Stochastic Quantization of Non-Abelian Gauge Field

20 Pages·2003·0.83 MB·English

1580
Progress of Theoretical Physics, Vol. 69, No.5, May 1983
Stochastic Quantization of Non-Abelian Gauge Field
-- Unitarity Problem and Faddeev-Popov Ghost Effects --
Mikio NAMIKI, Ichiro OHBA, Keisuke OKANO
and Y oshiya Y AMAN AKA D
o
w
n
Department of Physics, Waseda University, Tokyo 160 lo
a
d
e
d
(Received November 8, 1982) fro
m
The stochastic quantization method is applied to the non-Abelian gauge field up to the h
second order perturbation. It is shown that the stochastic quantization method automatically ttp
s
rperqoudiurceess toth ein straomdeu cceo rarretcifti crieaslulylt sa nays ggihvoesnt fbiyel dt.h e Twheel l-gkanuogwe nf iFxiandgd epervo-bPleompo vin trtihciks mbuett hnoedv eisr ://ac
a
also examined in detail. Finally preliminary discussions are given as to whether the physical d
e
state condition is automatically satisfied. m
ic
.o
u
§ 1. Introduction and summary p.c
o
m
Recently Parisi and Wu1 have proposed a new quantization method, that is, /p
) tp
the stochastic quantization different from the conventional ones, the canonical /a
and path-integral methods. They have described the stochastic quantization by rticle
means of the Langevin equation governing a hypothetical stochastic process of -ab
s
the Wiener type with respect to an additional and fictitious time. The stochastic tra
c
process is so designed as to give the ordinary field theory at the stationary limit. t/6
9
Our main interests in their theory are the following: The first interest is in their /5
/1
emphasis that the stochastic quantization method enables us to quantize gauge 58
0
fields without gauge fixing. The second is in their conjecture that the stochastic /1
8
6
quantization method, when applied to the non-Abelian gauge field, will 0
5
0
automatically lead us to the same correct results as given by the Faddeev-Popov 9
b
trick in the conventional field theory without resort to artificial input of any ghost y g
u
field. Zwanziger et aF) has discussed the two problems on the basis of the e
s
equivalent Fokker-Planck equation but not directly on the Langevin equation. t o
n
In this approach the Fokker-Planck operator is so modified with additional terms 10
A
as to keep expectation values of gauge invariant quantities unchanged. Their p
conclusion is that the stationary distribution given by the modified Fokker ril 2
0
1
Planck equation is equivalent to the well-known results brought about by the 9
Faddeev-Popov ghost field. However, one may point out that their rather heavy
modification of the Fokker-Planck operator is nothing other than a sort of
artificial input of the gauge fixing and the ghost field, or at least, may say that his
approach steps out of the original line given by Parisi and Wu. The stochastic
quantization is attractive to us for its simplicity in the principle.
Stochastic Quantization of Non-Abelian Gauge Field 1581
In this paper we return to the original Langevin equation to discuss the above
two problems through straightforward perturbation calculations up to the second
order. The main purposes of this paper are: First we show that Parisi and
Wu's conjecture is certainly justified, that is to say, that the stochastic quantiza
tion method automatically yields the Faddeev-Popov ghost effects without resort
to artificial input of any ghost field or artificial modification of the Fokker-Planck
operator. Second we show that the stochastic quantization method implicitly
D
o
introduces a sort of gauge fixing mechanism to give a gauge parameter in the field w
n
theoretical propagator, even though the method is free from the usual gauge lo
a
d
fixing to reduce the degree of freedom of fields by modifying the original field e
