Field-strength formulation of gauge theories: Transformation of the ...

PHYSICAL REVIE%

Field-strength

0 formulation

15 SEPTEMBER 1982

VOLUME 26, NUMBER 6

of the functional integral

of gauge theories: Transformation Loyal Durand and Eduardo Mendel

Physics Department,

adison, Madison, of Wisconsin M— (Received 10 March 1982)

University

Wisconsin 53706

We present a formulation of gauge field theories in which the gauge potentials A„(x) are eliminated in a simple way in terms of the field strengths Our results are closely related to, but are much simpler than, Halpern's dual variable formulation of gauge theories in the axial gauge. We work in the coordinate gauge xI'A„(x)=0, and show both analytically and geometrically that the potential A„can be determined uniquefor a suitable class of F 's, A ~A [F]. We show, furtherly from the field strengths more, that a tensor I"z,(x) is a coordinate-gauge field tensor if and only if it satisfies the restricted set of Bianchi identities d' x Dt'[F] ~Fei=O, D=tl+[A [F], ]. These results amplitude Z[J] permit us to transform the functional integral for the vacuum-to-vacuum for the gauge theory to a form in which the potentials are completely eliminated in terms of the field strengths. When the Bianchi constraints are eliminated using a set of Lagrange multiplier fields A, (x), the F s can be integrated out completely to obtain a form of the theory which appears to be useful for strong coupling.

I'„(x).

F„„

I.

INTRODUCTION

In this paper, we present a formulation of gauge field theories in which the usual potentials A&(x) are eliminated in a simple way in terms of the (x) That on. e should be gauge field strengths able to reformulate gauge theories in this way is perhaps evident for electrodynamics, where the field strengths are gauge-invariant observables, hence logical candidates for dynamical variables. It is less evident for non-Abelian theories, where the F's are gauge covariant rather than invariant, and the potentials are not always uniquely determined by the field strengths. ' (It was a study of this "copy problem" for the coordinate gauge 0 which led to the present work. ) The exx"A& — istence of a field-strength formulation for gauge theories was first shown by Halpern, who proposed a number of possible applications, e.g. , to the calculation of Wilson loop integrals, the study of monopole configurations, and the construction of confining states in non-Abelian theories. Our results are closely related to Halpero's, but involve a much simpler choice of gauge, the introduction of matter currents, and a somewhat different overall emphasis. They appear, in particular, to provide a natural starting point for the investigation of the strong-coupling limit of the continuum gauge theory and the Wilson loop integral. Results very similar to ours have also been obtained by Itabashi, a fact of which we became aware only recently.

F„,

In this paper, we will concentrate on the Lagrangian formulation of gauge theories in terms of the vacuum-to-vacuum amplitude or generating functional Z [J], and will show that the functional integral which determines this quantity can be transformed from an integral over gauge potentials A&(x) to one over the field strengths F&„(x).In the second paper in this series, we adopt a quite different approach, introducing the field strengths ab initio as canonical variables, and considering the gauge theory as the theory of a constrained Hamiltonian system. The extra canonical variables which appear in the Hamiltonian theory will appear here as l.agrange multiplier fields for constraints. These are analogs of Halpern's dual potentials. For simplicity, we will anticipate some results which are derived in the second paper, and postpone discussion of some questions concerning constraints on the physical Hilbert space in the quantum theory which can only be addressed properly in a Hamiltonian formalism. The generating function Z [J] for the Green's functions of a quantized gauge-theory is given (schematically) by the functional integral

Z[J]=X

'

J &A e' ("'s)detM5(G(A)),

(l)

J]

where S [A, is the action functional, G (A) is a gauge-fixing condition, detM is the measure of the gauge group for the gauge condition G (A) =0, and the integration is over all gauge-inequivalent potentials A„'(x),with a being a group index.

1982

The American Physical Society

FIELD-STRENGTH FORMULATION OF GAUGE THEORIES: Our objective is to transform the integral in Eq. (1) to an integral in which the field strengths appear as the independent dynamical fields. This requires as a first step that we be able to determine A given F,A ~A [Fj. We show both analytically and geometrically in Sec. II that this inversion can be accomplished essentially trivially in the coordinate =0, with the result gauge x "A„(x)

A&[F](x)=

f daax"F„&(ax).

I"„

We show, furthermore, that a tensor is a coordinate-gauge field strength, and that A [F] is consequently a potential for F, if and only if F satisfies a restricted set of Bianchi identities

x DP[F] «FR=0.

