given by 



X  Qpµ ie  X   Θ( − ω) − jµ (k) = (2π)3 Qi ,pi p · k Qf ,pf

.

(2.2)

Here Q (= Qi or Qf ) is the electric charge (in units of e) and p (= pi or pf ) the momentum of an initial or final state particle. is an arbitrary infrared cut-off which should cancel in any physical process. Θ represents the Heavyside unit-step function. The isotropic density matrix ρ(∆E) describing a final state |f i with soft photon emission with total energy less then ∆E is now given by ρ(∆E) =

Z

d4 pδ 4 (p − pˆ)Θ(∆E − p0 )|f ihf | ,

(2.3)

where pˆ is the four-momentum operator and where the final state |f i does not contain soft photons. It is then a tedious but straightforward calculation to find the probability p(∆E) for soft photon emission in a scattering process, i.e. p(∆E) = hij|ρ(∆E)|iji|hf |S() |ii|2

,

(2.4)

where |iji = U (ij)|0i is the multi-component coherent-state generated by the classical current jµ (x) with no soft photons present in the initial state |ii and where the soft part of the S-matrix has been separated out, i.e. S = U (ij)S() . Since the infrared cut-off is arbitrary it must cancel between the two factors in Eq.(2.4), i.e. p(∆E) = f (

∆E ¯ 2 , αC)|hf |S|ii| E

,

(2.5)

¯ is free of infrared divergences (but has, of course, where the S-matrix S in general to be renormalized due to the presence of ultraviolet divergences). The correction factor f ( ∆E , αC) in QED has the following E form, valid to all orders in perturbation theory: ∆E ∆E f( , αC) = E E

!αC

exp[−γαC] Γ(1 + αC)

,

(2.6)

f ( ∆E E , αC) 1.0 0.8 0.6 0.4 0.2 0

1.0 0.5 100

∆E E

200

Energy (GeV )

300

0.0

Figure 2: The correction factor f( ∆E , αC) in QED. E

where γ(= 0.57721...) is Eulers constant and C is a kinematical function of the initial and final charges and momenta. We observe that as ∆E tends to zero, the probability p(∆E) also tends to zero. To any finite order in perturbation theory we would, however, obtain an infrared divergent answer. Strictly speaking, we can therefore not have any elastic cross-section in QED. This must, of course, be the case since in any scattering process involving charged particles, classical physics tells us that we must have emission of radiation. As an illustrative example, let us consider e+e− -annihilation. At very high energies as compared to √ the rest-mass of the electron, we can then write αC ≈ 4α log( s/m)/π. The result of a simple numerical evaluation is presented in Figure 2 at typical LEP-energies. For high-energy resolutions, the correction factor can be considerable and is important in e.g. the precise determination of the parameters of the standard electro-weak model (a recent discussion can e.g. be found in Ref.[20]).

The expression

αC ¯ S = U (ij) S (2.7) E formally summarises much more precise work on the infrared problem in QED by in particular Yennie, Frautschi and Suura [21], Chung [22], Greco and Rossi [24], Kibble [23], Eriksson [19], Faddeev and Kulish [25], Zwanziger [26] and the subject is nicely reviewed by Papanicolaou [27]. Since the infrared problem in non-Abelian gauge theories and in particular in QCD has been discussed in a clear manner by C. A. Nelson at this symposium I will not enter much into it here. We, however, observe that the picture discussed above in QED in terms of soft-photon emission from a classical current cannot directly be carried over to the case of non-Abelian current [35, 47]. One essential and successful ingredient in the perturbative approach to the infrared problem in non-Abelian gauge theories is the idea that instead of the conventional splitting of the total Hamiltonian H into one free part H0 , which is quadratic in fields, and one interaction HI , one now considers the combination !

H = (H0 + HIR,C ) + (HI − HIR,C ) .

(2.8)

Here HIR,C = HIR + HC

,

(2.9)

where HIR describes the “infrared ” part of the interaction as alluded to above, and HC describes the “collinear ” divergences of massless (and in general non-Abelian) charged particles. For large separations of the scattered particles, the interaction HI − HIR,C then becomes unimportant and the asymptotic states are determined by the Hamiltonian H0 + HIR,C . It is a non-trivial property of gauge theories that HIR,C can be divided into an infrared part and one collinear part. The precise statement is that the commutator [HIR , HC ] is suppressed (details can be found in Ref.[28]). Recently Contopanagos and Einhorn [29] have discussed a possible interpretation of the asymptotic S-matrix in terms of physical transitions expressed directly by means of Fock space states. In the end, the treatment of collinear divergences are in agreement with

the Lee-Nauenberg treatment of mass-singularities in degenerate perturbation theory [30]. Even though we do not know, at present, how to deal with a realistic quantum theory of gravity, it is well-known that a conventional perturbative quantization of Einsteins theory is perfectly well-defined in the infrared. As was first shown in a brilliant analysis by Weinberg [31], the structure of all Feynman diagrams involving infrared real and virtual gravitons is in its form similar to the analogues structure in QED. The physical picture we have mentioned above in terms of soft particle emission from a classical charged current is also valid in quantum gravity. The current in question then corresponds to the energy-momentum tensor (see especially in this context Ref.[32]). These methods are, however, based on an expansion around a flat space-time background and, in particular, the cosmological constant Λ is put equal to zero which, of course, is the correct phenomenological thing to do [34]. It has, however, recently been pointed out by Tsamis and Woodard [33] that an initial non-zero cosmological constant makes the infrared problem much more severe in quantum gravity. In order to illustrate the basic idea, let us first consider the quantum field theory of a massless scalar particle with an interaction describing a non-zero classical vacuum, i.e. 1 L = ∂µ ϕ∂ µ ϕ − V (ϕ) (2.10) 2 where the potential V is of the form V (ϕ) = a + bϕ3 + cϕ4 , and where a and b are positive constants. If we develop a quantum theory around the classical vacuum ϕ = 0 we will, of course, encounter strong infrared divergences. What we then will find is that the scalar field develops a non-zero expectation value corresponding to a non-zero minimum of V (ϕ). So if we instead develop a quantum theory around this new classical vacuum, a non-zero mass emerges and hence no infrared divergences then appear. In quantum gravity with Λ 6= 0 one could imagine a similar process. The expansion around the flat space-time background may therefore not be correct and one can imagine that a De Sitter space-time background may be more appropriate. In gravity, involving a massless spin two particle, we can, however, not generate a

massive graviton in this case to cure the infrared divergences. Instead, as was observed by Tsamis and Woodard, the infrared divergences are amplified. In fact, it was argued in Ref.[33] that as a consequence of this, quantum gravity in the infrared may drive itself to the situation where, as a finial state, we reach Λ = 0. If this idea survives further theoretical studies, it would lead to a completely new physical understanding and solution of the cosmological constant problem [34].

