- Email: [email protected]
Towards a nonperturbative path integral in gauge theories
3 June 1999
Physics Letters B 456 Ž1999. 38–47
Towards a nonperturbative path integral in gauge theories Sergei V. Shabanov 1, John R. Klauder
2
Department of Mathematics, UniÕersity of Florida, GainesÕille, FL 32611, USA Received 10 February 1999; received in revised form 9 April 1999 Editor: P.V. Landshoff
Abstract We propose a modification of the Faddeev–Popov procedure to construct a path integral representation for the transition amplitude and the partition function for gauge theories whose orbit space has a non-Euclidean geometry. Our approach is based on the Kato–Trotter product formula modified appropriately to incorporate the gauge invariance condition, and thereby equivalence to the Dirac operator formalism is guaranteed by construction. The modified path integral provides a solution to the Gribov obstruction as well as to the operator ordering problem when the orbit space has curvature. A few explicit examples are given to illustrate new features of the formalism developed. The method is applied to the Kogut–Susskind lattice gauge theory to develop a nonperturbative functional integral for a quantum Yang–Mills theory. Feynman’s conjecture about a relation between the mass gap and the orbit space geometry in gluodynamics is discussed in the framework of the modified path integral. q 1999 Published by Elsevier Science B.V. All rights reserved.
1. Motivations In what follows we consider only gauge theories of a special ŽYang–Mills. type where gauge transformations are linear transformations in the total configuration space. Let X be a configuration space which is assumed to be a Euclidean space R N unless specified otherwise. It is a representation space of a gauge group G so the action of G on X is given by a linear transformation x ™ V Ž v . x. A formal sum
1
On leave from: Laboratory of Theoretical Physics, JINR, Dubna, Russia. 2 Also Department of Physics.
over paths for the action invariant under gauge transformations would diverge. To regularize it, Faddeev and Popov have proposed to insert the identity w1x 1 s D FP Ž x .
H d m Ž V . dŽ x Ž V x . . ,
Ž 1.
where x Ž x . s 0 is a gauge condition and D FP is the corresponding Faddeev–Popov determinant, into the divergent path integral measure so that after a change of variables x ™ V x the Ždivergent. volume of the gauge group can be factorized out. The determinant D FP is calculated by doing the group averaging integral Ž1.. This formalism has provided a solution to two important problems in the perturbatiÕe path integral quantization of gauge theories: The unitarity of the
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 4 9 3 - 1
S.V. ShabanoÕ, J.R. Klauderr Physics Letters B 456 (1999) 38–47
perturbative S-matrix and the construction of a local effective gauge fixed action. Later on it has been observed that the procedure becomes ill-defined beyond perturbation theory w2x. A geometrical reason for the Gribov obstruction is that there is no gauge condition which would provide a global parameteriG by a set zation of the gauge orbit space M s XrG of affine coordinates without singularities w3x. This situation is often rendered concrete in the vanishing or even sign changing of the Faddeev–Popov determinant. An additional problem expected in the nonperturbative regime is the operator ordering ambiguity w4x. When the kinetic energy x˙ 2 is projected on ˙ g p h h˙ ., where h the gauge orbit space M , one gets Ž h, are local coordinates on M and g p h is an induced metric on M . The metric is not flat w5x. In order to establish a correspondence between the Dirac operator formalism and the gauge-fixed path integral, the latter must be appropriately defined to take into account the operator ordering in the physical kinetic energy operator. It is known that the formal path integral for motion on a curved manifold is ambiguous w6x. Up to the second order of perturbative Yang–Mills theory, operator ordering corrections have no effect on the physical S-matrix w4x. Their role in a nonperturbative description is unknown. A standard argument used in dimensional regularization, i" d 3 Ž0. s 0, Ža commutator of canonically conjugated field variables at the same spacetime point, which typically emerges in the operator ordering terms. does not seem to be applicable because the validity of dimensional regularization beyond perturbation theory is not justified w7x. Finally, the space M may have a nontrivial topology w3x. Due to the locality of the Faddeev–Popov construction, global properties of M are lost in the path integral. On the other hand the topology of the orbit space may be important for the spectrum of physical excitations. Feynman conjectured that the mass gap in gluodynamics in Ž2 q 1. spacetime might be caused by a compactification of certain directions in the configuration space upon an identification of gauge equivalent configurations w8x. In this letter we propose a modification of the procedure Ž1. which gives a solution to the above problems and leads to a well-defined path integral on the orbit space M . Its validity is demonstrated with a few examples.
39
2. A modified Kato–Trotter product formula Let Hˆ and sˆa be self-adjoint Hamiltonian and operators of constraints, respectively. We also asˆ sˆa x s 0 which holds for a sufficiently large sume w H, class of physically interesting gauge models. Let the gauge group be compact. Consider a projection operator Pˆ s
H dmŽ v .e
i v a sˆ a
,
H dmŽ v . s1 ,
Pˆ s Pˆ † s Pˆ 2 .
Ž 2.
The operator Ž2. projects the total Hilbert space onto the Dirac subspace of gauge invariant states. If the gauge group generated by sˆa is not compact, we take a sequence cd Pˆd of rescaled projection operators Pˆd , where Pˆd projects on the subspace Ý sˆa2 F d 2 , and then consider the limit d ™ 0 in a weak sense. The coefficients cd can always be chosen so that the limit exists and defines a Hilbert space isomorphic to the Dirac one w9x. Let the Hamiltonian ˆ where V s V Ž x . is an have the form Hˆ s Hˆ0 q V, everywhere regular potential and Hˆ0 s pˆ 2r2 is a kinetic energy operator. The operators Hˆ0 and Vˆ are assumed to be gauge invariant, i.e., they commute with the constraint operators 3. Consider the evolution operator UˆT s expŽyiTHˆ . in the total Hilbert space. The physical evolution operator is then UˆTD s UˆT Pˆ. Making use of the SincKato–Trotter product formula w11x for the evolution operator UˆT we get the following product formula for the gauge invariant evolution operator n
ˆ ˆ ˆ ˆ UˆTD s Ž eyi e H 0 eyi e V . Pˆ s Ž eyi e H 0 Pˆeyi e V . ˆ
n
n
n
' Ž Uˆe0 D eyi e V . ' Ž UˆeD . ,
Ž 3.
where e s Trn and n ™ `. In our previous work w10x we have shown that in the continuum limit the averaging gauge variables v play the role of the Lagrange multipliers for the constraints in the classi-
3
This assumption is justified for gauge theories of the Yang– Mills type. More general cases will be considered elsewhere because they are technically more involved and require a phasespace representation of quantum theory and a Wiener measure regularization of the corresponding path integral as proposed in Ref. w10x.
S.V. ShabanoÕ, J.R. Klauderr Physics Letters B 456 (1999) 38–47
40
cal action. By making use of the classical theory of Kolmogorov, we gave an explicit construction of a countably additive probability measure for the gauge functions v Ž t .. One natural choice was such that any set of values of v Ž t . at any set of distinct times is equally likely. Now we show that this concept leads to a modification of the Faddeev–Popov procedure. First we observe that by construction UTD
X
Ž x,V x .
s UTD
X
Ž V x, x .
s UTD
X
Ž x, x .
Ž 4.
that is, the projected amplitude determines a genuine transition amplitude on the orbit space M . To obtain its relation to the conventional gauge fixed amplitude, we introduce a set of local coordinates h on M via a change of variables xsV Ž v . h ,
UTD Ž h,hXs . s UTD Ž h s ,hX . s UTD Ž h,hX . .
Ž 7.
It determines an S-inÕariant continuation of the transition amplitude to the coÕering space H of the modular domain K. The Faddeev–Popov determinant specifies the volume element on M w5x, so for any integral involving only functions on M , f Ž x . s f Ž V x ., one has H d xf Ž x . s V G HK d h D FP Ž h. f Ž h., where V G is the volume of the gauge group. Therefore the folding of two projected infinitesimal evolution operators reads
Ž 5.
where the configurations x s h satisfy a gauge condition x Ž x . s 0, i.e., h g H ; X and without loss of generality we may assume H ; R M Žunless specified otherwise. with M being the number of physical degrees of freedom. The gauge invariant amplitude Ž4. can be reduced on the gauge fixing surface H : UTD Ž x, xX . s UTD Ž h,hX .. This leads to a necessary modification of the procedure Ž1.: Instead of averaging the gauge condition over the gauge group, we propose to average an infinitesimal evolution operator kernel for a free motion as follows from the modified Kato–Trotter product formula Ž3. Ue0 D Ž h,hX . s Ž 2p i e .
x s h on the gauge fixing surface x Ž x . s 0. If S is a collection of all residual gauge transformations, the modular domain K is the quotient HrS. The relation Ž5. determines a one-to-one correspondence if h g K and v parameterizes a group manifold. If h,hX g K, then the following relation holds
yN r2
H dmŽ v .e
X
iŽ hy V Ž v . h . 2 r2 e
.
Ž 6.
Since e ™ 0, the averaging integral can be done to proper order in e explicitly by the stationary phase approximation. The kernel UeD Ž h,hX . is obtained by adding yi e V Ž h. to the exponential in Ž6. in accordance with Ž3. because V Ž x . s V Ž h. thanks to the gauge invariance of V. Suppose there exist transformations of the new variables v and h in Ž5., v ™ v q v s and h ™ h s , such that x Ž v ,h. s x Ž v q v s ,h s ., where all configurations x s s h s also satisfy the gauge condition x Ž h. s x Ž h s . s 0. From V Ž v q v s . h s s V Ž v . h follows that h s s V s h, i.e., h s is a Gribov copy of
U2De Ž h,hX . s
HK d h D 1
FP
Ž h1 . UeD Ž h,h1 . UeD Ž h1 ,hX . . Ž 8.