(3)

These are just the usual Bianchi identities for the covariant derivative D =8+ [A [Fj, ] and directions orthogonal to x. The inversion is unique if it is required that A [Fj and all allowed gauge transformations be well defined at the origin. If the conditions of Eq. (3) are satisfied, the remaining Bianchi identity is satisfied automatically. In Secs. III A and III B, we consider the properties of the functional integral for Z[J] in the coordinate gauge, and show that it can be written in terms of F as

Z [J]

+

i

f ~F eis[F J)g~~vaixoDP«F

algebra valued, A& =A&T„where the T's are the generators of the gauge group in the adjoint representation. We use the normalization TrT'Tb 25 . The Lagrangian for a general gauge theory is invariant under local gauge transformations of the vector potential of the form

=—

A„'(x)= V 'A„-(x)V + V 'a„V-,

where S is the nonlocal action obtained by the replacement A ~A [Fj. The 5 function can be eliminated by introducing subsidiary fields A&(x) in the functional integral. The result is a generalized and much-simplified version of Halpern's dual variable formulation of gauge theories, with

x"A„'(x)=v 'x"A„(x)U+U 'x"d„U=O, or equivalently,

xPa„V(x)=-xPA„(x)V(x) .

L

Gauge theories are usually described in terms of vector potential fields These fields are Lie-

A„.

A "(x)dxz

(9)

with the integration path the straight line which connects x" to the origin. The transformed potential A& can be written using Eqs. (8) and (9) as A p(x)

= U '(A„— xpx. A) U

+v '(a„— x„xa)-v

&(exp(iS[F J]+iA~Dp «F~) .

A. Existence of the gauge

f

—

U(x) =Pexp

f

II. THE COORDINATE GAUGE

ex-

ponential'

Z[J]=X-' ~F~X The F's appear only quadratically in this expression and can be integrated out to obtain an expression for Z [J] in terms of the A, 's and J. We show in the final sections that the fieldstrength form of the gauge theory gives the correct classical equations of motion, and note possible applications of the integrated expression for Z [J].

(6)

where U(x) is a group element. This means that all the potentials on a group orbit are physically equivalent. We have to choose one representative on each orbit in the functional integration over fields, Eq. (1), therefore fixing the gauge. We will =0 in our use the coordinate gauge ' x "A„(x) transformation to the field-strength formulation of gauge theories. We show first that it is possible to transform a general potential Az(x) which is nonsingular at x=0 into the coordinate gauge. This amounts to finding a gauge function U(x) such that the transformed potential A&(x) satisfies the gauge condition x&A& =x-A'=0. Thus we require that

A solution to this equation is the path-ordered (4)

1369

(10)

and clearly satisfies the gauge condition x.A'=0. The transformation matrix U(x) exists for potentials A&(x) which are less singular than x ' on the path (0~"), corresponding to field strengths less singular than x . It can also be defined with the path of integration running from x" to infinity along the line which connects x" to the origin provided A& vanishes more rapidly than x at infinity. The transforination does not exist for the special case of potentials which are singular of degree 1 at the origin, corresponding in general to field strengths which are singular of degree — 2. We will omit such configurations in the functional integration: the action is infinite, and the configura-

—

LOYAL DURAND AND EDUARDO MENDEL

1370

tions are presumably of zero measure in the space of all potentials. We note finally that the vector potentials have a residual gauge freedom within the coordinate gauge given by group elements satisfying x&t}„Uo— 0, as it is clear from Eq. (8). The solutions Uo to this equation are functions of x& homogeneous of degree 0, that is, functions of angles in four dimensions. We will require that all gauge functions be well defined at the origin. The allowable functions Uo are then independent of the angles, so are just global gauge transformations.

B.

Construction

of A„from

=t}qA„(x)—t}„A„(x)+ [A„,A„], tion

x"A„(x)=0, we x

the coordinate-gauge find that '

condi-

F q(x)=x (dQq de +[A— ,Aq]) =x t}Q~ —x 8~A~ =(1+x t} )Aq(x) .

around a closed path I to a well-defined nonAbelian flux through the path. It will be sufficient for present purposes to apply the theorem to a triangular path with vertex at the origin, straight sides x& and x"+M", and infinitesimal base )bc", as shown in Fig. 1(a). In the coordinate gauge, 3 dx=0 along the sides of the triangle, and

4[6,A] =P exp

To relate

= [D„[A],D„[A]) F„„(x) —i}„+ . [A&, ], and

&[I,A]=P exp( (IIr A.dx)

f A&(x)dx"

=1+A&(x)M"+O(M

F„„

We next prove a crucial property of the potentials A& in the coordinate gauge, namely, that they can be expressed in terms of the field strength F&„. By using the definition for the field strength in terms of the vector potential,

Dz

ordered phase factor

to F, we note that

)

.

(14)

4

is not changed original triangular path by adding segments which are traversed in both'directions (this amounts to inserting the unit operator in the definition of 4&), and use the construction shown in Fig. 1(b) to divide the triangular area into a set of infinitesimal trapezoids. The product of phase factors around each trapezoid (and the remaining vertex triangle} can be expressed in terms of F&,(ax) (the field at the position ax "}and the area of the trapezoid H, ' A

if we deform the

'

4[0~,A (ax)] = I +F~&(ax )(x b a)(aM") +O(M ), (15) 0&a &1. The extra phase factors for the line seg(12)

We assume again that the potential is less singular than x ' at the origin and on the line (0~"), hence that the field strength is less singular than x . We can then integrate Eq. (12) by changing +ax", recognizing that the right-hand scale, x"— side of the scaled equation is just (d lda)[aA&(ax)], and integrating over a, with the result

ments which connect each trapezoid to the origin refer F to the origin by parallel transport, F— +F ~(t.

