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The problem of gauge fixing for spherically symmetric Yang-Mills fields
ANNALS
OF PHYSICS
215, 63-80 (1992)
The Problem of Gauge Fixing for Spherically Symmetric Yang-Mills Fields D. SCH~TTE
Received June 7, 1991
The Yang-Mills theory restricted by spherical symmetry is investigated in view of the problem of gauge fixing. We show that-in contrast to the Coulomb gauge-an extended Schwinger gauge provides a completely consistent representation of gauge orbits with trivial winding number without any Gribov singularities, The structure of the classical gauge fixed theory is revealed showing a discrepancy to the standard expectation. The temporal gauge quantization with extended Schwinger gauge fixing in the sense of Christ and Lee is shown to display the “e-parameter” symmetry in a manifest way. lr 1992 Academx Press, Inc.
I. INTRODUCTION
AND OVERVIEW
As an alternative to the Feynman path integral method, the Hamiltonian formulation of QCD is becoming an important tool for obtaining insight into the structure of field theories of that kind [ 1 ]. It is usually considered to be necessary for the construction of such a formulation that the classical theory--whose quantization yields the Hamiltonian of interest-is canonicui. For a gauge field theory, such a canonical structure is expected to hold only for the gauge orbits. The theory has therefore to be formulated with respect to these orbits, the representatives of which are usualiy characterized by some gauge fixing condition (like the Coulomb, the axial, or the Schwinger gauge). For a non-abelian gauge field theory, however, a non-trivial problem arises: Does the gauge fixing fullil the condition that for any gauge held the existence and the uniqueness of the representative on the gauge orbit is guaranteed? Here, the existence has to be guaranteed on the gauge orbits generated by gauge transformations with tr~vi~i ~~~~ing n~~~er on/-v. This is necessary in order to incorporate into the gauge fixed formulation the “topological symmetry” related to the O-parameter; i.e., only then topologically non-trivial gauge transformations will appear as residual symmetry of the gauge fixed theory allowing a standard quantum mechanical treatment. Very little is known in relation to a solution of this problem: E.g., for the Coulomb gauge, both the uniqueness [2] and the existence fail [6] (see also Section 2.4). For the axial or the Schwinger gauge, the existence fails [3, 41; 63 ooo3-4916/92 $7.50 595:215i1-5
Copyright ‘(1 1992 by Academic Press, Inc. All rights of reproductzon m any form reserved.
64
D. SCHijTTE
extensions of these gauges have been proposed [4, 51 but the knowledge of the detailed structure of the quantized field theory within such frameworks is only fragmentary. As a consequence, it should be valuable to investigate a simple, non-trivial example where these gauge fixing problems can be studied in a complete way. The hope is to gain insights which might be important for the full QCD. The restriction of the SU(2) Yang-Mills theory to spherically symmetric fields [6] provides such a simplified framework which we call “SO(3) gauge field” model. Here, within the Coulomb gauge, the gauge fixing problem (showing all its complexities) has been considered by Jackiw et al. [6]. Some general structures of the (classical) theory have also been investigated [7, S]. It is the purpose of this paper to reconsider this model in view of an alternative gauge fixing which allows to elucidate some structures of the corresponding classical and quantized theories [9, lo]. We will be able to demonstrate that there exists (within the model) another gauge, the “extended Schwinger gauge” [4], which does not show any of the pathological structures of the Coulomb gauge-existence and uniqueness of the representative are guaranteed, if only gauge transformations with trivial winding number are used for the definition of the orbit. Subsequently, the structure of the classical gauge fixed theory is investigated. We are able to show that the gauge fixed Hamiltonian constructed according to Faddeev and Slavnov [IO] fails to give the correct gauge fixed equations of motion. The reason for this failure may be traced back to certain boundary condition problems which cannot occur in the finite dimensional example of Ref. [lo]. Despite of this peculiarity the standard temporal gauge quantization is well defined. Here, Gauss’s law is imposed on the quantum mechanical states, and-as demonstrated by Christ and Lee [91-a gauge fixing amounts to the introduction of a suitable (curvilinear) coordinate system in functional space. We find for our model that the quantized Hamiltonian in the extended Schwinger gauge displays manifestly the “topological symmetry” with respect to gauge transformations with non-vanishing winding number. This symmetry-related to the 0 vacuum-is well known from a non-gauge-fixed temporal gauge formulation [ll], although an incorporation of this structure into a gauge fixed formulation, however, was up to now not possible. (It is also missing in the rather advanced calculations of van Baa1 et al. [l, 161.) The extended Schwinger gauge version of our model is a first example where this symmetry is explicit. It should also be remarked that the Hamiltonian within this gauge does not show any singularities-like the Gribov horizon singularities in the Coulomb gauge. There is no horizon in the sense of Gribov [2]. Only the wave functionals have to describe representations of the topological symmetry characterized by the O-parameter. We now give a survey of the content of the paper. In Section 2, we will define the model using a suitably complexified notation. In Section 3, we give the equations of motion and the boundary conditions guaranteeing normalizable solutions. The problem of gauge fixing is then discussed in Section 4. The structure of the
GAUGE
FIXING
FOR YANG-MILLS
FIELDS
65
Coulomb gauge, displayed in Ref. [6], is shortly repeated. The most important case, the extended Schwinger gauge, is then presented in detail. In Section 5, the consistency of our extended Schwinger gauge approach is demonstrated for the instanton solutions [12]. The peculiar structure of the classical gauge fixed theory is discussed in Section 6. The temporal gauge quantization is presented in Section 7. Introducing the extended Schwinger gauge as a special functional coordinate system, the quantized Hamiltonian is given. The symmetry with respect to gauge transformations having winding number it is revealed. It should be mentioned that there are still questions which deserve investigation within our simplified model: The structure of the restriction of the theory to a finite volume (bag with radius R) which should be interesting in view of the calculations of Ref. [ 1, 161, and a discussion of the quantized theory with fermions (the classical version is discussed in Ref. [ 131). This generalized model may perhaps serve as a simplified tool to study the structure of the U( 1)-anomaly within a Hamiltonian formalism. 2.
THE
DEFINITION
OF THE
MODEL
We consider SU(2) gauge fields in Minkowski space-time, i.e., functions A,(x, t) (p = 0, .... 3) with values in the Lie algebra of SU(2) obeying the standard transformation rule A,+A;=g(A,-a,)g-’ (1) with respect to any element g(x, t) of the gauge group SU(2). Given a rotation REXI( the fields A,= (A, A,) may be “rotated” according to AcR'(x, t)=RA(R-'x, t)
AbR'(x,t)=A,(R-'x,
t).
A, -+Ay’
(2)
A gauge field is called “spherically symmetric,” if Ay’ is gauge equivalent to A,, i.e., if there exists a gauge transformation g, such that AIR),A”R w P.
(3)
The possible types of spherically symmetric gauge fields are known [8, 141. There are two inequivalent classes characterized by the structure of g, within a “canonical” gauge [S]: (i)
g, = 1. Then A, is rotation invariant itself, i.e., A = xH(r, t),
& = &(r, f), r=
1x1.
(4)
66
D.
SCHtiTTE
(ii) g, = ;I, where R = p(A) when p: SU(2) -+ SO(3) is the standard covering map. (;1 is only defined up to a sign, but g’ = g -“. ) In this case, A, may be written in the form [6]
A, = d(r, t) s, T,
(5)
(~7~are Pauli matrices.) In both cases, the restriction Lagrange density
of the full Yang-Mills
theory defined by the
L? = itr(Fp,,Fp”‘), (6)
F,,=a,A,.-a,,A,+[A,,A,,]
to fields of type (i) or (ii) yields a consistent “gauge field model” in the sense that the spherical symmetry is preserved under the Euler-Lagrange equations; i.e., given H(r, to) and A,(r, to) in (4), the solutions of the full Yang-Mills equations are again of the type (i) (the same is true for the type (ii) [S]. For the discussion of physically interesting problems, the model (i) is irrelevant; in fact, it is easily shown that the solutions of the equations of motion are then gauge equivalent to the trivial solution, i.e., A, = g Jflg-’ for some spherically symmetric gauge transformation g( r, t ). Gauge fields of the type (ii), called SO(3) gauge fields and given in terms of the four real functions a, b, c, d of r and t in (5), define the model we shall consider in this paper. The standard gauge still allows non-trivial gauge transformations respecting the type (5) of the fields. These transformations are related to a function a(r, t) by
g, = fw
(
--c((r, t)s, xk . r >
(7)
Using the complex field z=b-l+ic
the result of a gauge transformation
(8)
(1) with g from (7) can be written in the form
a + a’= a-r
=+f=
uz,
d+d”=d-iJ,a,
a,a, (9)
67
GAUGE FIXING FOR YANG-MILLSFIELDS
where U(r, t) is the complex number U = exp( icr).
