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A superspace formulation of the stochastic quantization method: fermions
Volume 140B, number 1,2
PHYSICS LETTERS
31 May 1984
A SUPERSPACE FORMULATION OF THE STOCHASTIC QUANTIZATION METHOD: FERMIONS
S. CHATURVEDI, A.K. KAPOOR and V. SRINIVASAN School o f Physics, University o f Hyderabad, Hyderabad 500 134, India Received 3 January 1984 Revised manuscript received 13 February 1984
A superspace formulation of the stochastic quantization method for fermions is presented.
a¢(x, t)/at = - 6 s/6~(x, t) + ~(x, t) ,
A straightforward extension of the Parisi-Wu stochastic quantization method [ 1] for scalar fields to the case of fermions proceeds through the following Langevin equations [2]
with
O~(x, t)/Ot = -6S/6 fJ(x, t) + rT(x, t) ,
(1)
is only a special case of a class of Langevin equations
O~(x, t)/at = 6S]6 ~(x, t) + ~(x, t) ,
(2)
(6)
(r?(x, t)rl(y, t')> = 26(x - y) 6(t - t') ,
a
a-i~(x,t)=- f dxK(x,y)~
6s
+
O(x,t),
(7)
(8)
where with
S[¢, ~] = - f d x d t Cp(x, t) (i~¢- m)t~(x,t) + V[~k, ~], (3) and the anticommuting noise fields 17 and ~ have the following stochastic properties
= -(riO(Y, t)rla(x, t)) = 26~06(x - y ) 6 ( t - t'), <7> = <~> = = < ~ > = 0 ,
(4) (5)
Higher n-point functions for r/and ~ are defined in analogy with the scalar case taking into account their anticommuting nature. This stochastic quantization method, however, has a major drawback. Owing to the non positive definiteness of (i~ - m) which appears in S[~k, ~] (and hence in the linear part of the drift termsin (1) and (2)) one can associate a properly defined steady state distribution to the Langevinequations (1) and (2) only in the case when m :/: 0 [2]. To overcome this difficulty, reverting back to the scalar case, one notes with Parisi and Wu that, as far as the steady state distribution is concerned, the Langevin equation 56
(O(x, t)O(y, t '))= 2K (x,y)f(t - t') .
(9)
Both (6) and (8) have the same steady state distribution. In the scalar case, since the bilinear part of S[¢] is already positive definite, one may make the simplest possible choice for r (x, y) viz. r (x,y) = 6(x - y ) . However, in the fermion case, this freedom in the choice of K(x,y) may be profitably exploited to overcome the shortcomings of the stochastic quantization method based on (1)-(5) [3,4]. Thus for fermions one starts with the Langevin equations a
a7 ¢(x, t) = a
~(x,t)=
-fdy~(x,y) a ~afsy , t ) + O(x,t), fdy ~
6s
K(.v,x) +O(x,t),
(aO)
(11)
with
= 28a#K(x,y)6(t - t'), and
(12)
Volume 140B, number 1,2
PHYSICS LETTERS
~ ( x , y ) = (iZx + m ) 8 ( x , y ) .
31 May 1984
03) Z[J,J] = f ~ C O ~ e x p ( -
The choice for r ( x , y ) in (13) makes the coefficient of the linear part of the drift terms in (10) and (11) to be (02 - m 2) and thus ensures the existence of a steady state distribution. We now give a superspace formulation for the Langevin equations (10) and (11) along the lines of our earlier work on scalar fields and gauge fields [5,6]. The steady state generating functional corresponding to the Langevin equations (10) and (11) may be written as
Z[J,J]
X {2~(~2/8~ O s ) K - l ~
\
Collecting the bilinear terms in the exponent in (20), the equation satisfied by the free superpropagator is found to be K -1 ( ( - [ ] + m 2 ) + 2 a2/a8 aa
[s a/as - (a/as)s] o/at} X D(x, t , u , & x ' , t ' , u ' , ~ ' ) =
\
(14)
where ff0,g and 50,g are solutions of (10) and (11) with initial conditions specified at t = -oo. Writing (10) and (11) equivalently as
K - I o = ( a l a t ) K - l ~ + 8S18 5 ,
(15)
OK -1 = ( O/Dt) 5 K -1 - 8S/8 ~ .