d
Lagrangian. fro
m
In § 2 we first outline the stochastic quantization of field and then apply it to h
the Abelian gauge field together with discussions on the gauge fixing problem. ttps
Section 3 is devoted to the central part of this paper in which we deduce the ://a
c
a
above-mentioned conclusion in the case of the non-Abelian gauge field. In § 4 we d
e
m
make preliminary discussions as to whether the physical state condition can be ic
.o
realized automatically in the stochastic quantization method. Concluding u
p
remarks are given in § 5. .co
m
/p
tp
§ 2. Abelian gauge field and gauge fixing problem /a
rtic
le
In the Euclidean field theory we have the well-known path-integral formula -ab
s
tra
L1 (x -y)_- Jd q)(pJ(dx ¢)e¢ (5y(9 1)) e-5[91) (2·1) ct/69/5
/1
5
8
for the propagator of a neutral scalar field ¢(x), where x and yare 4-dimensional 0
/1
Euclidean coordinates and 5 [¢] stands for the action integral. The stochastic 86
0
quantization method gives us a simple prescription to derive L1 (x - y ) based on 50
9
the Langevin equation b
y
g
u
e
s
Tat ¢(X,t)_- 7 oSo¢[¢ ] I +1J(x,t) (2·2) t on
1
91=91(x,t) 0
A
p
which describes a hypothetical stochastic process of a random field ¢(x, t) with ril 2
0
respect to fictitious time t (different from the real time). In (2·2) 1J(x, t) is a 1
9
Gaussian random source field characterized by the statistical properties
where <f(1J »n means the average over 1J given by
1582 M. Namiki, l Ohba, K. Okano and Y. Yamanaka
1 4
f dT/f(T/ )exp[ -4fd xdt{T/(x, t W]
<f(T/»~ [-! (2-4)
4
fdT/exp fd xdt{T/(x, tp]
Since (2-2) gives us its solution ¢(x, t) as a functional of T/ (and also initial
distribution), we can obtain the two point correlation function by the averaging
procedure D
o
w
n
D(x, tly, n=<¢(x, t)¢(y, t'»~. (2-5) lo
a
d
e
Recalling the fact that the Langevin equation (2-2) together with (2-3) or (2'4) d
leads us to the stationary distribution exp {- 5 [¢]} as t goes to infinity, we can from
easily show http
s
LI (x -y )=1imD(x, tly, t). (2·6) ://a
c
t-a> a
d
e
This is the central prescription of the stochastic quantization. m
ic
N ow let us apply the stochastic quantization method to the Abelian gauge .o
u
p
field Ap(x) whose action integral is given by .c
o
m
/p
tp
/a
rtic
le
In terms of Fourier transforms*) Ap(k, t) and T/p(k, t), we have the Langevin -a
b
equation s
tra
c
t/6
Ap(k, t)= -k2( opv- kk~v )Av(k, t)+T/p(k, t), (2·8) 9/5
/1
5
8
where Ap(k, t)=(J/Jt )Ap(k, t) and 0
/1
8
6
<T/p(k, t»~=0, <T/p(k, t)T/v(k', t'»~=20pv04(k+k')o(t-t'). (2·9) 05
0
9
Solving (2'8) under a special initial condition Ap(k, 0)=0, we can easily get by
g
u
e
(2'10) st o
n
1
0
where Gpv(k: t-t') is the Green function given by A
p
ril 2
(2'11) 01
9
which is the solution of the equation
*) Note that
Stochastic Quantization of Non-Abelian Gauge Field 1583
(2·12)
under the condition G"v(k: +O)=o"v and G"v(k: -0)=0_ The prescription of
the stochastic quantization requires us to prepare the correlation function in the
following way:
D<;1~(k, tlk', t')=<A,,(k, t)AAk', t'»~ D
o
w
n
= l"'dt"l"'dt"'G"IC(k: t-t")Gv.(k': t'-t"')<7JIC(k, t")r;.(k', t"'»~ loa
d
e
d
=o4(k+k')D<;1~(k: t, t'), (2·13a) fro
m
h
(2·13b) ttps
://a
c
a
where t< stands for the smaller one of t and t', and d
e
m
ic
A ("VO )T(k)=k2l ( ~ "V _ kk"2k v) , (2·14) .ou
,(.J U p
.c
o
m
which is nothing but the ordinary field propagator with the Landau gauge. /p
Consequently, the transverse component of the equal time correlation function tp
/a
D<;1~(x, tlx', t) tends to .J<;1V(x-x') as t goes to infinity, while the longitudinal rtic
le
component Uk"kj k2 will be cancelled out in gauge invariant quantities as already -a
b
observed in the Abelian gauge theory.I).3) This is the basic idea of the stochastic stra
quantization. ct/6