(10)
Within a canonical formulation of the theory, E, = Fok is canonically conjugate to Ax-. This field has the gauge transformation property E=gEg-‘.
(11)
In the X?(3) gauge field model E has the same structure as A, yk+ r
Introducing
r
k
r
u(r, t) r
-
the complex notation w = t4 + iv,
we have the gauge transformation
(13)
property for the canonically conjugate fields (14)
Of course, any arbitrary gauge transformation (respecting boundary conditions [6]) will also define a spherically symmetric gauge field in the original mathematical sense. However, only with the gauge transformations (7) does the spherical symmetry remain “manifest”; in general the symmetry will become hidden. We have an analogous situation in general relativity: Also there, spherically symmetric (Schwarzschild type) solutions of the Einstein equations are of special interest. Using coordinate transformations that keep the manifest spherically symmetry, different forms of the Schwarzschild solutions (Kruscal, Finkelstein) can be obtained. But of course, also here the Schwarzschild solution may be transformed by a general coordinate transformation hiding the spherical symmetry in a complicated way. The 50(3) gauge field model is defined in terms of the functions a, 6, c, d (for A, A,), h M, v (for E), and a (for gauge transformations g) which will be restricted by certain boundary conditions to be introduced in the next section.
3. DYNAMICS
AND BOUNDARY
CONDITIONS
Using the complex forms (8) and (13), the Lagrangian obtains the form for SO(3) gauge fields
L = (l/471) j d3x Y
68
D. SCHijTTE
L =i
+ 2 Re(w a,~*) + d(d,(rf)
!” dr(fa,a
+ 2 Im(zw*)))
(15) h(a,z,f;w)=iJdr(t
f2 +ww*+$
(lz1*-1)*+JD(a)zl’
>
,
(16)
where D(a)=f-ia,. r
The equations of motion become the dynamical equations (lzl”-1)z+D2(a)z,
-D,(d)w=$
(18) -r 3, f
= 2 Re(z*D(a)z),
D,(d)= = w,
d,u=f
(19)
+riJ,d
with D,(d) = a, + id.
(20)
In addition, we have the Gauss constraint
a,(rf) + 2 Im(zw*) = 0.
(21)
Using the relations D,(d”)U= D(d)
UD,(d),
u = u D(a),
(22)
these equations are obviously gauge covariant. (L and h are gauge invariant.) The equations of motion (18)-(21) have the same complexity as in the full YangMills theory. A, characterized by d plays the role of a Lagrange multiplier for the Gauss constraint (21). The theory fulfills the condition of Dirac’s generalized Hamiltonian dynamics, which means especially that if the constraint is fulfilled for t = 0, the equations of motion (18), (19), putting d= 0, guarantee that it is valid for all times. Due to this structure, the theory is not canonical in the usual sense for (A, El.
GAUGE
FIXING
FOR YANG--MILLS
69
FIELDS
In view of a later discussion of quantization, it is necessary to restrict the fields A by boundary conditions such that they are normalizable with respect to the norm (A, A) =&
1 d3x tr(AkAk)
(23)
= i j dr(a’ + 2b’ + 2~~).
We shall impose conditions at the origin (r = 0) and at infinity (r = co): (i) r =O. In standard quantum mechanics, it is known that a sufficiently large class of functions for a given angular momentum 1 is given by @= f(r) Y,m, where the radial part f has the expansion f(r)=r’(uo+a,r’+
.-.)
(24)
near the origin. This is just the weak condition that $ should (locally near r=O) allow a Weierstrass expansion I) = -jJ CnmkX;x~x:.
(25)
In our case, we shall assume the existence of such a Weierstrass expansion for the fields A, E, and A, leading to the following behaviour near the origin: a(r, t)=~,(t)r+ff~(t)r3+
+..,
b(r, t)=b,(t)r’+b,(t)r4+
...,
c(r, f)=c,(t)r+c,(t)r3+
..-;
f(r, t)=fi(t)r+f3(t)r3+
...,
u(r, t)=~2(f)r2+~4(f)r4+
-*-,
o(r, t)=u,(t)r+u,(t)r3+
. .. .
dfr, t)=d,(t)~+d~(t)r3+
-.-.
(26)
(27)
(28)
In addition, the possibility of an expansion of the type (25) for the gauge fields implies constraints on the lowest order coefficients given by
a,(t) = c*(t), f,(t) = ul(t).
(29)
(ii) r = co. Normalizability of the fields A, E amounts conditions for the asymptotic behaviour of the radial functions: a, b, c; f, u, v --+0 _f 0 r’ ’ for r -) ~3.
1 s>---, 2
to the following
(30)
70
D. SCHijTTE
In addition, we also impose a similar condition on the function d, 1 s > -, 2
(31)
for Y+ co, (iii) Gauge transformations have to be restricted by the condition that they do not change the boundary conditions of the gauge fields. This leads to the following boundary conditions for the function c((Y,t): CI(Y,t)=a,(t)r+~,(t)r~+ a(r, t) = 2nn + 0
L 0 rs
... nearr=O, for
r+rx),
(32)
n = integer, (s > f).
An additional constant 2mz could be added changing the r = 0 behaviour, but this can be disregarded without loss of generality since such an overall additive term does not change the gauge transformation g,. The integer number n in (32) is the winding number W(g,) defined by
w(p)=&j d3x ciiktrAiA,Ak, Aj= g-’
ajg,
(33)
and leads to the O-parameter of the quantized theory (see below). We assume the validity of these boundary conditions as part of the definition of our model. This guarantees, that the SO(3) gauge field theory fulfills the following requirements : (i) For a fixed time, the boundary conditions allow the definition of a suffkiently large class of functions, Especially complete basis sets exist obeying these boundary conditions. This appears necessary and sufficient for a quantization, since, e.g., for a scalar pair of canonically conjugate fields (@, Z7) the quantization [Z?(x), 6(y)] = i6(x -y) is fulfilled by the standard expansion 6=x fnqn, I?= C f,p, (p, = -ia/aq,) just in the case when the set of functions fn(x) is complete. (ii) The boundary conditions for the Lagrange multiplier d guarantee the existence of the temporal gauge d = 0; according to (9) the gauge transformation to this gauge is given by a(r, t) = j’ d( r, T) dz + N,,(r) to
(34)
GAUGE
FIXING
FOR YANG-MILLS
FIELDS
71
which is only unique up to a time independent part g,,. Putting t = to implies that c(~has to fulfil the relations (32). For arbitrary times, it follows then from (34) that d must obey the same boundary conditions as a. The existence of the temporal gauge is necessary since only then is the interpretation of the function d as Lagrange multiplier guaranteed and only then is a quantization definition possible (see below). Also the equations of motion are consistent only with these boundary conditions for d; see ( 19), especially also the equation for i3,c, 1).
a,c=u-d(b-
(35)
(iii) The boundary conditions-as conditions to be valid for all times-are consistent with the complete set of equations of motion (lS)-(20). It is sufficient to discuss this in the temporal gauge d = 0. Since here a,A = E, the fields A and E have to fulfill the same conditions, which indeed is the case. The non-trivial check is to prove that--due to (18)-the time derivative a,E obeys the boundary conditions. Therefore, we decompose (18) in terms of real and imaginary parts, which yields
-a,“=-$
(b2+C2-b+a*-r2af)c
+;a,h+; -a,f=;
(b-
(b-1)
l)(ab-a+
+ -$ (UC’ - cr f3,b).
a/,
r
-ra,c) (36)
The Gauss constraint becomes a,(rf)+(b-l)u-cu=O.
(37)
The first observation related to the conditions (26)-(28) is that Eqs. (36) (37) are obviously consistent with the assumption-dictated by the validity of the Weierstrass expansion-that a, c, .f, u are odd and 6, u are even in r. Inserting the expansion (26) with the restriction (29) into (36) and (37) shows that the radial functions indeed keep their structure during the time evolution due to several cancellations which are necessary because of the singular factor l/r* appearing in (36). This shows the consistency of the boundary conditions at the origin. On the other
72
D.SCHtiTTE
hand, the structure of Eqs. (36) also clearly guarantees that the same consistency holds for the conditions at infinity [S]. For these considerations, the constraint (37) is irrelevant since it is sufficient to fullil it at a fixed time. (iv) The boundary conditions at the origin are also necessary, because a more general expansion (like allo.wing terms constant in Y for b, U, or disregarding the condition (29), or abandoning the even-odd structure of the functions) induces singularities that destroy the square integrability near Y = 0. This is because the above-mentioned cancellations then do not take place. We skip here the eiementary, but technically lengthy proofs.
