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The propagator of spinless and spinning non-relativistic particles from the BRST-invariant path integral
Volume 207 number
PHYSICS
3
LETTERS
THE PROPAGATOR OF SPINLESS AND SPINNING FROM THE BRST-INVARIANT PATH INTEGRAL
B
23 June 1988
NON-RELATIVISTIC
PARTICLES
J. GOMIS and J. ROCA Departament d’Esfructura i Constituents de la Matkria, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain Received
I 1 February
The BRST-invariant istic particles.
1988
path integral
formalism
is used in order to calculate
1. Introduction
the propagator
for spinless and spinning
non-relativ-
m
In a recent paper the unification of BRST, Parisi Sourlas and Galilei symmetries was considered for the case of spinless [ 1 ] and spinning [ 21 non-relativistic particles. In this paper we want to calculate the first quantized propagator for spinless and spinning nonrelativistic particles using the BRST-invariant path integral formalism [ 3,4].
where v/is a ghost-number - 1 arbitrary gauge fixing function and R the BRST charge. The explicit expressionforQandvis [l] Q= -iPn+C@
w=PA,
(2.5)
with q”=(-iP,C),
2. The spinless non-relatvistic particle
Insertion The action for a non-relativistic invariant under reparametrizations
spinless s+f(
7)
S=jdr+n$.
.Ya=(iC,P). of (2.5 ) into (2.4) gives
particle is (2.1) -Pt?+@--A(2mE-p’)-iPP] .
The canonical points is
action with p and E fixed at the end-
We consider tions [4]
(2.6)
the BRST invariant
boundary
condi-
7-o
SC,, =
dr( -xjj+t&A@)
I7=C=C=O
,
@=2mE-p2
0370-2693/88/$ (North-Holland
7=Ta,
P(~,)=P,,
E(7,)=&>
P(~I,) =PL,,
E(7b)=&.
where @ is the first class constraint
7b
invariance
of the
(PEEL,; ~I,IPJG; [ 3,4]
03.50 0 Elsevier Science Publishers Physics Publishing Division )
(2.7)
Now we can compute mentum space
(2.3)
associated to the reparametrization action (2.1). We consider an effective action
at
(2.2)
B.V.
7,
> =
the Feynman
s
IdpI
exp(&f)
kernel in mo-
,
(2.8)
309
volume 207 number 3
where [ dp] is the Liouville measure of the extended phase space. The contribution of the ghost part of (2.8) is
s
[dP]
[dP]
23 June 1988
PHYSICS LETTERS B
[dC]
[dC]
3. The spinning non-relativistic
The action for a spinning [ 2 ] is given by
non-relativistic
X exp
S= j dr [ fm(i+xe)*/(i+xq)
U
dr( -P?+
i
&-ipP)
=
s
[dC]
in
>
*a
- rnxv-ii
[dC] exp *.
=-(7b-7a)
(2.9)
>
(3.1)
(nG+@j)+ii&]
where X, t are the usual spacetime variables and e, x, q, Q are anticommuting Grassmann variables. The canonical analysis gives rise to live second-class constraints
which is the same as in the relativistic case [ 5 1. The functional integration over l7, t and x is straightforward and gives &functions of the derivatives of their conjugate variables:
#=ZZ+fie,
(&%;
@, =Ev-pe+mij-ip@-im$,-iE@,,
7b h&i;
=
-
x
exp
(7b
7a >
-7a)
j
,
[&I
[ml
[&I a(j) d@:) h(b)
Tb (
-i
s
@,=I&-tirf,
,
(2.10)
>
ra
7b IA&; d’hJ(7b-7a)
7,
cD2=2mE-p*.
(3.3)
{x’,p’}=6U,
{x,z&}=-1,
{t,E}=-1,
(A,&}=l,
{rl,K~}=-l,
d(Eb-Ea)
[ 61
{&‘,z7’}=-&J,
iii,aj>=--1.
