Canonical formalism for gauge theories with application to monopole

Nuclear Physics B114 (1976) 61-99 © North-Holland Publishing Company

CANONICAL FORMALISM FOR GAUGE THEORIES WITH APPLICATION TO MONOPOLE SOLUTIONS * Norman H. CHRIST, Alan H. GUTH and Erick J. WEINBERG Columbia University, N e w York, iV. Y. 10027

Received 21 June 1976

We develop a canonical formalism, using the method of Dirac brackets, which may be applied to classical Yang-Mills theories with 't Hooft-Polyakov type magnetic monopole solutions. We argue that the axial gauge, W3 = 0, is superior to other common gauges. However, it is necessary to modify the conventional treatment of the axial gauge in order to avoid anomalous commutation relations among the transformations generated by the angular momentum. We discuss the implications for the quantum theory, including the appearance of fermion states in a field theory containing only Bose fields.

1. Introduction There has recently been considerable interest in strong field solutions to relativistic non-linear field equations [ 1]. An important class of such solutions is obtained from Yang-Mills theories containing scalar particles and exhibiting spontaneous symmetry breaking [ 2 - 4 ] . Among the novel properties of these solutions is the possibility of obtaining quantum states with half-integral angular momentum without introducing fermion fields [5]. In this paper we will develop a canonical formalism which can be applied to such non-perturbative Yang-Mills solutions and which provides an unambiguous procedure for dealing with such questions. Thus, particular emphasis will be placed on the conserved quantities in these theories and the symmetry transformations which they canonically generate. An understanding of these symmetry transformations and their effect on the minimum energy, static soliton solutions allows one to generate further time-dependent solutions possessing non-zero values of the corresponding conserved quantities. The time-dependent classical solutions obtained in this way are closely related to the spectrum of states in the quantized version of tile theory. An interesting example of this procedure, discussed in detail in sect. 4, stems from the conserved electromagnetic charge in the SU(2) gauge theory with isovector Higgs field. The transformation generated by this conserved charge does not leave the * This research was supported in part by the U.S. Energy Research and Development Administration. 61

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N.H. Christ et al. / Gauge theories

't Hooft-Polyakov monopole solution [3,4] invariant and in fact can be used directly to generate the time-dependent, electrically charged Julia-Zee generalization [6-8] of that static monopole solution *t. Of course, a given function of the canonical variables will generate a unique transformation on phase space in a Yang-Mills theory only if a particular gauge has been specified. The most physically familiar choice of gauge is the "unitary gauge" in which certain components of the Higgs scalar field o i are choosen to be zero [10]. However, such a choice is in general incompatible with the special boundary conditions obeyed by the Yang-Mills soliton solutions. For example in the SU(2) theory mentioned above, two of the components of the isovector field o i are set equal to zero, ai(r) = o(r)8i3, a configuration not related by continuous gauge transformation to the 't Hooft-Polyakov boundary conditions oi(r) ~ r i. Consequently this unitary gauge can be obtained only by a singular gauge transformation. One might attempt to avoid this problem by choosing conditions on the Higgs fields which vary with position in a manner compatible with the boundary conditions. In the 't Hooft-Polyakov example one might make the "radial gauge" requirement oi(r) = o(r)(ri/lrh ). Unfortunately such gauges are also singular for almost all field condigurations. For example the above radial gauge possesses a clear singularity at r = 0 unless N3=1[oi(r)] 2 vanishes for r = 0. Wishing to avoid the string singularities characteristic of these unitary gauges we might be tempted to adopt the ordinary Coulomb gauge, DiW~ = 0 ,

l~
(1.1)

where the W uk are the n Yang-Mills vector fields. However, as we will show in appendix A, not all Yang-Mills configurations can be transformed into Coulomb gauge: if the Julia-Zee dyon solution is transformed into Coulomb gauge (satisfying boundary conditions compatible with the usual variational principle) the resulting solution is a singular function of the time. Consequently, for the bulk of the paper we will work in axial gauge [11] W3k=0,

l<~k<,n.

(1.2)

As it is usually introduced, axial gauge, like Coulomb gauge, is "holonomic". The canonical variables which are conjugate to those eliminated by the gauge choice or vanishing because of the singularity of the Lagrangian, can be explicitly determined in terms of the other canonical variables using the equations of motion. The resulting reduced set of canonical variables then form a standard, unconstrained Hamiltonian system. Unfortunately, the long-range behavior of these Yang-Mills soliton solutions and the non-locality of the usual axial gauge are such that products of operations generated by the conserved angular momenta do not obey angular momentum corn'~ For a different treatment of the Julia -Zee dyon see ref. [9]. ~( Note added in proof'. P. Hassenfratz and D.A. Ross in a March 1976 Utrecht preprint have also given a canonical discussion of charge and angular momentum in this theory, using the unitary gauge (Nucl. Phys. B108 (1976) 462).

N.H. Christ et al. / Gauge theories

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mutation relations. In the usual axial gauge (axial gauge I), z-independent gauge transformations, not fixed by eq. (1.2), are determined by requiring limz__ ~ ~.(x, y, z) = 0. A consistent discussion of angular momenta can be given if, instead~ these z-independent gauge transformations are fixed by conditions imposed on a finite plane, z = z 0 *. The constraints in the resulting gauge (axial gauge lI) are nonholonomic and the corresponding constrained Hamiltonian system must be treated using the method of Dirac [13]. In fact we find Dirac's method simpler in many respects than the usual Hamiltonian treatment of gauge theories. An outline of this paper is as follows: In sect. 2 we review Dirac's constrained Hamiltonian formalism as it is applied to a general Yang-Mills theory. In sect. 3 we discuss tile question of gauge choice in greater detail, in particular the transformation to axial gauge. Also tile details of Dirac's method are worked out for the SU(2) gauge theory with isovector Higgs field in axial gauge. The conserved electromagnetic charge in this theory is examined in sect. 4. Tile canonical transformation generated by this charge is determined and is used to construct solutions with non-zero electric charge. Finally in sect. 5 we consider angular momentum for the SU(2) example, examining the pathologies of axial gauge I and demonstrating their absence in the second version. Sect. 6 contains concluding remarks, particularly about the quantum version of the SU(2) theory.

2. Canonical formalism We wish to transform the Yang-Mills theory with Lagrangian density ** . ~ = - g t~u[ F i u 12 , - 51[ ( O u o ) i ] 2 - V ( o ) ,

(2.1)

into a canonical, Hamiltonian formalism. Here i =Og W vi - - 3 v W ui + e t i / k W uj W vk , Guy

(2.2)

where wiu(r) is a vector field with group index i belonging to the adjoint representation of the gauge group ~ and ti/k, 1 <~ i, f k <~ n, the antisymmetric structure constants of the group. Similarly the scalar (Higgs) field o i belongs to an N-dimensional orthogonal representation R of ~ (perhaps reducible) with - . k l~ik) _uj , ( D u o ) i = 3uai + e W~

(2.3)

where the n antisymmetric matrices T k represent the group generators on R. The * The problem of defining rotations applicable to monopole solutions in Yang-Mills theories shares many features with the analogous question in the older magnetic theory of point-like, Dirac monopoles. In both cases "surface terms" are important in the definition of angular momentum and, because of the strings associated with monopoles, rotations involve explicit gauge transformations [ 12]. * * We use the metric ( - 1 , l, 1, 1) writing contravariant vectors with lowered indices.

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potential V(o) is a fourth-order polynomial in the components 0 i, invariant under the action of ~ . As is well known, the Yang-Mills Lagrangian is singular in the sense that the definition of canonical momenta and the equations of motion impose constraints on the Hamiltonian variables, wiu(r), oi(r) and their conjugate momenta

7r~i(r)-=

6L _=GOu, i,

fvi (r)

7ri(r)_

6L

_ [DOo(r)]i.

(2.4)

66i(r)

These constraints are

¢i+n (r) =- nio(r) = O,

(2.5a)

•

l =0

(2.5b)

The primary constraints, eq. (2.5a), follow the definition of rr~(r) in eq. (2.4), while the secondary constraints, eq. (2.5b), are required by eq. (2.5a) and the EulerLagrange equations: 0

- ~rD(r)

8L

awL(,)

- ~i(r)

(2.6)

We define the usual Poisson brackets computed without regard to any of the constraints, and denote them by { , }. Using these Poisson brackets, the constraints (J = - 6 L / 6 Wio generate local, time-independent gauge transformations and obey {¢i(r), ~J(r')) = e63(r - r') ti/kCk(r).

(2.7)

The most common approach to this singular Lagrangian adopts a gauge and then eliminates the variables W0 by solving tile Euler-Lagrange equations (2.6). Finally a reduced set of coordinates is introducec] (e.g. in Coulomb gauge, the transverse Fourier coefficients) which determine the variables W i in such a way that the gauge conditions are automatically obeyed. In terms of these new variables the Lagrangian is no longer singular and tile Hamiltonian formalism can be obtained without further diffichlty. However, in our case this standard procedure has a serious drawback. As we will see, the choice of boundary conditions which allows tile introduction of a set of variables automatically satisfying the constraints imposed by axial gauge produce strong dependence of the derived fields 7r~(r) and W~(r) on the remaining dynamical variables at distant points. Unfortunately this behavior is so non-local that the usual procedure for discussing angular momentum and rotations breaks down. The Jacobi identity fails, and the transformations generated by the conserved angular momenta do not form a rotation group when applied to solutions with magnetic charge. We would therefore like to consider other "non-holonomic" choices of boundary conditions which do not permit the introduction of coordinates automatically satisfying the constraints (2.5b).

N.H. Christ et al. / Gauge theories

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This can be done by using the general method for dealing with constrained Hamiltonian systems introduced by Dirac [13]. In fact we will find that this method of Dirac is considerably simpler in a number of respects than the usual procedure since it retains a close connection with the underlying gauge symmetry of the theory. Let us begin by recalling that Hamilton's equations for this constrained system can be obtained by minimizing the action subject to the "primary" constraints ~i(r) = O,

n
n 7rla'i~tils+Trioi--~--i~l[42~i"=

2n ~ ~ki~i] = 0 ,

(2.8)

i=n+l

where we have separated the term linear in W~, ~'i=1 wn vv0v', wi.d from the Hamiltonian density leaving the remainder ( r ) = l r r ii r rs { + ~' -t %i,2 ) + z~(Gk)2 + ~[(Dio)J]2+ V(cr) .