4. GAUGE FIXING The standard mathematical procedure to characterize gauge orbits, i.e., equivalence classes of gauge fields identified by (1) is by representatives defined by a suitable gauge fixing. However, in view of the fact that the set of gauge transformations is divided into different classes according to the winding number (33) we also have to distinguish the gauge orbits correspondingly. Therefore we define O(A, n) = (A 1 there exists a ge SU(2) with W(g) = n and A = A”).
(38)
After quantization, the (quantized) Gauss constraint amounts to the equivariance condition Y(A)= Y(A) only for 2~ O(A, 0) [ll]. A formulation with representatives should, therefore, be performed on the orbits O(A, 0). The existence of the orbits with non-trivial winding number leads then to a residual symmetry of the gauge fixed problem (existence of the O-parameter is the quantized version). We want to elucidate this point a little more in detail, whereby we may have in mind a general non-abelian gauge field theory. Let us denote the representative of A by F(A). Then we have that F(A) = F(Ag) for W(g)=0 by construction. However, F(Agn) (W(g,) #O) will in general not be equal to F(A); we construct below a counterexample. F(Agn) depends only on the winding number, so we may define F(Agn) = F”(A). If the gauge fixing would yield F” = F the topological symmetry would act trivially on the representatives. This excludes having a non-trivial representation of this symmetry characterized by a O-angle different from zero. For a consistent description of this symmetry we therefore should have F” # F for n # 0. Especially, this inequality should hold for A = 0; i.e., the representative of vacuum should not be unique when topological non-trivial gauge transformations are included. In summary, we are faced with the problem that having chosen a gauge fixing condition, one has to guarantee that-for a given field A-there exists a representative on O(A, 0) fulfilling this condition and that it is unique. We already discussed the temporal gauge d= 0 related to the gauge transforma-
GAUGE
FIXING
FOR
YANG-MILLS
FIELDS
73
tion (34). It exists but is not unique. This gauge is important for the definition of quantization (see Section 6). In addition, we shall consider two other choices, the Coulomb and the Schwinger (radial) gauges. 4.1. Coulomb Gauge This gauge fixing, given by the condition div A =O, has been thoroughly investigated in Ref. [6]. It appears to be not only inconvenient, since the uniqueness is in general not guaranteed [2], but it is also inconsistent because the existence of the representative a given field A on U(A, 0) in general fails. Details are discussed in Ref. [6], here we only want to stress the following points. For fields A in the neighbourhood of the classical vacuum A = 0 both existence and uniqueness hold, if only normalizabie fields are allowed. This is even true when including gauge transfo~ations with arbitrary winding number. This “uniqueness of the vacuum” already excludes the possibility of incorporating the topological symmetry in the Coulomb gauge in a non-trivial way (see above) and hints at a deficiency. The problematic structure of the Coulomb gauge then clearly shows up when considering large fields-generated by replacing a small A by iA and increasing A. Then not only the uniqueness will in general fail if L is large enough. In addition, there will be a critical A = i,, so that also the existence of the Coulomb gauge representative fails on O(A, 0) for iL> J.,. (There are representatives on O(A, n) with n # 0.) The field &A lies on the Gribov horizon where the Coulomb gauge Hamiltonian becomes singular. Examples are discussed in detail in Ref. [6], see especially Fig. 2 of this reference. This shows that the Coulomb gauge does not meet our consistency requirements. 4.2. Schwinger Gauge
A “natural” gauge in our model is the Schwinger (radial) gauge (l/r) x,A, = a = 0. Formally, this gauge is easily obtained by the gauge transformation g,,4 by putting (39)
The function tl, fulfils the boundary conditions at r = O--this unique-but its behaviour for r -+ co is in general wrong, because
makes it
Y,(f) = %(=A t) (40)
will usually not be equal to 2m. (With the boundary conditions (30) y, is always finite; if y would equal to 2nn, CIwould have the correct behaviour (32).) This leads us to the conclusion that the Schwinger gauge does not exist.
74
D. SCHiiTTE
However, this deficiency for r + cc is easily cured by extending this gauge in a suitable way [4]. Herefore, we choose a smooth function S(Y) with the properties s(0) = 0, p’(O) = 1,
(41)
s(r)= l+O
1 ( rm>
for
r-cc
(m>l).
(p # 0 is some arbitrary, fixed length scale.) For any SU(3) gauge field, the gauge transformation Li,=U,-y,S
(42)
fulfills all the boundary conditions (32) and the winding number is zero. Inserting 8, into (9) yields uau= y,(t) rs’(r).
(43)
We conclude that any solutions of the equations of motion (18)-(21) can be brought into a form such that the radial part a has the structure (43). We call this gauge fixing the extended Schwinger gauge. This gauge fixing does not only always exist-due to the explicit construction -but it is also unique on O(A, 0). Suppose yrs’ - r
a,a= y”rs’
(44)
then (keeping t fixed) o! = (y - 7)s + const, a(0) = 0 =z-const = 0, cc(co)=2nn*y-~=2n7c,
(45)
This calculation also shows that the transition to the orbits O(A, n) (n # 0) is provided by c1= 2nm mapping y -+ y + 2nn. In contrast to the Coulomb gauge, the representatives of A and Agz (CI= 2nm) are different in the extended Schwinger gauge. Thus there the vacuum is not unique if gauge transformations with arbitrary winding number are considered. A = 0 is gauge equivalent to a = 2nrcrs’, b = c = 0 on O(0, n). This leads to the possibility to incorporate within the extended Schwinger gauge the topological symmetry in a non-trivial way. An important structure of the gauge fixing (43) is given by the fact that the function y is not independent of the other fields. (Otherwise, the theory would fail to be canonical in the sense that the canonically conjugate to the variable y would not
GAUGEFIXING
FOR YANGMILLS
75
FIELDS
occur; see below.) This is due to the boundary condition (29) at r = 0 which is only fulfilled if we have Ht)=p d,O, t). (46) Thus also in the extended Schwinger gauge, the function a is completely fixed. Instead of being zero, however, it is a function of the other fields. 5. INSTANTON
SOLUTIONS WITHIN THE EXTENDED
S~HWINGER
GAUGE
As a consequence of the complex nature of the Coulomb gauge, smooth solutions to the equation of motion become, in general, discontinuous when transformed to this gauge. For the instanton solution [12] this has been demonstrated in Ref. [S], Such discontinuities do not occur in the extended Schwinger gauge. Within our notations, this instanton solution (which is a solution to the Euclidian EulerLagrange equation obtained by replacing E2 -B’ by E2 + B2) has the simple form ff=I’=rtN, b = r2N,
(47)
d= -rN,
where N=2(rZ+tZ+n’)-‘. Integration formation
(48)
of (39) and insertion into (42) yields the “allowed”
gauge trans-
c(= - tM(arc tan(rM) - 7w(r)), (49)
M=(p+n”)-‘!2,
and ‘J(t) = nthcf.
(50)
The fact that v(t) --+ i: n for r -+ fnci shows the well-known property of instanton solutions that they interpolate between fields related by a gauge transformation with winding number one. Obviously, this gauge fixing is completely continuous.
6. STRUCTURE OF THE EQUATIONS OF MOTION THE EXTENDED
IN
SCHWINGER GAUGE
Using the formal considerations of Faddeev and Slavnov [lo] we expect that the equations of motion for the independent fields of the extended Schwinger gauge, (z, w), should be the canonical equations with respect to the Hamiltonian h*(z, w) = h(rr(,-), 2, f(z, w), w),
(51)
76
D. SCHtiTTE
where a(z) = p d,c(o, t) rs’
(52)
is given by the gauge fixing and f(z, w) guarantees that Gauss’s constraint fulfilled.
is
f(Z, w) = - f K’(Im(,-w*)), where (54)
This elimination of the function f with the help of (21) is always willdefined and unique, since the lower limit of integration must be r = 0 because of the boundary condition (27), and also f is square integrable because f + 0( l/r) for r + co. In addition, (53) also guarantees the boundary condition (29) for the E-field, f'(0) = u’(O). (From (53) it follows that f'(0) = -F'(O), if F(0) =O, where F= Im(zw*). But F= (b- l)v--cu, so that the boundary conditions (26) for b, c with corresponding conditions for U, u yield F(0) = 0 and F’(0) = -v’(O).) Thus the Hamiltonian in the extended Schwinger gauge becomes explicitly h*(z,w)=i
j dr(ww*+$
(zz*-
l)‘+
ID(z)ri’+-$
(K’(Imzw*))2
, > (55)
D(z) = p d,c(O, t) s’(r) - 2,.