At first sight the above approach seems to be able to quantize the gauge field 9/5
as if the gauge fixing procedure were not necessary. Really it is true that the /15
8
stochastic quantization method never requires any modification of the Lagran 0/1
8
gian to fix gauge, that is to say, that the method is free from the usual gauge fixing 6
0
5
in this sense. However, we have to pay attention to the special choice of the 0
9
initial condition A,,(k, 0)=0, which is closely related to the gauge parameter by
g
fixing problem in the following sense. To examine this problem, it is convenient u
e
s
to decompose the Langevin equation (2·8) into two parts, transverse and lon t o
n
gitudinal ones, as follows: 1
0
A
p
(2·15a) ril 2
A· "L _ (2·15b) 019
-
where A"T(k, t)=(o"v-k"kv/k2)Av(k, t) and A"L(k, t)=(k"kv/k2)Av(k, t) stand
for the transverse and longitudinal parts, respectively, and r;"T and r;/ for the
corresponding parts of r;". Speaking in general, (2·15a) has the solution
(2·16)
1584 M. Namiki, L Ohba, K. Okano and Y Yamanaka
where ApT(O)(k, t) is the solution under the special initial condition A/(O)(k, 0)=0
and A/(k, 0) represents an arbitary initial value of ApT(k, t). It is then obvious
that ApT(k, t) tends to ApT(O)(k, t) irrespectively of the choice of initial condition
as t goes to infinity, in other words, that we can use the initial condition
ApT(k, 0)=0 without lack of generality. This fact comes from the presence of
2
the 'damping force' (-k Ap T) in (2 ·15a). Contrary to this, (2 ·15b) has no
D
'damping force', so that the initial value never disappears even for very large t. o
w
If we put the initial condition ApL(k, 0)= (kp/k2)if;(k) in which if;(k) is a scalar nlo
a
function, then we have another solution d
e
d
fro
(2·17) m
h
ttp
s
where ApL(O)(k, t)=!oOOdt"f)(t-t")7Jp(k, t") is the solution of (2·15b) under the ://a
c
initial condition ApL(O)(k, 0 )=0. Corresponding to this choice of initial condition, ad
e
we obtain another correlation function m
ic
.o
u
p
(2·18) .c
o
m
/p
It is, however, noted that the simple product if;(k)if;(k') never yields the important tp/a
factor (j4(k+k') which is required from the translational invariance or the rtic
le
uniformity of the space·time. Thus if;(k)=O as far as we adhere to put a sharp -a
b
functional distribution, if;(k), for the initial condition. However, we have to stra
remark that, in the theory of stochastic processes, a sharp functional distribution ct/6
is exceptional but a functional probability distribution over random field is 9/5
usually set down as its initial condition. Therefore, what we must have as the /15
8
correlation function is not (2 ·18) but its functional average over if; , namely 0/1
8
6
0
<Ap(k, t)AII(k', t'»/'= !i!~: ([)(k, k')+ D<;Jt(k, tlk', n, (2·19) 50
9
b
y
g
where ([)(k, k')= if;(k)if;(k')(> stands for the functional average of if;(k)if;(k') over u
e
s
if; with a given distribution. It is now easy to show that an adequate distribution t o
n
around zero field can give us ([)(k, k')=-a(j4(k+k'), a being a dimensionless 1
0
positive parameter. Thus, we get the correlation function A
*) p
ril 2
0
1
*) From the reality condition A"L(-k, n=A,p(k, t) we get r/l(-k)=-r/l*(k) and then (j)(k, k') 9
= - <f;(k)r/l*( - k' )'. If we expand r/l(k)= L:,iCiUi(k) in a complete orthonormal set {ui(k)}, Ci being real
variable, and average <f;(k)r/l*(-k')=L:,i.jCiCju,(k)u/(-k') with the distribution functional W[r/l]
=exp[-(l/2a)L:,ic,']/rrJ~:exp[-(l/2a)c!]dci, then we get (j)(k, k')=-aL:,iui(k)Ui*(-k')
=-a/5'(k+k'). Note that a>O. A more general form compatible with the translational invariance
would be (j)(k, k')=y(k')/5'(k+k'), but y(k') should simply be a constant unless we introduce par
ameters with dimension into the distribution functional.