3. Photon-Detection Theory “If it was so, it might be; And if it were so, it would be. But as it isn’t, it ain’t. ” Lewis Carrol The quantum-mechanical description of optical coherence was developed in series of beautiful papers of Glauber [38]. Here we will only touch upon some elementary considerations. Consider an experimental situation where a beam of particles, in our case a beam of photons, hits a beam-splitter. Two photon-multipliers measures the corresponding intensities at times t and t + τ of the two beams generated by the beam-splitter. The quantum state describing the detection of one photon at time t and another one at time t + τ is then of the form E +(t + τ )E +(t)|ii, where |ii describes the initial state and where E +(t) denotes a positive-frequency component of the second-quantized electric field. The total detection-probability, obtained by summing over all final states, is then proportional to the second-order correlation function g (2) (τ ) given by hi|E −(t)E −(t + τ )E +(t + τ )E +(t)|ii g (τ ) = (hi|E −(t)E +(t)|ii)2 (2)

.

(3.1)

The classical theory of radiation then leads to 1 Z dIP (I)(I − hIi)2 g (0) = 1 + 2 hIi (2)

,

(3.2)

where I is the intensity of the radiation and P (I) is a quasi-probability distribution (i.e. not an apriori positive definite function). What we call classical coherent light is described by Glauber-Klauder coherent states. These states leads to P (I) = δ(I − hIi). As long as P (I) is a positive definite function, there is a complete equivalence between the classical theory of optical coherence and the quantum field-theoretical description [39]. Incoherent light, as thermal light, leads to a secondorder correlation function g (2) (τ ) which is larger than one. This feature is referred to as photon bunching. Quantum-mechanical light is, however, described by a second-order correlation function which may be

smaller than one. If the beam consists of N photons, all with the same quantum numbers, we easily find that g (2) (0) = 1 −

1 N

.

(3.3)

Another way to express this form of photon anti-bunching is to say that in this case the quasi-probability P (I) distribution cannot be positive, i.e. it cannot be interpreted as a probability (for an account of the early history of anti-bunching see e.g. Ref.[40, 41]). In particular, a onephoton beam must have the property that g (2) (0) = 0, which simply corresponds to maximal photon anti-bunching. One would, perhaps, expect that a sufficiently attenuated classical source of radiation, like the light from a pulsed photodiode, would exhibit photon anti-bunching in a beam splitter. This sort of reasoning is, in one way or another, explicitly assumed in many of the beautiful tests of single-photon interference in quantum mechanics. It has, however, been argued by Aspect and Grangier [42] that this reasoning is incorrect. Aspect and Grangier actually measured the second-order correlation function g (2) (τ ) by making use of a beam-splitter and found this to be greater or equal to one even for an attenuation of a classical light source below the one-photon level. The conclusion, I guess, is that the radiation emitted from e.g. a monochromatic laser is always classical, i.e. even for such a strongly attenuated source below the one-photon flux limit the corresponding radiation has no non-classical features. As already mentioned in the introduction, it is, however, possible to generate photon beams which exhibit complete photon anti-bunching. This has been shown in the beautiful experimental work by Aspect and Grangier [42] and by Mandel and collaborators [13]. Roger, Grangier and Aspect in their study verified that their one-photon states generated exhibit one-photon interference in accordance with the rules of quantum mechanics as we, of course, expect. In the experiment by the Rochester group [13] beams of one-photon states, localized in both space and time, were generated. A quantum-mechanical description of such relativistic one-photon states will be discussed below.

4. Coherent-State Path-Integrals “It is not an integral, it is a linear functional.” J. R. Klauder As was already discussed in the pioneering thesis work of Klauder [43], coherent states are very useful in constructing a path-integral representation of quantum dynamics. In a multi-dimensional canonical coherent-state basis, the quantum-mechanical probability-amplitude hz00 , t00 |z0, t0 i can then be written in the following path-integral form D 2 z(t) 1 hz , t |z , t i = exp (zα∗ (t00)zα (t00 ) + zα∗ (t0)zα (t0)) exp[iS] , π 2 (4.1) where the (quantum-) action S is given by 00

00

S=

0

0

Z z00 ,t00 z0 ,t0

"

Z

#

z˙α∗ (t)zα (t) − zα∗ (t)z˙α (t) hz(t)|H|z(t)i   dt − 2i hz(t)|z(t)i 



.

(4.2)

For a derivation of this result as well as some further details on this path-integral representation, see e.g. the introduction by Klauder and Skagerstam in Ref.[5]. Characteristic features of Eq.(4.2) are i) the complex structure, i.e. the appearance of the components zα and zα∗ and ii) a symplectic form directly inferred from the kinetic term of the action S, i.e. from the terms involving a time-derivative of the complex phase-space parameters zα (t) and zα∗ (t). (The unnormalized coherent-state exp[z ∗ z/2]|z; φ0 i only depend on z, i.e. it is holomorphic in the variable z. Such states can conveniently be used in quantization schemes referred to as holomorphic quantization [15]. Holomorphic quantization of the Maxwell radiation field has e.g. recently been discussed in great detail by Ashtekar and Rovelli [37].) A natural connection then emerges between coherent-state path-integrals and geometric quantization methods (see e.g. [44]). The factors z∗ (t), z(t) at t = t00 , t0 in Eq.(4.1) are important when considering stationary phase approximations using the coherent-state path integral [16, 17], in which case one in general encounters complex-valued trajectories (for a recent discussion see e.g. Ref.[18]). [z∗00 should then not be identified with the

complex conjugate of z(t00 ) even though z(t0) = z0, and z∗ (t0) should not be identified with the complex conjugate of z0 even though z∗ (t00 ) = z∗00 . By making use of these rules one can then verify that the stationaryphase approximation of the coherent-state path-integral gives the exact answer for Hamiltonians of the form H = ω(t)a∗ a+f (t)a∗+f ∗ (t)a+β(t), in which case one can obtain the exact answer by standard and elementary operator methods (see e.g. Ref.[14])]. One can easily arrive at such path-integrals by considering concrete physical processes. One may also consider a path-integral representation of Greens functions. For reasons of simplicity, let us consider a multi-component scalar-field propagator, where the scalar field may couple to a non-Abelian internal degree of freedom charaterized by a d-dimensional hermitian matrix representation T a of a symmetry group G such that [T a , T b ] = ifabcT c , i.e. formally we consider the operator (ˆ pµ − igAaµ T a )2 − m2 1 i Z∞ ˆ   , G= = dτ exp −iτ a a 2 2 0 (ˆ pµ − igAµ T ) − m 2m 2m (4.3) where τ is a “proper-time” parameter (by a rescaling of the parameter τ the massless limit m = 0 can easily be obtained) and Aaµ represents a non-Abelian external gauge field. In order to obtain a path-integral ˆ we can make use of the conventional representation of the propagator G, Nambu-Schwinger treatment of the canonical four-momentum operator pˆµ , i.e. we use a complete set of plane waves hp|xi = exp[ipµ xµ ], and for the internal degrees of freedom we can make use of Schwingers bosonization technique (see e.g. Ref.[80] for a recent account), i.e. we a aβ , where aα (a∗α ) are d bosonic annihilation represent T a by Tâ = a∗β Tαβ (creation) operators. We then consider states of the form |x, zi = |xi ⊗ |zi. One can then show by means of standard manipulations that [45]: 