It is an unexpected fact that the equations of motion, obtained from the canonical equations with respect to (55) do not coincide with the equations (18)-(21), when in these the gauge fixing (43) is introduced and when the functions f and d are eliminated within these equations. Let us demonstrate this. The critical point is the elimination of the “Lagrange multiplier” d through (19) with the result (recall that the function a is given by (43)) that
d= -K’ Here, f is given by (53). The equations for (z, w) are obtained by inserting these expressions for f and d. However, the equation for c?,c, given in (35) does not determine a,?(t) = p a,a,c(O, t)-because of the boundary conditions (26), (27) one obtains only the trivial relation dry = pu, + (at? - pu,). Instead, 8,~ is fixed by (56): In general, d taken from (56) will not be normalizable, but in (35) only normalizable functions d are acceptable. Therefore, the value of d,y is provided by the square integrability condition yielding (using s( co ) = 1) 8,~ = K”(f/r).
(57)
GAUGE FIXING FOR YANG-MILLS
FIELDS
77
Thus the correct equation of motion for the function c is
a$=*-(l?- l)(sK,”(g-K(;)). This equation is different from the canonical equation for c from (48) which has the form
It is easily checked that for the instanton solution in the extended Schwinger gauge Eq. (58) and not (59) holds. (E.g., only (58) gives the correct boundary conditions at the origin.) Also the relation (57) is fulfilled. The reason for this discrepancy lies in the fact that, in our case, the operator (l/r) K’ leading to the elimination of the function f, the canonically conjugate to a (see (53)), and the operator K’( l/r) - s(r) K ‘“( l/r) fixing the Lagrange multiplier d (see (56) and (57)), are not related by Hermitian conjugation. However, only if this is true, the arguments for the derivation of the canonical gauge fixed theory hold (compare Eqs. (2.23) and (2.27) of Ref. [lo]). The point is that in our case the inverse of the operator a,r is well defined, whereas that of the Hermitian conjugate -ra, is not. The additional prescription (57) in (56) is needed because of the boundary conditions. We remark that this yields just the characteristic operator of the extended Schwinger gauge fixing which is quite plausible because d can be interpreted as a generating function for gauge transformations. In Ref. [lo], only finite dimensional examples are discussed. In that case, for any matrix A, det A = (det At)*, so if A is invertible, the same holds for At. We remark that the gauge fixed equations of motion still display the topological symmetry: If z(r, t), w(r, t) are solutions, the same holds for the “topologically” transformed functions Unz, U, w, where U, = exp( i2nrc.s). Also the “Hamiltonian” h*(z, w) in (55) is gauge invariant with respect to the action of U,.
7. Q~ANTIZATION
AND ~-VACUUM
Because of the non-standard structure of the classical theory discussed above, a direct quantization of our model in the extended Schwinger gauge does not appear to be well defined. However, the standard definition of canonical quantization goes along another path[9]: Disregarding the constraint (21) and putting d= 0, Eqs. (18), (19) for (A, E) are canonical with respect to the classical Hamiltonian h from (16).
78
D. SCHfiTTE
Denoting quantized entities with a “circumflex” quantisation is therefore given by [L?(r), f(r’)] ^
and putting
= i6(r - r’),
[b(r), li(r’)] = i&r - r’), [t(r),
t = 0, a natural
(60)
t?(r’)] = i&r-r’).
(All other commutators vanish.) Equation (60) is just the standard condition [A,,(x), ~~,u(x’)] = iS3(x -x’) S,,,Sjj, when the fields (A, E) are truncated according to (5) and (12). The standard “Bargmann space” realization of (60) is by functionals y(u(a,z), where ci, 6, 6 are multiplication operators and f= -i(6/6a), ti= -i(6/66), ti = - i( S/SC). This disregards the Gauss constraint (21). The quantized version of this constraint is imposed on the physical states: GY = (a,($) + 2 Im(N*))
Y= 0.
(61)
Like in the full theory, this condition is equivalent to F(A) = Y(Ag)
for
W(g) = 0.
(62)
(Here, of course, A and g are restricted to (5) and (7).) The operator 6 commutes with the quantized Hamiltonian H = hfci, 2, i G)
(63)
because H is gauge invariant. Within the full Yang-Mills theory, the same would hold true for the other generators of the Poincare group, and the temporal gauge quantization would also guarantee the (formal) validity of the correct commutation relations providing a representation of the Poincare group on the gauge invariant states fulfilling (62). States Y(A) obeying Gauss’s law (62) may be constructed by introducing a special functional coordinate system via gauge fixing. Hereby, the strict conditions related to the existence and uniqueness of representatives are also needed in order to have a mathematically consistent coordinate transformation. For the general case, this is, e.g., displayed in Refs. [9, IS]. Within our SO(3) gauge field model, we transform
(4 :I-
(a, Z)
(64)
with the help of the extended Schwinger gauge, i.e., a(a, z) = d,, Z(a, z) =exp(Z,)z.
(65)
GAUGE
FIXING
FOR
YANG-MILLS
FIELDS
79
The “equivariance” condition (62) is then trivially fulfilled by functionals Y(Z), independent of a. Since the Hamiltonian is gauge invariant, the action of H on gauge invariant states is equivalent to that of a suitable other operator fi(Z, I@) (I@= -i(6/6Z)). In Ref. [9] it is shown that this gauge fixed Hamiltonian is just (see (55)) B=h*(2, with the modification that the Faddeev-Popov such a way that the “Laplacian” (f)‘+2
I@,‘=
Fe)
(66)
determinant J has to be inserted in
-($)‘-2
13
(67)
is correctly evaluated with respect to the variables (a, Z). This guarantees that observables become independent of the choice of the gauge. In our case, J is the determinant of the Jacobian Sa sa
u=f)
= K’ ;-s(r)
K”
‘. r
(68)
Since this operator is independent of Z, one may put J= 1 without loss of generality, and the Hamiltonian is just given by (66) and (55). A most important property of this Hamiltonian is that it shows the “topological symmetry” with respect to winding number n gauge transformations g,=exp(i2nn
“; skXk), (69)
U = exp( i2nm) very explicitly. (The important point in (55) is that also for D(z) = ~~‘(0)s - ia, the relation D( Uz) Uz = UD(z)z holds because (9) yields PC’(O) -+ PC’(O)- 2nn for a gauge transformation U from (69) when the boundary conditions (26) are taken into account.) Thus the eigenstates of A may be chosen to be simultaneous eigenstates of the (Abelian) symmetry g,, characterized by Y(Agn) = exp(in@) Y(A).
(70)
This seems to be the first example of an explicit “topological symmetry” for the gauge fixed Hamiltonian of a non-abelian gauge field theory. Another interesting structure of the Hamiltonian fi is that it has no singularities; there is no horizon in the sense of Gribov. Especially the “Coulomb term” 1 dr 2(K’(Im zw*))‘/(r2) is regular. We finally remark that an apparent necessary condition for these results is the specific structure of the extended Schwinger gauge fixing-xistence and uniqueness of representatives on the orbit O(A, 0). Especially we do not see any way how the topological structures could be incorporated into a Coulomb gauge description.
80
D.SCHtiTTE ACKNOWLEDGMENTS
The author thanks Roman Jackiw for pointing out the importance Anselm, Janos Polonyi, and Pedro Paulo Schirmer for stimulating paper and Bernhard Metsch for carefully reading the manuscript.
of the topological symmetry, discussions on the subject
Alexej of this
REFERENCES 1. J. KOLLER AND P. VAN BAAL, Ann Phys. (N.Y.) 174 (1987) 299; Nucl. Phys. B 303 (1988), 1. 2. V. N. GRIBOV, Nucl. Phys. B 139 (1978) 1. 3. See, e.g., “Physical and Nonstandard Gauges” (P, Gaigg, W. Kummer, and M. Schweda, Eds.), Lecture Notes in Physics, Vol. 361, Springer-Verlag, New York/Berlin, 1990. 4. P. A. M. GUICHON AND D. SCHWTE, On the boundary condition problem for the Yang-Mills theory restricted to a spherical cavity, Phys. Rev., to appear. 5. H. YABUKI, Phys. Letr. B 231 (1989) 271. 6. R. JACKIW, I. MUZINICH, AND C. REBBI, Phys. Rev. D 17 (1978) 1576. 7. A. ACTOR, Rev. Mod. Phys. 51 (1979) 461; M. CARMELI, KH. HULEIHIL. AND E. LEIBOWITZ, “Gauge Fields, Classification and Equations of Motion,” World Scientific, Singapore, 1989. 8. P. P. SCHIRMER, “Global Existence for Spherically Symmetric Yang-Mills Fields on 3 + 1 SpaceTime Dimensions,” Doctoral dissertation, New York University, 1990; “Decay Estimates for Spherically Symmetric Yang-Mills Fields in Minkowski Space-Time,” University of Bonn, preprint, 1991. 9. N. H. CHRIST AND T. D. LEE, Phys. Rev. D 22 (1980) 939. 10. L. D. FADDEEV AND A. A. SLAVNOV, “Gauge Fields: Introduction to Quantum Theory,” p. 72 ff, Benjamin, London, 1980. 11. R. JACKIW, Topological Investigation of quantized gauge field theories, in “Current Algebra and Anomalies,” World Scientific, Singapore, 1985. 12. A. BELAVIN, A. POLYAKOV, A. SCHWARTZ, AND Y. TYUPKIN, Phys. Lett. B 59 (1975), 85. 13. D. SIVERS, Phys. Rev. D 35 (1986). 14. V. N. ROMANOV. A. S. SCHWARZ, AND Y. S. TYUPKIN, Nucl. Phys. B 130 (1977), 209. 15. P. BESTING AND D. SCH~TTE, Phys. Rev. D 42 (1990). 595. 16. P. VAN BAAL, Nucl. Phys. e 351 (1991’). 183.