Stochastic Quantization of Non-Abelian Gauge Field 1585
<Ap(k, t)Av(k', t'»~~=a t;A~ 04(k+k')+ D~t(k, tlk', t')
t~oo 04(k+k')[ 12 {opv-(l-a )kk~V }+2t kk~v J- (2·20)
This equation implies that the choice of initial state distribution just corresponds
to fixing of the gauge parameter. Hence w~ cannot assert that the stochastic
D
quantization method is quite free from the gauge fixing. It is, however, empha ow
n
sized that the stochastic quantization method can determine the gauge field lo
a
propagator without resort to the usual gauge fixing procedure of introducing a de
d
gauge fixing term like (2a )-1(aA)2 into the Lagrangian,*) that is to say, to reduce fro
m
the degree of freedom of fields by the constraint condition. h
The reason why the usual gauge fixing procedure was not necessary in the ttp
s
above approach can be understood from the character of the operator [0 ptea/ at ://a
c
+ (k20pte- kpkte)] in (2· 8) or (2·12). Contrary to the conventional theory in which ad
e
we have not Optea/at but only the projection operator (k2opte-kpkte), this operator m
ic
is not singular and hence has its inverse explicitly given by GpAk: t-t'). The .ou
p
situation will become clear if we describe (2·8) or (2·12) in terms of the Fourier .c
o
m
transform with respect to the fictitious time. Introducing /p
tp
/a
(2·21) rtic
le
-a
b
then we have s
tra
c
(2·22) t/6
9
/5
Due to the presence of - iw, the matrix II( - iw+ k2)oPte-kpktell can have its inverse /15
8
explicitly written down as 0/1
8
6
0
5
(2·23) 0
9
b
y
g
where - iw behaves as if it were the gaugeon mass squared. It is easily observed u
e
s
that introduction of the fictitious time and the longitudinal random source field t o
n
has given an additional degree of freedom to the field into the present theory. In 1
0
fact, our correlation function (2·13) or its limit limt-ooD~t(x, tlx', t) contains the Ap
longitudinal part besides the transverse part L1 ~tT (x - x'). ril 2
0
Since we can use the above method for the zeroth-order propagator in the 1
9
perturbation theory, we can easily guess that the same prescription of stochastic
*) In this case we have the 'damping force' -a-lk2A~ on the right· hand side of (2'15b) and then
can obtain the usual correlation function with gauge parameter a at t -> 00, irrespectively of initial
conditions. As is well known, however, such a modification of the Lagrangian should be followed by
introduction of Faddeev-Popov ghost field in the case of the non-Abelian gauge field.
1586 M. Namiki, l Ohba, K. Okano and Y. Yamanaka
quantization can be applied to the non-Abelian gauge field without resort to the
usual gauge fixing procedure. On the other hand, it is well-known that the usual
gauge fixing should be followed by artificial input of the Faddeev-Popov ghost
field. For these reasons we are now led to a natur:> I conjecture that the non
Abelian gauge field can be quantized without help of any ghost field within the
framework of stochastic quantization. In the next section we justify this con
jecture by showing that the stochastic quantization method automatically pro D
o
duces the Faddeev-Popov ghost effects without resort to artificial input of any wn
lo
ghost field. The natural occurrence of the Faddeev-Popov ghost effects is closely a
d
e
related to the above-mentioned additional degree of field brought from introduc d
tion of the fictitious time and the longitudinal random source field as mentioned from
above. The above averaging procedure over initial-state distribution and also h
ttp
the occurrence of additional degree of freedom will be more clearly described in s://a
the framework of the operator theory of stochastic quantization.4 c
) a
d
e
m
§ 3. Non-Abelian gauge field and Faddeev-Popov ghost effects ic.o
u
p
.c
A non-Abelian gauge field in the D-dimensional Euclidean space is character o
m
ized by the action integral /p
tp
/a
(3·1a) rticle
-a
b
s
(3·1b) tra
c
t/6
where a, band c stand for the color indices. From (3· 1) we get the Langevin 9
/5
equation /1
5
8
0
(3·2a) /1
8
6
0
5
where 0
9
b
y
D D g
Ypa(k, t)= Tjpa(k, t)+ (2:)D/2 jd kld k20D(k-k1- k2) ue
s
t o
x Vpa",b{(k, - kl' - k2 )A",b(k1, t)A/(k2, t) n 1
0
A
+ (2~)D jdDkldDk2dDk30D(k-kl-k2-k3) pril 2
0
1
9
(3·2b)
in which Vpa:{(k, -kl, -k2) and W:,:.,cl are, respectively, the three point and four
point vertex factors listed in Table I together with the corresponding diagrams.