i Z∞ Z dτ 2m 0



ˆ 0 , z0 i hx00 , z00 |G|x " # 2 1 ∗ 4 D z(τ ) ∗ D x exp (zα (∞)zα(∞) + zα (0)zα (0)) exp[iS] , π 2 (4.4)

where the action S then takes the following form: S=

Z τ 0



m dxµ (τ 0) dxµ (dτ 0 ) z˙α∗ (τ 0 )zα (τ 0) − zα∗ (τ 0)z˙α (τ 0) − + 2 2 dτ 0 dτ 0 2i  µ 0 dx (τ )  . − gIa (τ 0)Aaµ (x(τ 0)) (4.5) dτ 0

m dτ 0 

Here we have introduced the classical counterpart Ia (τ ) of the nonAbelian generators Tâ , i.e. a Ia (τ ) = hz(τ )|Tâ |z(τ )i = zα∗ (τ )Tαβ zβ (τ ) .

(4.6)

The action given in Eq.(4.5) leads to [92] the following classical Wong equations of motion [93] describing the interaction of a non-Abelian charged particle with an external gauge field: d a ν x˙ µ (τ ) = gIa Fµν x˙ (τ ) , dτ I˙a (τ ) = gfabc x˙ µ (τ )Abµ(x(τ ))Ic(τ ) , m

(4.7)

where a Fµν = ∂µ Aaν − ∂ν Aaµ − gfabc Abµ Acν

(4.8)

Z

(4.9)

is the non-Abelian field-strength tensor. The first of the equations in Eq.(4.7) is a generalization of the Lorentz force equation in classical electrodynamics and the second equation in Eq.(4.7) states that the classical non-Abelian charge Ia (τ ) is covariantly conserved. These equations of motion naturally enter into the analysis of effective actions in quantum field theory. Another object of much current interest in the context of quantum field theory is the Wilson-loop W (C), which is an object which depends on a space-like curve C and the gauge-field connection Aaµ T a , i.e. W (C) = P exp i

C

Aaµ T a dxµ

,

where P denotes a path-ordering of the T a -matrices along the spacelike curve C. By making use of Schwingers boson-representation of the

T a −matrices, Eq.(4.9) thus has the same formal structure as a timeevolution. A path-integral representation can therefore be found in ˆ in Eq.(4.3) as discussed above. the same way as for the propagator G By making use of such a path-integral representation of the Wilsonloop operator it has e.g. been shown by Gervais and Neveu [96] that the Wilson-loop operator is multiplicatively renormalizable in quantum field theory. One can use coherent-state path-integral representations in the form of the Eq.(4.4) as a starting point for studying various quantization procedures by writing down a suitable form for the action S. It is then not necesssary to restrict one-selve to situations where the coherent states are explicitly known, i.e. one can e.g. consider various non-compact groups and/or one can formally extend the path-integral to the infinitely many degrees of freedom in quantum field theory. Let us for the moment, however, restrict ourselves to a finite number of degrees of freedom and compact groups. The action appearing in e.g. Eq.(4.5) can then be quantized by canonical methods (see e.g. Ref.[92]). It is a matter of fact that one then, in general, finds reducible representations of the symmetry group G after canonical quantization. It is, however, possible to derive a coherent-state path-integral representation explicitly based on irreducible representations such that the canonical quantization of the corresponding action S yields the irreducible representation in question. One then makes use of conventional group-theoretical coherent states [3, 5] of the form |gi = U (g)|φ0i, g ∈ G. After straightforward manipulations and neglecting the space-time degrees of freedom, one obtains the action S = −i

Z t00 t0

dthK, g −1 (t)

D g(t)i Dt

(4.10)

where h, i denotes the trace-operation in the Lie-algebra of G and a G−covariant derivative, i.e. µ D d a dx g(t) = g(t) − igAµ T a g(t) . Dt dt dt

D Dt

is

(4.11)

The action in Eq.(4.10) was actually written down and canonically

quantized by Balachandran, Borchardt and Stern [94] without any reference to group-theoretical coherent states. It was then also shown that this action leads to Wongs equations of motion Eq.(4.7) as well. For a particular choice of K, which characterize the stability group H of the group coherent-state fiducial vector |φ0i, one indeed obtains one definite irreducible representation of the symmetry group G after canonical quantization of the action Eq.(4.10). It is interesting to observe that one may construct a coherent-state path-integral representation of a “universal” propagator which does not depend on the fiducial vector |φ0i at all, i.e. the universal propagator is representation independent (see Ref.[95] and references cited therein). We now observe that if we enlarge the one-dimensional parameterspace, which describes the trajectory of the particle in space-time, with one additional parameter to a space D 2, we can define the following closed 2-form: ωW Z = ihK, (g −1 dg)2 i = dΩ , (4.12)

where Ω = −ihK, g −1 dgi. We can then write, apart from boundary terms which do not contribute to the equations of motion and in the absence of external fields, the action S in the form S=

Z

D2

ωW Z = −i

Z

∂D2

hK, g −1 dgi ,

(4.13)

where ∂D 2 then plays the role of the time-parameter. ωW Z is referred to as the Wess-Zumino action for the internal degrees of freedom. A canonical analysis then reveals that no dynamical degrees of freedom are associated with the interior of D 2 . The structure exhibited in the actions given by Eq.(4.10) and Eq.(4.13) suggest immediate generalisation to non-compact groups and/or to quantum field theory. As an illustrative example one may consider the Poincare group ISO(2, 1) in 2+1 dimensions [112]. In this case the trace over the Lie-algebra is not defined. But as noticed by Witten [113], the bilinear form h., .i defined on the Lie-algebra used in the definition of the action Eq.(4.13) can be replaced by an invariant, non-degenerate form which shares all the properties of the trace-operation for compact Lie-groups. One can then write down a Wess-Zumino point-particle