(16)
Changing variables in (14) from O, 0 to if, ff with the help of 8-functions representing the constraints (15) and (16) following our earlier papers [6,7] we may write (14) as
z [ z 2] =fq
× exp [2c3K-lco + iff~K-lD~/~t + iSK-l~co/~t
+ i (8S/8 5) co - ~oI (8 2S/6 t~8 5) ~o1 (17 )
where co, ~Ol, ~02 etc., are auxiliary fields arising in the representation of the 8 functions and the associated functional determinants. Defining the superfields qz a~d • as follows
= ~ + S~l + ~ o 2 + i ~ s c 3 , (17) may be cast in the following form
t')8(s- s')8(~- ~').
(21)
It is easily checked that the solution of (21) written in terms of the Fourier transform of D with respect to x and x '
D ( x , t , a , 6l;x',t ', s',~') _
1
['dkdk'exp(ik'x+ik"x')
(2rr)4 a
× D(k,t,s, &k',t',s',6~')
(22)
is
D(k,t,s,~';k',t',~',~') = 6(k+k')Q( - m ) ( k 2 + m2) -1
-
s'(~-
~')O(t
-
t')l] .
(23)
It can be shown that when the Green's functions of the ~O, 5 fields are taken into account, the superfield propagator (23) may be replaced by
- ~ l K - 1 ~ 1 / D t - ~ 2 K - l ~ 2 / ~ t + i~SS/Sgg
xI, = ~ + ~ o I + s~o2 - i ~ c o ,
8(x-x')8(t-
X exp [ - ( k 2 + m 2) It - t' - s(~ - ~')O(t' - t)
(all fields)
+ ~o2 (8 2S/8 ~ ¢55) ~o2 ] ,
(20)
-- ~ [a a / a s - ( a / a s ) s ] ( a l a t ) K - l ~ } - S[@. C~]J ,
=f c90 ~Oexp(-~1 f d x d t ~ K _ l o
+ i 50 ,gJ + iJ~O0,g)
f dxdtd~da
(18) (19)
D(k,t,s,s';k',t',s',~') = 6(k+ k')Q( - m ) ( k 2 + m 2 ) -1 X exp [ - ( k 2 + m 2) It - t' - s~O(t' - t)
+ s'~x'O(t - t')I] •
(24)
With (24) as the propagator and using the methods of Nakazato et al. [8], we can prove the equivalence of our superspace formulation to the euclidean field theory to all orders. Finally, we note that in (13) the RHS is the Klein-Gordon divisor for the Dirac field 57
Volume 140B, number 1,2
PHYSICS LETTERS
acting on 5(x - y ) . For massive high spin fermion fields the above method can be generalized b y using the corresponding K l e i n - G o r d o n divisor [7] in the expression for r ( x , y ) . Finally the supersymmetry transformation
t' ~ t ' = t +aa, Ol " + O~r = Ol - - ( 1 ,
which lead to the following transformations on the component fields: a ~ = --~¢Pl + atP2), I =
~tp2 = --ia6o, 8 w = ia~2 , a~ =-~o
5~2 =
@ a - ia~,
6~ = -i~1
58
2 +a~l),
,
31 May 1984
leave the super action in (20) invariant, provided S [ ~ , ~] contains only Yukawa couplings i.e. only bilinears in ff and ~.
References [ 1 ] G. Parisi and Yong ShiWu, Scientia Sinica 24 (1981) 483. [2] T. Fukai, H. Nakazato, I. Ohba, K. Okano and Y. Yamanaka, Prog. Theor. Phys. 69 (1983) 1600. [3] J.D. Breit, S. Gupta and A. Zaks, Institute for Advanced Study preprint (Princeton, March 1983). [4] P.H. Damgaard and K. Tsokos, preprint MD-TP-218 (Maryland University, June 1983). [5 ] S. Chaturvedi, A.K. Kapoor and V. Srinivasan, Hyderabad University preprint. [6 ] S. Chaturvedi, A.K. Kapoor and V. Srinivasan, Hyderabad University preprint. [7] Y. Takahashi, An introduction to field quantization (Pergamon, London, 1969). [8] H. Nakagoshi, M. Namiki, I. Ohba and K. Okano, Prog. Theor. Phys. 70 (1983) 326.