(3.4)
the algebra of the constraints
is
{$‘, @I}= -iS”,
.
A(z)=-i/2m
(2.11)
to the ca-
(2.12)
Therefore, the integration on A0 must be carried out over the interval - co < & < 0. S(E’-E)&p’-p) 2mE_p2+ie
(3.5)
i@,, @,l=i.
d(Pb--Pa)
Using the equations of motion associated nonical action (2.2) we get
(2.13)
Notice that the Feynman kernel is independent of the unphysical parameter (rb- 7,); this result is due to the nontrivial contribution of the ghost part in eq. (2.10). 310
structure
>
x exp[ -&(2mE,-pa’)]
IPE) = -
(3.2)
and three first-class ones
{@,, @,}=iQ2, (P&b;
@,=17V-fir.
With respect the graded symplectic
dzl(2mE-p*)
The Lagrange multiplier 1 is not restricted at the endpoints, then S(i) selects the constant paths A(7) =&, while the integration on E, p has fixed values at the endpoints; we get
(P’E’
particle
m m
=-
particle
The appearance of the first-class constraints is associated to the reparametrization invariance of the action (3.1). The spurious degrees of freedom 17Vand II, are eliminated using the Dirac bracket associated to the second class constraints @Vand @,+ {a rl>n =i ,
(3.6)
while the other fundamental brackets remain unchanged. We could also have eliminated the variable IZ. However, in the quantum theory we cannot compute the propagator having e fixed at both endpoints. In order to avoid this problem we could still have performed the change of variables (e) --t (8, e, e3 ):
e=;
Jz (e’+ie*),
&+
Jz
(EL-ie*)
(3.7)
Volume 207, number
PHYSICS
3
and followed the procedure described in ref. [ 7 1, having 8 and 0 fixed at ra and ri,, respectively. However this would have broken explicit invariance under rotations because in such a case the variable a3 has to be singled out. Then 17 will be carried throughout the calculation. Since we have used the Dirac bracket for q and ?j the constraint @, is written as 6, =Eq-pe+
mfj+- ip@ .
(3.8)
LETTERS G=
B
23 June 1988
(i’&bnbqb;
=
7b 1
[dpl exp(i&f) ,
(3.13)
where the measure [ d,u] is the Liouville measure of the extended phase space. Performing the integration over the variables x, p, t, E, I&, 1, LT,, x and over the ghosts we get -Ed&P,
G= -d(Et, x
sZ=P,I&-iP,II,,+-C,6,
X exp[-&(7b-7a)
-Pd
I Gdxo [ml [de1[WI [drll (2mEa-~:)l
(3.9)
where P, , P, , C, , C, and Pz, P2, C,, C, are even and odd ghost variables, respectively. The gauge fixing function lJ/=P,x+P*n
7, >
s
Now the effective action is obtained in the same way as in the spinless case. The BRST charge is
+C,@,ti(C’)*P,,
~aEae,va;
)
(3.10)
leads to the relativistic gauge [ 8 ] x =i= If we consider the BRST-invariant conditions I&=I+C”=Ca=O,
0. boundary
r=r,,rb
(3.11a)
and
x
exp[flbe(7b)
x
cd-i
1
(3.14)
[rfbv(7b)+~(7ahl),
where A0 and x0 are the zero modes of the variables A and x. Let us explicitly compute the ghost contribution to the path integral
s
[@aI [fl,l Idcal IdCal m
~(7,)
=~a,
p(7b)
=pb,
q(b)
=&
E(7a)
=&,
E(7b)
e(7a)
=Eb,
q(7bb)=qb
=ea
n(7b)
U
3
x exp i
=17,
dr(P,C2+C2P2-iP
2
P2
7a
,
(3.11b)
>
-P,C’+C’P,
the effective action is [ 3,4] rb
=
s
-PIP,
[dCa]
-ix0C’P2) >
[dCa] m
xexp +,i17,+j17,-x& +C’P, +flbe(rb)
-M+P,C’+P*?