(2.0)

The ?t/(r), n < j <~ 2n, are Lagrange multipliers introduced to enforce the constraints (2.5a). For simplicity we will use the notation W~(r) = X/(r),

1 ~
(2.10)

The resulting Hamilton's equations are then eq. (2.5b) and 2n

~(r) = {~, H) + ~ f d3r' X/(r'){~(,'), ~/(r')},

(2.11)

]=1

where the constraints (2.5) have been used to eliminate terms of the form {~(r), Xi(r')}4fl(r'), 1 ~<] ~< n. The field ~(r) represents any one of tile 8n + 2N canonical fields and momenta. The Lagrange multipliers X/(r, t), n
~k = {0~/}} + ~ fdgr'xJ(r'){cj, cbk}=O, ]=1

n
(2.12)

The right-hand side of eq. (2.12) vanishes automatically bacause £r and OJ(r), 1 K f ~< 2n are independent of W~. Thus, as should be expected, the first-order Hamiltonian equations of motion (2.11) contain undetermined time-dependent functions XJ(r, t), n < ] K 2n, reflecting the possibility of including in the time development of the system an arbitrary gauge transformation. Thus we will now impose the additional gauge constraints

xk(~) = 0 ,

(2.13)

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66

where the index k may be composed of discrete and continuous variables. When combined with the equations of motion, eq. (2.13) requires 2,v/

=

o

+

=

¢/(r')?.

]--1 ~ Observing that eq. (2.11) implies Wd = -X/+n , we can rewrite eq. (2.14) as

(2.14)

n

+

o = {xk,

3,

/,

k

"/, k Wo(r)(x

/,

•

(2.15)

If the functions Xk(~) are properly chosen this equation will uniquely determine t) as a function of either the other canonical variables or the initial values W~(r, to), thereby eliminating all ambiguity in the time development of the system. Let us now restrict ourselves to gauge conditions xk(~) not involving W~ for which the matrix-coordinate space operator

W (r,

1 <~j<~n,

(X k, O/(r')},

(2.16)

is invertible, i.e. there exists a function Gi(r, k) obeying

Gi(r, k){x k, ~bJ(r')} = 8/]. 83(/" - r ' ) , k n

i=1

f da r {× k, ~i(r)} Gi(r, k') = 6 kk' .

(2.17)

With this choice of Xk, {X k, 7r~} = 0 and eq. (2.15) becomes simply a constraint uniquely determining W6. Thus, at this point bcth W~(r, t) and zrto(r, t) are fixed directly interms of the other canonical variables at the time t by the equations of constraint (2.5a) and (2.15); they are not determined by integrating Hamilton's equations (2.11) with respect to the time. We therefore choose to view W~ an..d 7r~ as dependent variables and reduce our phase space accordingly. As a result, ~ completely drops out of the problem while W~ enters the equations of motion (2.11)just as a Lagrange multiplier. The "secondary" constraints (2.5b) which were obtained as equations of motions by varying W~ are now viewed as constraints on phase space and the Hamiltonian equations of motion now follow by minimizing the action subject to these n constraints. If we also impose our gauge constraint (2.13) we find n

= {L 12t1 + ~ f d 3 r ' X/(r')(L (~/(r')) + ~ Xk{~, Xk), j=l

k

(2.18)

where the X/are our former variables W~ and both X/ and ~k are determined by the requirement ~" = ~k = 0. Of course the condition d / = 0 immediately requires ~k = 0 so that the last term has no effect in eq. (2.18).

N.H. Christ et al. / Gauge theories

67

The right-hand side of eq. (2.18) is an example of what is conventionally defined as a Dirac bracket. I r A is an arbitrary function of the canonical variables, then its Dirac brackets with the canonical fields can be viewed as an infinitesimal canonical transformation which leaves invariant the constraints X k and qg. The change 6~(r) i is given by in ~(r), one of the canonical fields W/, rr/,• 01,' or rro, 6 ~(r) : [~(r), A ] D ,

(2.19a)

/7

: {~(r),A) + ~fd3r'k]A(r'){~(r ),%](r')}+ ~ j=l "~ k

~kA {~(r), xk}.

(2.19b)

Clearly the Dirac bracket of A with a second function of the canonical variables B is /7

[B,A]D={B,A) + ~

j=l J

fd3r'X/A(r'){B,~/(r')} + ~ , k { B , xk } . k

A

(2.20)

To complete the definition (2.19) we must specify the quantities ~ and ~ . Just as in the case of transformations generated by H, our choice of ~J and X k insures that ~A and ~-A can be uniquely determined if we demand that these constraints are unchanged:

[Oi(r),A]D = [Xk, AID = 0 .

(2.21)

In fact, we will choose our gauge constraints X k so that

(2.22)

{xk, ×k'} = O, causing eq. (2.21) to reduce to

{~i(r),A) + ~ xkA {~i(r), X k} : O, k

(2.23a)

{X~ A} + ~

(2.23b)

fd3r'?tJA(r'){Xk, ~](r')} = 0 .

j=l ~,

These equations, (2.23), can be explicitly solved for Xfl(r) and xk using the Green function Gi(r, k) of eq. (2.17): t/

X~(r) = -- ~ GJ(r, k){x k, A ) . k

(2.24)

If these values of xJ(r) and XA k are substituted in the definition (2.20), one obtains

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68

the general result n

[B, A]D = {B, A) - ~ - {A, SJ(r)}

~ (d3r[{B, 0/(r))GJ(r, k i=l °

GJ(r, k){x k, B}I

•

k){?(k, A} (2.25)

Let us now examine in slightly greater depth the steps leading up to this definition of the Dirac bracket. In general, we need only require that tire "matrix" (X k, qS/(r)} be invertible in the sense that eqs. (2.23) have solutions. As a result, the inverse, G/(r, k) may not be uniquely determined even after the appropriate boundary conditions have been imposed on the variations (2.19a) in the canonical variables generated by Dirac brackets defined using G, as in eq. (2.25). We may nevertheless have a unique definition of Dirac brackets and satisfactory equations of motion ~/(r) = [~/(r), HID if the ambiguity in G does not affect the definition (2.25). We will find that this is precisely the case for axial gauge provided one considers Dirac brackets of quantities A and B which depend on tire field variables ~i(r) for r lying in a finite region. Dirac brackets with quantities written as integrals over all space of local densities can then be determined by first integrating over only a finite volume and at the end taking the limit of infinite volume. Finally it should be noted that if the Green function G/(r, k) is ambiguous then it may be possible to add a homogeneous term to the solutions XA, ~'A of eq. (2.24). We will always omit such a homogeneous term in order to retain the bilinear form of the definition (2,25). From the form of eq. (2.25) it is clear that [B, A]D is antisymmetric in A and B. It can also be shown that multiple Dirac brackets obey the Jacobi identity [13]. Tlms the infinitesimal transformation (2.19) generated by Dirac brackets with a quantity A is canonical in tile sense that it preserves the Dirac bracket through first order: [~i + ~ i , ~j + 6~j]D = [~i, ~j]D + O [ ( ~ ) 2] -

(2.26)

We conclude this section with tile very useful observation that if the quantities A and B are gauge-invariant functions of tile canonical variables, then their Dirac and Poisson brackets coincide:

[B, A]D

=

{B, A) .

(2.27)

This follows at once from eq. (2.25) if we recall that Poisson brackets with the quantities ~J(r) generate local gauge transformations and that the two final terms in eq. (2.25) contain the factors {B, 4fi(r)} or {~/'(r), A } which must therefore vanish.

3. Axial gauge If we are to use tire general formalism introduced in tile previous section to discuss the symmetries generated by classically conserved quantities it is necessary to

N.H. Christ et al. / Gauge theories

69

adopt appropriate gauge conditions. We are interested in discussing configurations of vector and scalar fields in a Yang-Mills theory which have finite energy and have gauge fields W~ falling at least as fast as i/r for large r * . In addition, the scalar fields are assumed to possess topologically non-trivial boundary values at infinity lim oi(r) = -d/(•), Irl ~

(3.1)

where ~ is the unit vector r~ Irl. We will assume that the asymptotic value of the ttiggs field 8i(t:), a minimum of the potential V(e), is annihilated by only one generator of the group ~ ,

ukff)

= 0,

(3.2)

n [uk(t~)]2 = 1. Thus tile underlying symmetry where u k ( r ) is normalized so that ~k=l group ~ is broken spontaneously down to U(1). Let us consider four reasonably familiar gauges:

(i)

d(r) L~ ~J(;o) = o,

1 .< k < n,

a~-[u k(~o) W.k(r)] = O,

where ro is a fixed direction; (ii)

cri(r) T t ~ ] ( ~ ) = O ,

(iii)

3 i W.k

(iv)

W3k = 0 ,

= 0,

1 <~k<~n,

3i[uk(r)Wlk(r)] = 0 ;

1 <~ k <~ n ;

1 ~
The first possibility is the usual unitarity gauge wher~ only Higgs fields ai(r) corresponding to physical massive scalar particles are retained and the gauge freedom associated with the single massless vector field is eliminated by the choice of Coulomb gauge [7,8]. However, the first of the conditions (i) requires that the Higgs field ai(r) approach the constant value ~i(~0) at infinity**. Since we began with a field configuration whose asymptotic form ~i(F) could not be continuously deformed to an angle-independent constant, such a gauge (i) can only be reached by singular transformation. As is well known, the fields in this singular gauge would include a massless gauge field with singularities identical, at least asymptotically, to those of * When we deal with axial gauge this requirement of I/r behavior of Wv(r) for large r will be relaxed. It will be imposed for all directions r except _+z,for which an asymptotic constant behavior, consistent with Hamilton's variational principle, will be allowed. ** The vanishing of (Dlo)i for large r implies that for such large r oi(r) is given by a gauge transformation of ~t(~o), and hence Z i {oi(r)} 2 = Z;~di(~o)}2 = v 2. It also implies that Ol[oi(~o)c/(r)] = -[~i(r0) TikiJ(r)]Wl k , which vanishes due to the gauge condition. Thus, [6i(F0)ol(r)] is c o n ] s t a n t for large r, and by considering r = rr 0 one sees that it equals v2. But this is possible only if oi(r) = ~i(~o).