The restriction of the integration domain in Ž8. does not lead to a formal restriction of the integration domain in the path integral resulting from Ž3. in the continuum limit as one might naively expect from Ž8.. An important point is that, if K / H Žthe Gribov problem is present., then Ue0 D Ž h,hX . has no standard form of an infinitesimal free transition amplitude on a manifold proposed in Ref. w6x as we now proceed to demonstrate. The stationary phase approximation can be applied before the reduction of UeD Ž x, xX . on a gauge fixing surface. A deÕiation from the conventional gauge-fixing procedure results from the fact that there may be more than just one stationary point. We can always shift the origin of the averaging variable v so that one of the stationary points is at the origin ˆ a be operators generating gauge transv s 0. Let d formations in X . Decomposing the distance Ž x y V Ž v . xX . 2 in the vicinity of the stationary point, we ˆ a xX . s 0. So in the formal continuum find Ž x y xX ,d limit we get the condition saŽ x, ˙ x . ' Ž x,d ˙ ˆ a x. s0 induced by the averaging procedure. This is nothing but the Gauss law enforcement for trajectories contributing to UTD Ž x, xX .. Suppose there exists a gauge condition x aŽ x . s 0, which involves no time derivatives, such that a generic configuration x s h satisfy-
S.V. ShabanoÕ, J.R. Klauderr Physics Letters B 456 (1999) 38–47
ing it also fulfills identically the Gauss law, i.e., ˙ ˆ a h. ' 0. We will call it a natural gauge. In this Ž h,d case all other stationary points in the integral Ž6. are v c s v s where V Ž v s . s V s g S. Therefore we get a sum over the stationary points in the averaging integral Ž6. if the Gribov problem is present. Still, in the continuum limit we have to control all terms of order e . This means that we need not only the leading term in the stationary phase approximation of Ž6. but also the next two corrections to it. Therefore the group element V Ž v . should be decomposed up to order v 4 because v 4re ; e as one is easily convinced by rescaling the integration variable v ™ 'e v . The measure should also be decomposed up to the necessary order to control the e-terms. The latter would yield quantum corrections to the classical potential associated with the operator ordering in the kinetic energy operator on the orbit space. We stress that the averaging procedure gives a unique ordering so that the integral is invariant under general coordinate transformations on M , i.e., does not depend on the choice of x Žcf. Ž14... Thus, Ue0 D Ž h,hX . s Ž 2p i e .
yM r2
Ý Dy1r2 Ž h,hXs . S
X
2
X
= ei Ž hyh s . r2 eyi e Vq Ž h , h s . q O Ž e 2 . ,
Ž 9. &
' Ý Dy1r2 Ž h,hXs . Ue Ž h,hXs . ,
Ž 10 .
S
where DŽ h,hX . is the conventional determinant arising in the stationary phase approximation, and by Vq we denote a contribution of all relevant corrections to the leading order. The amplitude UeD Ž h,hX . is obtained by adding yi e V Ž h. to the exponential in Ž9.. Note that V Ž h. s V Ž h s . thanks to the gauge invariance of the potential. In general, the equations saŽ x, ˙ x . s 0 are not integrable therefore the functions x a discussed above do not exist. In this case we consider two possibilities. Let V c Ž h,hX . be the group element at a stationary point in Ž6.. Decomposing the distance in the vicinity of the stationary point we get Ž h y ˆ a V c hX . s 0. Let x be such that the latter V c hX ,d condition is also satisfied if hX is replaced by hXs . In this case the sum over the stationary points is again
41
given by Ž9. where in the first term of the exponential hXs is replaced by V c Ž h,hXs . hXs . In the most general case, the sum over stationary points does not coincide with the sum over Gribov copies. However for sufficiently small e , the averaged short-time transition amplitude can always be represented in the form Ž10.. Note that as e goes to zero, Ue0 Ž x, xX . ™ d N Ž x y xX .. If we change the variables in the delta function according to Ž5. and then do the gauge group average, we get, in general, a sum of delta functions on the gauge fixing surface. From the symmetry of the change of variables Ž5. it follows that the support of the averaged delta function is formed by points h s hXs , hX g K and h g H , which means that the sum over S emerges again for sufficiently small e . We will explain below how to calculate relevant corrections to the leading order of the limit e ™ 0. Let us replace UeD Ž h,h1 . in Ž8. by the sum Ž10. and make use of Ž7. applied to the second kernel in Ž8.: UeD Ž h1 ,hX . s UeD Ž h1 s ,hX .. Since the measure on the orbit space does not depend on a particular choice of the modular domain, d h s D FP Ž h s . s d h D FP Ž h., we can extend the integration to the entire covering space by removing the sum over S U2De Ž h,hX . s
Hdh < D 1
FP
Ž h1 . < Dy1 r2 Ž h,h1 .
&
=Ue Ž h,h1 . UeD Ž h1 ,hX . . Ž 11 . The absolute value bars accounts for a possible sign change of the Faddeev–Popov determinant on the gauge fixing surface. The procedure can be repeated from left to right in the folding Ž3., thus removing the restriction of the integration domain and the sum over copies in all intermediate times t g Ž0,T .. The sum over S for the initial configuration hX remains in the integral. Now we can formally take a continuum limit with the result UTD Ž h,hX . s Ý D FP Ž h . D FP Ž hXs .
y1 r2
S h ŽT .sh
=
H
(
D h det g p h X
h Ž0 .sh s T
˙
˙
=e iH0 d t wŽ h , g p h h .r2yVq Ž h .yV Ž h .x , Ž 12 . where g p h is the induced metric on M . The local density Ł t det g p h should be understood as the
(
S.V. ShabanoÕ, J.R. Klauderr Physics Letters B 456 (1999) 38–47
42
result of the integration over the momenta in the corresponding time-sliced phase-space path integral . where the kinetic energy is Ž p h , gy1 p h p h r2 with p h being a canonical momentum for h. To obtain Ž12., one usually sets hX s h y D for h s hŽ t . and hX s hŽ t y e . in each intermediate moment of time t and makes a decomposition into the power series over D. According to the relation, obtained in Ref. w5x, between the volume of a gauge orbit through x s h, the induced metric g p h , and the Faddeev–Popov 2 Ž X. determinant we get D Ž hX q D, hX . s D FP h r X det g p hŽ h . q O Ž D .. This relation explains the cancellation of the absolute value of the Faddeev–Popov determinant in the folding Ž3. made in accordance with the rule Ž11.. The term D 2re in the exponential Ž9. gives rise to the kinetic energy Ž D, g p h D .r2 e q O Ž D 3 .. The most difficult part to calculate are the operator ordering corrections Vq Ž h. in the continuum limit. Here we remark that DŽ h,hX . has to be decomposed up to order D 2 , while the exponential in Ž9. up to order D 4 because the measure has support on paths for which D 2 ; e Ži.e., D 4re ; e .. There is a technique, called the equivalence rules for Lagrangian path integrals on manifolds, which allows one to convert terms D 2 n into terms e n and thereby to calculate Vq w6,12x
D j1 PPP D j 2 k™ Ž i e " . =
Ý
k
Ž gy1 ph .
j1 j 2
PPP Ž gy1 ph .
j 2 ky 1 j 2 k
,
Ž 13 .
p Ž j1 , . . . , j 2 k .
where the sum is extended over all permutations of the indices j to make the right-hand side of Ž13. symmetric under permutations of the j’s. Following Ž13. one derives the Schrodinger equation for the ¨ physical amplitude Ž12.. The corresponding Hamiltonian operator on the orbit space has the form Hˆp h s y1r2 Dy1 FP E j
ž
gy1 ph
jk
D FP E k q V Ž h . ,
/
Ž 14 . where E j s ErE h j . Observe that the kinetic energy in Ž14. does not coincide with the Laplace–Beltrami operator on M because wdet g p h x1r2 / D FP as shown
in Ref. w5x. The operator Ž14. is hermitian with respect to the scalar product where the volume element is d h D FP . It is also invariant under general coordinate transformations on M , i.e., its spectrum does not depend on the choice of local coordinates on M and, therefore, is gauge invariant. We skip technical details of the derivation of Ž14. because they are standard in the path integral formalism and instead turn to examples to illustrate the main features of the new projection formalism. But before we do so, let us comment that the effects on the energy spectrum caused by the modification of the path integral can be found from the pole structure of the trace of the resolvent tr Rˆ Ž t . s tr Ž t y iHˆ . tr UˆTD s
HK d h D
FP
`
y1
s
H0
dT ey t T tr UˆTD ,
Ž h . UTD Ž h,h . .
Ž 15 . Ž 16 .
3. Examples Let us take the simplest example w4x where x g R 3 and the gauge transformations are rotations of x about the origin. The orbit space is a space of concentric spheres. The natural gauge is x i s d i1 h. This gauge is not complete because there are rotations that change the sign of h: h ™ "h. These transformations form the group S s Z 2 . The modular domain is an open half-axis K s Ž0,`.. The measure on the orbit space is obtained by introducing spherical coordinates on R 3 so that h s < x <, while the spherical angles parameterize the gauge group manifold SOŽ3.rSOŽ2.. It is noteworthy to observe that in the projection formalism there is no principal difference between reducible and irreducible gauge theories. Here we obviously have a reducible case because x has a stationary group SOŽ2.. The Faddeev–Popov determinant is defined with respect to a subset of independent constraints. In this case, the constraints are three components of the angular momentum. So we can take the angular momenta about the axes i s 2,3 as independent constraints and find that D FP s h 2 in the full accordance with the measure resulting from the spherical coordi-
S.V. ShabanoÕ, J.R. Klauderr Physics Letters B 456 (1999) 38–47
nates. The angular measure must be normalized by 4p to fulfill the condition in Ž2.. So Eq. Ž6. assumes the form Ue0 D Ž h,hX . s Ž 2p i e . p
=
H0
y3 r2
1r2
d v sin v e iŽ h
2
X2
X
qh .r2 e yi h h cos v r e
.