action for ISO(2, 1) which after canonical quantization leads to an irreducible representations [112]. For spinless point-particles coupled to gravity in 2+1 dimensions such a particle action has e.g. been used by Witten in his study of quantum gravity in 2+1 dimensions [114]. In 3+1 dimensions the corresponding action will not have the form of a Wess-Zumino point-particle action but is, nevertheless, simple in its form (for a reviews see e.g. Ref. [116, 97]). After quantization, all physical interesting representations of the Poincare group in 3+1 dimensions can be obtained. In particular one can then develop a relativistic and quantum-mechanical description of a localized one-photon state and derive the existence of the geometrical phase of a photon directly in terms of a relativistic point-particle action [118]. In general terms, it turns out that one is quite naturally lead to actions expressed in terms of the canonical symplectic form that emerges from the geometrical structure of the coadjoint orbits induced by the group action on the dual of the Lie-algebra of the group under consideration (for recent discussions see e.g. Refs.[119, 115]). There are many related applications in quantum field theory of coherent-state methods. The considerations involving a classical source can e.g. be used in order to derive the generating functional for Greens functions. A compact expression in terms of coherent states for the S-matrix in quantum field theory can, furthermore, be written down as has so beautifully been discussed by Faddeev and Slavnov [16, 17]. The discussion of the point-particle Wess-Zumino action in terms of a coherent-state representation and the corresponding geometric quantization procedure can easily be extended to higher dimensions and to other internal degrees of freedom. In 2+1 dimensions and gauge-field theory this defines the Chern-Simons gauge theory (see e.g. Ref.[97] for a rewiew) for compact as well as for non-compact gauge groups (see e.g. Refs.[69, 70, 71, 72, 73]). This gauge theory has many interesting connections to knot theory [68], which plays an important rule in statistical mechanics [81], and to conformal field theory in 2 dimensions [82]. For a compact and simple group, the action SCS of the Chern-Simons

gauge theory is of the form SCS

k µνρ Z 3 2i = d xTr Fµν Aρ + Aµ Aν Aρ 4π 3

!

,

(4.14)

where topological arguments can be used to show that the parameter k is an integer (see e.g. Ref.[97]) In the case of loop-groups, i.e. mappings from circle to a group with a point-wise composition rule, coherent-state methods have successfully been applied. One can then, in a rather natural manner, derive the bosonized action of chiral fermions in (1+1 dimensions) [66], which, when canonically quantized [67], leads to a conventional current algebra. 5. Variational Principles in Quantum Field Theory “It is futile to employ many principles when it is possible to employ fewer. ” W. Ockham To our knowledge, the first application of coherent states in the context of quantum-mechanical variational principles goes back to the pioneering work by Lee, Low and Pines in 1953 [46]. These authors studied electrons in low-lying conduction bands. This is a strong-coupling problem due to interactions with the longitudinal optical modes of lattice vibrations. The Hamiltonian in question has the form H=

X k

(ω(k)a∗k ak

+ Vk ak +

Vk∗ a∗k )

+

(p −

P

2 ∗ k kak ak )

2m

,

(5.1)

where p is the total momentum of the system. The variational states used have the form |f i =



X exp  a∗k fk k



− ak fk∗  |0i ,

(5.2)

i.e. multi-component canonical coherent states even though they were not referred to as this in Ref.[46].

The basic idea behind the use of coherent states as a tool for a variational principle is to consider normal-ordered objects, i.e. : G(a∗ ; a) := a∗ ...a∗ a...a, which have the property that hz| : G(a∗; a) : |zi = z ∗ ...z ∗ z...z

,

(5.3)

and to treat the complex number z as a variational parameter. In D = 2 space-time dimensions and for a one-component scalar field, we consider the Hamiltonian density H as given by 1 1 H =: Π2 + (∇φ)2 + V (φ) : 2 2

,

(5.4)

which is free from ultraviolet divergences. V (φ) is an arbitrary interaction polynomial. If we split the complex field z into time-independent real and imaginary parts , i.e. |zi ≡ |f, gi, we can write 1 2 f + (∇g)2 + V (g) . (5.5) 2 By varying the arbitrary functions f and g we can find a stationary point, i.e. a static solution to the classical equations of motion. This method was applied by e.g. Cahill [55] in the case of a φ4 -coupling and in order to calculate radiative corrections to a classical soliton with a result which did not agree with the WKB-considerations. A timedependent variational principle has, however, recently been discussed by Tsue and Fujiwara [56] which is such that this variational principle gives an exact agreement with the WKB-quantization results for the φ4 and the sine-Gordon theories. One then considers a time-dependent variational state of the form

hz|H|zi =

|f (t)i = N (t) exp[S(t)] exp[T (t)]|0i , where S(t) = i and

Z ∞

−∞

dx(D(x, t)φ(x) − C(x, t)Π(x)) ,

Z ∞ 1Z∞ T (t) = dx dyφ(x)Ω(x, y; t)φ(y) . −∞ 2 −∞

(5.6) (5.7) (5.8)

The time-dependent variational principle then reads δ

Z t00 t0

hf (t)|i

∂ − H|f (t)i = 0 , ∂t

(5.9)

and determines the unknown functions D(x, t), C(x, t) and Ω(x, y; t). The exponential involving S(t), which is linear in the quantum field φ, generates a canonical coherent-state. The exponential involving T (t), which is quadratic in the quantum field φ, generates what we call “squeezing ”, a fundamental concept in modern optics (see e.g. Refs.[41, 57, 58, 59]). The method illustrated above can be shown [60] to be equivalent to the variational method of Jackiw and Kerman [61] using general Gaussian wave-functionals in the Schrödinger picture. Time-dependent variational principles, as illustrated avove, have been applied in many areas of theoretical physics and has already been discussed a lot during this conference. A recent application concerns e.g. surface adsorption processes [62]. At strong-coupling in quantum field theory, we expect the vacuum to be different from the naive perturbative vacuum. In scalar QED one can quite easily verify that the perturbative vacuum is unstable if the coupling is sufficiently large. This argument is due to Ni and Wang [63] and makes use of U (1)-conserved charged coherent states [36]. One then neglects the transverse photon degrees of freedom and considers the Hamiltonian H = H0 + HI , where H0 =

Z

d3 k|k| (a∗k ak + b∗k bk )

(5.10)

is the free Hamiltonian for particles and anti-particles, and where the interaction Hamiltonian is given by HI =

1Z 3 Z 3 1 d x d y : ρ(x) ρ(y) : 2 4π|x − y|

.

(5.11)

The charge density ρ is given by

ρ = −ie(Π∗ φ − Πφ∗ ) .