OF PHYSICS
215, 63-80 (1992)
The Problem of Gauge Fixing for Spherically Symmetric Yang-Mills Fields D. SCH~TTE
Received June 7, 1991
The Yang-Mills theory restricted by spherical symmetry is investigated in view of the problem of gauge fixing. We show that-in contrast to the Coulomb gauge-an extended Schwinger gauge provides a completely consistent representation of gauge orbits with trivial winding number without any Gribov singularities, The structure of the classical gauge fixed theory is revealed showing a discrepancy to the standard expectation. The temporal gauge quantization with extended Schwinger gauge fixing in the sense of Christ and Lee is shown to display the “e-parameter” symmetry in a manifest way. lr 1992 Academx Press, Inc.
I. INTRODUCTION
AND OVERVIEW
As an alternative to the Feynman path integral method, the Hamiltonian formulation of QCD is becoming an important tool for obtaining insight into the structure of field theories of that kind [ 1 ]. It is usually considered to be necessary for the construction of such a formulation that the classical theory--whose quantization yields the Hamiltonian of interest-is canonicui. For a gauge field theory, such a canonical structure is expected to hold only for the gauge orbits. The theory has therefore to be formulated with respect to these orbits, the representatives of which are usualiy characterized by some gauge fixing condition (like the Coulomb, the axial, or the Schwinger gauge). For a non-abelian gauge field theory, however, a non-trivial problem arises: Does the gauge fixing fullil the condition that for any gauge held the existence and the uniqueness of the representative on the gauge orbit is guaranteed? Here, the existence has to be guaranteed on the gauge orbits generated by gauge transformations with tr~vi~i ~~~~ing n~~~er on/-v. This is necessary in order to incorporate into the gauge fixed formulation the “topological symmetry” related to the O-parameter; i.e., only then topologically non-trivial gauge transformations will appear as residual symmetry of the gauge fixed theory allowing a standard quantum mechanical treatment. Very little is known in relation to a solution of this problem: E.g., for the Coulomb gauge, both the uniqueness [2] and the existence fail [6] (see also Section 2.4). For the axial or the Schwinger gauge, the existence fails [3, 41; 63 ooo3-4916/92 $7.50 595:215i1-5
Copyright ‘(1 1992 by Academic Press, Inc. All rights of reproductzon m any form reserved.
64
D. SCHijTTE
extensions of these gauges have been proposed [4, 51 but the knowledge of the detailed structure of the quantized field theory within such frameworks is only fragmentary. As a consequence, it should be valuable to investigate a simple, non-trivial example where these gauge fixing problems can be studied in a complete way. The hope is to gain insights which might be important for the full QCD. The restriction of the SU(2) Yang-Mills theory to spherically symmetric fields [6] provides such a simplified framework which we call “SO(3) gauge field” model. Here, within the Coulomb gauge, the gauge fixing problem (showing all its complexities) has been considered by Jackiw et al. [6]. Some general structures of the (classical) theory have also been investigated [7, S]. It is the purpose of this paper to reconsider this model in view of an alternative gauge fixing which allows to elucidate some structures of the corresponding classical and quantized theories [9, lo]. We will be able to demonstrate that there exists (within the model) another gauge, the “extended Schwinger gauge” [4], which does not show any of the pathological structures of the Coulomb gauge-existence and uniqueness of the representative are guaranteed, if only gauge transformations with trivial winding number are used for the definition of the orbit. Subsequently, the structure of the classical gauge fixed theory is investigated. We are able to show that the gauge fixed Hamiltonian constructed according to Faddeev and Slavnov [IO] fails to give the correct gauge fixed equations of motion. The reason for this failure may be traced back to certain boundary condition problems which cannot occur in the finite dimensional example of Ref. [lo]. Despite of this peculiarity the standard temporal gauge quantization is well defined. Here, Gauss’s law is imposed on the quantum mechanical states, and-as demonstrated by Christ and Lee [91-a gauge fixing amounts to the introduction of a suitable (curvilinear) coordinate system in functional space. We find for our model that the quantized Hamiltonian in the extended Schwinger gauge displays manifestly the “topological symmetry” with respect to gauge transformations with non-vanishing winding number. This symmetry-related to the 0 vacuum-is well known from a non-gauge-fixed temporal gauge formulation [ll], although an incorporation of this structure into a gauge fixed formulation, however, was up to now not possible. (It is also missing in the rather advanced calculations of van Baa1 et al. [l, 161.) The extended Schwinger gauge version of our model is a first example where this symmetry is explicit. It should also be remarked that the Hamiltonian within this gauge does not show any singularities-like the Gribov horizon singularities in the Coulomb gauge. There is no horizon in the sense of Gribov [2]. Only the wave functionals have to describe representations of the topological symmetry characterized by the O-parameter. We now give a survey of the content of the paper. In Section 2, we will define the model using a suitably complexified notation. In Section 3, we give the equations of motion and the boundary conditions guaranteeing normalizable solutions. The problem of gauge fixing is then discussed in Section 4. The structure of the
GAUGE
FIXING
FOR YANG-MILLS
FIELDS
65
Coulomb gauge, displayed in Ref. [6], is shortly repeated. The most important case, the extended Schwinger gauge, is then presented in detail. In Section 5, the consistency of our extended Schwinger gauge approach is demonstrated for the instanton solutions [12]. The peculiar structure of the classical gauge fixed theory is discussed in Section 6. The temporal gauge quantization is presented in Section 7. Introducing the extended Schwinger gauge as a special functional coordinate system, the quantized Hamiltonian is given. The symmetry with respect to gauge transformations having winding number it is revealed. It should be mentioned that there are still questions which deserve investigation within our simplified model: The structure of the restriction of the theory to a finite volume (bag with radius R) which should be interesting in view of the calculations of Ref. [ 1, 161, and a discussion of the quantized theory with fermions (the classical version is discussed in Ref. [ 131). This generalized model may perhaps serve as a simplified tool to study the structure of the U( 1)-anomaly within a Hamiltonian formalism. 2.
THE
DEFINITION
OF THE
MODEL
We consider SU(2) gauge fields in Minkowski space-time, i.e., functions A,(x, t) (p = 0, .... 3) with values in the Lie algebra of SU(2) obeying the standard transformation rule A,+A;=g(A,-a,)g-’ (1) with respect to any element g(x, t) of the gauge group SU(2). Given a rotation REXI( the fields A,= (A, A,) may be “rotated” according to AcR'(x, t)=RA(R-'x, t)
AbR'(x,t)=A,(R-'x,
t).
A, -+Ay’
(2)
A gauge field is called “spherically symmetric,” if Ay’ is gauge equivalent to A,, i.e., if there exists a gauge transformation g, such that AIR),A”R w P.
(3)
The possible types of spherically symmetric gauge fields are known [8, 141. There are two inequivalent classes characterized by the structure of g, within a “canonical” gauge [S]: (i)
g, = 1. Then A, is rotation invariant itself, i.e., A = xH(r, t),
& = &(r, f), r=
1x1.
(4)
66
D.