PHYSICS LETTERS
31 May 1984
A SUPERSPACE FORMULATION OF THE STOCHASTIC QUANTIZATION METHOD: FERMIONS
S. CHATURVEDI, A.K. KAPOOR and V. SRINIVASAN School o f Physics, University o f Hyderabad, Hyderabad 500 134, India Received 3 January 1984 Revised manuscript received 13 February 1984
A superspace formulation of the stochastic quantization method for fermions is presented.
a¢(x, t)/at = - 6 s/6~(x, t) + ~(x, t) ,
A straightforward extension of the Parisi-Wu stochastic quantization method [ 1] for scalar fields to the case of fermions proceeds through the following Langevin equations [2]
with
O~(x, t)/Ot = -6S/6 fJ(x, t) + rT(x, t) ,
(1)
is only a special case of a class of Langevin equations
O~(x, t)/at = 6S]6 ~(x, t) + ~(x, t) ,
(2)
(6)
(r?(x, t)rl(y, t')> = 26(x - y) 6(t - t') ,
a
a-i~(x,t)=- f dxK(x,y)~
6s
+
O(x,t),
(7)
(8)
where with
S[¢, ~] = - f d x d t Cp(x, t) (i~¢- m)t~(x,t) + V[~k, ~], (3) and the anticommuting noise fields 17 and ~ have the following stochastic properties
(4) (5)
Higher n-point functions for r/and ~ are defined in analogy with the scalar case taking into account their anticommuting nature. This stochastic quantization method, however, has a major drawback. Owing to the non positive definiteness of (i~ - m) which appears in S[~k, ~] (and hence in the linear part of the drift termsin (1) and (2)) one can associate a properly defined steady state distribution to the Langevinequations (1) and (2) only in the case when m :/: 0 [2]. To overcome this difficulty, reverting back to the scalar case, one notes with Parisi and Wu that, as far as the steady state distribution is concerned, the Langevin equation 56
(O(x, t)O(y, t '))= 2K (x,y)f(t - t') .
(9)
Both (6) and (8) have the same steady state distribution. In the scalar case, since the bilinear part of S[¢] is already positive definite, one may make the simplest possible choice for r (x, y) viz. r (x,y) = 6(x - y ) . However, in the fermion case, this freedom in the choice of K(x,y) may be profitably exploited to overcome the shortcomings of the stochastic quantization method based on (1)-(5) [3,4]. Thus for fermions one starts with the Langevin equations a
a7 ¢(x, t) = a
~(x,t)=
-fdy~(x,y) a ~afsy , t ) + O(x,t), fdy ~
6s
K(.v,x) +O(x,t),
(aO)
(11)
with
(12)
Volume 140B, number 1,2
PHYSICS LETTERS
~ ( x , y ) = (iZx + m ) 8 ( x , y ) .
31 May 1984
03) Z[J,J] = f ~ C O ~ e x p ( -
The choice for r ( x , y ) in (13) makes the coefficient of the linear part of the drift terms in (10) and (11) to be (02 - m 2) and thus ensures the existence of a steady state distribution. We now give a superspace formulation for the Langevin equations (10) and (11) along the lines of our earlier work on scalar fields and gauge fields [5,6]. The steady state generating functional corresponding to the Langevin equations (10) and (11) may be written as
Z[J,J]
X {2~(~2/8~ O s ) K - l ~
\
Collecting the bilinear terms in the exponent in (20), the equation satisfied by the free superpropagator is found to be K -1 ( ( - [ ] + m 2 ) + 2 a2/a8 aa
[s a/as - (a/as)s] o/at} X D(x, t , u , & x ' , t ' , u ' , ~ ' ) =
\
(14)
where ff0,g and 50,g are solutions of (10) and (11) with initial conditions specified at t = -oo. Writing (10) and (11) equivalently as
K - I o = ( a l a t ) K - l ~ + 8S18 5 ,
(15)
OK -1 = ( O/Dt) 5 K -1 - 8S/8 ~ .
