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The gluon propagator in temporal gauge: A simple derivation based on stochastic mechanics
Volume 149B, number 1,2,3
PHYSICS LETTERS
13 December 1984
THE GLUON PROPAGATOR IN TEMPORAL GAUGE: A SIMPLE DERIVATION BASED ON STOCHASTIC MECHANICS S.C. LIM
Department of Physics, Fakulti &tinsFizis & Gunaan, UniversitiKebangsaan Malaysia, Bangi, Selangor, Malaysia Received 2 July 1984
The stochastic quantization scheme based on Nelson's stochastic mechanicsis used to derive the non-abeliangauge field propagator. The result for the longitudinal part of the gluon propagator is consistent with that obtained by Caracciolo, Curci and Menotti.
Temporal gauge has proved to be useful in the perturbative calculation of QCD mainly because of the absence of Faddeev-Popov ghosts and the fact that the hamiltonian in this gauge has a very simple form. It is also very suitable for the study of the topological properties of non-abelian gauge theories. However, the temporal gauge condition Ag = 0 does not fix the gauge completely. The residual gauge invariance under the time-independent gauge transformations gives rise to an ambiguous 1/p~ pole in the gluon propagator. The usual way of regularizing this singularity is by using the principal value prescription [ 1]. Recently, several authors have raised doubts on the validity of this regularlzation method. Calculations by Mtiller and Rtihl [2] on the small couphng of a non-abelian gauge field on a lattice in temporal gauge have shown that the correct longitudinal part of the propagator differs from that derived from the principal value prescription. A similar conclusion has been obtained by Caracciolo et al. [3] by considering the Wilson loop in temporal gauge. They have found that the principal value prescription does not lead to the correct exponentiation of the time-dependence of the Wilson loop operator. In order to give a result which is in agreement, up to order g4, with that of Feynman and Coulomb gauges the longitudinal component of the gluon propagator needs to be of the form
(01T(AaL(x, t)Ab L(X', t'))10) =
--}i8 ab [It - t'l + (t + t') + a]
x (aio//v2)
(1)
(3)(x - x'),
where t~ is a constant. In comparison, the principal value prescription gives the following result: (01T(AaL(x, t)A b L(x', t'))l 0)
= - ~ i S a b l t - t'l(aiOl/V2)8(3)(x-x').
(2)
In this paper we shall apply the quantization method of Nelson's stochastic mechanics [4] to the classical gauge field in minkowskian space-time with the aim of obtaining a result for the free gluon propagator which is in agreement with eq. (1). First, recall the basic equations of the classical non-abelian gauge field A a in temporal gauge. The lagrangian of a pure nonabelian gauge theory without fermions is
z = f(eg
7 -
g87) d3x,
(3)
where the colour electric and magnetic fields are given by
E(]=-aoAOi,
Ba=--~eiikqk .
(4)
The space components of the field strength tensor are
a a)
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Volume 149B, number 1,2,3
PHYSICS LETTI- RS
where g is the coupling constant and fabe are the structure constants of the non-abelian group. The colour magnetic field can also be expressed as
B a = V X A a + ~gfabcAb X A e .
13 December 1984
E~) = +[coA a + co-i V(V.Aa)], into eq (9) to obtain (6)
dAaV(x, t) = -COAaT(x, t) dt + dW al (x, t),
The equation of motion is
and
a2A a
dAaL(x, t) = dWaL(x, t),
_ V 2 A a + V(V.A a)
= --gfabc(Ab X (V X AC)),
(lS)
(7)
where V and 7 2 respectively denote the gradient and laplacian in R 3. For the determination of the free gauge field propagator one needs to consider only the linear part ofeq. (7):
(l 6)
(17)
where co - ( - V 2 ) 1/2. Eq. (16) has a genre al soluhon of tile form _ aT AaT(x, t) -A(o)(X, t) +AaW(x) exp(-cot), (18)
Nelson's stochastic quantization requires A a be promoted to a Markov process satisfying the following stochastic differential equation of Langevin type:
whme Aaf~(x, t)is the solution satisfying the mmal condmon Aa(~(x, 0) = 0, and AaT(x) denotes an albltraJy mmal value of AaT(x, t). Clearly,AaT(x, t) is mdependent of the imtlal value AaT(x) for large t. By takmgAaT(x) = 0 one gets for the solution ofeq. (16) a gaussmn random field with mean zero and covanance gwen by
dAa(x, t) = D(+)Aa(x, t) dt + dWa(x, t ) .