Aq(x) =Aq[F] (x) 1

0

daax F &(ax) .

This expression is unique for the class of potentials and gauge functions considered. It satisfies the gauge condition x A =0 automatically, and permits the existence of the Aharonov-Bohm effect because of its nonlocal dependence on the field strengths. We have shown elsewhere that Eq. (13) can also be derived by a simple geometrical argument which we repeat here. The argument is based on our generalization of Stokes's theorem for non-Abelian gauge theories. ' This theorem relates the path-

"

(o) (b) FIG. 1. (a) The triangular path considered in the path-ordered phase factor C&[E,A ] = P exp( A„dx"). hx~ is infinitesimal. (b) Modification of the triangular

f

path by one insertion traveled in both directions. be is taken infinitesimal. The remaining triangle is subdivided in the same way to obtain an expression for in terms of the non-Abelian flux P (Stokes's theorem).

4

FIELD-STRENGTH FORMULATION OF GAUGE THEORIES: These factors are unity in the coordinate gauge, so the complete phase factor for the triangular path can be written as

to be in the coordinate gauge. Given a tensor F, but not A, we cannot test the gauge directly. We can, however, construct both the coordinate-gauge potential A [F] defined by the integral in Eq. (13), and the corresponding covariant derivative D [F]= d+ [A [F]], ]. We will show in the following that F is the field strength for the potential A [F] (the reconstruction theorem), hence is in the coordinate gauge, if and only if

F =F~~ and

4[6„A]= P [1+F z(ax}(x~ha}(aM") +O(M }] ha~0

f daaF „(ax)xM" 1

1+

+O(M

).

1371

(16)

d'"'x. Dq[F] 'F,, =0 .

Comparison of Eqs. (14) and (16) leads at once to the expression for A in terms of F given in Eq. (13),' and establishes the uniqueness of that result for potentials which are integrable in a neighborhood of the line (0~").

This is just the requirement that the dual field ~Fqi„satisfy the Bianchi identity Dq[F] ~Fqi =0 for the three directions orthogonal to x". If it is satisfied, F is expressible in terms of the potential A [F], and the fourth Bianchi identity holds automatically. The field strength which corresponds to the potential A [F] is given by

C. The reconstruction theorem We have shown so far that it is possible to find a (unique) potential for field strengths F(A) known

A„[F]] Fq, (A [F])=BqA„[F]—B„Aq[F]+[A„[F],

=f da 1

+

ax

f da f 1

1

Bax"

8 Fz (ax) Bax

F „(ax)+

+—2aF&,

(ax)

d13a13x x [Fi„z(13x}, F~„(ax)].

The last term in this expression can be rearranged in a more familiar form by splitting the range of the P integration at P=a, rescaling P on the interval 0 &P& a,P — +ay, changing the order of integration on the remaining interval a &P & 1 and rescaling a, and finally, relabeling the variables. The result is 1

0

daa x~

1

dyyax Fi&(yax),

~

0

F,(ax) =

—(p~v)

f daa x 1

.

The commutators in Eq. (19) combine with the derivatives in the variable ax, and we find that

F»(A [F])=

F,

I[A„[F](ax), (ax)]+[A„[F](ax), F& (ax)]] . in

(19)

Eq. (18) to give covariant derivatives D [F]

f daI a x [(D&F~„)(ax)+(D,F&~)(ax)]+2aF»(ax) 1

0

da a x

aax

J

(208)

F„„(ax)+2aF&„(ax)

f daa2x [(D F q)(ax)+(DqF )(ax)+(D Fq )(ax)] = f da [a F&„(ax)]+ daa x e„&~i„(Dq[F] )(ax) da =F~„(x)+f daa x e„„ )(ax) . i„(Dq[F] 1

+

1

1

*Fq

0

Fq

(20b)

(20c)

(20d)


1372

In the transition from Eq. (20a) to (20b), we have added and subtracted D I' z and used the relation x A [F]=0 which holds automatically. Note for the again that I' must be less singular than x line integrals to exist. Thus a F»(ax)~0 for a— &0, a property used in going from Eqs. (20c) to (20d). We conclude from Eq. (20d) that the potential A [F] reproduces the field F to which it supposedly corresponds if and only if the last integral vanishes for all choices of x. This is only possible if the in-

26

tegrand vanishes. A sufficient condition for this vanishing would be the validity of the complete set of Bianchi identities, D~[F] ~F~ (x) =0 for all x. It is only necessary that the restricted identities in Eq. (17) hold. If they do, the construction above shows that F is expressible in terms of A [F] in the usual way, hence is the field strength for a coordinate-gauge potential, and of course satisfies all the Bianchi identities. Conversely, A [F] is the unique potential for I' which is less singular than

x-'.