Ž 17 . Doing the averaging integral exactly and substituting the result into Ž3. we obtain UTD Ž h,hX . s w hhX x
y1
UT Ž h,hX . y UT Ž h,y hX . 4 , Ž 18 .
where the amplitude UT Ž h,hX . is given by a standard path integral for a one-dimensional system with the gauge-fixed action S g f w h x s H0T d t w h˙ 2r2 y V Ž h.x. There is no operator ordering correction. The stationary phase approximation leads to the same result if one takes into account all the stationary points, i.e., v s s 0 and v s s p . The second stationary point is associated with the only nontrivial residual gauge transformation hX ™ yhX . Here one also gets Vq s 0 Žcf. Ž9... The same path integral can be derived from the Schrodinger equation projected ¨ on the orbit space w13x. It is not hard to calculate the partition function Ž16. for a harmonic oscillator V s x 2r2 s h2r2 and find the resolvent Ž15.. The poles determine the spectrum En s 2 n q 3r2. The distance between the energy levels is 2, while the oscillator frequency in the Hamiltonian is 1. The effect of the doubling would be lost, had we ignored the redundant discrete gauge symmetry and calculated the partition function by a standard path integral for the gauge-fixed action S g f . A simple model to illustrate the effect of the residual gauge symmetry in systems with several physical degrees of freedom is a Yang–Mills theory in Ž1 q 0. spacetime w13x. A total configuration space is a Lie algebra X, and the gauge group acts in the adjoint representation x ™ V x Vy1 . The quadratic form in the exponential Ž6. is taken with respect to the Killing form on X: Ž x, x . ' tr xˆ 2 , where xˆ y ' w x, y x for any x, y g X and the commutator stands for the Lie bracket in X. If a matrix representation is assumed, for compact groups it can always be nor-
43
malized to the ordinary matrix trace. The Gauss law is s s w x, ˙ x x s 0, therefore the natural gauge condition to parameterize M is x s h g H where H is a Cartan subalgebra Ža maximal commutative subalgebra of X .. The system has r s dim H physical degrees of freedom. The invariant measure d m Ž v . is proportional to d v det wŽexp vˆ y 1. vˆ y1 x. In the exponential in Ž6. we get Ž h y expŽ vˆ . hX . 2 . An adjoint transformation of x to H is not unique and determined modulo the Weyl group W acting in H. The Weyl transformations are inequivalent compositions of reflections in the hyperplanes orthogonal to simple roots in H w19x. So S s W and M ; HrW s Kq, where Kq is the Weyl chamber in H. The stationary points of the integral Ž6. are v c s v W where the adjoint action of the group element exp v W on h transforms the latter by a Weyl reflection. Introducing the Cartan-Weyl basis w19x one can find the Jacobian of the change of variables Ž5. Žthe Faddeev–Popov determinant. D FP Ž h. s det hˆ s k 2 Ž h. where k Ž h. s Ł a ) 0 Ž a ,h.; the product is extended over all positive roots in H. The operator hˆ acts in the orthogonal complement of the Cartan subalgebra. Note that the Cartan group is the stationary group of h g H, so v should take its values in the orthogonal supplement of the Cartan subalgebra. The factor DŽ h,hX . is given by the determinant of the operator ˆˆX . Doing the stationary phase approximation in the hh averaging integral, and then taking the folding Ž3. we find UTD Ž h,hX . s Ý k Ž h . k Ž hXW .
y1
UT Ž h,hXW . .
W
Ž 19 . Here the amplitude UT Ž h,hX . is given by the conventional path integral Žno restriction of the integration domain to the Weyl chamber. with the action H d t Ž h˙ 2r2 y V Ž h... There is no operator ordering correction, and the physical metric is Euclidean. The same integral follows from a direct solution of the Schrodinger equation w13x. The Faddeev–Popov de¨ terminant vanishes at the hyperplanes Ž h, a . s 0. The boundary of the modular domain Žthe Weyl chamber. is formed by these hyperplanes when a runs over the simple roots. The amplitude Ž19. remains finite at the singular points Žthe Gribov horizon. because k Ž hW . s p W k Ž h. where p W s "1 is
S.V. ShabanoÕ, J.R. Klauderr Physics Letters B 456 (1999) 38–47
44
the parity of the corresponding Weyl transformation Žcf. also Ž18.: It remains finite at h s 0 or hX s 0... Contributions of paths hitting the horizon have a dramatic effect on the spectrum of the isotropic oscillator V s x 2r2 s h2r2. Doing the path integral for UT Ž h,hX . in Ž19. and calculating the poles of the resolvent Ž15. we find w13x En s n 1 n1 q PPP qnr n r q Nr2, where n i are non-negative integers, n i are orders of invariant irreducible symmetric tensors of the Lie algebra, and r and N are, respectively, the rank and dimension of the group Že.g., for SUŽ r q 1., n i s i q 1, i s 1,2, . . . , r .. Thus, the physical frequencies of the isotropic oscillator appear to be n i / 1. The partition function calculated by a standard path integral for the gauge fixed action of the model would lead to the spectrum of the isotropic oscillator Ži.e., n i ™ 1.. To illustrate the effects of curvature of the orbit space, we consider a simple gauge matrix model. Let x be a real 2 = 2 matrix subject to the gauge transformations x ™ V Ž v . x where V g SOŽ2.. An invariant scalar product reads Ž x, xX . s tr x T xX with x T being a transposed matrix x. The total configuration space is R 4 . Let ´ be a generator of SOŽ2., i.e., ´ i j s y´ ji , and ´ 12 s 1. Then V Ž v . s expŽ v ´ .. The Gauss law enforced by the projection is s s tr x˙ T ´ x s 0. It is not integrable. We parameterize M by triangular matrices h 21 ' 0 Žthe gauge x 21 s 0.. The residual gauge transformations form the group S s Z 2 : h ™ "h. The modular domain is a positive half-space h11 ) 0. Calculating the Jacobian of the change of variables Ž5. for this model we find the Faddeev–Popov determinant D FP Ž h. s h11. The plane h11 s 0 is the Gribov horizon. The averaging measure in Ž6. reads Ž2p .y1 d v , where v g w0,2 p .. The quadratic form in the exponential in Ž6. is v
X 2
X
X
X
Ž h y e ´ h . s Ž h,h . q Ž h ,h . y 2 Ž h,h . cos v y 2 Ž h,´ hX . sin v .
Ž 20 .
A distinguished feature of this model from those considered above is that the stationary point is a function of h and hX . Taking the derivative of Ž20. with respect to v and setting it to zero, we find
v c s tany1
Ž h,´ hX . , v cs s v c q p . Ž h,hX .
Ž 21 .
The second stationary point v cs is associated with the Gribov transformation h ™ yh. A geometrical meaning of the transformation hX ™ expŽ v c ´ . hX is transparent. The distance wŽ h y hX . 2 x1r2 between two points on the gauge fixing plane is greater than the minimal distance between the two gauge orbits through x s h and xX s hX . By shifting xX along the gauge orbit to xXc s expŽ v c ´ . hX a minimum of the distance between the orbits is achieved. In such a way the metric on the orbit space emerges in the projection formalism. Its explicit form can be found as has been explained in Section 2. We substitute v s v c Ž h,hX .into Ž20., set hX s h y D and decompose Ž20. in a power series over D. The quadratic term Žthe leading term. determines the metric. We get Ž D, g p hŽ h. D . s Ž D, D . q Ž D,´ h.Ž ´ h, D .rŽ h,h.. The metric is not flat. The scalar Riemann curvature of M is R s 6rŽ h,h. s 6rŽ x, x .. Note that it is gauge invariant as it should be because the curvature R does not depend on the choice of local coordinates on M . In the stationary phase approximation the cosine and sine in Ž20. should be decomposed up to fourth order in the vicinity of the stationary point. In this model quantum corrections do not vanish. The infinitesimal transition amplitude on M assumes the form UeD Ž h,hX . s Ž 2p i e .
y3 r2
X
Dy1r2 Ž h,hX . e i Se Ž h , h . X
qDy1r2 Ž h,y hX . e i Se Ž h ,y h . ,
Ž 22 .
Se Ž h,hX . s Ž h,h . q Ž hX ,hX . y 2 D Ž h,hX . r Ž 2 e . y e 8 D Ž h,hX .
y1
y e V Ž h. ,
Ž 23 .