E[AaT(x, t)Ab T(x ', t')]
~4 a = V2A a - V(V "Aa).
(8)
(9)
D(+) denotes the forward time derivative as defined in ref. [4]. Wa is the brownian disturbance field specified by the following expectation values: E[dW~/(x, t)l = 0 ,
(lOa)
E [dWa(x, t) dWf(x', t)] = 6abso6(3)(X - x ' ) dt. (10b)
becomes
(lI)
i.e. the colour electric field splits into a pair of functionals R a ('D(_) is the backward time derivative). Eq. (8) takes the following stochastic form:
~(D(+)D(_) + D(_)D(+))A a = V2A a - V (V "Aa), (12) or
~(D(+)E(_)' a +D(_)E&))=-V2A a +V(V-Aa).
(13)
It is convenient to resolve A a into the transverse and longitudinal components A aT and A aL with V'AaT = 0 ,
VXAaL=0.
(14)
Following the same argument as given in ref. [5] one can substitute
202
X exp [-co(p)lt - t'[ + lp(x - x ' ) ] ,
(19)
with co(p) = (p2)1/2. By using the Fourier reversion formula
e x p [ - w ( p ) l t - t't] = 1 f 2co(p)
The stochastic form of the equation aOAa = - E a
D(+)Aa(x,t)= -E~(+)(A],x, t),
= (27r)-38abf d3p gg.. 2co(p) "v'l - PiP]/P2)
e x p 0 p 4 ] t - t'[) oo2 + p2
dP4, (20)
one can identify eq. (19) with the two-point function of the euchdean field AaT(x), x =--(x, x4) provided t IS taken as x4, just as in the scalar case [5]. This interpretation of the stochastic field also explains the absence of the factor i m the covanance (l 9) as compared with the propagator obtained by using the conventional quanhzation method. Due to the presence of a damping drift term in the stochastic differential equation for A aT, it does not depend on the mitml distribution for large t. However this is not the case for A aL. The stochastic differential eq. (17) for the longitudinal components A aL does not possess a damping drift term. This means that the longitudinal modes of the gauge field would just drift endlessly since there are no frictmn terms to stabilize their random motion. A aL is now dependent on the initial value even for t -+ ~. The general solution of eq. (17) can be expressed as
Volume 149B, number 1,2,3
AaL(x, t) = Aa(oL)(x,t) + V qb(x)/V 2 .
PHYSICS LETTERS (21)
AaL(x, O) = V~(x)/~7 2 denotes the initial condition,
13 December 1984
E [AaL(x, t)A~L(x ', t')] = - ~ 8 ab[It- t'l + (t + t ' ) - a ]
where q~ is a scalar function;
X (aia]/v2)5(3)(x - x ' ) ,
oL t) = f O(t - s) dlCaL(x, s), A(O)(X, 0
(22)
aL , 0) = 0. Note that Aa(~)(x, t) is the solution with A(0)(x = A ~ L, t / > 0, is a Wiener process with mean zero and the following covariance aL bL , , E[A(o)i(x, t)A(o)i(x , t )]
= ~ abmin(t, t')(~ a//V2)8(3)(x -- x').