+C2P2-P,P, +
-iP2P2-ixC’P1]
ii[~bv(%)+4(c?hl
=,
(3.12)
where the terms at the endpoints follow from the fact that e, U, q, q satisfy a first order differential equation of motion. The Feynman kernel in momentum space is given
(I -i
1.
dr [iC2(C2+ix0CL)+C’C’] SC
> (3.15)
Notice that in the spinning case the contribution of the ghosts is a constant as in the relativistic case [ 5 1. After integration over q, q, IZand e we are left with
by
311
PHYSICS
Volume 207. number
3
G= -6 t&-E,)
d(~, -A)
x exp( inbea)exp( -
23 June 1988
B
{E’, &‘}D = -id”,
{q, q}b =i , (3.21)
{E’, rl>D = i&l, em =o .
vbI],)
Then
”
X
LETTERS
s
~oexp{-i(zb_T,)~o(2mE,-p~)
e’=i
s&K,
rj=bK,
$=Kb,
(3.22)
where b and b verify [b, 6]+ = - 1 and K is a Klein
x
(3.16)
[E~I],-iip,(~b-~ie~)+m~bl)a
which is clearly independent of the unphysical parameter ( rb - ~,a), as it must be. Now we would like to express (3.16 ) as the matrix element of some operator between the states 1.s~~) and 1nb rfb) . Using the scalar products (ZZle> = exp(iZZe); (4lv> we rewrite the Feynman (PbEbnb)?b
I~aEa~ava
= ew(-rlvl)
,
factor given by K= exp ( - iI7bb). The physical states can be written as
Ix>=@Io>+wm
(3.23)
>
where 4 and v are bispinors. (3.20) as
Then we can rewrite eq.
(p’E’ !P’ IpEY)
(3.17)
kernel as
= - 4 &iS(E’
-E)6(p’
xp/‘ys If $
(YO-l)E
-vp+t
>
Jz
-p)
(Yo+l)ml~‘,
(2mE-p’+ic)-’
(3.24)
where Y is a four-spinor containing both @and I+V. Therefore we have obtained the Levy-Leblond [ 91 propagator for a non-relativistic spin 4 particle. (3.18) Imposition of the second-class constraints @as weak equations on the states defines the physical subspace (3.19)
Ct’ I@lX> =0 and finally gives
-p)
tx’ I (m-Pe^+G)
Ix> (3.20)
In the physical subspace (3.19) the anticommutation relations are no longer those we used in the whole space. We will implement 4, 4 and 6 on the physical subspace fulfilling the anticommutation relations which follow from the Dirac brackets:
312
[ 11R. Casalbuoni, [2] [ 31
CP’E’Y IPEx) =i6(E’ -E)d(p’ 2mE-p’+it
References
[4] [ 51 [6] [ 71
[ 81 [9]
D. Dominici, R. Gatto and J. Gomis, Phys. Lett. B 198 (1987) 165. J. Gomis and M. Novell, Phys. Rev. D 33 (1986) 2212. ES. Fradkin and G.A. Vilkovisky, Phys. Lett. B 55 ( 1975 ) 224. M. Henneaux, Phys. Rep. 126 (1985) I. C. Bathe, J. Gomis and J. Rota, Barcelona preprint UBECMPF-2188. R. Casalbuoni, Nuovo Cimento A 33 (1976) I1 5. A. Barducci, F. Bordi and R. Casalbuoni, Nuovo Cimento B 64 (1981) 287. C. Teitelboim, Phys. Rev. D 25 ( 1982) 3 159. J.M. Levy-Leblond, Commun. Math. Phys. 6 (1967) 286.