(5.12)

Let |f1i and |f2i be multi-component canonical coherent states for particle and anti-particle degrees of freedom. One then considers a sta-

tionary variational state with fixed electric charge q, i.e. [36] dϕ exp[iqϕ]| exp[iϕ]f1 i ⊗ | exp[−iϕ]f1 i (5.13) 0 2π where N is a normalization constant. One then finds that in the vacuum sector q = 0 [63] |f1, f2 ; qi = N

Z 2π

e2   hf ; 0|H|f ; 0i = αC − hNop i , 4π 



(5.14)

where αC = O(1), Nop is the number operator and where |f ; qi = |f , f ; qi. Eq.(5.14) indicates an instability of the perturbative vacuum if the coupling e is sufficiently large. If the charge e is larger than a critical value, screening effects may become important (see e.g. Ref.[64]). The coherent states constructed above in the case of a conserved U (1) charge can easily be extended to non-Abelian charges (see e.g. Refs.[47, 48]). For an application of such non-Abelian charged coherent states in the context of strong interaction physics and to Skyrmions (see e.g. Ref.[97] for a review) which leads to an elimination of unwanted zero-mode pole terms in πN scattering amplitudes see Ref.[65]. The examples mentioned in this section illustrate that coherent-state methods unifies many different aspects of quantum-mechanical variational principles. 6. Particle Physics “The standard model is at once totally successful and manifestly incomplete.” R. N. Cahn and G. Goldhaber The use of photon-detection theory as mentioned in Section 3 goes historically back to Hanbury-Brown and Twiss [74] in which case the second-order correlation function was used in order to extract information on the size of distant stars. The same idea has been applied in high-energy physics. The two-particle correlation function C2 (p1 , p2 ), where p1 and p2 are three-momenta of the (boson) particles considered, is in this case given by the ratio of two-particle probabilities

P (p1 , p2 ) and the product of the one-particle probabilities P (p1 ) and P (p2 ), i.e. C2 (p1 , p2 ) = P (p1 , p2 )/P (p1 )P (p2). For a source of pions where any phase-coherence is averaged out, corresponding to what is called a chaotic source, there is an enhanced emission probability as compared to a non-chaotic source over a range of momenta such that R|p1 − p2 | ' 1, where R represents an average of the size of the pion source. For pions formed in a coherent-state one finds that C2 (p1 , p2 ) = 1. The width of the experimentally determined correlation function of pions with different momenta, i.e. C2 (p1 , p2 ), can therefore give information about the size of the pion-source. A lot of experimental data has been compiled over the years and the subject has recently been discussed in detail by e.g. Boal et al. [75]. A recent experimental analysis has been considered by the OPAL collaboration in the case of like-sign charged track pairs at a center-off-mass energy close to the Z 0 peak. 146624 multi-hadronic Z 0 candidates were used leading to an estimate of the radius of the pion source to be close to one fermi [76]. Similarly the N A 44 experiment at CERN have studied π + π +-correlations from 227000 reconstructed pairs in S + P b collisions at 200GeV /c per nucleon leading to a space-time averaged pion-source radius of the order of a few fermi [77]. The impressive experimental data and its interpretation has been confronted by simulations using relativistic molecular dynamics [78]. In heavy-ion physics the measurement of the second-order correlation function of pions is of special interest since it can give us information about the spatial extent of the quark-gluon plasma phase, if it is formed. Recently, it has been suggested that one may make use of photons instead of pions when studying possible signals from the quark-gluon plasma. In particular, it has been suggested [79] that the correlation of high transverse-momentum photons is sensitive to the details of the space-time evolution of the high density quark-gluon plasma. A related example is the electromagnetic production of particles in heavy-ion collisions. One can then regard the heavy-ions as classical source of radiation leading to a coherent-state of the radiation field. The strong radiation fields produced can then be a source of particle

production as e.g. pion production simply by considering the inverse of the two-photon decay of π0 or similar processes like the electromagnetic production of Higgs particles [99, 100, 98]. The prediction for such Higgs particle production is that in P b + P b collisions the production cross-section can be as large as O(4 − 100) nb at LHC or SSC energies. The strong electromagnetic fields produced in e.g. high-energy colliding beams [83] or in cosmological phase-transitions [84] has stimulated much interesting research on electro-weak magnetism (for a review of the subject see e.g. Ref.[85]). At magnetic fields B such that eB ' m2W /2 ' 1024 gauss, where mW is the mass of the charged electroweak boson of the standard electro-weak model, W ± bound states with zero energy can be formed, which may then lead to a Bose-Einstein condensation of these non-Abelian charged bosons. Some of this analysis has been carried out in terms of charged bosonic coherent states [86, 87], similar in structure to the states mention in Section 5. A variational principle can then be used and one obtains an estimate of the energy-density E of the electro-weak condensate, at least in the case when mZ < mH . One finds [85] that E increases much less than the classical energy-density of the classical background field. If such a condensate can be generated in the laboratory, it may generate an anomalous production of vector bosons, quarks and leptons when this metastable field-configuration decays. It has been observed that the electro-weak model provides for a mechanism of baryon-number violation [88]. The basic idea is simple. The vacuum in a non-Abelian gauge theory is determined by a pure gauge field. This pure gauge field can, however, in general not be deformed into a trivial gauge field configuration since non-trivial windings are possible (due to the fact that Π3(SU (2)) = Z). The vacuum is therefore degenerate. Electro-weak instantons then describe tunnelling between these degenerate vaccua. Since the electro-weak model is chiral, there is an anomaly in the baryon and lepton currents JBµ and JLµ , i.e.   02 2 g g µ µ a a ∂µ JB = ∂µ JL = nf  E·B− E ·B  , (6.1) 16π 2 16π 2

E[A]

NCS [A]

∆B 6= 0

−2

−1

0

Tunneling

1

2

Figure 3: The classical field-energy functional E[A] as a function of NCS [A].

where g and g 0 are the SU (2)L and U (1)L gauge couplings, Ea , Ba and E, B are the corresponding electric and magnetic field-strengths and nf is the number of quark flavours (e = g 0 cos θ = g sin θ, where θ is the Weinberg angle). One then finds that dNCS [A] dQB dQL = = −nf dt dt dt

,

(6.2)

where the Chern-Simons topological winding number NCS [A] of the SU (2)L gauge field Ai = Aai T a is given by Z g2 2ig 3 NCS [A] = d xTr F A + Ai Aj Ak ijk ij k 16π 2 3

!

.