where DŽ h,hX . s Ž h,hX .cos v c q Ž h,´ hX .sin v c . Observe that the middle term in Ž23. is a quantum correction to the classical potential in the short-time action Ž23. Žit is proportional to " 2 , if one restores the Planck constant.. The two first terms in Ž23. determine Ž uniquely!. the operator ordering in the kinetic energy operator on the orbit space in accordance with Ž14.. Finally, we consider the case of an infinite number of stationary points in the averaging integral Ž6.. This would have an effect that the modular domain K may become compact, which, in turn, leads to a
S.V. ShabanoÕ, J.R. Klauderr Physics Letters B 456 (1999) 38–47
discrete spectrum regardless of the details of the potential and therefore gives a mechanism for the gap between the ground state and the first excited state. A suitable soluble model is the Yang–Mills theory in cylindrical space time w14–16x where space is compactified into a circle. A Yang–Mills theory without matter in two dimensions has no physical degrees of freedom unless the spacetime has a nontrivial topology. The latter is exactly our case. The total configuration space X is the space of all periodic connections AŽ x q 2p . s AŽ x . taking their values in the Lie algebra of a compact Lie group. The gauge group G acts on X as A ™ VA s V A Vy1 y i VEVy1, and V Ž x q 2p . s V Ž x . Ži.e., the gauge group elements V can be continuously deformed towards the group unit w17,18x.. The exponential in Ž6. assumes the form H02 p d x trŽ A yVAX . 2 for any two configurations AŽ x . and AX Ž x .. It is the distance between AŽ x . and VAX Ž x . introduced by Feynman w8x. A rigorous definition of the averaging integral over G can be given within the Kogut–Susskind lattice gauge theory, to which we turn shortly. Note, for example, that N in Ž6. is infinite because the number of degrees of freedom is determined by the number of Fourier modes of AŽ x .. Here we assume the existence of a normalized averaging measure on G . The gauge group average enforces the Gauss law s s E A˙y iw A, A˙x ' =Ž A. A˙ s 0. The orbit space M can be parameterized by constant connections AŽ x . s h taking their values in the Cartan subalgebra H w14,16x. This is the natural ˙ . ' 0. The number of the physgauge because s Ž h,h ical degrees of freedom equals the rank of the gauge group. We set V Ž x . s expŽ i v Ž x ... The averaging function v takes its values in the Lie algebra and its constant Žindependent of x . Fourier mode does not have any component in the Cartan subalgebra because the connection AŽ x . s h is inÕariant under constant gauge transformations from the Cartan subgroup. The Gaussian integral over such functions can formally be done in Ž6. yielding Dy1 r2 Ž h,hX . s dety1 r2 wy=Ž h.=Ž hX .x which is easy to compute in the Cartan-Weyl basis w16x. In the lattice formulation described below this procedure would not involve any infinite factors in DŽ h,hX . as in the formal continuum limit. Here we draw attention to the following feature. A projection of any connection AŽ x . into the subspace of constant Cartan connec-
45
tions h is not unique and determined modulo the affine Weyl transformations w16x. The affine Weyl group WA is a semidirect product of the Weyl group W and the group of translation along the group unit lattice in the Cartan subalgebra w19x. Thus, the number of stationary points in the averaging integral Ž6. is infinite because the number of elements WA is infinite: h ™ h s ' hW q 2 a narŽ a , a ., where hW is a Weyl transformation of h, a any positive root, na an integer. The modular domain K s HrWA is called the Weyl cell w19x. It is compact in H. This has a significant effect on the spectrum of the system. In the continuum limit the transition amplitude has the form Ž19. where the sum is extended over the affine Weyl group Žover an infinite number of stationary points hXs . w16x, the Faddeev–Popov determinant reads D FP Ž h. s k 2 Ž h. and k Ž h. s Ł a ) 0 sinŽ h, a .. There is no potential energy in 1 q 1 dimensional Yang–Mills theory Ž V Ž h. ' 0 in Ž19... Thus, we get an amplitude for a free particle symmetrized over the affine Weyl group. In the simplest SUŽ2. case, the system has only one physical degree of freedom. The Weyl transformations are reflections hW s "h and the translations over the group unit lattice are shifts h ™ h q 2 n. The amplitude Ž19. appears similar to the one for a particle in an infinite well or on a circle. The spectrum of such a system is discrete. In general, we can calculate the poles of the resolvent Ž15. for the 2D Yang–Mills theory using our path integral formalism and find w16x the spectrum En s E0 wŽgn ,gn . y Ž r , r .x where E0 is a constant, r is a half the sum of all positive roots, and gn ranges over a lattice in the Weyl chamber that labels all irreducible representation of the gauge group. Thus, an infinite number of stationary points in the averaging integral Ž6. is, in fact, evidence that the physical configuration space Žor its subspace. is compact. This is the mathematical structure of the path integral which is to be sought when studying the relation between the mass gap in gluodynamics and the compactness of the orbit space. Physically, the existence of an infinite number of the stationary points separated by different distances Žlarge and small. implies that physical eigenstates of the Hamiltonian should exhibit a periodicity in some directions in X , which may cause the mass gap as has been argued by Feynman w8x.
S.V. ShabanoÕ, J.R. Klauderr Physics Letters B 456 (1999) 38–47
46
4. Kogut–Susskind lattice gauge theory To fulfill our program in realistic Yang–Mills theory, we need a regularization at short distances to give a rigorous meaning to the averaging integral Ž6. which is a functional integral. The simplest possibility is to set the system on a lattice. Since we still want to have a Hamiltonian formalism, the time should remain continuous. With such a choice of the regularization we arrive at the Kogut–Susskind lattice gauge theory w20x. Let y be a three-vector whose components are integer-valued in the lattice spacing unit a, i.e., y labels the lattice sites. With each link connecting the lattice sites y and y q ia, where i is the unit vector in the direction of the ith coordinate axis, we associate a group element u y,i called the link variable. The collection of all values of all link variables forms the configuration space X of the system. The gauge group G acts on X by the rule u y,i ™ V y u y,i Vy1 yqi a . Therefore the averaging measure is a product of the normalized Haar measures at each site: d m s Ł y d m H Ž V y .. The Kogut–Susskind Hamiltonian is H s H0 q V where the potential V is nothing but the color magnetic energy on the lattice. The kinetic energy is the sum of kinetic energies of a rigid rotators associated with each link variable 2
H0 s 1r2 Ý tr Ž u˙ y ,i uy1 y ,i . .
Ž 24 .
y,i
An important observation is that the coupling of the rotators occurs only through the potential V. Therefore the ‘‘free’’ amplitude UT0 Ž u4 , uX 4., where u4 is a collection of configurations of rotators u y,i , is factorized into a product of the amplitudes for each rotator UT0 Ž u y,i ,uXy,i .. An explicit form of the latter is well-known w21x Ža transition amplitude of a particle moving on a group manifold.. This replaces the free motion amplitude in a Euclidean space in the averaging integral Ž6.. The averaging integral Ue0 D Ž u4 , uX 4 . s
H Ły
d mH Ž V y .
= Ł Ue0 Ž u y ,i , V y uXy ,i V yqai . y,i
Ž 25 .
can be calculated via both the stationary phase approximation and the decomposition over the characters of the irreducible representations of the amplitudes UT0 Ž u,uX . proposed in Ref. w21x. Thus, we have reduced the problem of constructing the transition amplitude on the orbit space to a standard mathematical problem in group theory. It is important that the effects of the Riemann structure Žmetric. and topology of the orbit space are completely determined by the averaging procedure proposed and do not depend on the details of the self-interaction V. Note that the average in Ž25. induces a nontriÕial coupling between the rotators because six link variables attached to the same site are transformed with the same group element. So the factorization of the amplitude on M is no longer possible. This is the result of the enforcement of the Gauss law in the physical amplitude. Thus, we get, in principle, a constructive and consistent procedure to study the effects of the nonEuclidean geometry of the orbit space on physical quantities like the spectrum of low energy excitations. In particular, in the strong coupling limit Ž g large. the potential V can be treated by perturbation theory w20x. To introduce an explicit parameterization of the orbit space, one can, for example, make use of Morse theory for the Kogut–Susskind lattice gauge theory as proposed in Ref. w22x. In this case, the initial configuration uX 4 in the averaging integral Ž25. is taken from the corresponding modular domain. The projected amplitude has a natural gauge invariant continuation to the covering space of the modular domain.
5. Conclusions We have proposed a self-consistent path integral quantization of gauge theories which resolves the Gribov obstruction and provides control for the operator ordering problem, and it does not rely explicitly on the Schrodinger equation on the orbit space. This ¨ was our main goal. The continuum limit in the Kogut–Susskind lattice gauge theory might still be a difficult analytical problem. However, it is believed that the techniques we have developed might be useful to study the effects of the non-Euclidean geometry and topology of the orbit space in some
S.V. ShabanoÕ, J.R. Klauderr Physics Letters B 456 (1999) 38–47
physically reasonable approximations. We have also demonstrated its effectiveness with several examples.
References w1x w2x w3x w4x w5x w6x w7x
L.D. Faddeev, V.N. Popov, Phys. Lett. B 25 Ž1968. 30. V.N. Gribov, Nucl. Phys. B 139 Ž1978. 1. I. Singer, Commun. Math. Phys. 60 Ž1978. 7. N. Christ, T.D. Lee, Phys. Rev. D 22 Ž1980. 939. O. Babelon, C.-M.Viallet, Phys. Lett. B 85 Ž1979. 246. B. De Witt, Rev. Mod. Phys. 29 Ž1957. 377. J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 2nd Edition, Clarendon Press, Oxford, 1990. w8x R. Feynman, Nucl. Phys. B 188 Ž1981. 479. w9x J.R. Klauder, Ann. Phys. ŽNY. 254 Ž1997. 419.
47
w10x J.R. Klauder, S.V. Shabanov, Phys. Lett. B 398 Ž1997. 116. w11x E. Nelson, J. Math. Phys. 5 Ž1964. 332; T. Kato, H.F. Trotter, Pacific Math. J. 8 Ž1959. 545. w12x L.V. Prokhorov, Sov. J. Part. Nucl. 13 Ž1982. 456. w13x L.V. Prokhorov, S.V. Shabanov, Phys. Lett. B 216 Ž1989. 292; Sov. Phys. Uspekhi 34 Ž1991. 108. w14x S.G. Rajeev, Phys. Lett. B 212 Ž1989. 203. w15x E. Langman, G.W. Semenoff, Phys. Lett. B 303 Ž1993. 303. w16x S.V. Shabanov, Phys. Lett. B 318 Ž1993. 323. w17x G.’t Hooft, Nucl. Phys. B 153 Ž1979. 141. w18x R. Jackiw, Rev. Mod. Phys. 52 Ž1980. 661. w19x S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978. w20x J. Kogut, L. Susskind, Phys. Rev. D 11 Ž1976. 395. w21x M.S. Marinov, M.V. Terentiev, Fortshcr. Phys. 27 Ž1979. 511. w22x D. Zwanziger, Nucl. Phys. B 485 Ž1997. 185.