III. TRANSFORMATION OF THE FUNCTIONAL INTEGRAL A. The functional integral in the coordinate gauge In this section, we begin the derivation of the field-strength formulation of gauge field theory by expressing the Feynman functional integral for the generating function Z[J] in the coordinate gauge. The preceding results will then allow us to replace the potentials by field strengths in this integral. We consider for simplicity a gauge theory coupled to an external source J. This classical current can be replaced later by fermion currents and will be assumed to be covariantly conserved at the classical level. The functional integral in the coordinate gauge is given formally by

I &A

'

Z[J]=N

exp i

fd x

—,

Tr

2F»F»(A)+A„J" P, 2 Pv

is given explicitly by

5(x.A

5(x "(A )„'(x)) 5~os

It is interesting to note that the gauge transformations which are eigenfunctions for the matrix M of Eq. (23) satisfy (22)

(y)

x BU(x)=kU(x),

(24)

3

is the gauge transform of 3 by U, with U(x) co, (x)T' an infinitesimal gauge transformation on the vector potential with parameters where

consequently no need to introduce ghost fields in the coordinate-gauge formulation of the theory in contrast, e.g. , to the Lorentz-gauge formulation. (M is also field independent in the axial gauge. ) %e will henceforth omit the factor detM in Eq.

(21) )

U=i

=det

g5(x"A„). X

The integral in the exponent is the action for the gauge theory in the normalization TrT'T = —25', and detM is the Faddeev-Popov measure for the gauge condition A x=o. This determinant

detM =det

detM

=1+

co, (x).

The matrix M is easily calculated,

M=

8

A'

D' 5(x — y) =5' x.B5(x — y),

P

where we have used the gauge condition x.3 =0 in the last step. The result is field independent, and the determinant can therefore be absorbed in the normalization of the functional integral. There is

hence correspond to functions homogeneous of degree (k) in the coordinates. The determinant is given by the product over all acceptable eigenvalues k. Negative k are not accepted if we restrict our Hilbert space by admitting only gauge transformations which are nonsingular at the origin. Zero eigenvalues, which would signal a residual gauge freedom, corresponds to gauge transformations homogeneous of degree zero. These reduce to global gauge transformations if we impose single valuedness at the origin. The residual global gauge transformations are irrelevant in the sense that they only change the overall normaliza-

FIELD-STRENGTH FORMULATION OF GAUGE THEORIES:

26

of the coordinate-gauge

tion for the functional integral and can be eliminated by parametrizing the integration over the gauge group properly. Note that the constraints k & 0, U(x) single-valued, guarantee the uniqueness

Z[J]=N

'

f

NA O'Fexp i

fd x

—,

Tr

1373

"

potential. We conclude by reexpressing Eq. (21) in the first-order formalism in which we consider A and I' to be independent fields,

2

—[A„,A F„„+A„J" B(„A„) g 5(A x)5(F„„— IJt&

]) . (25)

If the F s are integrated out using the constraints, we obtain Eq. (21) with detM=1. It will be our objective, however, to integrate out the A s to obtain a new expression for Z [J] in terms of a functional integral over field strengths rather than potentials. B.

Transformation

to the field-strength

description

The first-order form of the functional integral given in Eq. (25) contains n +6n =7n constraints on the 4n potentials A& and 6n independent fields I"z,p & v for a gauge group with n generators. In Sec. II, we showed that the 4n coordinate-gauge potentials can be expressed in terms of the field strengths through Eq. (13), and that the result was consistent provided the F s satisfied the 3n restricted Bianchi identities of Eq. (17). These conditions again give 7n constraints on the A's and F's. ' This suggests that it should be possible to rewrite Eq. (25) in the form

Z[J]=

f &A NFexp i f d x Tr Fq„F"+AqJ" X g 5 A„— f da ax F „(ax) 5(e""" x„DI' —,

4g22

g [A,F]

1

~F~

),

(26)

X

g

[A, F] a functional Jacobian determinant. We will first evaluate the Jacobian [A, F] and show that it is independent of the fields and so can be absorbed in the normalization. Notice that the constraint 5(A.x) appears in both expressions for Z [J] so the Jacobian matrix is just 6n X6n. The transformation is taken on the fields F&„the functions of A being considered as constants in the transformation. Furthermore, to get a block determinant, it is convenient to all independent. group the 6n independent F's into 3n fields x"F&„and 3n fields x" [These are generalizations of the field E and B, and coincide with them for x =(1,0).] In this basis, the Jacobian is given with

g

~F„„,

by

f da ax F~~(ax) 1

5

5(z

g [A, F]=

Fi, (z))

5(x&(DqF v+D~Fq„+DvFq )) 5(z Fi„(z))

f daax 1

5

F z(ax)

5(z ~Fi„~(z))

(27)

5(xl'(DqF v+D Fv~+DvF~ ))

5(z"~Fi „(z))

where

Dz

Dz[F] =5&+

f

1

— F z(ax), .

da—ax

(28)