SCHtiTTE
(ii) g, = ;I, where R = p(A) when p: SU(2) -+ SO(3) is the standard covering map. (;1 is only defined up to a sign, but g’ = g -“. ) In this case, A, may be written in the form [6]
A, = d(r, t) s, T,
(5)
(~7~are Pauli matrices.) In both cases, the restriction Lagrange density
of the full Yang-Mills
theory defined by the
L? = itr(Fp,,Fp”‘), (6)
F,,=a,A,.-a,,A,+[A,,A,,]
to fields of type (i) or (ii) yields a consistent “gauge field model” in the sense that the spherical symmetry is preserved under the Euler-Lagrange equations; i.e., given H(r, to) and A,(r, to) in (4), the solutions of the full Yang-Mills equations are again of the type (i) (the same is true for the type (ii) [S]. For the discussion of physically interesting problems, the model (i) is irrelevant; in fact, it is easily shown that the solutions of the equations of motion are then gauge equivalent to the trivial solution, i.e., A, = g Jflg-’ for some spherically symmetric gauge transformation g( r, t ). Gauge fields of the type (ii), called SO(3) gauge fields and given in terms of the four real functions a, b, c, d of r and t in (5), define the model we shall consider in this paper. The standard gauge still allows non-trivial gauge transformations respecting the type (5) of the fields. These transformations are related to a function a(r, t) by
g, = fw
(
--c((r, t)s, xk . r >
(7)
Using the complex field z=b-l+ic
the result of a gauge transformation
(8)
(1) with g from (7) can be written in the form
a + a’= a-r
=+f=
uz,
d+d”=d-iJ,a,
a,a, (9)
67
GAUGE FIXING FOR YANG-MILLSFIELDS
where U(r, t) is the complex number U = exp( icr).
(10)
Within a canonical formulation of the theory, E, = Fok is canonically conjugate to Ax-. This field has the gauge transformation property E=gEg-‘.
(11)
In the X?(3) gauge field model E has the same structure as A, yk+ r
Introducing
r
k
r
u(r, t) r
-
the complex notation w = t4 + iv,
we have the gauge transformation
(13)
property for the canonically conjugate fields (14)
Of course, any arbitrary gauge transformation (respecting boundary conditions [6]) will also define a spherically symmetric gauge field in the original mathematical sense. However, only with the gauge transformations (7) does the spherical symmetry remain “manifest”; in general the symmetry will become hidden. We have an analogous situation in general relativity: Also there, spherically symmetric (Schwarzschild type) solutions of the Einstein equations are of special interest. Using coordinate transformations that keep the manifest spherically symmetry, different forms of the Schwarzschild solutions (Kruscal, Finkelstein) can be obtained. But of course, also here the Schwarzschild solution may be transformed by a general coordinate transformation hiding the spherical symmetry in a complicated way. The 50(3) gauge field model is defined in terms of the functions a, 6, c, d (for A, A,), h M, v (for E), and a (for gauge transformations g) which will be restricted by certain boundary conditions to be introduced in the next section.
3. DYNAMICS
AND BOUNDARY
CONDITIONS
Using the complex forms (8) and (13), the Lagrangian obtains the form for SO(3) gauge fields
L = (l/471) j d3x Y
68
D. SCHijTTE
L =i
+ 2 Re(w a,~*) + d(d,(rf)
!” dr(fa,a
+ 2 Im(zw*)))
(15) h(a,z,f;w)=iJdr(t
f2 +ww*+$
(lz1*-1)*+JD(a)zl’
>
,
(16)
where D(a)=f-ia,. r
The equations of motion become the dynamical equations (lzl”-1)z+D2(a)z,
-D,(d)w=$
(18) -r 3, f
= 2 Re(z*D(a)z),
D,(d)= = w,
d,u=f
(19)
+riJ,d
with D,(d) = a, + id.
(20)
In addition, we have the Gauss constraint
a,(rf) + 2 Im(zw*) = 0.
(21)
Using the relations D,(d”)U= D(d)
UD,(d),
u = u D(a),
(22)
these equations are obviously gauge covariant. (L and h are gauge invariant.) The equations of motion (18)-(21) have the same complexity as in the full YangMills theory. A, characterized by d plays the role of a Lagrange multiplier for the Gauss constraint (21). The theory fulfills the condition of Dirac’s generalized Hamiltonian dynamics, which means especially that if the constraint is fulfilled for t = 0, the equations of motion (18), (19), putting d= 0, guarantee that it is valid for all times. Due to this structure, the theory is not canonical in the usual sense for (A, El.
GAUGE
FIXING
FOR YANG--MILLS
69
FIELDS
In view of a later discussion of quantization, it is necessary to restrict the fields A by boundary conditions such that they are normalizable with respect to the norm (A, A) =&
1 d3x tr(AkAk)
(23)
= i j dr(a’ + 2b’ + 2~~).
We shall impose conditions at the origin (r = 0) and at infinity (r = co): (i) r =O. In standard quantum mechanics, it is known that a sufficiently large class of functions for a given angular momentum 1 is given by @= f(r) Y,m, where the radial part f has the expansion f(r)=r’(uo+a,r’+
.-.)
(24)
near the origin. This is just the weak condition that $ should (locally near r=O) allow a Weierstrass expansion I) = -jJ CnmkX;x~x:.
(25)
In our case, we shall assume the existence of such a Weierstrass expansion for the fields A, E, and A, leading to the following behaviour near the origin: a(r, t)=~,(t)r+ff~(t)r3+
+..,
b(r, t)=b,(t)r’+b,(t)r4+
...,
c(r, f)=c,(t)r+c,(t)r3+
..-;
f(r, t)=fi(t)r+f3(t)r3+
...,
u(r, t)=~2(f)r2+~4(f)r4+
-*-,
o(r, t)=u,(t)r+u,(t)r3+
. .. .
dfr, t)=d,(t)~+d~(t)r3+
-.-.
(26)
(27)
(28)
In addition, the possibility of an expansion of the type (25) for the gauge fields implies constraints on the lowest order coefficients given by
a,(t) = c*(t), f,(t) = ul(t).
(29)
(ii) r = co. Normalizability of the fields A, E amounts conditions for the asymptotic behaviour of the radial functions: a, b, c; f, u, v --+0 _f 0 r’ ’ for r -) ~3.
1 s>---, 2
to the following
(30)
70
D. SCHijTTE
In addition, we also impose a similar condition on the function d, 1 s > -, 2
(31)
for Y+ co, (iii) Gauge transformations have to be restricted by the condition that they do not change the boundary conditions of the gauge fields. This leads to the following boundary conditions for the function c((Y,t): CI(Y,t)=a,(t)r+~,(t)r~+ a(r, t) = 2nn + 0
L 0 rs
... nearr=O, for
r+rx),
(32)
n = integer, (s > f).
An additional constant 2mz could be added changing the r = 0 behaviour, but this can be disregarded without loss of generality since such an overall additive term does not change the gauge transformation g,. The integer number n in (32) is the winding number W(g,) defined by
w(p)=&j d3x ciiktrAiA,Ak, Aj= g-’
ajg,
(33)
and leads to the O-parameter of the quantized theory (see below). We assume the validity of these boundary conditions as part of the definition of our model. This guarantees, that the SO(3) gauge field theory fulfills the following requirements : (i) For a fixed time, the boundary conditions allow the definition of a suffkiently large class of functions, Especially complete basis sets exist obeying these boundary conditions. This appears necessary and sufficient for a quantization, since, e.g., for a scalar pair of canonically conjugate fields (@, Z7) the quantization [Z?(x), 6(y)] = i6(x -y) is fulfilled by the standard expansion 6=x fnqn, I?= C f,p, (p, = -ia/aq,) just in the case when the set of functions fn(x) is complete. (ii) The boundary conditions for the Lagrange multiplier d guarantee the existence of the temporal gauge d = 0; according to (9) the gauge transformation to this gauge is given by a(r, t) = j’ d( r, T) dz + N,,(r) to
(34)
GAUGE
FIXING
FOR YANG-MILLS
FIELDS
71
which is only unique up to a time independent part g,,. Putting t = to implies that c(~has to fulfil the relations (32). For arbitrary times, it follows then from (34) that d must obey the same boundary conditions as a. The existence of the temporal gauge is necessary since only then is the interpretation of the function d as Lagrange multiplier guaranteed and only then is a quantization definition possible (see below). Also the equations of motion are consistent only with these boundary conditions for d; see ( 19), especially also the equation for i3,c, 1).
a,c=u-d(b-
(35)
(iii) The boundary conditions-as conditions to be valid for all times-are consistent with the complete set of equations of motion (lS)-(20). It is sufficient to discuss this in the temporal gauge d = 0. Since here a,A = E, the fields A and E have to fulfill the same conditions, which indeed is the case. The non-trivial check is to prove that--due to (18)-the time derivative a,E obeys the boundary conditions. Therefore, we decompose (18) in terms of real and imaginary parts, which yields
-a,“=-$
(b2+C2-b+a*-r2af)c
+;a,h+; -a,f=;
(b-
(b-1)
l)(ab-a+
+ -$ (UC’ - cr f3,b).
a/,
r
-ra,c) (36)
The Gauss constraint becomes a,(rf)+(b-l)u-cu=O.