Physics Letters B 456 Ž1999. 38–47
Towards a nonperturbative path integral in gauge theories Sergei V. Shabanov 1, John R. Klauder
2
Department of Mathematics, UniÕersity of Florida, GainesÕille, FL 32611, USA Received 10 February 1999; received in revised form 9 April 1999 Editor: P.V. Landshoff
Abstract We propose a modification of the Faddeev–Popov procedure to construct a path integral representation for the transition amplitude and the partition function for gauge theories whose orbit space has a non-Euclidean geometry. Our approach is based on the Kato–Trotter product formula modified appropriately to incorporate the gauge invariance condition, and thereby equivalence to the Dirac operator formalism is guaranteed by construction. The modified path integral provides a solution to the Gribov obstruction as well as to the operator ordering problem when the orbit space has curvature. A few explicit examples are given to illustrate new features of the formalism developed. The method is applied to the Kogut–Susskind lattice gauge theory to develop a nonperturbative functional integral for a quantum Yang–Mills theory. Feynman’s conjecture about a relation between the mass gap and the orbit space geometry in gluodynamics is discussed in the framework of the modified path integral. q 1999 Published by Elsevier Science B.V. All rights reserved.
1. Motivations In what follows we consider only gauge theories of a special ŽYang–Mills. type where gauge transformations are linear transformations in the total configuration space. Let X be a configuration space which is assumed to be a Euclidean space R N unless specified otherwise. It is a representation space of a gauge group G so the action of G on X is given by a linear transformation x ™ V Ž v . x. A formal sum
1
On leave from: Laboratory of Theoretical Physics, JINR, Dubna, Russia. 2 Also Department of Physics.
over paths for the action invariant under gauge transformations would diverge. To regularize it, Faddeev and Popov have proposed to insert the identity w1x 1 s D FP Ž x .
H d m Ž V . dŽ x Ž V x . . ,
Ž 1.
where x Ž x . s 0 is a gauge condition and D FP is the corresponding Faddeev–Popov determinant, into the divergent path integral measure so that after a change of variables x ™ V x the Ždivergent. volume of the gauge group can be factorized out. The determinant D FP is calculated by doing the group averaging integral Ž1.. This formalism has provided a solution to two important problems in the perturbatiÕe path integral quantization of gauge theories: The unitarity of the
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 4 9 3 - 1
S.V. ShabanoÕ, J.R. Klauderr Physics Letters B 456 (1999) 38–47
perturbative S-matrix and the construction of a local effective gauge fixed action. Later on it has been observed that the procedure becomes ill-defined beyond perturbation theory w2x. A geometrical reason for the Gribov obstruction is that there is no gauge condition which would provide a global parameteriG by a set zation of the gauge orbit space M s XrG of affine coordinates without singularities w3x. This situation is often rendered concrete in the vanishing or even sign changing of the Faddeev–Popov determinant. An additional problem expected in the nonperturbative regime is the operator ordering ambiguity w4x. When the kinetic energy x˙ 2 is projected on ˙ g p h h˙ ., where h the gauge orbit space M , one gets Ž h, are local coordinates on M and g p h is an induced metric on M . The metric is not flat w5x. In order to establish a correspondence between the Dirac operator formalism and the gauge-fixed path integral, the latter must be appropriately defined to take into account the operator ordering in the physical kinetic energy operator. It is known that the formal path integral for motion on a curved manifold is ambiguous w6x. Up to the second order of perturbative Yang–Mills theory, operator ordering corrections have no effect on the physical S-matrix w4x. Their role in a nonperturbative description is unknown. A standard argument used in dimensional regularization, i" d 3 Ž0. s 0, Ža commutator of canonically conjugated field variables at the same spacetime point, which typically emerges in the operator ordering terms. does not seem to be applicable because the validity of dimensional regularization beyond perturbation theory is not justified w7x. Finally, the space M may have a nontrivial topology w3x. Due to the locality of the Faddeev–Popov construction, global properties of M are lost in the path integral. On the other hand the topology of the orbit space may be important for the spectrum of physical excitations. Feynman conjectured that the mass gap in gluodynamics in Ž2 q 1. spacetime might be caused by a compactification of certain directions in the configuration space upon an identification of gauge equivalent configurations w8x. In this letter we propose a modification of the procedure Ž1. which gives a solution to the above problems and leads to a well-defined path integral on the orbit space M . Its validity is demonstrated with a few examples.
39
2. A modified Kato–Trotter product formula Let Hˆ and sˆa be self-adjoint Hamiltonian and operators of constraints, respectively. We also asˆ sˆa x s 0 which holds for a sufficiently large sume w H, class of physically interesting gauge models. Let the gauge group be compact. Consider a projection operator Pˆ s
H dmŽ v .e
i v a sˆ a
,
H dmŽ v . s1 ,
Pˆ s Pˆ † s Pˆ 2 .
Ž 2.
The operator Ž2. projects the total Hilbert space onto the Dirac subspace of gauge invariant states. If the gauge group generated by sˆa is not compact, we take a sequence cd Pˆd of rescaled projection operators Pˆd , where Pˆd projects on the subspace Ý sˆa2 F d 2 , and then consider the limit d ™ 0 in a weak sense. The coefficients cd can always be chosen so that the limit exists and defines a Hilbert space isomorphic to the Dirac one w9x. Let the Hamiltonian ˆ where V s V Ž x . is an have the form Hˆ s Hˆ0 q V, everywhere regular potential and Hˆ0 s pˆ 2r2 is a kinetic energy operator. The operators Hˆ0 and Vˆ are assumed to be gauge invariant, i.e., they commute with the constraint operators 3. Consider the evolution operator UˆT s expŽyiTHˆ . in the total Hilbert space. The physical evolution operator is then UˆTD s UˆT Pˆ. Making use of the SincKato–Trotter product formula w11x for the evolution operator UˆT we get the following product formula for the gauge invariant evolution operator n
ˆ ˆ ˆ ˆ UˆTD s Ž eyi e H 0 eyi e V . Pˆ s Ž eyi e H 0 Pˆeyi e V . ˆ
n
n
n
' Ž Uˆe0 D eyi e V . ' Ž UˆeD . ,
Ž 3.
where e s Trn and n ™ `. In our previous work w10x we have shown that in the continuum limit the averaging gauge variables v play the role of the Lagrange multipliers for the constraints in the classi-
3
This assumption is justified for gauge theories of the Yang– Mills type. More general cases will be considered elsewhere because they are technically more involved and require a phasespace representation of quantum theory and a Wiener measure regularization of the corresponding path integral as proposed in Ref. w10x.
S.V. ShabanoÕ, J.R. Klauderr Physics Letters B 456 (1999) 38–47
40
cal action. By making use of the classical theory of Kolmogorov, we gave an explicit construction of a countably additive probability measure for the gauge functions v Ž t .. One natural choice was such that any set of values of v Ž t . at any set of distinct times is equally likely. Now we show that this concept leads to a modification of the Faddeev–Popov procedure. First we observe that by construction UTD
X
Ž x,V x .
s UTD
X
Ž V x, x .
s UTD
X
Ž x, x .
Ž 4.
that is, the projected amplitude determines a genuine transition amplitude on the orbit space M . To obtain its relation to the conventional gauge fixed amplitude, we introduce a set of local coordinates h on M via a change of variables xsV Ž v . h ,
UTD Ž h,hXs . s UTD Ž h s ,hX . s UTD Ž h,hX . .
Ž 7.
It determines an S-inÕariant continuation of the transition amplitude to the coÕering space H of the modular domain K. The Faddeev–Popov determinant specifies the volume element on M w5x, so for any integral involving only functions on M , f Ž x . s f Ž V x ., one has H d xf Ž x . s V G HK d h D FP Ž h. f Ž h., where V G is the volume of the gauge group. Therefore the folding of two projected infinitesimal evolution operators reads
Ž 5.
where the configurations x s h satisfy a gauge condition x Ž x . s 0, i.e., h g H ; X and without loss of generality we may assume H ; R M Žunless specified otherwise. with M being the number of physical degrees of freedom. The gauge invariant amplitude Ž4. can be reduced on the gauge fixing surface H : UTD Ž x, xX . s UTD Ž h,hX .. This leads to a necessary modification of the procedure Ž1.: Instead of averaging the gauge condition over the gauge group, we propose to average an infinitesimal evolution operator kernel for a free motion as follows from the modified Kato–Trotter product formula Ž3. Ue0 D Ž h,hX . s Ž 2p i e .
x s h on the gauge fixing surface x Ž x . s 0. If S is a collection of all residual gauge transformations, the modular domain K is the quotient HrS. The relation Ž5. determines a one-to-one correspondence if h g K and v parameterizes a group manifold. If h,hX g K, then the following relation holds
yN r2
H dmŽ v .e
X
iŽ hy V Ž v . h . 2 r2 e
.
Ž 6.
Since e ™ 0, the averaging integral can be done to proper order in e explicitly by the stationary phase approximation. The kernel UeD Ž h,hX . is obtained by adding yi e V Ž h. to the exponential in Ž6. in accordance with Ž3. because V Ž x . s V Ž h. thanks to the gauge invariance of V. Suppose there exist transformations of the new variables v and h in Ž5., v ™ v q v s and h ™ h s , such that x Ž v ,h. s x Ž v q v s ,h s ., where all configurations x s s h s also satisfy the gauge condition x Ž h. s x Ž h s . s 0. From V Ž v q v s . h s s V Ž v . h follows that h s s V s h, i.e., h s is a Gribov copy of
U2De Ž h,hX . s
HK d h D 1
FP
Ž h1 . UeD Ž h,h1 . UeD Ž h1 ,hX . . Ž 8.