+
t
Note that V:J'f(k, -k1, -k2)= V~~)rC(k, -kl, -k2) and Wpa,:"cl= W~V",'ibCd in
terms of the conventional Feynman rule. Equation (3·2a) under the initial
Stochastic Quantization of Non-Abelian Gauge Field 1587
Table 1. Diagrams of Green function, propagator and vertices.
DIAGRAM NOTATION FORMULA
}eu-
t~tl G~t(k: t-t') oab{ (oPv- kk~v )e-·'lt-t'l+ kk~v t')
)A lJ
oab{-k..2L ( 0 pv _ kpk2k v)( e -''It-t'l_ e -''It+t'l)
t~t' mt(k: t, t') Do
)I. y w
+2t< kk~v} n
lo
a
d
e
d
:~~ (- ~)grbC fro
gVpalf(k, kl' k2) m
h
k;l "c- [(k-k1 ),oPK+(k1- k2)POd+(k2-k)KOP'] ttp
s
V - ~ g2[jabelede(op./Jv,-Op,OvK) ://ac
a
~~". d
a c g2Wpa:~cl + laeelbde(/JI'v/J.,-OP'Ov.) em
i'- + lade Ie be (/J I'./J v, - 0 pvO d)] ic.o
u
p
.c
o
m
condition Al'a(k, 0)=0 is equivalent to the integral equation /p
tp
• /a
(3·3) rtic
le
-a
b
s
where G~t(k: t-t')= aabGI',,(k: t-t') is the zeroth order Green function discus tra
c
sed in detail in § 2. It may be convenient to write (3·3) in a symbolic way as t/6
9
/5
A= G(lJ+gVAA +g2 WAAA). (3·4) /1
5
8
0
Solving (3·3) or (3·4) by means of the iteration method, we obtain its solution in /1
8
6
a perturbation expansion as follows: 0
5
0
9
A = GlJ + gG V( GlJ )( GlJ )+ g2 G V {G V( GlJ )( GlJ )}( GlJ ) by
g
u
e
+g2GV( GlJ ){GV( GlJ)( GlJ )}+g2 W( GlJ)( GlJ)( GlJ )+ ... , (3·5) s
t o
n
which is graphically represented in Fig. 1. Here, following Parisi and Wu1 we 1
), 0
A
shpaevcet ivreeplyr,e sbeyn tae dth rGe e1'1 .1 abnyd aa fwouarv-ypo liinnte v, elJr It'e xb.y a cross and g V and g2 W, re pril 2
0
Our first task is to calculate the correlation function <Al'a(k, t )A"b(k', t'). 19
As anticipated by discussion in § 2, we shall have in the correlation function those
Fig. 1. Perturbative expansion of Apa(k, t).