(37)
The first observation related to the conditions (26)-(28) is that Eqs. (36) (37) are obviously consistent with the assumption-dictated by the validity of the Weierstrass expansion-that a, c, .f, u are odd and 6, u are even in r. Inserting the expansion (26) with the restriction (29) into (36) and (37) shows that the radial functions indeed keep their structure during the time evolution due to several cancellations which are necessary because of the singular factor l/r* appearing in (36). This shows the consistency of the boundary conditions at the origin. On the other
72
D.SCHtiTTE
hand, the structure of Eqs. (36) also clearly guarantees that the same consistency holds for the conditions at infinity [S]. For these considerations, the constraint (37) is irrelevant since it is sufficient to fullil it at a fixed time. (iv) The boundary conditions at the origin are also necessary, because a more general expansion (like allo.wing terms constant in Y for b, U, or disregarding the condition (29), or abandoning the even-odd structure of the functions) induces singularities that destroy the square integrability near Y = 0. This is because the above-mentioned cancellations then do not take place. We skip here the eiementary, but technically lengthy proofs.
4. GAUGE FIXING The standard mathematical procedure to characterize gauge orbits, i.e., equivalence classes of gauge fields identified by (1) is by representatives defined by a suitable gauge fixing. However, in view of the fact that the set of gauge transformations is divided into different classes according to the winding number (33) we also have to distinguish the gauge orbits correspondingly. Therefore we define O(A, n) = (A 1 there exists a ge SU(2) with W(g) = n and A = A”).
(38)
After quantization, the (quantized) Gauss constraint amounts to the equivariance condition Y(A)= Y(A) only for 2~ O(A, 0) [ll]. A formulation with representatives should, therefore, be performed on the orbits O(A, 0). The existence of the orbits with non-trivial winding number leads then to a residual symmetry of the gauge fixed problem (existence of the O-parameter is the quantized version). We want to elucidate this point a little more in detail, whereby we may have in mind a general non-abelian gauge field theory. Let us denote the representative of A by F(A). Then we have that F(A) = F(Ag) for W(g)=0 by construction. However, F(Agn) (W(g,) #O) will in general not be equal to F(A); we construct below a counterexample. F(Agn) depends only on the winding number, so we may define F(Agn) = F”(A). If the gauge fixing would yield F” = F the topological symmetry would act trivially on the representatives. This excludes having a non-trivial representation of this symmetry characterized by a O-angle different from zero. For a consistent description of this symmetry we therefore should have F” # F for n # 0. Especially, this inequality should hold for A = 0; i.e., the representative of vacuum should not be unique when topological non-trivial gauge transformations are included. In summary, we are faced with the problem that having chosen a gauge fixing condition, one has to guarantee that-for a given field A-there exists a representative on O(A, 0) fulfilling this condition and that it is unique. We already discussed the temporal gauge d= 0 related to the gauge transforma-
GAUGE
FIXING
FOR
YANG-MILLS
FIELDS
73
tion (34). It exists but is not unique. This gauge is important for the definition of quantization (see Section 6). In addition, we shall consider two other choices, the Coulomb and the Schwinger (radial) gauges. 4.1. Coulomb Gauge This gauge fixing, given by the condition div A =O, has been thoroughly investigated in Ref. [6]. It appears to be not only inconvenient, since the uniqueness is in general not guaranteed [2], but it is also inconsistent because the existence of the representative a given field A on U(A, 0) in general fails. Details are discussed in Ref. [6], here we only want to stress the following points. For fields A in the neighbourhood of the classical vacuum A = 0 both existence and uniqueness hold, if only normalizabie fields are allowed. This is even true when including gauge transfo~ations with arbitrary winding number. This “uniqueness of the vacuum” already excludes the possibility of incorporating the topological symmetry in the Coulomb gauge in a non-trivial way (see above) and hints at a deficiency. The problematic structure of the Coulomb gauge then clearly shows up when considering large fields-generated by replacing a small A by iA and increasing A. Then not only the uniqueness will in general fail if L is large enough. In addition, there will be a critical A = i,, so that also the existence of the Coulomb gauge representative fails on O(A, 0) for iL> J.,. (There are representatives on O(A, n) with n # 0.) The field &A lies on the Gribov horizon where the Coulomb gauge Hamiltonian becomes singular. Examples are discussed in detail in Ref. [6], see especially Fig. 2 of this reference. This shows that the Coulomb gauge does not meet our consistency requirements. 4.2. Schwinger Gauge
A “natural” gauge in our model is the Schwinger (radial) gauge (l/r) x,A, = a = 0. Formally, this gauge is easily obtained by the gauge transformation g,,4 by putting (39)
The function tl, fulfils the boundary conditions at r = O--this unique-but its behaviour for r -+ co is in general wrong, because
makes it
Y,(f) = %(=A t) (40)
will usually not be equal to 2m. (With the boundary conditions (30) y, is always finite; if y would equal to 2nn, CIwould have the correct behaviour (32).) This leads us to the conclusion that the Schwinger gauge does not exist.
74
D. SCHiiTTE
However, this deficiency for r + cc is easily cured by extending this gauge in a suitable way [4]. Herefore, we choose a smooth function S(Y) with the properties s(0) = 0, p’(O) = 1,
(41)
s(r)= l+O
1 ( rm>
for
r-cc
(m>l).
(p # 0 is some arbitrary, fixed length scale.) For any SU(3) gauge field, the gauge transformation Li,=U,-y,S
(42)
fulfills all the boundary conditions (32) and the winding number is zero. Inserting 8, into (9) yields uau= y,(t) rs’(r).
(43)
We conclude that any solutions of the equations of motion (18)-(21) can be brought into a form such that the radial part a has the structure (43). We call this gauge fixing the extended Schwinger gauge. This gauge fixing does not only always exist-due to the explicit construction -but it is also unique on O(A, 0). Suppose yrs’ - r
a,a= y”rs’
(44)
then (keeping t fixed) o! = (y - 7)s + const, a(0) = 0 =z-const = 0, cc(co)=2nn*y-~=2n7c,
(45)
This calculation also shows that the transition to the orbits O(A, n) (n # 0) is provided by c1= 2nm mapping y -+ y + 2nn. In contrast to the Coulomb gauge, the representatives of A and Agz (CI= 2nm) are different in the extended Schwinger gauge. Thus there the vacuum is not unique if gauge transformations with arbitrary winding number are considered. A = 0 is gauge equivalent to a = 2nrcrs’, b = c = 0 on O(0, n). This leads to the possibility to incorporate within the extended Schwinger gauge the topological symmetry in a non-trivial way. An important structure of the gauge fixing (43) is given by the fact that the function y is not independent of the other fields. (Otherwise, the theory would fail to be canonical in the sense that the canonically conjugate to the variable y would not
GAUGEFIXING
FOR YANGMILLS
75
FIELDS
occur; see below.) This is due to the boundary condition (29) at r = 0 which is only fulfilled if we have Ht)=p d,O, t). (46) Thus also in the extended Schwinger gauge, the function a is completely fixed. Instead of being zero, however, it is a function of the other fields. 5. INSTANTON
SOLUTIONS WITHIN THE EXTENDED
S~HWINGER
GAUGE
As a consequence of the complex nature of the Coulomb gauge, smooth solutions to the equation of motion become, in general, discontinuous when transformed to this gauge. For the instanton solution [12] this has been demonstrated in Ref. [S], Such discontinuities do not occur in the extended Schwinger gauge. Within our notations, this instanton solution (which is a solution to the Euclidian EulerLagrange equation obtained by replacing E2 -B’ by E2 + B2) has the simple form ff=I’=rtN, b = r2N,
(47)
d= -rN,
where N=2(rZ+tZ+n’)-‘. Integration formation
(48)
of (39) and insertion into (42) yields the “allowed”
gauge trans-
c(= - tM(arc tan(rM) - 7w(r)), (49)
M=(p+n”)-‘!2,
and ‘J(t) = nthcf.
(50)
The fact that v(t) --+ i: n for r -+ fnci shows the well-known property of instanton solutions that they interpolate between fields related by a gauge transformation with winding number one. Obviously, this gauge fixing is completely continuous.