The first block in g is linear in the fields so its functional derivative gives a field-independent matrix. The second block, to the right, involves functional derivatives of independent fields and is zero. The third block is quadratic in the fields because of the covariant derivatives. Its functional derivative is therefore linear in the fields and is field dependent. The last block in g is also field independent. The covariant derivative x D„[F]in the first term can be

1374


replaced by an ordinary derivative x"8& because x"A&[F] =0. This term is therefore linear in the F's, and its functional derivative is field independent. The commutators in the covariant derivatives in the second have vanishing functional derivatives with respect and third terms involve only the fields of type x to the independent fields x F~„.The remaining pieces involve only ordinary derivatives and are linear in the F s, so again have field-independent functional derivatives. To summarize, has the form

F~„so

g

f

where and g do not depend on the field strengths. The Jacobian of the functional transformation from Eq. (25) to Eq. (26} is therefore field independent, and can be absorbed in the normalization factor in Eq.

(26). We can now integrate over the potentials

F's

A„in

Eq. (26) to obtain an expression for Z [J] in terms of the

alone,

Z[J]=

f &Fexp i f d x

—,

Tr

4

2'„F""+A„[F]J"g 5(e""" x„DI'[F]'Fp ), 2

(30)

I

where A [F] is given everywhere in terms of the field strengths by Eq. (13). This expression is the analog of the usual functional integral in Eq. (21). It expresses the gauge theory in terms of the six independent field strengths F&, p g v, instead of the potentials 3&. The gauge condition x "3& 0 in Eq. (21) is replaced in Eq. (30) by the three Bianchi identities for directions orthogonal to x". These constraints, and our restriction that be less singular than x at the origin, guarantee that F&„is a coordinate-gauge field tensor with the unique potential A&[F]. However, the potentials have completely disappeared from the theory as independent dynamical variables. This, of course, is at the cost of the introduction of nonlocal expressions in the F 's, as is necessary to ensure the existence of the Aharonov-Bohm effect.

—

F„,

C. Elimination of the constraints and integration

over the fields

The "action" in the exponent in Eq. (30) has the peculiar feature that it contains no time derivatives of the F's, so is nondynamical. All the dynamics is hidden in the Bianchi constraints. It will therefore be convenient to write the constraints in exponential form by expressing the 5 function as a

Z[J]=N

'

f &F&iexp i f d x

—,

Tr

functional Fourier integral. This requires the introduction of new Lie-algebra-valued fields which act as Lagrange multiplier fields for the constraints. In this construction, we can use either a tensor field iP"(x) or (to simplify notation) a vector field A, (x) =e """x&A,„&(x)which satisfies the constraint x k =0. This constraint is actually superfluous. It limits the functional Fourier transform for each x to three independent A, 's and leads to the three-dimensional 5 function in Eq. (30), thus enforcing only the Bianchi identities in the restricted set of Eq. (17). However, the "fourth" Bianchi identity holds automatically for fields F less singular than x which satisfy Eq. (17). We can therefore integrate without restriction over all four components of the vector field A, at the expense of introducing a functional 5(0) for the fourth (automatic) Bianchi identity. This factor can again be absorbed in the normalization. (We give an alternative treatment of the constraints in the Appendix, and will discuss this transformation in more detail in a second paper on the Hamiltonian formulation of the theory. } We conclude from the foregoing observations that we can eliminate the constraints in Eq. (30) using 4n independent Lagrange multiplier fields A, (x), and rewrite Z[J] as

,

F„„F""+A„[F]J" +iPD~[F] *Fp

The action in this equation is quadratic in the field strengths and can therefore be integrated over the independent F 's. It will be convenient, as a next step, to extract the F dependence of the source term in Eq. (31),

(31)


1375

f d x A~[F] Jl'(x)= f d x f daax"Fvq(ax)J"(x) = f d x f d y f day"F„&(y)5(y —ax)J"(x) = f d yy"Fz„(y)f daa J"(y/a) = f F„„(y)J""(y), 1

d y

—,

where

J""(y) is

J""(y)=

a nonlocal tensor current,

dPP'[y"J" (13y) y"J"— (Py)]

f,

(33)

With this notation,

f NF&A,

Z[J]=N

exp i

fd x

—,

Fq„F"'+A,

Tr

Z[J]=N

'

f

NA,

&Fexp i

&&exp

i

fd

of F&„with p

1

—~(B"A,

& v as the independent

„(y)

"b'~"(x,y)F~ (x)—K,

[J," —*(8"A,, )]Fz

x—

(34)

fields, and rewrite Eq. (34) as

f d x f d ,'F„', y

*Fp (x)

— )]F„(x).

+ , [J" We will take the six components

f da axsF"~(ax),

(x)

(35)

where KI'b

~"(x,y)=

Integration

—1 5,&gl'~g"s5(x —y)+2f' 'A, , (x) g'

over the

Z[J]=N

'

F 's

f da5(y 1

now gives an expression for

f &A(detK)

'~ exp i

f d'x f

y"d"' ]— , p&~, p&ri .