1588 M. Namiki, L Ohba, K. Okano and Y. Yamanaka
(a) (b) (e)
( d) ( e)
D
o
w
n
lo
a
d
e
d
(f) fro
m
Fig. 2. Graphical representations of D~e(k: t, n. h
ttp
s
terms which diverge as t or t' goes to infinity. Those divergent terms are ://a
c
a
considered to be characteristic to gauge non-invariant quantities such as d
e
m
<A,..a(k, t )A'/(k, t» and to be cancelled out if we make gauge invariant or ic
.o
covariant quantities. Indeed we can show by straightforward calculations that u
p
this is the case for limt~=<Fffv(x, t )Fff).(x, t» for example. In what follows, .co
m
consequently, we keep only those terms which remain finite as t or t' goes to /p
tp
• infinity. Using (3·5) and averaging over TJ by (2,9), we get the second-order /a
pertubation terms as follows: rtic
le
-a
<A,..a(k, t)Avb(k', t'»=oD(k+k')D~e(k: t, t') (3·6a) bs
tra
c
with t/6
9
/5
D~e(k: t, t')=(a)+2(b)+2(c)+2(d)+3(e)+(f), (3·6b) /15
8
0
/1
where (a) ~ (f) stand for the contributions from the corresponding diagrams 86
0
shown in Fig. 2. Diagram (a) (wavy line with a cross) represents the free 5
0
9
propagator (2,13), and diagrams (b), (c), (d) and (e), respectively, correspond to b
y
the following contributions: g
u
e
s
t o
n
1
0
A
p
ril 2
0
1
X21=dt"'Idt"G,..~(k: t-t") 9
xD~Ot(kl: t", t''')D~J(k2: t"', t")Gv,,(k: t'-t"'), (3.7)
Stochastic Quantization of Non-Abelian Gauge Field 1589
I I
Xgv~a/l(k, -kl, -k2)gVJft(-k, kl' k2)[2 dt'" dt"
X{D~k(k: t,t")D~OJ(kl: t",t''')Ge~(k2: t"'---:t")G <1(k: t'-t"')
V
I,
+ Gp~(k: t - f" )DW(kl : f", t"') Ge~(k2: f" - t'" )DISJ(k: t', f"')}
kt+-->k2)
+ A+-->~ J. (3-8) Do
( w
lJ+-->S nlo
a
d
e
3( e )-- (2J1l" )D fdDk 19 2 Wavb).Ce,d d fro
m
x [3 I dt" D~k(k: t, t")D~OJ(kl: f", f"')G<1Ak: f' - f")+(;:~) J. https
://a
c
(3-9) ad
e
m
Diagram (f) gives vanishing contribution due to the anti-symmetry of V~~c with ic.o
respect to a, band c, namely, Vv,!:~CObC=O. Here it should be noted that we can up
.c
discard longitudinal components of external lines for the following reasons: (i) o
m
After integrating over loop momenta, we can write the cross terms from lon /p
tp
gXi tOu~d-Vin<ka)l, two hterraen sverse corresponding to Fig. 3(a) in the form 0 ~~(k)II~<1(k) /article
-a
b
s
OL<1 v (k)= k<k1k2v ' (3-10) tra
c
t/6
9
Because of the transformation property, n~<1(k) is a linear combination of O~<1 and /5
/1
k~k<1, so that the above cross terms should vanish. (ij) The longitudinal-lon 58
0
gitudinal terms corresponding to Fig. 3(b) can be written as O~,,(k)n/C<1(k)O~v(k). /1
8
6
These terms are propotional to O~v(k) and hence cancelled out in gauge covar 0
5
iant combinations like limH,,<F~v(k, t )Ff<1(k', f», as we shall explicitly see later. 09
b
On the other hand, those longitudinal components which take place in internal y
g
u
lines will produce important contributions just corresponding to the Faddeev e
s
Popov ghost effects in the conventional theory. Recall that those longitudinal t o
n
components originate in an additional degree of freedom coming from introduc 10
A
tion of the ficititious time as remarked at the end of the previous section. We p
summarize finite contributions in the limit f = f' --+ 00 in Table II. The contri- ril 20
1
9
~ ~
(a) (b)
Fig. 3. (a) Cross term of propagator. T and L stand for the transverse and Ion·
gitudinal components, respectively.
(b) Longitudinal component of propagator.