70

N.H. Christ et aL/ Gauge theories

the electromagnetic vector potential in the presence of a Dirac magnetic monopole. A natural generalization of unitarity gauge, more appropriate in a situation with direction-dependent asymptotic values of the Higgs field, replaces the constant ~i(F0) in (i) by ~i0¢ ), varying with the direction r~. The resulting gauge (ii) is now compatible with the specific asymptotic behavior of oi(r). Nevertheless, as was noted in the introduction, imposition of the first condition in (ii) will introduce a string singularity if Ni]o il 2 does not vanish at the origin, at least in the SU(2) case with isovector oi(r). Wishing to avoid the use of such possibly ambiguous gauge transformations, we will not consider gauges (i) or (ii) further. The Coulomb gauge (iii) might appear at first sight to be a less controversial choice It possesses spherical symmetry and, for example, is a condition automatically obeyed by the 't H o o f t - P o l y a k o v monopole solution as it was originally written down. However, this gauge has one very serious defect: an arbitrary configuration of non-Abelian gauge fields can not necessarily be transformed into a solution of the condition (iii). In fact, as will be shown in appendix A, if the dyon solution of Julia-Zee is transformed into Coulomb gauge, there will be a certain time at which 14,'0 will become infinite throughout space; after this time there will be no natural continuation to the time evolution. We are left with (iv), the axial gauge. Although this gauge is not rotationally symmetric it has the great virtue that an arbitrary field configuration Wig(r, t) can be transformed into it by a gauge transformation G(r, t) obeying

~3G(r, t) = eG(r, t) W~ T i ,

(3.3)

an equation that can be formally integrated, determining G up to an overall z-independent transformation. Here G(r, t) is viewed as an element of the gauge group in a particular unitary representation while T i, 1 <~i <.N are the generators in that representation. Eq. (3.3) is solved by the ordered integral Z

G(r, t) = O {exp[+e / W~(x, y, z', t) ri dz'l} ,

(3.4)

2O

where, in an expansion of the right-hand side, the products of group generators are arranged so that factors associated with arguments z' closer to z are arranged farther to the right. The action of the gauge transformation (3.4) can be described quite succinctly. Consider for example the Higgs field oi(r). The gauge-transformed value of oi(r), G(r, t)i i oJ(r), can be described as the value obtained by "parallel transport" of oi(x, y, z) along the z-axis from z to z 0. That is, one considers a sequence of values oil(x, y, z'; z), z 0 <~z' <~z, satisfying

a[i(x, y, z ; z ) : oi(x, y, z) , [ 6ij a - 3 z ' + e W 3 ( x ' y ' z ' ) k T k ] ° ~ ( x ' y ' z ' ; z ) = O '

(3.5)

N.H. Christ et al. / Gauge theories

71

the function eli(x, y, z';z) is said to be obtained from oi(x, y, z) by parallel transport along the z-axis and

o~(x, y, Zo; z) = G(x, y, z)i/ o](x, y, z) .

(3.6)

Let us illustrate this procedure by examining the 't Hooft-Polyakov monopole in axial gauge. In its original form the vector and scalar fields for this example are written

,~ (!") = r H( O ,

Wi(r) = (~~ × r) F( r) ,

(3.7)

where we will use vector notation to represent only SU(2) indices. Since the SU(2) vector e 3 X r appearing in N 3 does not depend on z, the ordering in eq. (3.4) is irrelevant and we obtain the simple expression 2'

G(r) = exp[+e f

F(x/x 2 +y2 + z ~ ) dz'(~ 3 X r)" T] ,

(3.8)

where, since for large r

F(r) ~ 1/er 2

(3.9)

we have, for convenience, taken z 0 to minus infinity. Although the expression (3.8) depends on the function F0") which does not have a known analytic form, its effects can be understood quite well by referring to the asymptotic form (3.9). In fact, since in regions far from the origin the covariant derivative off(r),

D i ~= Oil+ el¢ i × ~,

(3.10)

vanishes exponentially fast, eq. (3.5) is obeyed by~ (r) itself and the parallel transport implied by the product Gq(r) oi(r) is trivial: if the line (x, y, z'), z' ~ z never passes near the monopole,

Gij(r ) oJ(x, y, z) = oJ(x, y, --oo) = _v6J 3 '

(3.1 1)

where v = H(°°). Thus for the 't Hooft-Polyakov solution in axial gauge a(r) is essentially the constant vector - v~3, except inside a tube extending from the location of the monopole up the z-axis to infinity having a diameter equal to that of the monopole (fig. 1). The vanishing of the covariant derivative (3,10) outside of this tube implies that the isovector I~i must be parallel to ~ and therefore just the vector potential of a magnetic monopole in axial gauge:

Wi(r) = 1 ^

eij3rj (l + z ) .

(3.12)

e e3 x 2 +y2 The string singularity of this vector potential (on the positive z-axis) lies in the tube

N.H. Christ et al. / Gauge theories

72

Z

I

Fig. 1. A schematic y-z cross section of the 't Hooft-Polyakov monopole in axial gauge. The cross-hatched disk represents a region of non-vanishing energy density while the vertical tube extending from the location of the monopole up the z-axis represents the region in which the Higg's field ~ is not simply a constant in the negative z-direction. Each arrow on the diameter D represents the SU(2) direction of the Higgs field ~ at the location of the base of the arrow. The entire configuration has axial symmetry about the z-axis.

in which the asymptotic form (3.12) is not valid. The exact gauge-transformed fields are without singularity. Of course, eqs. (3.11) and (3.12) can be obtained by explicitly performing the integral in eq. (3.8) using the asymptotic form for F(r), eq. (3.9), and then applying the resulting G(r) to the asymptotic values o f f ( r ) and ~¢i(r). Up to this point we have not completely specified axial gauge. The condition (iv), W~ = 0, 1 ~ i ~< n, is not changed by a gauge transformation G(r, t) which is independent of z. We will analyze two alternative procedures for fixing this remaining, unconstrained, z-independent gauge transformation. The first, axial gauge I, eliminates this gauge ambiguity by imposing the boundary conditions lim

Wi(x, y, z) = 0 ;

(3.13)

this is the conventional form of axial gauge. For the second, axial gauge II, we re-

N.H. Christet aL/ Gaugetheories

73

quire that the fields satisfy the conditions of unitarity gauge on the plane z = z 0 :

ai(x, y, zo) T k 8/(-z) i=1,2

= 0,

3i[uk(--z) W.k(x, y,

(3.14a)

Zo) ] = O,

(3.14b)

where z 0 will be taken large and negative. If we begin with a configuration of Y a n g Mills and Higgs fields with finite energy and for which the gauge fields fall off as 1/r at infinity, then the gauge transformation (3.4) with z 0 replaced by --oo changes our original configuration into axial gauge 1. We can then transform to the second version of axial gauge if we next carry out a gauge transformation G(x, y) obeying

OI(x, y, zo)Gli(x, y )

T/; a f ( - z ) = 0 ,

(3.15)

where al(x, y, z) is the Higgs field resulting from the initial transformation into axial gauge I. The existence of such a transformation is guaranteed for sufficiently negative z 0 if al(x, y, Zo) approaches ~l(_~) as z 0 -+ -oo, uniformly in x and y. Finally the condition (3.14b) requires a gauge transformation about the direction Uk(-Z,) through an angle e~'(x, y) solving 2

i=1

2

32"y(x, y) = ~ 3i [Uk(--:~)Wig(x, y, i=1

z0) ] ,

(3.16)

or

"/(x, y)= ~-~ f dx'dy '

(r - r)i [(x - x') 2 + ( y - y,)21

, , Uk(-Z)Wik (x,Y, Zo).

(3.17)

The convergence of the x' and y ' integrals in eq. (3.17) is assured by the asymptotic form, similar to that in eq. (3.12), possessed by Uk(--z,) wik(x, y, z) for any axial gauge I solution. Thus we can transform any solution of interest into axial gauge II. Finally, let us complete the discussion of axial gauge by determining for each version of axial gauge the Green function Gi(r, k) entering the Dirac bracket introduced in the previous section. First consider axial gauge I. For that case our gauge constraint can be written x J ( r ) = wJ3(r) = o ,

1
(3.18)

Recalling that the constraints ~i(r), eq. (2.5b), generate a local gauge transformation, we find

(Xf(]~,), i])i(~?)}__~ly ~

(~(i? _. I"),

(3.19)

N.H. Christ et aL / Gauge theories

74

so that the equations for Gi/(r, r'), (2.17), become a

Oz'

G iJ(r, r') = - ~T G"/ (r, r') = gJif 6'(r - r') ,

(3.20)

with solution

Gii(r, r') = -6(x - x')6(y

y')O(z - z').

(3.21)

In the above formulae we have replaced the generic index k labeling the gauge constraints Xk in sect. 2 by a position r and an adjoint representation index I ~
[Wki(rl,~((r')lD=Sij6XtS(r-r'),

l<~i~<2,

1~<]~<2,

(3.22a)

while

1 ~< i~< 2 .

(3.22b)

The particular solution (3.21) has been uniquely determined by the requirement that the transformation of the canonical fields, eq. (2.19a), generated by Dirac brackets obey the appropriate boundary conditions. Taken by itself, the differential equation (3.20) determines G up to an arbitrary added function of x, y, x', y'. However, if we require that the Dirac bracket of W/(r) with an arbitrary function A vanish as z -+ --0% then this undetermined term in G must be independent o f x and y. Furthermore, if Dirac brackets of rr~(r') withA are to vanish for z' -+ +co this additional term must vanish identically *. In the more conventional treatment of axial gauge, one views

W/(r), 7r{(r), okfr), rrk(r),

i = 1, 2 ,

(3.23)

as independent canonical variables and uses the constraint (2.5b) to solve for rr~(r) in terms of them: 2 rrk = f Z

d z ' [ ~ (0irr/k + eW[ tkl / n D -- eo l Tl)rr~]. i=1

(3.24)

The resulting Poisson brackets among the dependent variables, irk(r), and indepen* Some treatments of axial gauge use aGreen function with O(z - z') replaced by the more symmetrical ~ e(z - z'). However, it is not possible to choose boundary conditions which completely fix the gauge and yet are consistent with this Green function. Furthermore, this choice does not eliminate the difficulties in the treatment of rotations in axial gauge I.