The restriction of the integration domain in Ž8. does not lead to a formal restriction of the integration domain in the path integral resulting from Ž3. in the continuum limit as one might naively expect from Ž8.. An important point is that, if K / H Žthe Gribov problem is present., then Ue0 D Ž h,hX . has no standard form of an infinitesimal free transition amplitude on a manifold proposed in Ref. w6x as we now proceed to demonstrate. The stationary phase approximation can be applied before the reduction of UeD Ž x, xX . on a gauge fixing surface. A deÕiation from the conventional gauge-fixing procedure results from the fact that there may be more than just one stationary point. We can always shift the origin of the averaging variable v so that one of the stationary points is at the origin ˆ a be operators generating gauge transv s 0. Let d formations in X . Decomposing the distance Ž x y V Ž v . xX . 2 in the vicinity of the stationary point, we ˆ a xX . s 0. So in the formal continuum find Ž x y xX ,d limit we get the condition saŽ x, ˙ x . ' Ž x,d ˙ ˆ a x. s0 induced by the averaging procedure. This is nothing but the Gauss law enforcement for trajectories contributing to UTD Ž x, xX .. Suppose there exists a gauge condition x aŽ x . s 0, which involves no time derivatives, such that a generic configuration x s h satisfy-
S.V. ShabanoÕ, J.R. Klauderr Physics Letters B 456 (1999) 38–47
ing it also fulfills identically the Gauss law, i.e., ˙ ˆ a h. ' 0. We will call it a natural gauge. In this Ž h,d case all other stationary points in the integral Ž6. are v c s v s where V Ž v s . s V s g S. Therefore we get a sum over the stationary points in the averaging integral Ž6. if the Gribov problem is present. Still, in the continuum limit we have to control all terms of order e . This means that we need not only the leading term in the stationary phase approximation of Ž6. but also the next two corrections to it. Therefore the group element V Ž v . should be decomposed up to order v 4 because v 4re ; e as one is easily convinced by rescaling the integration variable v ™ 'e v . The measure should also be decomposed up to the necessary order to control the e-terms. The latter would yield quantum corrections to the classical potential associated with the operator ordering in the kinetic energy operator on the orbit space. We stress that the averaging procedure gives a unique ordering so that the integral is invariant under general coordinate transformations on M , i.e., does not depend on the choice of x Žcf. Ž14... Thus, Ue0 D Ž h,hX . s Ž 2p i e .
yM r2
Ý Dy1r2 Ž h,hXs . S
X
2
X
= ei Ž hyh s . r2 eyi e Vq Ž h , h s . q O Ž e 2 . ,
Ž 9. &
' Ý Dy1r2 Ž h,hXs . Ue Ž h,hXs . ,
Ž 10 .
S
where DŽ h,hX . is the conventional determinant arising in the stationary phase approximation, and by Vq we denote a contribution of all relevant corrections to the leading order. The amplitude UeD Ž h,hX . is obtained by adding yi e V Ž h. to the exponential in Ž9.. Note that V Ž h. s V Ž h s . thanks to the gauge invariance of the potential. In general, the equations saŽ x, ˙ x . s 0 are not integrable therefore the functions x a discussed above do not exist. In this case we consider two possibilities. Let V c Ž h,hX . be the group element at a stationary point in Ž6.. Decomposing the distance in the vicinity of the stationary point we get Ž h y ˆ a V c hX . s 0. Let x be such that the latter V c hX ,d condition is also satisfied if hX is replaced by hXs . In this case the sum over the stationary points is again
41
given by Ž9. where in the first term of the exponential hXs is replaced by V c Ž h,hXs . hXs . In the most general case, the sum over stationary points does not coincide with the sum over Gribov copies. However for sufficiently small e , the averaged short-time transition amplitude can always be represented in the form Ž10.. Note that as e goes to zero, Ue0 Ž x, xX . ™ d N Ž x y xX .. If we change the variables in the delta function according to Ž5. and then do the gauge group average, we get, in general, a sum of delta functions on the gauge fixing surface. From the symmetry of the change of variables Ž5. it follows that the support of the averaged delta function is formed by points h s hXs , hX g K and h g H , which means that the sum over S emerges again for sufficiently small e . We will explain below how to calculate relevant corrections to the leading order of the limit e ™ 0. Let us replace UeD Ž h,h1 . in Ž8. by the sum Ž10. and make use of Ž7. applied to the second kernel in Ž8.: UeD Ž h1 ,hX . s UeD Ž h1 s ,hX .. Since the measure on the orbit space does not depend on a particular choice of the modular domain, d h s D FP Ž h s . s d h D FP Ž h., we can extend the integration to the entire covering space by removing the sum over S U2De Ž h,hX . s
Hdh < D 1
FP
Ž h1 . < Dy1 r2 Ž h,h1 .
&
=Ue Ž h,h1 . UeD Ž h1 ,hX . . Ž 11 . The absolute value bars accounts for a possible sign change of the Faddeev–Popov determinant on the gauge fixing surface. The procedure can be repeated from left to right in the folding Ž3., thus removing the restriction of the integration domain and the sum over copies in all intermediate times t g Ž0,T .. The sum over S for the initial configuration hX remains in the integral. Now we can formally take a continuum limit with the result UTD Ž h,hX . s Ý D FP Ž h . D FP Ž hXs .
y1 r2
S h ŽT .sh
=
H
(
D h det g p h X
h Ž0 .sh s T
˙
˙
=e iH0 d t wŽ h , g p h h .r2yVq Ž h .yV Ž h .x , Ž 12 . where g p h is the induced metric on M . The local density Ł t det g p h should be understood as the
(
S.V. ShabanoÕ, J.R. Klauderr Physics Letters B 456 (1999) 38–47
42
result of the integration over the momenta in the corresponding time-sliced phase-space path integral . where the kinetic energy is Ž p h , gy1 p h p h r2 with p h being a canonical momentum for h. To obtain Ž12., one usually sets hX s h y D for h s hŽ t . and hX s hŽ t y e . in each intermediate moment of time t and makes a decomposition into the power series over D. According to the relation, obtained in Ref. w5x, between the volume of a gauge orbit through x s h, the induced metric g p h , and the Faddeev–Popov 2 Ž X. determinant we get D Ž hX q D, hX . s D FP h r X det g p hŽ h . q O Ž D .. This relation explains the cancellation of the absolute value of the Faddeev–Popov determinant in the folding Ž3. made in accordance with the rule Ž11.. The term D 2re in the exponential Ž9. gives rise to the kinetic energy Ž D, g p h D .r2 e q O Ž D 3 .. The most difficult part to calculate are the operator ordering corrections Vq Ž h. in the continuum limit. Here we remark that DŽ h,hX . has to be decomposed up to order D 2 , while the exponential in Ž9. up to order D 4 because the measure has support on paths for which D 2 ; e Ži.e., D 4re ; e .. There is a technique, called the equivalence rules for Lagrangian path integrals on manifolds, which allows one to convert terms D 2 n into terms e n and thereby to calculate Vq w6,12x
D j1 PPP D j 2 k™ Ž i e " . =
Ý
k
Ž gy1 ph .
j1 j 2
PPP Ž gy1 ph .
j 2 ky 1 j 2 k
,
Ž 13 .
p Ž j1 , . . . , j 2 k .
where the sum is extended over all permutations of the indices j to make the right-hand side of Ž13. symmetric under permutations of the j’s. Following Ž13. one derives the Schrodinger equation for the ¨ physical amplitude Ž12.. The corresponding Hamiltonian operator on the orbit space has the form Hˆp h s y1r2 Dy1 FP E j
ž
gy1 ph
jk
D FP E k q V Ž h . ,
/
Ž 14 . where E j s ErE h j . Observe that the kinetic energy in Ž14. does not coincide with the Laplace–Beltrami operator on M because wdet g p h x1r2 / D FP as shown
in Ref. w5x. The operator Ž14. is hermitian with respect to the scalar product where the volume element is d h D FP . It is also invariant under general coordinate transformations on M , i.e., its spectrum does not depend on the choice of local coordinates on M and, therefore, is gauge invariant. We skip technical details of the derivation of Ž14. because they are standard in the path integral formalism and instead turn to examples to illustrate the main features of the new projection formalism. But before we do so, let us comment that the effects on the energy spectrum caused by the modification of the path integral can be found from the pole structure of the trace of the resolvent tr Rˆ Ž t . s tr Ž t y iHˆ . tr UˆTD s
HK d h D
FP
`
y1
s
H0
dT ey t T tr UˆTD ,
Ž h . UTD Ž h,h . .
Ž 15 . Ž 16 .
3. Examples Let us take the simplest example w4x where x g R 3 and the gauge transformations are rotations of x about the origin. The orbit space is a space of concentric spheres. The natural gauge is x i s d i1 h. This gauge is not complete because there are rotations that change the sign of h: h ™ "h. These transformations form the group S s Z 2 . The modular domain is an open half-axis K s Ž0,`.. The measure on the orbit space is obtained by introducing spherical coordinates on R 3 so that h s < x <, while the spherical angles parameterize the gauge group manifold SOŽ3.rSOŽ2.. It is noteworthy to observe that in the projection formalism there is no principal difference between reducible and irreducible gauge theories. Here we obviously have a reducible case because x has a stationary group SOŽ2.. The Faddeev–Popov determinant is defined with respect to a subset of independent constraints. In this case, the constraints are three components of the angular momentum. So we can take the angular momenta about the axes i s 2,3 as independent constraints and find that D FP s h 2 in the full accordance with the measure resulting from the spherical coordi-
S.V. ShabanoÕ, J.R. Klauderr Physics Letters B 456 (1999) 38–47
nates. The angular measure must be normalized by 4p to fulfill the condition in Ž2.. So Eq. Ž6. assumes the form Ue0 D Ž h,hX . s Ž 2p i e . p
=
H0
y3 r2
1r2
d v sin v e iŽ h
2
X2
X
qh .r2 e yi h h cos v r e
.