Z [J] in terms of the Lagrange multiplier fields

d4y

In the Abelian case, E is essentially an identity operator (exactly so for the Euclidean theory), the exponent in Eq. (37) is quadratic in the A, 's, and the remaining functional integration can be perforrned. The result for Z[J] will be shown in our paper on. the Hamiltonian formulation of the theory to give the usual propagators, though Z [J] itself appears in a somewhat disguised form. For non-Abelian theories, E depends on the A, 's, and the final integration cannot be performed explicitly. An analog of the functional integral for Z [J] in Eq. (31) was obtained some time ago by Halpern using the axial gauge (with proper reinterpretation of the inversion A [F] and the ){,'s in that gauge), and results equivalent to ours were obtained more It seems likely that similar recently by Itabashi. results can be obtained in any gauge in which a unique inversion A [F] can be defined. The coordi-

"~ ax)[y~d—

(36) A,

,

'[J —*(a~)]K-'[J—~(W)]

(37)

—,

nate gauge appears to be the most convenient as the inversion formula is simple, has a straightforward geometrical interpretation, and fixes the gauge completely for well-behaved fields.

D. Equations of motion As a first application of our formalism, we will derive the usual equations of motion for the classical Yang-Mills field from the action in Eq. (31),

S[J]= f d

x

—,

Tr

4

+ The Euler-Lagrange

2

A,

F„,F""+A„[F]JI' Jtlv

Dl'[F] ~Fp

.

(38)

equations for this system are


1376

dix for details. ] The variation of S with respect to the I' 's is somewhat more subtle because of the nonlocality of A [F]. Using the relation

obtained by requiring the action to be an extremum with respect to variations of the A, 's and I' 's. The variation with respect to the A, 's is straightforward and gives the Bianchi identities (the original constraints)

5S =Dl'[E] p ~Ep 5A, (x)

(x)=0.

5Ex (x')

"5~)5(x' =5, (5x5" 5— 5F„'„(x)

(39)

used

[F] as &„[E](x)= d'x'

A,

5F„„(x)

g~

F""(x) &"~ D— [E]& (x)+

f

f dax'"F„„(x')5(x' ~x), 1

(4l) we find that

f d f da5(x 1

y

(40)

and writing A

's subject to the constraint x~A, (x) =0, we would obtain only the three Bianchi identities in the restricted set in Eq. (17). The remaining identity follows automatically from the reconstruction theorem in Sec. II C; see the Appen-

[If we

x— )

0

"J (y)+x"[~F" (y), A(y)] ay)[x—

,

(p~v—)I =0. (42)

Integrating

over y and changing variables to

F""(x)=d'"~Dz[F] A, (x}

f

P=a, we obtain

the equation of motion

[x"J"(Px)+xI'[*F (Px), A, (Px)] (p~v) I — . dPP —

(43)

I

To obtain an equation of motion for I' which does not involve A, , we take a covariant divergence of this equation. The reconstruction theorem of Sec. II C may be used to show that d'"~ Dq[E]Dq[E]A, (x) = — [~F'~(x), A,

(x)],

(44)

and we find that

D~[F]FI'"(x)-= —[~F~~(x),A, (x)] D~[E]

f

d— l3P

The covariant derivative of the integral on the right-hand

D [E]x~(

~

)

(45)

We note that

~

~

(46a)

xl'A„[E]=0, and that

Dq[E]x'( )=5q( )+x'Dq[E] ( and we write the result

dPP

side can be calculated explicitly.

=a„x~( ) =(4+x~a„)()

because of the identity

—f,

[x"J"(13x)+x"[~F (Px), A, (Px}]—(p~v)I .

),

(46b)

of the differentiation

as

f

(3+x"d„)IJ'(Px)+[ F" (13x), (Px)]]+x"D„[F] A,

d13P

[J"(lax)+[*F"(Px), A,

(13x)]I

.

The first integral is just

f

—

d13

13

[J'(Px)+[~E" (Px), A, (13x)][=J"(x)+[*F"(x), A, (x)],

(47b)

and we find that Eq. (45) reduces to

Dz[E]F" (x)=J"(x)+x"D&[F]

f

dl3P

[J"(Px)+[*F"(Px), A~(Px)]] .

(48)

We show finally that the integral term in Eq. (48) vanishes. This is automatic by current conservation in the Abelian case. In the non-Abelian case, we calculate the covariant divergence of Eq. (48), use the covari-


26

ant current conservation

condition

D„[F]J'=0,the

relation

= ——,[ Fq„(x), F""(x) ] =0 D„[F]D~[F]F""(x) which follows from the results of Sec.

(4+x

II C,

1377

(49)

and Eq. (46a) to show that

"d„)D„[F] J dPP [J"(Px)+[*F"(Px), (Px)]] =0.

(50)

A,

The current density, field strengths, and Lagrange multiplier functions must be less singular than x 3, x and x ' for consistency of the field-strength formalism. As a consequence, the homogeneous operator (4+ x i)„)cannot annihilate the remaining expression, and we conclude that

Dq[F]

J

dPP (J"(Px)+[~F" (Px), A,

(Px)]]=0.