N.H. Christ et al. / Gauge theories

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dent variables (3.23) are identical to the corresponding Dirac brackets discussed above, e.g. eq. (3.22). Thus the Dirac formalism is not required for a canonical treatment of axial gauge I. However, it is very helpful to recognize that the conventional Poisson bracket of two gauge-invariant quantities, defined using the independent variables (3.23), can be directly obtained by evaluating a corresponding Poisson bracket in which all 6n + 2 N canonical variables ~j(r) are treated as independent. As will be seen in our discussion of angular momentum, axial gauge I has seriously divergent behavior at infinity. As a portent of things to come let us observe that the variations generated by Dirac brackets in this gauge do not respect the finite-energy condition lim

Try(x, y, z) = 0 .

(3.25)

This can be deduced, for example, from eq. (3.22b) if tile right-hand side is viewed as the effect on 7r3(r' ) of tile transformation generated by wik(r). (Note that the condition (3.25) is not re,~uired for the variational derivation, eq. (2.8), of the equations of motion provided Wo(X, y, z) = X,it(x ,l y, z) approaches zero as z tends to minus infinity.) To avoid such serious non-locality we turn to axial gauge II. In order to simplify the derivation of the Green function appropriate to axial gauge II we restrict ourselves to the SU(2) theory with isovector Higgs field *. In addition to the gauge constraints

xi(r) = W~(r) = 0 ,

l ~< i ~< 3 ,

(3.26a)

we also impose X]+3(r) = o/(r)6(z - z0) = 0 ,

] = 1,2,

x6(r) = 3 i W.3 (r)6(z - z0) = 0 .

(3.26b) (3.26c)

The equation (2.17) obeyed by Gi/(r, r') (again the index k is replaced by r' and 1 ~< j ~< 6) now becomes 2

~z'

Gi/(r, r') + e ~ Gi'k+3(r, r')ekf3 o3(r')~(z ' - Zo) k=l 2 1=1

OrI

/ OrI

=6qS(r-r'),

. * For the triplet representation

1~
i

Tjk = - t i / k = -el~ k.

1<]<3,

(3.27a)

N.H. Christ et aL / Gauge theories

76

~-fGii(r, r') + ee i-3'l' 3o3(r) GO(r, r') 6(z - Zo)

_~i6 0 I ~0 63l +eWka(r)e3kl1 GlJ(r,r')6(Z-Zo) =6iJ6(r-r'),

1~
l~
(3.27b)

Here Gi/(r, r') is assumed to vanish for 4 ~< i ~< 6, 1 ~< / ~< 6 and eq. (3.27b) applies when 4 <~ ] ~< 6 only at the point z' = z 0. These equations are solved by

GiJ(r, r') = 6iJ 6(x - x')~(.): -X')[0(z' - z) - O(z' - z0) ]

- { 6ig6(x - x') 8(y - / ) ek, f - 3,3 × eo3(r')

_(L nI(x - x')'- + ( y -

y,)2])W~(r,)eu3k6i3}

,+Tr \Or'l

1

~-Trln[(x-x')2+(y-y')2]~i36j6.

(3.28)

The particular solution (3.28) is not unique. As is suggested by the presence of the logarithm, the defining equation (3.27) is still obeyed if we add to Gii(r, r') tile term

C6 i3 6j6 ,

(3.29)

where C is a constant, Our gauge condition requires no further boundary conditions (and Poisson brackets with Try(r) now vanish automatically at z = +_oo) so that tile arbitrary constant C represents an actual ambiguity in G~J(r, r'). However, it is not difficult to see that if the extra term (3.29) is added to the Green function in the definition (2.25) of the Dirac bracket, then that definition will not be changed. The added term is simply

fd3r{A, ~3(r)}Cfdx, dy, (~;w i3 (x,, y,, z0), 8) - [A - ~ 1 .

(3.30)

The integral over x' and y' will vanish automaticaUy if A and B depend on the canonical fields ~i(r) for r restricted to some finite region. I r A or B involves an integral of a local density over all space, then the Dirac bracket may be defined by first restricting the integration to a finite volume and then taking the infinite volume limit at the end of the calculation. This concludes our general discussion of axial gauge. We will now apply it to discuss specifically the operations generated by the conserved charge and angular momentum in the SU(2) theory.

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77

4. Charge Using the formalism developed in the preceeding sections, we now consider the electric charge and the transformations generated by it (which we will call charge rotations) in an SU(2) gauge theory with isovector Higgs field. In particular we find that neither the 't Hooft -Polyakov solution nor any other monopole solution is invariant under these transformations but instead belongs to a one-parameter family of static solutions. Furthermore, if a time-dependent charge rotation is applied to a static monopole solution, a family of charged solutions can be generated. We obtain an expression for the electric charge by noting that the charge density is the divergence of the electric field. Using the gauge-invariant definition of tile electromagnetic field [3], 1

Ft*v : I~--T"G~z'

e{o13 a - [ D ~ o × Dr,* ] ,

(4.1)

and writing all quantities in terms of canonical field variables, we obtain

~ "hi + el~l--1~ o'[DioXita]} 0 =fdrrai { ~g[

(4.2)

In order to determine the transformations generated by (2, we calculate the Dirac brackets of tile canonical variables with (2. Since (2 is gauge-invariant, tile last term in eq. (2.25) vanishes. The first term (i.e. the ordinary Poisson bracket) also vanishes, as is easily seen by considering (2 as a surface integral. Thus the entire Dirac bracket is given by tile second term:

[~/(r), Q]D = f

d 3r ,, {~/(r),

Oa(r")} X~7(r,,)

(4.3)

.

The result is just an infinitesimal gauge transformation with gauge function

x~(r) = -fdrr'aab(r, r'){Xb(/), - fd3r '' ~

[ ¢ d 3 r 'Gab(r, crrl., ,x

Q}

r'){xb(r'), F°i(r")}] ,

where for axial gauge lI the Green function lating the Poisson bracket we find

Gab(r, r') is given by

° b ( r I-(r")l ") X~o(r)=- fdBr"aarl , It3)~=1Gab(r, r ")~i3 1

5

+ - ~ Gab(r, r") [¢r(r") × Diu(r")]" eb- 3 6(z" - Zo) eb=4

(4.4) eq. (3.28). Calcu-

N.H. Christ et al. / Gauge theories

78

a Ga6(r, r ) o~3 ( r ) 6 ( z " - z 0 )1 . ,,

(4.5)

"

0 ri-' Viewing the integral over r" as a surface integral, we see that the first two terms vanish: in the first the Green function vanishes as z " ~ +_~, while in the second Di~ decreases rapidly at large transverse distances. Since - o 3 / I w I is equal to one on the plane z = z o, the last term gives simply

~aQ(r) = -fd3r"[V2Ga6(r, r")] 6(z" - z0) = --6 a3

.

(4.6)

Charge rotation in axial gauge II is thus the same as global isospin rotation about the third isospain axis. If we had used axial gauge I, tile last two terms in eq. (4.5) would have been absent. Furthermore, the Green function Gab(r, r") for 1 ~< a ~< 3, 1 ~< b ~< 3 would have been

Gab(r, r") = - 6 ( x - x") 6(y - y") O(z - z") ~ab .

(4.7)

It is easy to see that the change in the Green function (4.7) exactly compensates for the absence of the last two terms; both axial gauges give the same result. Having obtained this simple expression for charge rotation in axial gauge, we can easily see that the 't H o o f t - P o l y a k o v solution nor any other non-singular monopole configuration can be invariant under charge rotations. These solutions correspond to non-trivial mappings of ~(r) onto the sphere at r = ~ and so cannot be invariant under any global isospin rotation, no matter what its axis. Consider a static monopole solution to the equations of motion, N(0)(r), ~(0)(r), with vanishing canonical momenta, n} 0) = n(o0) = 0 (i.e. a static solution with W0 = 0). In general, we expect such a static solution to be a minimum of the Hamiltonian H, eq. (2.9), subject to the constraint (2.5b) and the requirement that tile Higgs field ~(0)(r) have a particular, topologically nbn-trivial behavior at infinity. Charge rotation of this electrically neutral solution will thus necessarily generate an entire degenerate family of solutions,

o(CO(r) = e- ic~r3/2 o(O)(r ) eiC~-3/2 , -- e-

W 0)(r)

(4.8)

where the symbols o and Wi represent the 2 X 2 hermitian matrices ~ - ~ and W/• x ; T1 , ~-2, r3 being the usual Pauli matrices. Clearly c~ + 21rn is equivalent to c~ for integral n. Related to this family of static solutions is a series of time-dependent, charged solutions whose time development is simply charge rotation. For example, it is not difficult to see that if ~ is replaced by cot in eq. (4.8), then for small w, o(c°t)(r) and W}t°t)(r) form a time-dependent, charged solution through first order in co.

N.H. Christ et al. / Gauge theories

79

Examining Hamilton's equations O"= [0, H I D ,

(4.9a)

~a = [no, t / ] D ,

(4.9b)

I~/ = Ill/, ./~rlD ,

Iti = [ni,/~/]D ,

we can view the first, eq. (4.9a), as determining the canonical momenta, which appear linearly on the right-hand side, in terms of (d/dt)W (wt) and (d/d0tr(wt) as they are given by eq. (4.8). Similarly both sides of eq. (4.9b) are of order co2 and hence equal through order co. Of course, if one wishes to find the canonical momenta gi and n o explicitly for this approximate @arge rotating solution, it is easiest to retain the Lagrange multiplier 3,~3(r) = W6(r ) in eq. (4.9a),

l~i=lti +eWoX Wi-- OiW0 ,

~=Tto+eWoX r~,

(4.10)

so that n i and n o are rewritten in terms of Wi, ~and W0. If0 can then be determined by solving the equation of constraint (2.5b), a second-order partial differential equation for W0 when written in terms of W0, Wi and ~. An exact @arge rotating monopole solution can be obtained if we minimize the Hamiltonian H subject to the constraint (2.5b) and the two requirements that the field" configuration have charge q and that the Higgs field have the same topological behavior at infinity as ~(°)(r),

6{fI+ fWo.iOi~ti +eWi×rti+eeo×n4)d3r+co(Q-q)}=O.