(16)
Changing variables in (14) from O, 0 to if, ff with the help of 8-functions representing the constraints (15) and (16) following our earlier papers [6,7] we may write (14) as
z [ z 2] =fq
× exp [2c3K-lco + iff~K-lD~/~t + iSK-l~co/~t
+ i (8S/8 5) co - ~oI (8 2S/6 t~8 5) ~o1 (17 )
where co, ~Ol, ~02 etc., are auxiliary fields arising in the representation of the 8 functions and the associated functional determinants. Defining the superfields qz a~d • as follows
= ~ + S~l + ~ o 2 + i ~ s c 3 , (17) may be cast in the following form
t')8(s- s')8(~- ~').
(21)
It is easily checked that the solution of (21) written in terms of the Fourier transform of D with respect to x and x '
D ( x , t , a , 6l;x',t ', s',~') _
1
['dkdk'exp(ik'x+ik"x')
(2rr)4 a
× D(k,t,s, &k',t',s',6~')
(22)
is
D(k,t,s,~';k',t',~',~') = 6(k+k')Q( - m ) ( k 2 + m2) -1
-
s'(~-
~')O(t
-
t')l] .
(23)
It can be shown that when the Green's functions of the ~O, 5 fields are taken into account, the superfield propagator (23) may be replaced by
- ~ l K - 1 ~ 1 / D t - ~ 2 K - l ~ 2 / ~ t + i~SS/Sgg
xI, = ~ + ~ o I + s~o2 - i ~ c o ,
8(x-x')8(t-
X exp [ - ( k 2 + m 2) It - t' - s(~ - ~')O(t' - t)
(all fields)
+ ~o2 (8 2S/8 ~ ¢55) ~o2 ] ,
(20)
-- ~ [a a / a s - ( a / a s ) s ] ( a l a t ) K - l ~ } - S[@. C~]J ,
=f c90 ~Oexp(-~1 f d x d t ~ K _ l o
+ i 50 ,gJ + iJ~O0,g)
f dxdtd~da
(18) (19)
D(k,t,s,s';k',t',s',~') = 6(k+ k')Q( - m ) ( k 2 + m 2 ) -1 X exp [ - ( k 2 + m 2) It - t' - s~O(t' - t)
+ s'~x'O(t - t')I] •
(24)
With (24) as the propagator and using the methods of Nakazato et al. [8], we can prove the equivalence of our superspace formulation to the euclidean field theory to all orders. Finally, we note that in (13) the RHS is the Klein-Gordon divisor for the Dirac field 57
Volume 140B, number 1,2
PHYSICS LETTERS
acting on 5(x - y ) . For massive high spin fermion fields the above method can be generalized b y using the corresponding K l e i n - G o r d o n divisor [7] in the expression for r ( x , y ) . Finally the supersymmetry transformation
t' ~ t ' = t +aa, Ol " + O~r = Ol - - ( 1 ,
which lead to the following transformations on the component fields: a ~ = --~¢Pl + atP2), I =
~tp2 = --ia6o, 8 w = ia~2 , a~ =-~o
5~2 =
@ a - ia~,
6~ = -i~1
58
2 +a~l),
,
31 May 1984
leave the super action in (20) invariant, provided S [ ~ , ~] contains only Yukawa couplings i.e. only bilinears in ff and ~.
References [ 1 ] G. Parisi and Yong ShiWu, Scientia Sinica 24 (1981) 483. [2] T. Fukai, H. Nakazato, I. Ohba, K. Okano and Y. Yamanaka, Prog. Theor. Phys. 69 (1983) 1600. [3] J.D. Breit, S. Gupta and A. Zaks, Institute for Advanced Study preprint (Princeton, March 1983). [4] P.H. Damgaard and K. Tsokos, preprint MD-TP-218 (Maryland University, June 1983). [5 ] S. Chaturvedi, A.K. Kapoor and V. Srinivasan, Hyderabad University preprint. [6 ] S. Chaturvedi, A.K. Kapoor and V. Srinivasan, Hyderabad University preprint. [7] Y. Takahashi, An introduction to field quantization (Pergamon, London, 1969). [8] H. Nakagoshi, M. Namiki, I. Ohba and K. Okano, Prog. Theor. Phys. 70 (1983) 326.