Ž 17 . Doing the averaging integral exactly and substituting the result into Ž3. we obtain UTD Ž h,hX . s w hhX x
y1
UT Ž h,hX . y UT Ž h,y hX . 4 , Ž 18 .
where the amplitude UT Ž h,hX . is given by a standard path integral for a one-dimensional system with the gauge-fixed action S g f w h x s H0T d t w h˙ 2r2 y V Ž h.x. There is no operator ordering correction. The stationary phase approximation leads to the same result if one takes into account all the stationary points, i.e., v s s 0 and v s s p . The second stationary point is associated with the only nontrivial residual gauge transformation hX ™ yhX . Here one also gets Vq s 0 Žcf. Ž9... The same path integral can be derived from the Schrodinger equation projected ¨ on the orbit space w13x. It is not hard to calculate the partition function Ž16. for a harmonic oscillator V s x 2r2 s h2r2 and find the resolvent Ž15.. The poles determine the spectrum En s 2 n q 3r2. The distance between the energy levels is 2, while the oscillator frequency in the Hamiltonian is 1. The effect of the doubling would be lost, had we ignored the redundant discrete gauge symmetry and calculated the partition function by a standard path integral for the gauge-fixed action S g f . A simple model to illustrate the effect of the residual gauge symmetry in systems with several physical degrees of freedom is a Yang–Mills theory in Ž1 q 0. spacetime w13x. A total configuration space is a Lie algebra X, and the gauge group acts in the adjoint representation x ™ V x Vy1 . The quadratic form in the exponential Ž6. is taken with respect to the Killing form on X: Ž x, x . ' tr xˆ 2 , where xˆ y ' w x, y x for any x, y g X and the commutator stands for the Lie bracket in X. If a matrix representation is assumed, for compact groups it can always be nor-
43
malized to the ordinary matrix trace. The Gauss law is s s w x, ˙ x x s 0, therefore the natural gauge condition to parameterize M is x s h g H where H is a Cartan subalgebra Ža maximal commutative subalgebra of X .. The system has r s dim H physical degrees of freedom. The invariant measure d m Ž v . is proportional to d v det wŽexp vˆ y 1. vˆ y1 x. In the exponential in Ž6. we get Ž h y expŽ vˆ . hX . 2 . An adjoint transformation of x to H is not unique and determined modulo the Weyl group W acting in H. The Weyl transformations are inequivalent compositions of reflections in the hyperplanes orthogonal to simple roots in H w19x. So S s W and M ; HrW s Kq, where Kq is the Weyl chamber in H. The stationary points of the integral Ž6. are v c s v W where the adjoint action of the group element exp v W on h transforms the latter by a Weyl reflection. Introducing the Cartan-Weyl basis w19x one can find the Jacobian of the change of variables Ž5. Žthe Faddeev–Popov determinant. D FP Ž h. s det hˆ s k 2 Ž h. where k Ž h. s Ł a ) 0 Ž a ,h.; the product is extended over all positive roots in H. The operator hˆ acts in the orthogonal complement of the Cartan subalgebra. Note that the Cartan group is the stationary group of h g H, so v should take its values in the orthogonal supplement of the Cartan subalgebra. The factor DŽ h,hX . is given by the determinant of the operator ˆˆX . Doing the stationary phase approximation in the hh averaging integral, and then taking the folding Ž3. we find UTD Ž h,hX . s Ý k Ž h . k Ž hXW .
y1
UT Ž h,hXW . .
W
Ž 19 . Here the amplitude UT Ž h,hX . is given by the conventional path integral Žno restriction of the integration domain to the Weyl chamber. with the action H d t Ž h˙ 2r2 y V Ž h... There is no operator ordering correction, and the physical metric is Euclidean. The same integral follows from a direct solution of the Schrodinger equation w13x. The Faddeev–Popov de¨ terminant vanishes at the hyperplanes Ž h, a . s 0. The boundary of the modular domain Žthe Weyl chamber. is formed by these hyperplanes when a runs over the simple roots. The amplitude Ž19. remains finite at the singular points Žthe Gribov horizon. because k Ž hW . s p W k Ž h. where p W s "1 is
S.V. ShabanoÕ, J.R. Klauderr Physics Letters B 456 (1999) 38–47
44
the parity of the corresponding Weyl transformation Žcf. also Ž18.: It remains finite at h s 0 or hX s 0... Contributions of paths hitting the horizon have a dramatic effect on the spectrum of the isotropic oscillator V s x 2r2 s h2r2. Doing the path integral for UT Ž h,hX . in Ž19. and calculating the poles of the resolvent Ž15. we find w13x En s n 1 n1 q PPP qnr n r q Nr2, where n i are non-negative integers, n i are orders of invariant irreducible symmetric tensors of the Lie algebra, and r and N are, respectively, the rank and dimension of the group Že.g., for SUŽ r q 1., n i s i q 1, i s 1,2, . . . , r .. Thus, the physical frequencies of the isotropic oscillator appear to be n i / 1. The partition function calculated by a standard path integral for the gauge fixed action of the model would lead to the spectrum of the isotropic oscillator Ži.e., n i ™ 1.. To illustrate the effects of curvature of the orbit space, we consider a simple gauge matrix model. Let x be a real 2 = 2 matrix subject to the gauge transformations x ™ V Ž v . x where V g SOŽ2.. An invariant scalar product reads Ž x, xX . s tr x T xX with x T being a transposed matrix x. The total configuration space is R 4 . Let ´ be a generator of SOŽ2., i.e., ´ i j s y´ ji , and ´ 12 s 1. Then V Ž v . s expŽ v ´ .. The Gauss law enforced by the projection is s s tr x˙ T ´ x s 0. It is not integrable. We parameterize M by triangular matrices h 21 ' 0 Žthe gauge x 21 s 0.. The residual gauge transformations form the group S s Z 2 : h ™ "h. The modular domain is a positive half-space h11 ) 0. Calculating the Jacobian of the change of variables Ž5. for this model we find the Faddeev–Popov determinant D FP Ž h. s h11. The plane h11 s 0 is the Gribov horizon. The averaging measure in Ž6. reads Ž2p .y1 d v , where v g w0,2 p .. The quadratic form in the exponential in Ž6. is v
X 2
X
X
X
Ž h y e ´ h . s Ž h,h . q Ž h ,h . y 2 Ž h,h . cos v y 2 Ž h,´ hX . sin v .
Ž 20 .
A distinguished feature of this model from those considered above is that the stationary point is a function of h and hX . Taking the derivative of Ž20. with respect to v and setting it to zero, we find
v c s tany1
Ž h,´ hX . , v cs s v c q p . Ž h,hX .
Ž 21 .
The second stationary point v cs is associated with the Gribov transformation h ™ yh. A geometrical meaning of the transformation hX ™ expŽ v c ´ . hX is transparent. The distance wŽ h y hX . 2 x1r2 between two points on the gauge fixing plane is greater than the minimal distance between the two gauge orbits through x s h and xX s hX . By shifting xX along the gauge orbit to xXc s expŽ v c ´ . hX a minimum of the distance between the orbits is achieved. In such a way the metric on the orbit space emerges in the projection formalism. Its explicit form can be found as has been explained in Section 2. We substitute v s v c Ž h,hX .into Ž20., set hX s h y D and decompose Ž20. in a power series over D. The quadratic term Žthe leading term. determines the metric. We get Ž D, g p hŽ h. D . s Ž D, D . q Ž D,´ h.Ž ´ h, D .rŽ h,h.. The metric is not flat. The scalar Riemann curvature of M is R s 6rŽ h,h. s 6rŽ x, x .. Note that it is gauge invariant as it should be because the curvature R does not depend on the choice of local coordinates on M . In the stationary phase approximation the cosine and sine in Ž20. should be decomposed up to fourth order in the vicinity of the stationary point. In this model quantum corrections do not vanish. The infinitesimal transition amplitude on M assumes the form UeD Ž h,hX . s Ž 2p i e .
y3 r2
X
Dy1r2 Ž h,hX . e i Se Ž h , h . X
qDy1r2 Ž h,y hX . e i Se Ž h ,y h . ,
Ž 22 .
Se Ž h,hX . s Ž h,h . q Ž hX ,hX . y 2 D Ž h,hX . r Ž 2 e . y e 8 D Ž h,hX .
y1
y e V Ž h. ,
Ž 23 .