(4.11)

This implies that [~/(r), HID = --co[~/.(r); Q ] D ,

(4.12)

where ~](r) is one of the 6n + 2 N = 24 canonical field variables and co is a Lagrange multiplier. Thus a particular minima, ~i(r, co), obeying eq. (4.12) can be used as initial values for a charged, time-dependent solution

~i(r, t; co) = e-C°tQ°P'[~i]!t=~(r;w) ,

.(4,13)

where the exponential of the charge operator Qop. applied to a function f(~) of the field variables U(r) is defined by the series

e -°°tQ [f(~)] = f ( ~ )

_

cot[f(~), Q]D + 3( cot ) 2 [[f(~), Q]D, Q]]D + .... (4.14) 1

The entire time dependence of the solution (4.13) is simply charge rotation. (Likewise the value of Wo(r ) at time t can be obtained by isospin rotation of its initial value WO(r, co) through an angle wt about the third isospin axis.) As co approaches zero, ~j(r; co) necessarily approaches a member of the family of static solutions of Hamilton's equations with which we began. It is interesting to

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80

note that time reversal symmetry implies that the quantities ~](r; w ) corresponding to W i and ~ contain only even powers of c+ so that the solution (4.13) agrees through first order in co with the approximate solution W}cor),~(~ut) discussed earlier. Finally, let us show that the charged solution (4.13) agrees with that obtained by Julia and Zee +'. As it was introduced, the charged solution (4.13) was to be obtained from the minima ~](r, co). However, it could also be obtained by searching for a set of fields ~](r, co) so chosen that when charge rotated with frequency co, eq. (4.13), they form a time-dependent solution to Hamilton's equations. This somewhat awkwardly phrased problem can be much simplified if we perform a time-dependent gauge transformation on eq. (4.13), eliminating the factor e wtQ°P'. Such a timedependent, global gauge transformation naturally produces a non-vanishing field 14/0(r) at infinity. In fact, t

(..t)

^

Wo(r; co) = Wo(r; co) - ~ - e 3 ,

(4.15)

where the prime indicates a gauge-transformed quantity. Since leo(r, w) is required to vanish asymptotically only outside of a vertical tube above tile monopole, the aseymptotic behavior of W'o(r, co) may vary from - ( c o / e ) ~ 3 inside that tube. In fact, the asymptotic form of No(r, co) inside tile tube is determined completely in terms of g(r; co) by the requirement that the original, time-dependent solution have finite energy, In particular,

n o(r, co) =~'(r, 0, co) - e Ie0(r; co) ×~ (r, co) = [¢u~3 - e le0(r; cu)] × ~(r, ¢u), (4.16) must vanish for large r so that

Wo(r, co) ~ - ~ (e3 + o / { a l ) ,

(4.17)

where the component of W0 parallel to g, not determined by the asymptotic vanishing of (4.16), is fixed by requiring that rt i -- li]i + e W i × W 0 + Oileo fall faster than 1/r as r --> oo and that W0 vanish outside the tube, where ~ = - u2 3. Thus a charged solution can be obtained by seeking a static solution for which the asymptotic behaviors of W0 and ~ are proportional. For a monopole with unit strength this is most easily done by choosing a spherically symmetric gauge in which g, and therefore WO, are proportional to t¢ at least for large r. This is precisely the case for the Julia-Zee ansatz:

a(r) - r H(r) e r2 '

wa(r ) _ 1 ell~r! (K(r) - I) e r2 '

* This connection between charge rotation and the electrically charged solutions of Julia and Zee has been discussed previously in unitary gauge by Jackiw, (ref. [7]) and Tomboulis and Woo (ref. [8]).

N.tt. Christ et al./ Gauge theories

w~(r) = L,~ J(r) e

81

(4.18)

r2

For their solution J(r) has the asymptotic form

J(r) ~ M + b . 1"

(4.19)

r

The constant b can be readily recognized as the electric charge of the solution while M is simply the frequency co of the corresponding time-dependent form (4.13) for the solution.

5. Rotations

In this section we study the angular momentum Ji and the infinitesimal transformation which it generates. In a gauge theory, we expect that this transformation will be a combination of a naive rotation (i.e., the usual angular derivative and spin terms) and a gauge transformation. It is well known that a gauge transformation may be needed to maintain the gauge conditions. However, this requirement does not uniquely determine the gauge transformation (e.g. in axial gauge it is only determined up to a charge rotation). As we shall see, more subtle considerations come into play. The Dirac bracket formalism has the advantage of providing a straightforward and unambiguous procedure for determining the transformation. Of course it is possible to choose a particular gauge in which the form of the rotation transformation is quite easy to deduce. The best example is "radial gauge", defined by the conditions

°a(r) - ra/r, I~ (r) I

0i

W~.(r) = 0

(5.1)

which are satisfied by the 't H o o f t - P o l y a k o v ansatz. In this gauge the transformation generated by Ji includes an infinitesimal global isospin transformation T i so that as transformations in radial gauge:

Ji = Jinaive + Ti"

(5.2)

The form of eq, (5.2) ensures that the gauge condition (5.1) is maintained. If further charge rotations were added to the Ji of eq. (5.2), those transformations would no longer obey angular momentum commutation relations. However, as was discussed in sect. 3, this gauge will contain singular strings unless [a[ vanishes at r = 0 and unless the total magnetic charge is -+l/e. In the remainder of this section we restrict our attention to axial gauge. Beginning with the usual gauge-invariant definition of angular momentum in terms of the symmetric energy-momentum tensor, we compute the Dirac brackets of the fields with

N.H. Christ et al. / Gauge theories

82

Ji" Axial gauges I and II are found to give significantly different results for the gauge transformation to be associated with rotations about the z-axis. In particular, the axial gauge I result implies that the 't Hooft-Polyakov monopole is invariant under rotations about the x- andy-axes but not the z-axis. This violation of the rotation group algebra is traced to a failure of the Jacobi identity which should be satisfied by double Dirac brackets of Jx, Jy and a field variable. This failure results from the significant non-locality inherent in the superficially simpler axial gauge I. We define the angular momentum Ji using the symmetrical energy-momentum tensor Our in the usual way:

Jk = eijk / d 3 r r i Ooj -

(5.3)

ei/k/d3rri{nl.G/l+rto.D/~}.

The gauge invariance of our definition (5.3) implies that tile Dirac brackets o f J i with J/or Ji with Q coincide with the simple Poisson brackets. It is then easy to show that

[Ji, Jj] D = eijk Jk '

[Ji, Q] D = 0 .

(5.4)

It is convenient to rewrite eq. (5.3) as

Jk

= --eijk/d3r(rtl

" ri O/W l

+ Tt i • W j + n o •

r i Oj~r}

-- ei/k f d 3 r r i W j • {Olltl + e W 1 Xn l + e ~ X n o } + eij k f d 3 r ~lfrirtl. Wj ) .

(5.5)

We recognize the first term as the standard expression for the angular momentum obtained from a naive application of Noether's theorem. The second term vanishes because of the constraint (2.5b). The final term is a surface term which we might expect to vanish; in axial gauge, we shall see, it need not. We can easily calculate the value of the surface term, using the asymptotic forms

wa -+ 7rla

6a3ei]3 n r ]

e x 2 +y2

"-->

8 a3 q ri/r3

(1+ z )

'

(n = integer) (5.6)

which follow from the assumption that the only long-range gauge fields correspond to radial electric and magnetic fields. (These forms are not valid in the tube extending from the monopole; however, it is easy to see that the contribution from the tube to the surface integral vanishes as the surface is taken to infinity.) The integral is

N.H. Christ et al. / Gauge theories

83

most easily done by integrating over the surface of a cylinder with axis along the z-axis, first letting the radius become infinite and then taking the top and bottom to z = _+oo.The result is Jksurface = --47r(n)q 6k3 .

(5.7)

Thus, when both electric and magnetic charges are present the symmetric energymomentum tensor and a naive application of Neother's theorem lead to different definitions of J 3 in axial gauge. The former, gauge-invariant definition is certainly the correct one. We should stress that this ambiguity in definition is not directly related to the phenomenon of static electric and magnetic charges giving rise to angular momentum through their mutual interaction. In our formalism this contribution to Ji comes also from the first term in eq. (5.5). The transformations of the canonical variables under rotations are given by their Dirac brackets with Jk" Here we will calculate only [W~(r), Jk ]D; the others are similar. Because ark is gauge-invariant, the Dirac bracket takes the particularly simple form [W;(r), Jk]D = {W~(r), Jk} -- [6ac Ol + eeabc Wb(r)] X}k(r) '

(5.8)

?~ajk(r) : -- ~ b

(5.9)

where

d3r'Gab( r, r ' ) ( x b ( r ' ) , Jk) "

Since the Dirac bracket of a constraint with any quantity vanishes, we omit the second term of eq. (5.5) in doing our calculations. We then find

[WT(r), Jk] D : - ~/k Pi a/WT(~) + ~. WT(r)] - S%

+ e , ~ e W~(,)] ~x(,),

(5. l o) ^

where only the first and third terms o f J k are to be used in computing X}k. As expected, the transformation of W~(r) is composed of a naive rotation plus a gauge transformation. Using the Green functions obtained in sect. 3, we can calculate the X~ for either k axial gauge. Writing the contribution from the first and third terms o f J k separately, we have 3

~,~k(r)= fd3r" d3r'{ ~

b=l

GaV(r,r')e3/k Wl.b(r")6(3)(r'--r")

5 + b=4

G ab ( r ., r ) e, i / k 6 ( Z - z 0 ) 6 ( 3 ) ( r

' - r . .)r . . .iOj . . . o b -3(r")

84

N.H. Christ et al. / Gauge theories tt

tt

tt

3

tt

+ Ga6(r, r') ei/k 6(z 0 - z') 3;, 6(3)(r ' - r )[r i 3 / W n ( r ) + 6in 14:/3(r")]) 3

-

f d3r''d3r'{~ Gab(r' r')ei/k33~" [r"W/b(r")6(3)(r' i b=l

+ G a 6 ( r , r ) 'e i ] t 6 ( z ' - ZO)3n3n[6(3)(r . .-.r. ). r . '

- r")]

W l3( r ),,] } .