(23)
Using the fact that Aa__L, t i> 0 is also a Wiener process, one can write the general form of the covariance for aL A(0 ) as aL bL , E[A(o)i(x, t')] t)A(o)i(x,
= ~6 ab [-It - t'l +-(t + t')](aib//72)6(3)(x - x ' ) ,
(26)
which is just the propagator obtained by Caracciolo et al. except for a factor i. This difference can again be explained by regarding A aL as an euclidean field as in the case of A aT. Note that the second term in the square brackets in eq. (26) is not translationally invariant. However this would not affect the translational invariance of the physical results [2,3]. In the above derivation of the gluon propagator, only the basic principles of Nelson's stochastic mechamcs are used, no complicated calculations are necessary to obtain the correct form of the longitudinal part of the propagator. Finally we remark that our result holds for QED as well as QCD, whereas Caracciolo et al. [3] could determine the coefficients of the (t + t')-term (m eq. (1)) only for the case of QCD.
(24) where the + signs correspond to A aL respectively. The covariance o f A aL can be obtained by modifying eq. (24) with an additional term a i ~/qb(x, .x-')/V2V '2, where qr~(x, x ' ) - q~(x)ff~(x'), the overbar denotes an average in the initial distribution (in this case an average over the scalar function ~). It is possible to choose a distribution such that ~(x, x ' ) is delta-correlated of the form q~(x, x ' ) = F(V2)6 (3)(x - x ' ) ,
(25)
where F ( V 2) is a non-negative operator polynomial in ~72. If we take F(~72) = ct~72/2, c~/> 0, then
References [1] w. Kummer, Acta Phys. Austriaea 41 (1975) 315; J Frenkel, Phys. Lett. 85B (1979) 63; A. Bumel, Nncl. Phys. B198 (1982) 531. [2] V B. Muller andW. Ruhl, Ann. Phys. (NY) 133 (1981) 240. [3] S. Caraceiolo, G. Curci and P. Menotti, Phys. Lett. l13B (1982) 311. [4] E. Nelson, Dynamical theories of brownian motion (Princeton U.P., Princeton, NJ, 1967). [5] F. Guerra, Phys. Rep. 77 (1981) 263.
203
PHYSICS LETTERS
13 December 1984
THE GLUON PROPAGATOR IN TEMPORAL GAUGE: A SIMPLE DERIVATION BASED ON STOCHASTIC MECHANICS S.C. LIM
Department of Physics, Fakulti &tinsFizis & Gunaan, UniversitiKebangsaan Malaysia, Bangi, Selangor, Malaysia Received 2 July 1984
The stochastic quantization scheme based on Nelson's stochastic mechanicsis used to derive the non-abeliangauge field propagator. The result for the longitudinal part of the gluon propagator is consistent with that obtained by Caracciolo, Curci and Menotti.
Temporal gauge has proved to be useful in the perturbative calculation of QCD mainly because of the absence of Faddeev-Popov ghosts and the fact that the hamiltonian in this gauge has a very simple form. It is also very suitable for the study of the topological properties of non-abelian gauge theories. However, the temporal gauge condition Ag = 0 does not fix the gauge completely. The residual gauge invariance under the time-independent gauge transformations gives rise to an ambiguous 1/p~ pole in the gluon propagator. The usual way of regularizing this singularity is by using the principal value prescription [ 1]. Recently, several authors have raised doubts on the validity of this regularlzation method. Calculations by Mtiller and Rtihl [2] on the small couphng of a non-abelian gauge field on a lattice in temporal gauge have shown that the correct longitudinal part of the propagator differs from that derived from the principal value prescription. A similar conclusion has been obtained by Caracciolo et al. [3] by considering the Wilson loop in temporal gauge. They have found that the principal value prescription does not lead to the correct exponentiation of the time-dependence of the Wilson loop operator. In order to give a result which is in agreement, up to order g4, with that of Feynman and Coulomb gauges the longitudinal component of the gluon propagator needs to be of the form
(01T(AaL(x, t)Ab L(X', t'))10) =
--}i8 ab [It - t'l + (t + t') + a]
x (aio//v2)
(1)
(3)(x - x'),
where t~ is a constant. In comparison, the principal value prescription gives the following result: (01T(AaL(x, t)A b L(x', t'))l 0)
= - ~ i S a b l t - t'l(aiOl/V2)8(3)(x-x').