where DŽ h,hX . s Ž h,hX .cos v c q Ž h,´ hX .sin v c . Observe that the middle term in Ž23. is a quantum correction to the classical potential in the short-time action Ž23. Žit is proportional to " 2 , if one restores the Planck constant.. The two first terms in Ž23. determine Ž uniquely!. the operator ordering in the kinetic energy operator on the orbit space in accordance with Ž14.. Finally, we consider the case of an infinite number of stationary points in the averaging integral Ž6.. This would have an effect that the modular domain K may become compact, which, in turn, leads to a
S.V. ShabanoÕ, J.R. Klauderr Physics Letters B 456 (1999) 38–47
discrete spectrum regardless of the details of the potential and therefore gives a mechanism for the gap between the ground state and the first excited state. A suitable soluble model is the Yang–Mills theory in cylindrical space time w14–16x where space is compactified into a circle. A Yang–Mills theory without matter in two dimensions has no physical degrees of freedom unless the spacetime has a nontrivial topology. The latter is exactly our case. The total configuration space X is the space of all periodic connections AŽ x q 2p . s AŽ x . taking their values in the Lie algebra of a compact Lie group. The gauge group G acts on X as A ™ VA s V A Vy1 y i VEVy1, and V Ž x q 2p . s V Ž x . Ži.e., the gauge group elements V can be continuously deformed towards the group unit w17,18x.. The exponential in Ž6. assumes the form H02 p d x trŽ A yVAX . 2 for any two configurations AŽ x . and AX Ž x .. It is the distance between AŽ x . and VAX Ž x . introduced by Feynman w8x. A rigorous definition of the averaging integral over G can be given within the Kogut–Susskind lattice gauge theory, to which we turn shortly. Note, for example, that N in Ž6. is infinite because the number of degrees of freedom is determined by the number of Fourier modes of AŽ x .. Here we assume the existence of a normalized averaging measure on G . The gauge group average enforces the Gauss law s s E A˙y iw A, A˙x ' =Ž A. A˙ s 0. The orbit space M can be parameterized by constant connections AŽ x . s h taking their values in the Cartan subalgebra H w14,16x. This is the natural ˙ . ' 0. The number of the physgauge because s Ž h,h ical degrees of freedom equals the rank of the gauge group. We set V Ž x . s expŽ i v Ž x ... The averaging function v takes its values in the Lie algebra and its constant Žindependent of x . Fourier mode does not have any component in the Cartan subalgebra because the connection AŽ x . s h is inÕariant under constant gauge transformations from the Cartan subgroup. The Gaussian integral over such functions can formally be done in Ž6. yielding Dy1 r2 Ž h,hX . s dety1 r2 wy=Ž h.=Ž hX .x which is easy to compute in the Cartan-Weyl basis w16x. In the lattice formulation described below this procedure would not involve any infinite factors in DŽ h,hX . as in the formal continuum limit. Here we draw attention to the following feature. A projection of any connection AŽ x . into the subspace of constant Cartan connec-
45
tions h is not unique and determined modulo the affine Weyl transformations w16x. The affine Weyl group WA is a semidirect product of the Weyl group W and the group of translation along the group unit lattice in the Cartan subalgebra w19x. Thus, the number of stationary points in the averaging integral Ž6. is infinite because the number of elements WA is infinite: h ™ h s ' hW q 2 a narŽ a , a ., where hW is a Weyl transformation of h, a any positive root, na an integer. The modular domain K s HrWA is called the Weyl cell w19x. It is compact in H. This has a significant effect on the spectrum of the system. In the continuum limit the transition amplitude has the form Ž19. where the sum is extended over the affine Weyl group Žover an infinite number of stationary points hXs . w16x, the Faddeev–Popov determinant reads D FP Ž h. s k 2 Ž h. and k Ž h. s Ł a ) 0 sinŽ h, a .. There is no potential energy in 1 q 1 dimensional Yang–Mills theory Ž V Ž h. ' 0 in Ž19... Thus, we get an amplitude for a free particle symmetrized over the affine Weyl group. In the simplest SUŽ2. case, the system has only one physical degree of freedom. The Weyl transformations are reflections hW s "h and the translations over the group unit lattice are shifts h ™ h q 2 n. The amplitude Ž19. appears similar to the one for a particle in an infinite well or on a circle. The spectrum of such a system is discrete. In general, we can calculate the poles of the resolvent Ž15. for the 2D Yang–Mills theory using our path integral formalism and find w16x the spectrum En s E0 wŽgn ,gn . y Ž r , r .x where E0 is a constant, r is a half the sum of all positive roots, and gn ranges over a lattice in the Weyl chamber that labels all irreducible representation of the gauge group. Thus, an infinite number of stationary points in the averaging integral Ž6. is, in fact, evidence that the physical configuration space Žor its subspace. is compact. This is the mathematical structure of the path integral which is to be sought when studying the relation between the mass gap in gluodynamics and the compactness of the orbit space. Physically, the existence of an infinite number of the stationary points separated by different distances Žlarge and small. implies that physical eigenstates of the Hamiltonian should exhibit a periodicity in some directions in X , which may cause the mass gap as has been argued by Feynman w8x.
S.V. ShabanoÕ, J.R. Klauderr Physics Letters B 456 (1999) 38–47
46
4. Kogut–Susskind lattice gauge theory To fulfill our program in realistic Yang–Mills theory, we need a regularization at short distances to give a rigorous meaning to the averaging integral Ž6. which is a functional integral. The simplest possibility is to set the system on a lattice. Since we still want to have a Hamiltonian formalism, the time should remain continuous. With such a choice of the regularization we arrive at the Kogut–Susskind lattice gauge theory w20x. Let y be a three-vector whose components are integer-valued in the lattice spacing unit a, i.e., y labels the lattice sites. With each link connecting the lattice sites y and y q ia, where i is the unit vector in the direction of the ith coordinate axis, we associate a group element u y,i called the link variable. The collection of all values of all link variables forms the configuration space X of the system. The gauge group G acts on X by the rule u y,i ™ V y u y,i Vy1 yqi a . Therefore the averaging measure is a product of the normalized Haar measures at each site: d m s Ł y d m H Ž V y .. The Kogut–Susskind Hamiltonian is H s H0 q V where the potential V is nothing but the color magnetic energy on the lattice. The kinetic energy is the sum of kinetic energies of a rigid rotators associated with each link variable 2
H0 s 1r2 Ý tr Ž u˙ y ,i uy1 y ,i . .
Ž 24 .
y,i
An important observation is that the coupling of the rotators occurs only through the potential V. Therefore the ‘‘free’’ amplitude UT0 Ž u4 , uX 4., where u4 is a collection of configurations of rotators u y,i , is factorized into a product of the amplitudes for each rotator UT0 Ž u y,i ,uXy,i .. An explicit form of the latter is well-known w21x Ža transition amplitude of a particle moving on a group manifold.. This replaces the free motion amplitude in a Euclidean space in the averaging integral Ž6.. The averaging integral Ue0 D Ž u4 , uX 4 . s
H Ły
d mH Ž V y .
= Ł Ue0 Ž u y ,i , V y uXy ,i V yqai . y,i
Ž 25 .
can be calculated via both the stationary phase approximation and the decomposition over the characters of the irreducible representations of the amplitudes UT0 Ž u,uX . proposed in Ref. w21x. Thus, we have reduced the problem of constructing the transition amplitude on the orbit space to a standard mathematical problem in group theory. It is important that the effects of the Riemann structure Žmetric. and topology of the orbit space are completely determined by the averaging procedure proposed and do not depend on the details of the self-interaction V. Note that the average in Ž25. induces a nontriÕial coupling between the rotators because six link variables attached to the same site are transformed with the same group element. So the factorization of the amplitude on M is no longer possible. This is the result of the enforcement of the Gauss law in the physical amplitude. Thus, we get, in principle, a constructive and consistent procedure to study the effects of the nonEuclidean geometry of the orbit space on physical quantities like the spectrum of low energy excitations. In particular, in the strong coupling limit Ž g large. the potential V can be treated by perturbation theory w20x. To introduce an explicit parameterization of the orbit space, one can, for example, make use of Morse theory for the Kogut–Susskind lattice gauge theory as proposed in Ref. w22x. In this case, the initial configuration uX 4 in the averaging integral Ž25. is taken from the corresponding modular domain. The projected amplitude has a natural gauge invariant continuation to the covering space of the modular domain.
5. Conclusions We have proposed a self-consistent path integral quantization of gauge theories which resolves the Gribov obstruction and provides control for the operator ordering problem, and it does not rely explicitly on the Schrodinger equation on the orbit space. This ¨ was our main goal. The continuum limit in the Kogut–Susskind lattice gauge theory might still be a difficult analytical problem. However, it is believed that the techniques we have developed might be useful to study the effects of the non-Euclidean geometry and topology of the orbit space in some
S.V. ShabanoÕ, J.R. Klauderr Physics Letters B 456 (1999) 38–47
physically reasonable approximations. We have also demonstrated its effectiveness with several examples.
References w1x w2x w3x w4x w5x w6x w7x
L.D. Faddeev, V.N. Popov, Phys. Lett. B 25 Ž1968. 30. V.N. Gribov, Nucl. Phys. B 139 Ž1978. 1. I. Singer, Commun. Math. Phys. 60 Ž1978. 7. N. Christ, T.D. Lee, Phys. Rev. D 22 Ž1980. 939. O. Babelon, C.-M.Viallet, Phys. Lett. B 85 Ž1979. 246. B. De Witt, Rev. Mod. Phys. 29 Ž1957. 377. J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 2nd Edition, Clarendon Press, Oxford, 1990. w8x R. Feynman, Nucl. Phys. B 188 Ž1981. 479. w9x J.R. Klauder, Ann. Phys. ŽNY. 254 Ž1997. 419.
47
w10x J.R. Klauder, S.V. Shabanov, Phys. Lett. B 398 Ž1997. 116. w11x E. Nelson, J. Math. Phys. 5 Ž1964. 332; T. Kato, H.F. Trotter, Pacific Math. J. 8 Ž1959. 545. w12x L.V. Prokhorov, Sov. J. Part. Nucl. 13 Ž1982. 456. w13x L.V. Prokhorov, S.V. Shabanov, Phys. Lett. B 216 Ž1989. 292; Sov. Phys. Uspekhi 34 Ž1991. 108. w14x S.G. Rajeev, Phys. Lett. B 212 Ž1989. 203. w15x E. Langman, G.W. Semenoff, Phys. Lett. B 303 Ž1993. 303. w16x S.V. Shabanov, Phys. Lett. B 318 Ž1993. 323. w17x G.’t Hooft, Nucl. Phys. B 153 Ž1979. 141. w18x R. Jackiw, Rev. Mod. Phys. 52 Ž1980. 661. w19x S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978. w20x J. Kogut, L. Susskind, Phys. Rev. D 11 Ž1976. 395. w21x M.S. Marinov, M.V. Terentiev, Fortshcr. Phys. 27 Ž1979. 511. w22x D. Zwanziger, Nucl. Phys. B 485 Ž1997. 185.