(5.11)

For axial gauge I, Gab(r, r') is given by eq. (3.21) if b = 1,2, 3 and vanishes if b = 4, 5, 6, while for axial gauge II the Green function is given by eq. (3.28). In accordance with the remarks made in sect. 3, the integration over r" should be done last; i.e., we first calculate the Dirac brackets with the angular momentum density, and then integrate the result over all space. We first consider the case k 4: 3. For axial gauge I, substitution of the form of the Green function gives simply r3

^

X~k(r) = --e3/k f

dz' Wja(rl , r2, z ' ) ,

k 4:3 (axial gauge I ) .

(5.12)

The result for axial gauge II is much more complicated; however, it simplifies considerably if z 0 -+ ---~. (Note that this limit can be taken only after having computed the Dirac brackets; otherwise one obtains the axial gauge I result.) We may write r3

X~k(r)----eBjk[f

dz' W;(rl, r2, z') + z0-depent t e r m s ] ,

20

k 4= 3 (axial gauge I I ) ,

(5.13)

where "z0-dependent terms" denotes terms which, because of the asymptotic behavior of the fields, vanish as z 0 -+ -- ~. We now turn to the case k ; 3. The invariance of the xb's under this rotation implies that the first integral in eq. (5.11) vanishes in either axial gauge. Because of the behavior of the Green functions and the fields at large Lz [, the first term of the second integral is also zero. Consequently, ),~t3 vanishes in axial gauge I, while in axial gauge II it is given by

Xas3(r)- ei/3 6a3f d 3r" 6(z" - Zo)3~ (r;'~.3(r")3; [InC(x- x") 2 + (y -/.)2)] } 47r (axial gauge II).

(5.14)

Substituting the asymptotic form of the gauge field, and integrating about a circle at X / ~ T ~ y 2 = ~o, we obtain ~a r - ~ a 3 [ ~n J3 ()~e]'

(axial gauge II)

(5.15)

N.H. Christ et al. / Gauge theories

85

Grouping these results together, we have

r3 XaJk(r) = -- e3/k f

dz' w/a(rl , r 2, z ' ) ,

(axial gauge I ) ,

(5.16a)

j~3 ~ajk(r) = -- e3].k

rig' Wl.a(rl , r 2 , z') + z0-dependent terms} ZO

+(~k3 (~a3( ne) '

(axialgauge II)

(5.16b)

The form of XJ1 and XJ2 are easily understood; they are exactly the gauge transformations required to restore the gauge and boundary conditions. On tile other hand, the ~'J3 which appears in axial gauge II is not needed to maintian any gauge conditions. To understand its significance, we recall that the commutator of infinitesimal rotations about two different axes should be a rotation about the third; e.g. 62(81~ ) -- (51((52~) = 63~.

(5.17)

From the calculations in appendix B, we see that the 't H o o f t - P o l y a k o v monopole is invariant under rotations about the x- or y-axes in either axial gauge, but that it is invariant under rotations about the z-axis only if the axial gauge II transformation (5.16b) is used. Thus the effect of the extra term in _?tr3 is to restore the proper rotation group algebra. It we look more carefully at the failure of the rotation group algebra in axial gauge I, we find that it cannot be traced to the Dirac brackets of the Ji with each other; the gauge invariance of the Ji guarantees that these brackets have the proper form, eq. (5.4). The failure lies instead in the arguments using the Jacobi identity to relate the transformation generated by the Dirac bracket of two quantities to the commutator of the two transformations generated by those quantities. To be more specific, consider the Jacobi identity involving Jl' J2 and any one of tile canonical variables:

[ [~(r), J1 ]D, J21D - [ [~(r), J2 ]D, J1] D = [~(r), [Jl, J2] D ]D

(5.18a)

= [~(r), J31D .

(5.18b)

Each term in the "identity" (5.18a) involves two integrations over infinite volume. In the presence of monopoles the long-range behavior of the fields is such that the left-hand side of eq. (5.18a) depends on the order of integration. The Jacobi identity will hold if each of the J's is regulated by integrating the density over a finite region Irl < R . The identity will then also hold in the limit R -~ 0% which corresponds to taking the two infinite volume limits simultaneously. However, the commutator of the transformations is obtained by taking the R ~ oo limit corresponding to the outer bracket first. It is shown in appendix C that when one performs the limits in

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this order (using axial gauge I), one finds a different value for tile left-hand side of eq. (5.18a) than one finds using the pre~ous order of limits. The difference is exactly equal to the gauge transformation ?~J3 which appears in axial gauge II. If one uses axial gauge II, ttre z0-plane gauge conditions regularize the integrals sufficiently well that the order of limits is irrelevant. If one couid write down a static field configuration which were not spherically symmetric, then the transformations derived here could be used to obtain time-dependent solutions with non-vanishing angular momentum. The procedure is analogous to that used in sect. 4 to obtain charged solutions. At the present time, however, no such field configurations are known. 6. Conclusion Let us now briefly examine the application of our methods to SU(2) theories containing additional scalar and spinor fields and discuss some of the properties of the corresponding quantum theories. We will assume that the additional fields do not produce any further symmetry breaking so that our theory retains a U(1) invariance. For simplicity of notation, we will continue to assume that h* I has a nonvanishing value in the ground state of the theory. The conserved electromagnetic charge is still given by eq. (4.2) and the conditions (3.26), defining axial gauge II, are still appropriate. On the other hand, the angular momenta, given by eq. (5.3), will receive additional contributions from the new fields that have been included. Also, the constrained quantities ~bl of eq. (2.5b) will contain additioiaal terms so that their Poisson brackets continue to generate local gauge transformations on all the fields in the theory. It is quite easy to see that the results of sect. 4 and 5 for the Dirac brackets of Q and Ji with the field variables apply also in this more general case. One need only recall that in addition to the naive transformation the Dirac bracket with Q or di contains a gauge transformation determined solely by the behavior of the original fields W i and a entering the gauge condition (3.26). Hence, even as they act on the additional fields, the Dirac bracket with Q continues to generate a global isospin rotation about the third axis while that with Ji generates a naive rotation plus the SU(2) gauge transformation (5.16b). More explicitly, if $j,i(mj, m 1, r) is one of the additional fields with spin J, z-component of spin rod, isotopic spin I and 3-component of isospin ml, then

[ ~ J , I ( m j , mI, r), Q]D -- i m I ~ j , I ( m j , mI, r) , [~bj, i(rn ], m I, r),di] D =(3 i ~d,i(rnj, m I, r) , + ~ ( J i ) m j , m'j ~J,I(m'J , mI, r) + e~k(r) ~ T k m) Ji mS mI, m?~j, 1(ms, mS,,)'

(6.1)

where Q~i = --CiikrjOk while the matrices (Ji)mj, m'j and Tkmi" m'I are the appropriate

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87

generators of spin and isospin. The gauge functions XSi(r) are those of eq. (5,16b). The Dirac bracket of tOj, I with J3 is particularly simple because ),k3 = (n/e)6 k3, where n/e is t h e t o t a l magnetic charge. A finite spatial rotation through 27r about the z-axis will change the field ~j, I by the phase factor ( - 1 ) 2J+2nI. The underlying rotational symmetry of the theory implies that a 27r rotation about any axis must produce the same phase change in ~j, 1" In a quantum mechanical treatment of this theory the quantities Q and Ji will become operators. An easy way to investigate their spectrum is to introduce collective coordinates parametrizing charge rotation or space rotation about a particular axis. For example, we might study the operator Q in the sector of magnetic charge 1/e by writing

~/(r) = eO°p'C~(~0)(r - R) + 8~j(r - R ) ) ,

(6.2)

where ~/(r) is one of the nine fields Wi(r ), ~(r) and ~}0)(r) is the 't H o o f t - P o l y a k o v solution (in axial guage lI) centered at the origin. The exponential in eq, (6.2) has the usual meaning as a transformation on the o n u m b e r fields ~0)(r) and 8~/(r). Here a and R are collective coordinates with conjugate momenta (2 and P, respectively, P being the total linear momentum of the system. The fluctuation term 6~/(r) is chosen to satisfy

~] f6~/(r)V~)O)(r)d3r = ~ f6~/(r)[~(O'(r), _

(6.3)

Similarly, the additional field if j, 1 would be written

6.1, I (m j, rnI, r) = e imla 6 6j, I (m j, m I, r -. R ) .

(6.4)

The field variables ~i and ~j, I are periodic in c~ with period 27r/e (or 47r/e if any one of the additional fields has half-integral/), Consequently when the system is quantized, the operator Q conjugate to a must have eigenvalues ke (or ~ke) for integral k - a familiar result. If, in a similar fashion, we expand about a classical solution that is not invariant under rotations about the ith axis (such as the example studied by Jackiw and Rebbi [5]), then we can introduce a collective coordinate conjugate to Ji" The fields ~/. and t)j, I would depend periodically on this coordinate with period 2rr (or 4rr if any one of the additional fields has half-integral J + t//). Therefore the quantum operator Ji has eigenvalues kh (or ~kh) for integral k. Thus by adding a single I = ~ scalar field we can generate a quantum theory containing particles with halfintegral spin, as was recently pointed out in this context by Jackiw and Rebbi and Hasenfratz and 't Hooft *. Since an I = 21 field implies particles of charge ½e, this anomalous half-integral angular momentum is just that associated classically with * Comparing the treatment of rotations given by these authors [51 and that presented here, we note that Hasenfratz and 't Hooft consider field configurations wittlq(r) parallel to r and de-