(2)
In this paper we shall apply the quantization method of Nelson's stochastic mechanics [4] to the classical gauge field in minkowskian space-time with the aim of obtaining a result for the free gluon propagator which is in agreement with eq. (1). First, recall the basic equations of the classical non-abelian gauge field A a in temporal gauge. The lagrangian of a pure nonabelian gauge theory without fermions is
z = f(eg
7 -
g87) d3x,
(3)
where the colour electric and magnetic fields are given by
E(]=-aoAOi,
Ba=--~eiikqk .
(4)
The space components of the field strength tensor are
a a)
(5) 201
Volume 149B, number 1,2,3
PHYSICS LETTI- RS
where g is the coupling constant and fabe are the structure constants of the non-abelian group. The colour magnetic field can also be expressed as
B a = V X A a + ~gfabcAb X A e .
13 December 1984
E~) = +[coA a + co-i V(V.Aa)], into eq (9) to obtain (6)
dAaV(x, t) = -COAaT(x, t) dt + dW al (x, t),
The equation of motion is
and
a2A a
dAaL(x, t) = dWaL(x, t),
_ V 2 A a + V(V.A a)
= --gfabc(Ab X (V X AC)),
(lS)
(7)
where V and 7 2 respectively denote the gradient and laplacian in R 3. For the determination of the free gauge field propagator one needs to consider only the linear part ofeq. (7):
(l 6)
(17)
where co - ( - V 2 ) 1/2. Eq. (16) has a genre al soluhon of tile form _ aT AaT(x, t) -A(o)(X, t) +AaW(x) exp(-cot), (18)
Nelson's stochastic quantization requires A a be promoted to a Markov process satisfying the following stochastic differential equation of Langevin type:
whme Aaf~(x, t)is the solution satisfying the mmal condmon Aa(~(x, 0) = 0, and AaT(x) denotes an albltraJy mmal value of AaT(x, t). Clearly,AaT(x, t) is mdependent of the imtlal value AaT(x) for large t. By takmgAaT(x) = 0 one gets for the solution ofeq. (16) a gaussmn random field with mean zero and covanance gwen by
dAa(x, t) = D(+)Aa(x, t) dt + dWa(x, t ) .
E[AaT(x, t)Ab T(x ', t')]
~4 a = V2A a - V(V "Aa).
(8)
(9)
D(+) denotes the forward time derivative as defined in ref. [4]. Wa is the brownian disturbance field specified by the following expectation values: E[dW~/(x, t)l = 0 ,
(lOa)
E [dWa(x, t) dWf(x', t)] = 6abso6(3)(X - x ' ) dt. (10b)
becomes
(lI)
i.e. the colour electric field splits into a pair of functionals R a ('D(_) is the backward time derivative). Eq. (8) takes the following stochastic form:
~(D(+)D(_) + D(_)D(+))A a = V2A a - V (V "Aa), (12) or
~(D(+)E(_)' a +D(_)E&))=-V2A a +V(V-Aa).
(13)
It is convenient to resolve A a into the transverse and longitudinal components A aT and A aL with V'AaT = 0 ,
VXAaL=0.