(6.3)

We see that the NCS [A] is nothing else than the Euclidean version of the Chern-Simons action Eq.(4.14) as mentioned in Section 4. In the vacuum sector this leads, in particular, to a possible change in the baryon number if we consider transitions between two topologically distinct classical field configurations as illustrated in the Figure 3. At low energies the rate of baryon-number violation process are exponentially

suppressed by a typical instanton suppression factor exp[−8π 2 /g 2 ], i.e. the typical time-scale for baryon-number violation processes is of the order 10137 tu , where tu is the age of our universe [88]. At high-energy the physics involved is different and, in particular, we expect abundant particle production and hence many degrees of freedom will be excited. One coherent-state approach to such a multi-particle phenomena is to use the compact coherent-state representation of the S-matrix in quantum field theory. We are therefore interested in S-matrix elements of the form hk1 , ..., kn |S|p1 , ..., pm i =

∂ ∂ ∂ ∂ ... ... ∗ hw|S|zi z = 0 ∗ i∗ ∂z1 ∂zn ∂w1 ∂wn w = 0 i

(6.4)

where hw| and |zi are canonical coherent states. In general we can write for any observable O that hw|O|zi =

Z

dϕi

Z

dϕf hw|ϕf ihϕf |O|ϕiihϕi |zi

(6.5)

where the amplitudes hw|ϕf i and hϕi |zi naturally can be written in an exponential form , i.e. hw|ϕf i = exp [Bf (ϕf , w ∗ )] or hϕi |zi = exp [Bi(ϕi , z)]. In the case of the S-matrix one obtains the following expression hw|S|zi = "

0

lim 00 t →∞ t0 → −∞ ∗

Z

dϕi

Z

00

dϕf

exp Bi(ϕi , z exp[−iωt ]) + Bf (ϕf , w exp[iωt ]) + i

Z

Dϕ

Z t00 t0

dtL(ϕ, ∂µ ϕ)

#

(6.6) ,

where L(ϕ, ∂µ ϕ) is denotes the classical Lagrangian of the physical system under consideration. Stationary phase approximation techniques, supplied with non-trivial boundary conditions, can then be used. This method has in particular been used by Khlebnikov, Rubakov and Tinyakov [89] in studies of B + L-violation in multiparticle processes (for a recent review see e.g. Ref.[90]). By making use of such or alternative methods, one can show that the total cross-section σtot for baryon and lepton number violation processes at a center-of-mass

energery

√

s is of the form σtot

√  4π s  ' exp  F( ) αW Esp 

,

(6.7)

9 F (x) = −1 + x4/3 + ... , 8

(6.8)

where the function F (x) is given by

and where αW = g 2 /4π = α/ sin2 θ. Here Esp = O(10)T eV stands for the height of the energy-barrier between the degenerate vacuum states (see Figure 3). There is an unstable field-configuration, called the spahleron, which interpolates between two degenerate vacuum configurations with a total field-energy Esp [91]. It is not yet clear what hap√ pens for large energies s as compared to the energy of the sphaleron, even though some estimates do suggest that the cross-section σtot can be sufficiently large to be observable. 7. Thermodynamics of Non-Abelian Degrees of Freedom “Thermodynamics is a phenomenological theory of matter.” K. Huang In their presentations at this Symposium, Lieb and Solovej have already, in a beautiful manner, illustrated the usefulness of coherent states in obtaining rigorous bounds on partition functions of spin systems in quantum statistical mechanics. Here we would like to point out some rather trivial applications of a slight generalization of canonical coherent states to SU (NC ) non-Abelian degrees of freedom [48], where NC denotes the strong-interaction colour degrees of freedom. We will argue that an ideal gas of such degrees of freedom in a finite volume V , and in the large NC -limit, exhibits an interesting structure of phase-transitions if a global constraint is imposed on the system, namely that the system is an SU (NC )-singlet. The motivation for such a constraint comes from physics. We believe that in e.g. heavy-ion collisions, hadronic matter can be converted into a quark-gluon plasma

phase as mentioned in Section 6. The physical process of the formation of the quark-gluon plasma phase suggest that it has finite extent in space (and time) which we also have briefly alluded to in Section 6. Because of confinement of colour degrees of freedom the quark-gluon plasma must be in a colour singlet state. Due to asymptotic freedom of the quark-gluon gas, we expect that the gas could be described by free particles at a sufficiently high temperature T as compared to the critical temperature Tc of the quark-gluon plasma phase-transition, i.e. if T >> TC = O(200)M eV . We therefore study the grand canonical partition function b

Z =

Z

D 2z hz| exp[−H/T ]|zi π

,

(7.1)

where the Hamiltonian H, in the weak-coupling limit, is given by H=

X

k,α

ω(k)a∗α (k)aα (k) ,

(7.2)

and where we have restricted ourselves to the gluon-like sector only and where we suppress polarization degrees of freedom. Let U (g) represent an element of G = SU (NC ). We then have that U †(g)aα (k)U (g) = Rαβ (g)aβ (k) ,

(7.3)

where Rαβ (g) denotes the adjoint representation of the SU (NC ) degrees of freedom. Strong interaction physics is described by NC = 3. Some dynamical features of the non-Abelian interactions are, however, supposed to survive in the limit NC → ∞. From the partition function Z b b we now would like to project out the singlet contribution Zsinglet . This can now easily be done by explicitly projecting out the singlet components of the multi-component canonical coherent-state |zi, i.e. we find that b Zsinglet

D 2z = dg1 dg2 hR(g1 )z| exp[−H/T ]|R(g2 )zi G G π   Z ∞ exp(−nω(k)/T ) X X = dg exp  χR (g n ) , (7.4) G n k,α n=1 Z

Z

Z

df π 2/2T 4 3.0 2.5

SU (2)

SU (∞)

2.0 1.5 1.0

2V T 3/π 2

0.5 0.0 1

2

3

4

Figure 4: The energy-density per degree of freedom df as a function of 2V T 3/π 2 for SU(2) (dashed line) and SU(∞) (solid line).

where χR (g) is the SU (NC )-character in the real representation R. If we approximate the Bose-Einstein statistics with classical statistics the singlet partition function reduces to b Zsinglet ≈

Z

G

dg exp [λb χR(g)]

.