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such a charge in the field of a magnetic monopole of strength 1/e. If the classical monopole solution about which we wish to expand is spherically symmetric, so that a non-singular collective coordinate conjugate to Ji cannot be defined, we can determine the spectrum o f J i by the usual methods of quantum field theory. For example the fields 6~j and 6 ~j, I, representing fluctuations about the initial classical solution, can be expanded in basis functions belonging to irreducible representations of the rotation group, as generated by Dirac brackets with the Ji" These representations will be integral for 6 ~i, while for 6 ~j, I the representations will be integral or half-integral according to which J + n I is. Because of the spherical symmetry of the initial classical solution, the piece of the quantum Hamiltonian quadratic in the fluctuations 6~/and 6 ¢ will not couple the coefficients of basis functions belonging to different representations. Thus when this quadratic Hamiltonian is diagonalized, these coefficients will be combinations of operators creating or destroying particle states belonging to a particular representation of the rotation group. I f J + n I is half-integral for some field J/j, I, then the theory will contain fermion states even if only Bose fields (integral J) are present. Finally, let us summarize the main result of this paper. As we have seen, even for the SU(2) theory, there are a number of different gauges which might be used to discuss monopole solutions to Yang-Mills theories. The radial gauge, eq. (5.1), permits a simple discussion of rotations for the ' t H o o f t - P o l y a k o v monopole, but lacks general applicability. It also obscures the physical particle content of the theory sincethe direction of the symmetry breaking varies continuously from point to point in space. On the other hand, the unitary gauge, in which the single conserved generator uk(r) T k is the same for all points in space, is the simplest for analyzing the charges of and symmetries among the distinct physical particles in the theory. However, the string singularities which it contains necessarily complicate rotations. In fact, in either of the above gauges a general field configuration will contain string singularities. These strings emanate from an arbitrarily chosen origin in the case of radial gauge and from zeros of [g(r)12 in both gauges. The presence of these singular strings introduces a new aspect to the task of constructing a canonical formalism: in order to fix the gauge, a particular path of the singular strings must also be spcifled. As we have demonstrated in the preceding sections the axial gauge, W3 = 0, offers an attractive alternative to these two other possibilities. It is more general than the radial gauge, being directly applicable to field configurations with arbitrary magnetic duce that rotations have the form (5.2) for that case (essentially radial gauge, eq. (5.1)). Jackiw and Rebbi consider a canonical description of unitary gauge and express the Ji of eq. (5.3) in terms of the independent fields and conjugate momenta. They show that when applied to an isodoublet scalar field, U(r), Ji generates a naive rotation plus a gauge transformation of the second kind. In particular, if the vector field A i -~, 1 ~
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charge and, after certain delicate points have been mastered, it provides an unambiguous formalism for discussing rotations. In addition, axial gauge has much of the physical simplicity of the unitary gauge, with tile Higgs field o pointing in an essentially constant direction in all of space outside tile vertical tubes characteristic of axial gauge. The string singularities of unitary gauge are thus replaced by well-defined, extended tubes in which o varies from its vacuum value; tile orientation of these tubes is completely determined by the initial choice of gauge and boundary conditions. We conclude that axial gauge is a promising choice for further study of the classical and quantum properties of the Yang-Mills soliton solutions.

Appendix A Singularities o f Coulomb gauge

The purpose of this appendix is to show explicitly that there exists at least one configuration of fields which cannot be transformed into Coulomb gauge without producing severe singularities. Specifically, we will show that if the dyon solution of Julia and Zee is transformed into Coulomb gauge, there will be a certain time at which the field W0 will become infinite throughout space. The field configuration will have no natural continuation beyond this critical time. The Julia-Zee solution can be written as follows:

1

6 = ;H(r)/er,

W 0 = ~J(r)/e r ,

(A. 1)

where tile functions K(r), J(r) and H(r) are chosen to give a static solution to tile field equations. The configuration (A. 1) can easily be seen to satisfy the Coulomb gauge condition 3 i Wi = 0 .

(1.2)

However, in the canonical Coulomb gauge formalism, W0 is expressed in terms of the other fields using the equations of motion and the boundary condition lira W0 = 0 .

(A.3)

The Julia-Zee solution has lira W0 -- ~M,

(A.4)

where M is a constant related to the electric charge of the dyon. Thus, one must make a gauge transformation which achieves the condition (1.3) while maintaining the condition (1.2).

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For large r, the required gauge transformation has the form lim ~2(r, t) = eiMr'~t/2 . The spherical symmetry evident in eqs. ( A . I ) suggests that the complete expression has the form

~(r, t) = e iO(r' t)r.T/2 ,

(A.5)

lira O(r, t) = Mt .

(A.6)

where r - - ~ ~o

The above gauge transformation yields a field 1¢~ given by

Wi = cosOW i + (1 - c o s 0 ) ( r " Wi)r + sinOW i X +el ~3i0 + s i n O 3 i r - ( 1 -

c o s 0 ) r X 3 i ~]

Then, inserting eq. (A.1), =r~ {r2V20-2K(r)

3iW i' er 2

sin0}.

Thus, the requirement that 3 i N; = 0 implies that

3---[

3-37~=2K(r)sinO .

(A.7)

The function K(r) behaves for small r as 1 - (const.) r 2, and for large r it decays exponentially to zero. The qualitative behavior of the solutions to the above equation is best analyzed by defining o - lnr, so

c320

O0 -

302

3o

÷ 2 K ( o ) sin0 .

(A.8)

In this form, the equation is an exact analogue for a viscously damped circular pendulum moving in a time-varying gravitational field g = 2 K(o); 0 is the angle measured from the top of the circle, and o is the t i m e ; g is equal to 2 at early times, and then falls to zero. As r-+ 0 (o -+ --oo), the regular solutions to eq. (A.8) approach 0 = 0. To understand this limit one can approximate sin0 ~ 0 and K(o) ~ 1, in which case eq. (A.8) is satisfied by 0 = ct e ° = c~r. (There are also irregular solutions, which we will ignore, which behave like e o = 1/r as r-+ O. These solutions can be seen by neglecting the last term of eq. (A.8).) The regular solutions O,(r) form a one-parameter class, char-

N.11. Christ et aL / Gauge theories

91

f

D

B A

-5

0

5 Cr

Fig. 2.0c~(O) is shown for c~ = 0.25, 0.5, 2.5, and 30, for curves A through D, respectively. The curves were obtained by numerically solving eq. (A.8). We used r/sinhr for K(r), a form which is valid in the limit Mo/M W . 0 [ 14]. (The scale of length has been set by defining M W ~ 1.) "Gravity" essentially turns off during the interval - 1 < o < 2. Note that curve C has an asymptotic value Of which is greater than ~.

acterized by c~. By writing cte a = e a+ln~, it can be seen that for early times these solutions are related to each other by o-translations. It is n o w possible to qualitatively sketch the family of solutions. For very small c~, the analogue p e n d u l u m starts to fall at a very late time o, when gravity has already started to turn off. The p e n d u l u m falls a very short distance and coasts to a stop at some small value 0f(c0, as shown on fig. 2, curve A. As c~ increases, so does 0f(c 0, as shown on curve B. F o r very large c~, on the other hand, the viscous damping will bring the p e n d u l u m nearly to a stop at 0 ~ 7r before gravity turns off. By linearizing eq. (A.8) about 0 = 7r one discovers that f o r g > ~ it is an u n d e r d a m p e d oscillator. Thus, the p e n d u l u m will oscillate about 0 = 7r a finite n u m b e r of times before gravity turns o f f and it coasts to a stop, as shown on curve D. The number of times the pendulum crosses 0 = n will increase with ct, and clearly 0 f ( a ) will be greater than when the n u m b e r o f crossings is odd *. As c~ ~ 0% it is clear that 0f(a) approaches 7r. The i m p o r t a n t point is that 0f(c 0 is a continuous f u n c t i o n of a which reaches its m a x i m u m at some finite value of a, which we will call a 0. The f u n c t i o n O(r, t) which we seek can now be written as

o(r, t) = o~(t)(~) , where c~(t) is determined from eq. (A.6), which implies 0f(c~(t)) = M t .

(a.9)

* One could worry that the n u m b e r of crossings might s o m e h o w always be even. This, however,

is impossible. The curves are continuous in c~, so one must imagine a continuous sequence of curves with no crossings, two crossings, etc. On the borderline of the set of curves with no crossings and the set of curves with two crossings must be a curve for which O(r) has a single maximum point O(ro) = ~. This curve corresponds to a pendulum which changes direction at the extreme bottom of its arc, and such motion is obviously inconsistent with the differential equation.

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This equation can be solved for a(t) only until a(t) = a O. For later times, no solution exists. For c~< a0, note that

O0(r, t) _ M aO ~ ( r ) / d Of (c0 at -~d-/ Td " Thus, as a approaches a 0 the denominator will tend to zero, and O0/Ot will become t infinite. It follows that W0 will become infinite as the critical time is approached. At this point the reader may be thinking that the problems could perhaps be bypassed by avoiding the spherically symmetric ansatz of eq. (A.5). However, if nonspherically symmetric solutions for ~(r, t) could also be found, the difficulties of Coulomb gauge would be exacerbated. Given the same initial conditions, the function we have just found would still satisfy tile equations of motion for all times earlier than the critical time. The existence of other solutions would mean that in addition to this problem, the equations of motion would be ambiguous. In fact the ambiguity would have to be very great, because any non-spherically symmetric solution could be rotated into a family of solutions.

Appendix B

Spherical s y m m e t r y o f the 't H o o f t - P o l y a k o v monopole As originally written, the 't Hooft--Polyakov solution is clearly spherically symmetric in the sense that the effect of a naive rotation can be cancelled by a gauge transformation. It follows that, in a gauge with an unambiguous rotation group Jacobi identity, the solution is invariant under the transformations generated by the angular momenta *. In this appendix we will show explicitly how this invariance is manifested in axial gauge II. We adopt a matrix notation for the fields: W i = raW?, o = 7 a o a , where the za are the usual Pauli matrices. Tildes will indicate axial gauge fields; untilded symbols indicate the 't Hooft -Polyakov gauge. These are related by

1~, = GWuG

1 _ 2i (buG)G_ 1 e

(B.1)

"d=Go G 1

* The proof begins by noting that [~(r), Ji]D must correspond to an infinitesimal gauge transformation which is consistent with the gauge conditions, and this means that it can be written as ci[~(r), Q]D for some constants e i. One then uses the Jacobi identity and the Dirac brackets of the Ji's with themselves and with Q to show that c i = O.

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where G = e- i M ~ / 2 , rar b M = e3ab

x f ~ +y2

t~ = f

dz'v~

+ y2 F ( v / x 2 + y2 + z'2) + tan- l ~

(B.2)

2 + y2

z0 ZO

(Recall that F(r), defined by eq. (3.7), has the asymptotic form - l / r 2 . ) The symmetry of the 't Hooft-Polyakov ansatz is expressed by 0 = ~ l W k + ejklW/ + ~i[W k, 7/] ,

(B.3)

0 = ~ 1 o + li[o, rl] ,

(B.4)

where we have defined % = - em r / a k .