(5l) I

Thus, from Eq. (48), 1 2

D [F]F""(x)=J"(x)

(52a)

provided that

J"(x)=0, D„[F]

(52b)

and we have reproduced the usual equations of motion for the Yang-Mills field. The condition of covariant current conservation is necessary as a consistency condition, as in the usual theory.

IV. SUMMARY AND COMMENTS Our objective in this paper has been to show that the content of non-Abelian gauge theories can be expressed entirely in terms of the gauge field strengths in a compact and manageable way using fields in the coordinate gauge x "A„=O. The existence of the field-strength formulation for gauge theories was first demonstrated by Halpern. However, Halpern's use of the axial gauge led to rather complicated, unsymmetrical expressions, a defect which is removed in the coordinate gauge. (The advantages of the coordinate gauge in this respect were also noted by Itabashi, who obtained results essentially equivalent to ours. ) In this work, we showed both analytically and geometrically that it is possible to construct a unique potential A [F] for a coordinate-gauge field F less singular than x on the path (0~&) from knowledge of the field strengths alone. We showed, furthermore, that a tensor F&„(x)is a coordinate-gauge field tensor if and only if E satisfies a restricted set of Bianchi identities, x~Di'[F] *F&i =0. These results allowed us to eliminate the potentials in the coordinate-gauge functional integral for Z [J], and obtain an expression for Z [J] entirely in terms of the F 's, Eq. (30). It was useful at that stage to introduce a set of

Lagrange multiplier fields A, (x) to eliminate the Bianchi constraints. The resulting action led through Hamilton's principle to the correct equations of motion for the Yang-Mills field. Moreover, the action was quadratic in the E's, so the functional integral on the F 's could be performed explicitly. The result was a new expression for Z [J] as a functional integral over the Lagrangemultiplier fields A. (x), Eq. (37). This expression seems to have interesting properties for the strongcoupling limit of the non-Abelian theory. In the second paper in this series, we will rederive the field-strength formulation of gauge theories in a Hamiltonian approach, treating the theory from the beginning as the theory of a constrained Hamiltonian system. This discussion will clarify some questions left in abeyance in this paper, e.g., the constraints on the Hilbert space of the F 's and A, 's which results from constraints on the canonical variables. We will also show how the usual results for Green's functions are recovered in Abelian theories, discuss quantization in the fieldstrength formalism, and possible applications of our results to the strong-coupling problem.

ACKNOWLEDGMENTS We would like to thank Dr. Cosmas Zachos for useful conversations on this work This resea. rch was supported in part by the U. S. Department of Energy under Contract No. DE-AC02-76ER00881.

APPENDIX In this appendix, we give an alternative treatment of Z [J] and the equations of motion for F&„ using the restricted Bianchi identity [Eq. (3)] rather than the full Bianchi identity used in Secs. III C and III D. The restricted Bianchi identity requires that for any vector AF orthogonal to x",


1378

0— — , x "A,„=O.

iPD~[F] ~Fp

26

" in our

(Al)

earlier discussion, at the expense of an infinite change in normalization of the functional integral. If we do not make this simplification, the Lagrange multiplier fields Ai'(x) in Eq. (31) must be restricted, and A,

As shown in Sec. II C, the "fourth" Bianchi identity follows automatically from the form of F»(A [F]). We therefore ignored the restriction on l

Z[J]=N

'

=N

'

f ~F~~exp f d'x Tr F„,FI'"+g„[F]JI+}„Dl'[F]*F +&{xi)„„{x)} f —SF&A, &riexp f d"x Tr, F„,F~ +g„[F]J~+g~DqF]*F +»~g '

i

1

—,

4g2

i

PV

—,

JMV

(A2)

Here ri(x) is a new Lagrange multiplier field used to enforce the constraint. [Itabashis used the constraint on A, explicitly in his treatment of the field-strength functional integral and integrated only over the three fields &(x), with &'= x. &/x . We used this construction in our early work, but found it to be much more cumbersome than the present method. ] The %nations of motion for F A and ~ are ~sily obtained from the modified action in Eq. (A2) using the methods of Sec. III D. We find that

"

5S M. (x)

5S S~(x)

=D~[F]*F~+x~ri=0,

(A3}

=x "A,„(x)=0,

(A4)

iF""(x) d'~

»x

g

+

A, (x) Dp[F]—

f d'y f da5(x

"{y)+x~[*F"(y), A.(y)] —(p~v) j =0 ay)Ix"J—

(A5)

It is now straightforward to derive the usual equation of motion and Bianchi identity for I'. We first take the scalar product of Eq. (A3) with the three independent vectors orthogonal to x", and recover the restricted Bianchi identity, Eq. (Al}. The reconstruction theorem in Sec. II C then shows that Fz can be written as

=[D,[F]» [F]]

(A6)

hence satisfying the complete Bianchi identity,

.

Di'[F] 'F,

=0 .