(14)
Following the same argument as given in ref. [5] one can substitute
202
X exp [-co(p)lt - t'[ + lp(x - x ' ) ] ,
(19)
with co(p) = (p2)1/2. By using the Fourier reversion formula
e x p [ - w ( p ) l t - t't] = 1 f 2co(p)
The stochastic form of the equation aOAa = - E a
D(+)Aa(x,t)= -E~(+)(A],x, t),
= (27r)-38abf d3p gg.. 2co(p) "v'l - PiP]/P2)
e x p 0 p 4 ] t - t'[) oo2 + p2
dP4, (20)
one can identify eq. (19) with the two-point function of the euchdean field AaT(x), x =--(x, x4) provided t IS taken as x4, just as in the scalar case [5]. This interpretation of the stochastic field also explains the absence of the factor i m the covanance (l 9) as compared with the propagator obtained by using the conventional quanhzation method. Due to the presence of a damping drift term in the stochastic differential equation for A aT, it does not depend on the mitml distribution for large t. However this is not the case for A aL. The stochastic differential eq. (17) for the longitudinal components A aL does not possess a damping drift term. This means that the longitudinal modes of the gauge field would just drift endlessly since there are no frictmn terms to stabilize their random motion. A aL is now dependent on the initial value even for t -+ ~. The general solution of eq. (17) can be expressed as
Volume 149B, number 1,2,3
AaL(x, t) = Aa(oL)(x,t) + V qb(x)/V 2 .
PHYSICS LETTERS (21)
AaL(x, O) = V~(x)/~7 2 denotes the initial condition,
13 December 1984
E [AaL(x, t)A~L(x ', t')] = - ~ 8 ab[It- t'l + (t + t ' ) - a ]
where q~ is a scalar function;
X (aia]/v2)5(3)(x - x ' ) ,
oL t) = f O(t - s) dlCaL(x, s), A(O)(X, 0
(22)
aL , 0) = 0. Note that Aa(~)(x, t) is the solution with A(0)(x = A ~ L, t / > 0, is a Wiener process with mean zero and the following covariance aL bL , , E[A(o)i(x, t)A(o)i(x , t )]
= ~ abmin(t, t')(~ a//V2)8(3)(x -- x').
(23)
Using the fact that Aa__L, t i> 0 is also a Wiener process, one can write the general form of the covariance for aL A(0 ) as aL bL , E[A(o)i(x, t')] t)A(o)i(x,
= ~6 ab [-It - t'l +-(t + t')](aib//72)6(3)(x - x ' ) ,
(26)
which is just the propagator obtained by Caracciolo et al. except for a factor i. This difference can again be explained by regarding A aL as an euclidean field as in the case of A aT. Note that the second term in the square brackets in eq. (26) is not translationally invariant. However this would not affect the translational invariance of the physical results [2,3]. In the above derivation of the gluon propagator, only the basic principles of Nelson's stochastic mechamcs are used, no complicated calculations are necessary to obtain the correct form of the longitudinal part of the propagator. Finally we remark that our result holds for QED as well as QCD, whereas Caracciolo et al. [3] could determine the coefficients of the (t + t')-term (m eq. (1)) only for the case of QCD.
(24) where the + signs correspond to A aL respectively. The covariance o f A aL can be obtained by modifying eq. (24) with an additional term a i ~/qb(x, .x-')/V2V '2, where qr~(x, x ' ) - q~(x)ff~(x'), the overbar denotes an average in the initial distribution (in this case an average over the scalar function ~). It is possible to choose a distribution such that ~(x, x ' ) is delta-correlated of the form q~(x, x ' ) = F(V2)6 (3)(x - x ' ) ,
(25)
where F ( V 2) is a non-negative operator polynomial in ~72. If we take F(~72) = ct~72/2, c~/> 0, then
References [1] w. Kummer, Acta Phys. Austriaea 41 (1975) 315; J Frenkel, Phys. Lett. 85B (1979) 63; A. Bumel, Nncl. Phys. B198 (1982) 531. [2] V B. Muller andW. Ruhl, Ann. Phys. (NY) 133 (1981) 240. [3] S. Caraceiolo, G. Curci and P. Menotti, Phys. Lett. l13B (1982) 311. [4] E. Nelson, Dynamical theories of brownian motion (Princeton U.P., Princeton, NJ, 1967). [5] F. Guerra, Phys. Rep. 77 (1981) 263.
203