(B.5)

In sect. 5 we found that the axial gauge II fields transform under rotations about tile/-axis according to 6 l W i = CDI'~Vi - eti/W/ -- 3iX l + lie[ffti, Xl] ,

(B.6a)

61~ = CT)l~ + ~ie[~, Xl] ,

(B.6b)

where Z

~,(r) = - e3f, /

,R._

dz' l~/(x, y, z') + ~'t3e 7.3.

(B.7)

(Since our purposes are illustrative, we have set z 0 = _oo.) We will show that the vanishing of 618 is a consequence of eq. (B.4); the vanishing of ~l~]i is obtained by a similar argument. Setting k = 3 in eq. (B.3), we obtain Wi = (rio 3 - zOi)W 3 - lieif3[7.], W3] ,

(i@ 3).

(B.8)

Recalling that 03G = - ½i03(M~b)e - i M 6 / 2 = - ~ieW3G ,

we find ~/i = G Wi G - 1 - 2 i (Oi G) G -I=-½iei/3 G[r/, W3]G -1 + riG(O 3 W3)G -1 e

(B.9)

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94

_ zG(3iW3)G_ 1

2i (3iG)G_ 1

-- ~-

=33{-leij3Gr/G-l+2~-ie

ri(33G)G-l-~-z(~iG)G-1},

(B.IO) (i:/=3),

(Note that the quantity in brackets vanishes as z -+ -oo). Next, we obtain by a straightforward computation

r3 = Gr 3 G-1 _ 2i(CO3G)G- t .

(B.1 1)

We may therefore write

~ I = I GrlG_I - - e( -2 i c-DIG)G_I .

(B.12)

Final/y,

c1)l~ = (Q) I G) oG -1 + G O( @l G-l) + G( CDlO)G -1 = [(OIG)G -1, o] - ½iG[o, rl]G -1= -~ie[~, XII .

(B.13)

Combining (B.6b) and (B.13), we see that 8l~ vanishes. Appendix C

The rotation group Jacobi identity The rotation group generators Ji have Dirac brackets of the usual form: [Ji,Jj] D = eiikJ k, Thus, one would expect to have a Jacobi identity,

eij k [[~(r), J]]D, Jk]D = [~(r), Ji]D.,

(C.1)

for any canonical variable ~(r), which would guarantee that the transformations generated by the Ji have the structure of the rotation group. In this appendix we will show that the Dirac brackets in axial gauge I are so non-local that the left-hand side of eq. (C.1) is ambiguous; as a result, the transformations do not have the proper algebra. Furthermore, we will show that there is no ambiguity in axial gauge II. As has been described in the text, the ambiguity in the Jacobi identity is associated with the infinite volume integration which occurs in the definition ofJi, eq. (5.3). To discuss the infinite volume limit, we first define the regulated quantity

jR _=f d 3r a R (r) ~ i(r) , where 9 i ( r ) = eijk 9 Ook(r) '

(C.2)

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95

and ~2R (r) is a regulating function. It is found that a sharp regulator will lead to ambiguities, so we smooth our regulator with a fixed length parameter 6, which may be arbitrarily small. Thus, 1,

ifr
O,

i f r > R +6 ,

R(r) =

(c.3)

with a smooth continuation for R < r < R + 6. A Dirac bracket with Ji is defined by taking the Dirac bracket with j R and then taking the limit R ~ oo. The Dirac brackets of the j R with each other can be obtained by first computing the brackets of the densities. (As Ook(r) is gauge invariant, one simply computes the Poisson brackets.) The result is [~i(r),

~j(/")]D= eijk ~k(r)53( r ' -

r)

+ elk l r'k ~ [ ~j (r) 63 (r' - r)] -- ejk lrk ~I [ ~i (r) 63 (r' - t)] .

(C.4)

(One uses the constraint eq. (2.5b) in obtaining the above result.) Then [:'

:

]D

= e i / k : d3r ~2 (r) ~ k ( r ) .

(C.5)

For a fixed value of R, the formalism guarantees that the Jacobi identity of the form ofeq. (C.1) will'hold. Since it holds for allR, it will also hold in the limit R -+ ,,o, so we can define the quantity

c(i l)[~(r)] =-R~lim eijk[[~(r),J Rlj ID, jR,kID = lira

[~(r),fd3r'~22R(r ') ffi(r')] D = [~(r), Ji]D.

(C.6)

However, if one is interested in the commutator of successive transformations, then one must calculate the double Dirac bracket in which the limit o f R -~ ~ is taken first for the outer bracket, and then for the inner bracket. That is, one needs

c:

lira

lira

(C.V)

One could imagine calculating C: 2) directly from the above definition, but such a calculation would be very tedious. We will instead directly calculate the difference

AC i : C[ 2~ - C: 1~ .

(C.8)

This calculation is not only shorter, but it shows directly that the value of this double Dirac bracket depends upon the manner in which the limits are taken.

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Subtracting eq. (C.6) from eq. (C.7), we obtain

AC i = lira

lira eijk[~(r),Jff]O, ( J ~ ' - Jff)]o

= lim eijk[[~(r),J]RlD, JkRID,

(C.91

where

jR - fd 3 r [1 -

aR(r)] ~ k(r).

(C.10)

To evaluate AC i we recall the meaning of the Dirac brackets:

[~(r), J/R] D--(~(r), ]ZR)+fd3r'(~(r),i~a(r')}~a(jf;r'),

(C.ll)

where xa(j/R;r)= -

fd3/aab(r,, r ) { x b,(r),Jj R }.

(C.12)

Then, adopting the convention that all terms are to be evaluated in the limit R -~ ~o, one can write

AC i = eij k [(~(r), J ]J, R2 jR1 kiD

+ ei]kfd3r,[(~(r), ~a,(r )), JXK] O xa(J]R;r')

+fag/{~(r), ~(r')} A~(r'),

(C.13)

+f d3r,,{Gab(r, r,)(Xl~(r,),j]R), ce (r" )) Zc(Jk~ ;r" )1 .

(C.14)

where

The first two terms of eq. (C.13) will vanish in the limit R -~ ~o. For the first, note that :[~(r), j R } will depend only on the fields at the point r, and the Dirac bracket of Jff with a field at a fixed point will clearly vanish in the limit. For the second terms, one makes the same argument for [(~(r), ¢fl(r')},JRk]D , and one notes that the factor k a ( j f ; r ') remains finite in the limit. Thus, only the last term, which is a gauge transformation, remains. The above equations are true for any gauge; we will use them later to discuss axial gauge II. We now restrict our attention to axial gauge I, with its Green function 1

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97

given by eq. (3.21). Eq. (C.14) then becomes Z

f

--oo

+.far ffw3(r),j.R}, ~b(r,,))Xb(j~;r,,)], 3

17

tt

(C.lS)

where x' =- x, y' = y and Z

x17(4R-., ' ) = f

a t dz P {W3(r),J ~/ ~ 3.

(c.16)

--oo

It is clear that the contributions to Aa(r) from finite values o f t ' will vanish in the limit R -+ ~, so we have only to consider contributions from r' ~ R. At large distances one has a radial magnetic field corresponding to magnetic charge n/e. Outside of the vertical tube,

G~.(r)

, (n/e)6a3 eij k rk/r3 ,

(C.17)

g ~oo

arid wia(r) is also proportional to 6 a3. Using eq. (5.3) for the angular momentum density, one has

{w~(~), j.R} = _ ~m ~k a~ (r) a R(~),

(C. 18)

and then

{G~. (r), cb (r')) = eabe G~. (r) 6 3 (r' - r) ,

(C. 19)

(G~ (r), Jff } = - ektm { 6aeai + e eabc Wib (r) } (r IGc/(r) [1 - g2R (r)] } - (i ++/) n ~173~

r ~ 2 -e-~U tOkj Oi - 6ki 3/]~2(r) .

(C.20)

One now has all the information necessary to evaluate A~(r). Note that the second term in eq. (C.15) will vanish for large r', as the double Poisson bracket is proportional to eab3, and ~b ~ 6b3. The first term becomes Aafr ) = _ 2n_ f 3 z z,r'i dfZRfr' ) e f dz' f2R(r' ) i 2 dr' If we had used a 0-function for a regulator, the above equation would have contained the ambiguous factor O(R - r)6(R - r). As written it can be evaluated unambiguously to give

AT(r)--" e 6;3 5173-

(c.21)

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Note that the above term is exactly the gauge transformation associated with J3 in axial gauge II. The above calculation shows that this additional gauge transformation is precisely what is needed to restore the rotation group commutation relations for the transformations generated by Ji in axial guage I. The rotation group is automatically satisfied by the transformations generated in axial gauge II, because there is no ambiguity in the Jacobi identity. To see this, one returns to eq. (C.14). The constraints xb(r) are listed in eq. (3.26), and the Green function is given by eq. (3.28). From the Green function one can see that there is no contribution when the index b = 1,2, or 3, as r p is not allowed to become arbitrarily large. For b = 4 or 5, the terms involve the Poisson bracket (oa(r,) j.R) :

, 1o) a ~2(r ,) -- e/k I rk(D

and Dlo(r' ), corresponding to a massive field, can be assumed to fall off very quickly at large distances. The same can be said for the double Poisson brackets in eq. (C.14). Thus, the only contribution which requires detailed analysis is the term b = 6, with x6(r) =

OiW3(r)6(z - Zo) ,

Ga6 (r, r') = - ~ - ~ 5a31n[(x' - x) 2 + ( y ' - y ) 2 ] . The second term in eq. (C.14) will not contribute for the same reason as before, and the calculation of the first term is very similar. The result is

A~.(r) = - ~ ~a3 f d 3 r ' 6(z' - z 0) ln[(x' - x) 2 + (y' - y)a] r

r

+~)kr r ~ " f ~ R ( r ' ) ~ l " As R -> oo the leading piece of the above expression would grow as lnR, except that it vanishes by symmetric integration about a large circle in the z 0 plane. Thus the expression vanishes as R -->0% and there is no ambiguity in the Jacobi identity in axial gauge II.

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