(7.5)

Here λb = 2V T 3 /π 2 if we consider a finite volume V in three spacedimensions, where we also have taken into account the spin-one polarization degrees of freedom. It is amazing to notice that the integral in Eq.(7.5) is just the one-link integral naturally appearing in lattice gauge theories with the Wilson lattice action in the adjoint representation. In particular, this is the one-link integral which describes the lattice version of Yang-Mills gauge theory in 1 + 1 space-time dimensions and has been studied in detail in large NC -limit of G = SU (NC ) by Gross and Witten in the case of the fundamental representation. Gross and Witten [101] then found a third-order phase-transition separating the weak- and strong-coupling regimes (at zero temperature). Such a third-order phase-transition will now appear, in our model, as a

thermal phase-transition when we consider fermions in the fundamental representation. It is, however, not very hard to carry out a similar analysis in the case of the adjoint representation [48]. One then obtains a first order thermal phase-transition in the large NC -limit at λb = 1. The energy-density per the NC2 − 1 degree of freedom df , is shown in Figure 4 together with a numerical evaluation of the similar quantity for G = SU (2). One can show that this phase-transition survives if we include all terms in Eq.(7.4) needed in order to fully take the Bose-Einstein statistics into account. In particular we then also have a thermal phasetransition in 1 + 1 space-time dimensions, which may be a surprising result. A related result have been discussed in Ref.[102]. In the case of fermions, which transforms according to the fundamental representation of SU (NC ), the analysis has to allow for complex representations. For an ideal quark and anti-quark gas the corresponding singlet partition function is 

∞ X X



exp(−nω(k)/T ) = dg exp  (−1)n+1 (χF (g n ) + χ∗F (g n )) , G n k n=1 (7.6) where χF (g) is the SU (NC )-character in the fundamental representation F . If we keep only the first term in the sum over n in Eq.(7.6) we obtain Z f Zsinglet ≈ dg exp [λf (χF (g) + χ∗F (g))] , (7.7)

f Zsinglet

Z

G 3

where now λf = 2nf V T /π 2 . Eq.(7.7) is precisely the one-link integral studied by Gross and Witten [101] mentioned above and hence we obtain a third order thermal phase-transition at λf = 0.5. In 1 + 1 space-time dimensions it turns out that fermion statistics is important. It can be shown [49] that by including all terms in the sum over n in Eq.(7.6) the third-order Gross-Witten phase-transition actually disappears. The technical reason for this is the same as the reason why the Wilson lattice action exhibits the Gross-Witten phase-transition but that the alternative Manton lattice action does not lead to such a phase-transition [50]. The same coherent-state technique can now

df π 2/2T 4 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0

x = 0.1 x = 0.05 x = 0.01 2V T 3/π 2

x ≡ nf /NC 1

2

3

4

Figure 5: The energy-density per degree of freedom df for the ideal quark-gluon colour singlet gas in the large NC -limit suitable normalized as a function of 2V T 3/π 2 for various values x of the ratio of the number of quark flavours nf to the number of colour degrees of freedom NC .

be used in order to derive an expression for the singlet partition function Zsinglet of an ideal quark-gluon gas. We then obtain for the colour singlet partition function in the Boltzmann statistics approximation Zsinglet ≈

Z

G

dg exp [λb χR(g) + λf (χF (g) + χ∗F (g))]

.

(7.8)

The energy-density df per degree of freedom normalized to the ideal, infinite volume limit, quark-gluon gas, is shown in Figure 5. The quarks then dictate the order of the thermal phase-transition and we therefore obtain a third-order transition at λb + 2λf = 1. The effect of a finite baryon-number has been studied in Ref.[51] in a micro-canonical approach. In a finite volume, the energy levels are actually discrete. It has been shown [52] that this fact does not change the qualitative picture of the effects discussed above due to the colour-singlet constraint condition on a canonical partition function of an ideal quark and/or gluon gas. 8. Quantum Cosmology

“The effort to understand the universe is one of the very few things that lifts human life a little above the level of farce, and gives it some of the grace of tragedy.” S. Weinberg Coherent-state methods appear in many applications in modern astrophysics and cosmology. One simple application of coherent states we have already mentioned is actually the calculation of thermodynamical partition functions. If we are considering free quantum fields at thermal equilibrium, the Hamiltonian of the system corresponds to a system of independent harmonic oscillators, one for each quantummechanical degree of freedom of the system. The Bose-Einstein distribution then almost trivially follows as was mentioned in Section 7. As has now so beautifully been confirmed by the COBE-satellite observations, the spectrum of the cosmic background radiation (CMB) is that of an ideal Bose-Einstein distribution with a temperature T = 2.73K, apart form a dipole asymmetry and primordial fluctuations such that ∆T /T ≈ O(10−5). If we are interested in corrections to the Planck distribution of the CMB caused by a spatially varying temperature at decoupling, we may as a model consider free quantum fields at local equilibrium. In order to illustrate what may happen under such circumstances let us consider [53] a scalar field described by the Lagrangian density L such that 1 L = (∂µ φ∂ µ φ − m2 φ2 ) . (8.1) 2 We then define the density operator ρl.eq. [β(x)] of local equilibrium at inverse temperature β(x) by R exp[− d3 xβ(x)T 00 (x)] ρl.eq. [β(x)] = . (8.2) R Tr(exp[− d3 xβ(x)T 00 (x)]) The field φ and its energy-momentum tensor T µν in a finite quantization volume V are given by X 1 √ φ(t, x) = [ak e−i(ωk t−kx) + a†k ei(ωk t−kx) ] , (8.3) 2V ωk k 1 T µν (t, x) = ∂ µ φ∂ ν φ − η µν (∂α φ∂ αφ − m2 φ2 ) , (8.4) 2

√ where ωk = m2 + k2 , and η µν = diag (1, −1, −1, −1). If β¯ describes the average of the inverse of the temperature T , then we can write ¯ eff Z = Tr exp −βH h

i

.

(8.5)

Here we have introduced the effective Hamiltonian Heff given by Heff =

X

ωk

l

−

X k,l

a†k ak

1 + 2

!

1 ∗ ∗ ak a†l ) + fk,la†k a†l + fk,l ak al (ωk,l a†k al + ωk,l 2

!

,(8.6)

where ωk,l and fk,l describes the spatial deviations from global equilibrium and are obtained from the Fourier transform of β(x). The form of the effective Hamiltonian appearing in Eq.(8.5) is not unusual in physics. Hamiltonians of this form generate a time-evolution leading to what in quantum optics is known as squeezed states and is by know a well-established concept (see e.g. Refs.[41, 57, 58, 59]) which has been alluded to in Section 5. A formal expression for a partition function of the form Eq.(8.5) can be obtained using coherent-state methods [54]. In our case Heff has the effect of increasing the mean number of excitations or in other words the entropy is increased. This effect, valid at least for small deviations from global equilibrium, may be described by an effective increase of the temperature to the extent that one has to allow for a slight momentum dependence, i.e. the inverse of the effective temperature is ¯ k 1 ¯ k βω βω βk = β¯ 1 − coth( ) 2 2 β¯2 V 

Z



2 ˜ d3 x(β(x))

,

(8.7)

˜ where β(x) describes the inverse of the temperature variations. In the ˜ derivation of Eq.(8.7) the only condition on β(x) is that its Fourier ˜ acts like a cut-off for large momenta l [53] . transform β(l) Recently, Hamiltonians leading to squeezed states have turned out to be important in an alternative quantum-mechanical description of amplifications of the primordial fluctuations in the early universe. These

primordial fluctuations, we now believe, eventually determines the largescale structure of our universe (for a recent review see e.g. [103]). In order to illustrate the essential idea [104, 106], let us consider a (0) Robertson-Walker space-time with a metric gµν in the following form (0) ds2 = gµν dxµ dxν = a2(η)(dη 2 − dx2 − dy 2 − dz 2 ) ,

(8.8)

where η corresponds to conformal time and a(η) corresponds to a cosmological scale factor from which we define the quantity σ(η) = ia(η)−1da(η)/dη. Let us then consider small fluctuations of the metric, i.e. we consider the new metric (0) gµν = gµν + hµν

,

(8.9)

where we can write hµν = N

Z

3

dk

2 X

λ=1

λµν (akλ (η) exp[ik · x] + h.c.)