(A7)

Equation (A3) thus reduces to the condition x&i}(x)=0 for all x", that is, g(x) =0. The remainder of the calculation is based on Eqs. (A5) and (A7) and follows the calculation in Sec. III D. The latter does not depend on any special properties of A,"(x), and so is independent of the extra constraint in Eq. (A4). We conclude that E satisfies the usual equations of motion

g

2

Dz[F]F" (x)=J"(x), D„[F]J(x)=0.

(A8)

k" is

determined by Eqs. (A4) and (AS). As a last point, we note that the functional integration over the field strengths I' necessary to obtain the expression for Z[J] in Eq. (37) is not affected by the extra factors in Eq. (A2). We therefore find that Z[J] can be expressed (using the restricted Bianchi identity) as

Z[J]=N

'

f

r

&A, &ri(detK)

'~ exp i

f d x f d y[

—,

[J —*(BA,)]E '[J —*(BA,)]+5 (x y)rix&A&I— (A9)


T. T. Wu

and C. N. Yang, Phys. Rev. D 12, 3843 (1975). For examples of field strengths for which the potential is not unique see, e.g., S. Deser and F. Wilczek, Phys. Lett. 65B, 391 (1976); M. Calvo, Phys. Rev. D 15, 1733 (1977); Xia Dao-Xing, Sci. Sin. 20, 145 (1977); M. B. Halpern, Nucl. Phys. 8139, 477 (1978). M. B. Halpern, Phys. Rev. D 19, 517 (1979). We were

unfortunately not aware of this paper until our own work was essentially complete. We would like to thank Dr. C. Zachos for pointing out this reference. K. Itabashi, Prog. Theor. Phys. 65, 1423 (1981). We would like to thank Dr. K. Seo for pointing out this reference. 4See, for example, L. D. Faddeev and A. A. Slavnov, Gauge Fields: Introduction to Quantum Theory (Benjamin, New York, 1980). 5C. Cronstrom, Phys. Lett. 908, 267 (1980). M. A. Shifman, Nucl. Phys. 8173, 13 (1980). L. Durand and E. Mendel, Proceedings of the 2nd Chilean Symposium on Theoretical Physics, 1980 (unpublished). 8M. Azam, Phys. Lett. 1018, 401 (1981). We will use the notation ~E~q for the dual of E,

origin, it cannot depend on the four-dimensional spherical angles (ratios of the components of x&), so must be a constant matrix, and B„=O.The conditions necessary for the uniqueness of A„[F]were not examined in earlier works (Refs. 5 and 6). The copy problem (Ref. 1) arises in the coordinate gauge precisely for potentials which contain components which are homogeneous of degree —1 We will discuss this problem elsewhere. L. Durand and E. Mendel, Phys. Lett. 858, 241 (1979). Equivalent generalizations of Stokes's theorem to the non-Abelian case have since been given by I. Ya Aref'eva, Phys. Lett 95B, 269 (1980); P. M. Fishbane, S. Gasiorowicz, and P. Kaus, Phys. Rev. D 24, 2324 (1981); M. Iyanaga, J. Math. Phys. 22, 2713 ~

(1981). 3S. Mandelstam,

Ann. Phys. (N. Y.) 19, 1 (1962); W. Ambrose and I. M. Singer, Trans. Am. Math. Soc. 75, 428 (1953). ~4In gauges other than the coordinate gauge, the parallel-transport factors do not vanish, and Eq. (13) is replaced by (Ref. 12) 1

A„{x)= 0 daax E „(ax) 1

' If we introduce four-dimensional spherical coordinates, the Euclidean version of Eq. (8) reads Br

U =— A„(r,gi)U

r =(x2)1/2

and

U(r, O;) =P exp

—

0

A, (s, g;)ds

The gauge condition x"A„=rA„(r, O; ) =0 requires simradial component. have no that ply A& ~~The general solution to Eq. (12) is 1

A„{x)= 0 daax E „(ax)+B„(x) with B„(x) homogeneous of degree {—1),

(1+x.a.)B„(x)=0.

E„„be

at less singular than x The condition that of the gauge form the origin then requires that U 'B„Ufor some U, while the gauge condition X~A„=Orequires that U be homogeneous of degree 0, X%&U=O. Finally, if U is to be well defined at the

B„be

1379

0

! k!(k+2)

X X

'

X

The right-hand sides of these expressions depend on A, so we do not obtain an explicit construction of A from E. The equality of the number of constraints in Eqs. (25) and (26) is not special to four dimensions. In N dimensions, the first-order formulation of the functional integral, Eq. (25), involves n constraints on the A' s from the gauge conditions x&A„'=0,and nN(N —1)/2 conditions relating the independent field strengths p p v, to the A's for a total of n (N — N +2)/2 constraints. Of the n (3 ) Bianchi identities satisfied by the E s in N dimensions, n (N — 2)/2 involve only directions orthogonal 1)(N — to x". If these restricted identities hold, the nN potentials can be expressed in terms of the field strengths by Eq. (13). Thus in the field-strength ap2)/2 proach, Eq. (26) involves nN +n (N —1)(N — =n(N — N+2)/2 constraints, the same number as in Eq. (25).

E„„,