6. STRUCTURE OF THE EQUATIONS OF MOTION THE EXTENDED
IN
SCHWINGER GAUGE
Using the formal considerations of Faddeev and Slavnov [lo] we expect that the equations of motion for the independent fields of the extended Schwinger gauge, (z, w), should be the canonical equations with respect to the Hamiltonian h*(z, w) = h(rr(,-), 2, f(z, w), w),
(51)
76
D. SCHtiTTE
where a(z) = p d,c(o, t) rs’
(52)
is given by the gauge fixing and f(z, w) guarantees that Gauss’s constraint fulfilled.
is
f(Z, w) = - f K’(Im(,-w*)), where (54)
This elimination of the function f with the help of (21) is always willdefined and unique, since the lower limit of integration must be r = 0 because of the boundary condition (27), and also f is square integrable because f + 0( l/r) for r + co. In addition, (53) also guarantees the boundary condition (29) for the E-field, f'(0) = u’(O). (From (53) it follows that f'(0) = -F'(O), if F(0) =O, where F= Im(zw*). But F= (b- l)v--cu, so that the boundary conditions (26) for b, c with corresponding conditions for U, u yield F(0) = 0 and F’(0) = -v’(O).) Thus the Hamiltonian in the extended Schwinger gauge becomes explicitly h*(z,w)=i
j dr(ww*+$
(zz*-
l)‘+
ID(z)ri’+-$
(K’(Imzw*))2
, > (55)
D(z) = p d,c(O, t) s’(r) - 2,.
It is an unexpected fact that the equations of motion, obtained from the canonical equations with respect to (55) do not coincide with the equations (18)-(21), when in these the gauge fixing (43) is introduced and when the functions f and d are eliminated within these equations. Let us demonstrate this. The critical point is the elimination of the “Lagrange multiplier” d through (19) with the result (recall that the function a is given by (43)) that
d= -K’ Here, f is given by (53). The equations for (z, w) are obtained by inserting these expressions for f and d. However, the equation for c?,c, given in (35) does not determine a,?(t) = p a,a,c(O, t)-because of the boundary conditions (26), (27) one obtains only the trivial relation dry = pu, + (at? - pu,). Instead, 8,~ is fixed by (56): In general, d taken from (56) will not be normalizable, but in (35) only normalizable functions d are acceptable. Therefore, the value of d,y is provided by the square integrability condition yielding (using s( co ) = 1) 8,~ = K”(f/r).
(57)
GAUGE FIXING FOR YANG-MILLS
FIELDS
77
Thus the correct equation of motion for the function c is
a$=*-(l?- l)(sK,”(g-K(;)). This equation is different from the canonical equation for c from (48) which has the form
It is easily checked that for the instanton solution in the extended Schwinger gauge Eq. (58) and not (59) holds. (E.g., only (58) gives the correct boundary conditions at the origin.) Also the relation (57) is fulfilled. The reason for this discrepancy lies in the fact that, in our case, the operator (l/r) K’ leading to the elimination of the function f, the canonically conjugate to a (see (53)), and the operator K’( l/r) - s(r) K ‘“( l/r) fixing the Lagrange multiplier d (see (56) and (57)), are not related by Hermitian conjugation. However, only if this is true, the arguments for the derivation of the canonical gauge fixed theory hold (compare Eqs. (2.23) and (2.27) of Ref. [lo]). The point is that in our case the inverse of the operator a,r is well defined, whereas that of the Hermitian conjugate -ra, is not. The additional prescription (57) in (56) is needed because of the boundary conditions. We remark that this yields just the characteristic operator of the extended Schwinger gauge fixing which is quite plausible because d can be interpreted as a generating function for gauge transformations. In Ref. [lo], only finite dimensional examples are discussed. In that case, for any matrix A, det A = (det At)*, so if A is invertible, the same holds for At. We remark that the gauge fixed equations of motion still display the topological symmetry: If z(r, t), w(r, t) are solutions, the same holds for the “topologically” transformed functions Unz, U, w, where U, = exp( i2nrc.s). Also the “Hamiltonian” h*(z, w) in (55) is gauge invariant with respect to the action of U,.
7. Q~ANTIZATION
AND ~-VACUUM
Because of the non-standard structure of the classical theory discussed above, a direct quantization of our model in the extended Schwinger gauge does not appear to be well defined. However, the standard definition of canonical quantization goes along another path[9]: Disregarding the constraint (21) and putting d= 0, Eqs. (18), (19) for (A, E) are canonical with respect to the classical Hamiltonian h from (16).
78
D. SCHfiTTE
Denoting quantized entities with a “circumflex” quantisation is therefore given by [L?(r), f(r’)] ^
and putting
= i6(r - r’),
[b(r), li(r’)] = i&r - r’), [t(r),
t = 0, a natural
(60)
t?(r’)] = i&r-r’).
(All other commutators vanish.) Equation (60) is just the standard condition [A,,(x), ~~,u(x’)] = iS3(x -x’) S,,,Sjj, when the fields (A, E) are truncated according to (5) and (12). The standard “Bargmann space” realization of (60) is by functionals y(u(a,z), where ci, 6, 6 are multiplication operators and f= -i(6/6a), ti= -i(6/66), ti = - i( S/SC). This disregards the Gauss constraint (21). The quantized version of this constraint is imposed on the physical states: GY = (a,($) + 2 Im(N*))
Y= 0.
(61)
Like in the full theory, this condition is equivalent to F(A) = Y(Ag)
for
W(g) = 0.
(62)
(Here, of course, A and g are restricted to (5) and (7).) The operator 6 commutes with the quantized Hamiltonian H = hfci, 2, i G)
(63)
because H is gauge invariant. Within the full Yang-Mills theory, the same would hold true for the other generators of the Poincare group, and the temporal gauge quantization would also guarantee the (formal) validity of the correct commutation relations providing a representation of the Poincare group on the gauge invariant states fulfilling (62). States Y(A) obeying Gauss’s law (62) may be constructed by introducing a special functional coordinate system via gauge fixing. Hereby, the strict conditions related to the existence and uniqueness of representatives are also needed in order to have a mathematically consistent coordinate transformation. For the general case, this is, e.g., displayed in Refs. [9, IS]. Within our SO(3) gauge field model, we transform
(4 :I-
(a, Z)
(64)
with the help of the extended Schwinger gauge, i.e., a(a, z) = d,, Z(a, z) =exp(Z,)z.
(65)
GAUGE
FIXING
FOR
YANG-MILLS
FIELDS
79
The “equivariance” condition (62) is then trivially fulfilled by functionals Y(Z), independent of a. Since the Hamiltonian is gauge invariant, the action of H on gauge invariant states is equivalent to that of a suitable other operator fi(Z, I@) (I@= -i(6/6Z)). In Ref. [9] it is shown that this gauge fixed Hamiltonian is just (see (55)) B=h*(2, with the modification that the Faddeev-Popov such a way that the “Laplacian” (f)‘+2
I@,‘=
Fe)
(66)
determinant J has to be inserted in
-($)‘-2
13
(67)
is correctly evaluated with respect to the variables (a, Z). This guarantees that observables become independent of the choice of the gauge. In our case, J is the determinant of the Jacobian Sa sa
u=f)
= K’ ;-s(r)
K”
‘. r
(68)
Since this operator is independent of Z, one may put J= 1 without loss of generality, and the Hamiltonian is just given by (66) and (55). A most important property of this Hamiltonian is that it shows the “topological symmetry” with respect to winding number n gauge transformations g,=exp(i2nn
“; skXk), (69)
U = exp( i2nm) very explicitly. (The important point in (55) is that also for D(z) = ~~‘(0)s - ia, the relation D( Uz) Uz = UD(z)z holds because (9) yields PC’(O) -+ PC’(O)- 2nn for a gauge transformation U from (69) when the boundary conditions (26) are taken into account.) Thus the eigenstates of A may be chosen to be simultaneous eigenstates of the (Abelian) symmetry g,, characterized by Y(Agn) = exp(in@) Y(A).
(70)
This seems to be the first example of an explicit “topological symmetry” for the gauge fixed Hamiltonian of a non-abelian gauge field theory. Another interesting structure of the Hamiltonian fi is that it has no singularities; there is no horizon in the sense of Gribov. Especially the “Coulomb term” 1 dr 2(K’(Im zw*))‘/(r2) is regular. We finally remark that an apparent necessary condition for these results is the specific structure of the extended Schwinger gauge fixing-xistence and uniqueness of representatives on the orbit O(A, 0). Especially we do not see any way how the topological structures could be incorporated into a Coulomb gauge description.
80
D.SCHtiTTE ACKNOWLEDGMENTS
The author thanks Roman Jackiw for pointing out the importance Anselm, Janos Polonyi, and Pedro Paulo Schirmer for stimulating paper and Bernhard Metsch for carefully reading the manuscript.
of the topological symmetry, discussions on the subject
Alexej of this
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