.

(8.10)

Here λµν represents the two independent polarization degrees of freedom of the gravitational field and N is a normalization factor. The Hamiltonian which determines the η-dependence of the system in the Heisenberg picture naturally splits into two parts, i.e. H = H0 + HI , where H0 =

X kλ

0 Hkλ

,

0 Hkλ = |k| (a∗kλ akλ + a∗−kλ a−kλ )

,

(8.11)

and HI =

X kλ

I Hkλ

,

I Hkλ = σ(η)a∗kλ a∗−kλ + h.c.

(8.12)

i.e. the interaction HI leads to squeezing. Let us consider an initial quantum state |0i with only ground-state quantum fluctuations, i.e. ak (η0 )|0i = 0 .

(8.13)

The η time-evolution operator can now be factorized in a form which 0 I contains operators of the form Hkλ or Hkλ by making use of the fact ∗ ∗ ∗ the that set of operators akλ akλ , akλ a−kλ and akλ a−kλ span a Lie-algebra [110]. We remark in passing that such methods have successfully been applied in the study of free electron lasers [111]. By extending these well-known methods from the theory of squeezing, one can then write the η time-evolution operator in the following form U (η, η0 ) = S[Rkλ ; Φk ]U0 [Θk ] ,

(8.14)

where the “free” η-evolution operator U0[Θk ] is given by U0[Θk ] =

Y kλ

exp [−iΘkλ (a∗kλ akλ + a∗−kλ a−kλ )]

,

(8.15)

.

(8.16)

and where the squeezing operator S is given by Rkλ −2iΦkλ S[Rkλ ; Φkλ ] = exp (e a−kλ akλ − h.c.) 2 kλ Y

"

#

The functions Θk, Rkλ and Φkλ then obey ordinary, coupled differential equations which can be solved for in terms of the conformal timeparameter η. The squeezing operator S[Rkλ ; Φkλ ] is the part of the ηevolution which generates momentum-conserving pair excitations and is responsible for the amplification of the quantum fluctuations inherent in the initial state |0i. A measure of the quantum-mechanical squeezing is the mean value of the number-operator Nˆ (kλ) of quanta with quantum numbers k and λ , i.e. hNˆ (kλ)i = sinh2 (Rkλ ) .

(8.17)

It can be shown (Ref.[105]) that on scales larger then the Hubble radius the squeeze phase Φkλ freeze out and the squeeze factor Rkλ grows, leading to a macroscopic density-perturbation. On smaller scales than the Hubble radius these parameters oscillate. The amplification of the quantum-mechanical fluctuations inherent in the state |0i into macroscopic perturbations, which occurs during the era of cosmic inflation, can therefore be thought of as a process of quantum squeezing. It has

been argued in particular in Ref.[105] that there are no new features in this quantum-mechanical description as compared to the conventional and classical treatment of density perturbations in inflationary cosmological models. It seems, however, natural to discuss the connection between quantum and classical properties of fluctuations in terms of quantum squeezing and the method shows in a transparent manner in what sense a quantum-mechanical coherence exists between classical density inhomogeneities. In a related work [106] an exact formula for the angular correlation functions of temperature fluctuations caused by squeezed gravitational waves has, furthermore, been derived. We have seen above, in connection with the effective Hamiltonian Heff describing departures from global thermal equilibrium, that quantum squeezing leads to an increase of entropy. The growth of entropy in the cosmological production of momentum-correlated pairs of particles in an expanding background space-time geometry has recently been discussed extensively in the literature (see in particular Refs.[107, 108, 109]). The entropy growth ∆Skλ of a particular mode with quantum numbers k and λ in such a cosmological process is then determined by the squeeze parameter Rkλ , i.e. ∆Skλ ≈ 2Rkλ for large values of the squeezing parameter. As has been argued by Gasperini and Giovannini [107] the total entropy generated in this manner may represent a substantial fraction of the entropy of our present universe, at least in a specific inflationary scenario. 9. Final Remarks “The more the island of knowledge expands in the sea of ignorance, the larger its boundary to the unknown.” V. F. Weisskopf In this excursion in the expanding space of applications of coherent states, we have said something about the past and something about the present. There are many more subjects which could have been discussed. The recent high-precision observations of the cosmic microwave background strongly support the existence of early quantum

fluctuations needed in order to form the observed large scale structure of the universe. Something non-standard is needed here like cosmological models based on inflation, since there are correlations which extends over horizon size at decoupling. At first sight it may be considered remarkable that coherent states can be used in a description of the origin of the large scale structure of our universe. In view of this remarkable achievement and in thinking about future applications of coherent states one should perhaps expect more surprises. A recent such a somewhat surprising application of coherent-state methods in the context of quantum field theory is connected with the Borel transform. The coefficients an in the perturbative expansion of a Wightman function G can be used to define a coherent-state |Ci after a Borel P transform, i.e. C(z) = hz|Ci = n an z n/n!. As shown in Ref.[120] this simple observation can be used to derive an interesting connection between the absorptive part of G and C(z), i.e. the Borel transform is the coherent-state representation of the abstract vector which describes the original perturbative series of G. It is hard to say anything intelligible concerning future applications and developments of coherent-state methods. In view of what has been said in this presentation, it seems, however, to us that one safe prediction for the future is to say that “coherent states are here to stay ” and that many more surprising applications are to come. 10. Acknowledgement We are very grateful to John R. Klauder for his continuous encouragement and enthusiastic support over the years in the struggle in finding a continuous path in the increasing space of applications of coherent states in physics and mathematical physics. We would like to thank John Ellis for the hospitality of the Theory Division at CERN, where some of this work has been initiated. We are grateful for useful discussion with Ling-Lie Chau, D. Ellinas, D. Husan Feng, E. Heller, R. Glauber, E. Lieb, M. Nauenberg, C. A. Nelson, M. Nieto, K. Kuratsuji, C. Stroud, W. A. Tomé, A. S. Wightman and W.-M. Zhang during the

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