PHYSICS
3
LETTERS
THE PROPAGATOR OF SPINLESS AND SPINNING FROM THE BRST-INVARIANT PATH INTEGRAL
B
23 June 1988
NON-RELATIVISTIC
PARTICLES
J. GOMIS and J. ROCA Departament d’Esfructura i Constituents de la Matkria, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain Received
I 1 February
The BRST-invariant istic particles.
1988
path integral
formalism
is used in order to calculate
1. Introduction
the propagator
for spinless and spinning
non-relativ-
m
In a recent paper the unification of BRST, Parisi Sourlas and Galilei symmetries was considered for the case of spinless [ 1 ] and spinning [ 21 non-relativistic particles. In this paper we want to calculate the first quantized propagator for spinless and spinning nonrelativistic particles using the BRST-invariant path integral formalism [ 3,4].
where v/is a ghost-number - 1 arbitrary gauge fixing function and R the BRST charge. The explicit expressionforQandvis [l] Q= -iPn+C@
w=PA,
(2.5)
with q”=(-iP,C),
2. The spinless non-relatvistic particle
Insertion The action for a non-relativistic invariant under reparametrizations
spinless s+f(
7)
S=jdr+n$.
.Ya=(iC,P). of (2.5 ) into (2.4) gives
particle is (2.1) -Pt?+@--A(2mE-p’)-iPP] .
The canonical points is
action with p and E fixed at the end-
We consider tions [4]
(2.6)
the BRST invariant
boundary
condi-
7-o
SC,, =
dr( -xjj+t&A@)
I7=C=C=O
,
@=2mE-p2
0370-2693/88/$ (North-Holland
7=Ta,
P(~,)=P,,
E(7,)=&>
P(~I,) =PL,,
E(7b)=&.
where @ is the first class constraint
7b
invariance
of the
(PEEL,; ~I,IPJG; [ 3,4]
03.50 0 Elsevier Science Publishers Physics Publishing Division )
(2.7)
Now we can compute mentum space
(2.3)
associated to the reparametrization action (2.1). We consider an effective action
at
(2.2)
B.V.
7,
> =
the Feynman
s
IdpI
exp(&f)
kernel in mo-
,
(2.8)
309
volume 207 number 3
where [ dp] is the Liouville measure of the extended phase space. The contribution of the ghost part of (2.8) is
s
[dP]
[dP]
23 June 1988
PHYSICS LETTERS B
[dC]
[dC]
3. The spinning non-relativistic
The action for a spinning [ 2 ] is given by
non-relativistic
X exp
S= j dr [ fm(i+xe)*/(i+xq)
U
dr( -P?+
i
&-ipP)
=
s
[dC]
in
>
*a
- rnxv-ii
[dC] exp *.
=-(7b-7a)
(2.9)
>
(3.1)
(nG+@j)+ii&]
where X, t are the usual spacetime variables and e, x, q, Q are anticommuting Grassmann variables. The canonical analysis gives rise to live second-class constraints
which is the same as in the relativistic case [ 5 1. The functional integration over l7, t and x is straightforward and gives &functions of the derivatives of their conjugate variables:
#=ZZ+fie,
(&%;
@, =Ev-pe+mij-ip@-im$,-iE@,,
7b h&i;
=
-
x
exp
(7b
7a >
-7a)
j
,
[&I
[ml
[&I a(j) d@:) h(b)
Tb (
-i
s
@,=I&-tirf,
,
(2.10)
>
ra
7b IA&; d’hJ(7b-7a)
7,
cD2=2mE-p*.
(3.3)
{x’,p’}=6U,
{x,z&}=-1,
{t,E}=-1,
(A,&}=l,
{rl,K~}=-l,
d(Eb-Ea)
[ 61
{&‘,z7’}=-&J,
iii,aj>=--1.
(3.4)
the algebra of the constraints
is
{$‘, @I}= -iS”,
.
A(z)=-i/2m
(2.11)
to the ca-
(2.12)
Therefore, the integration on A0 must be carried out over the interval - co < & < 0. S(E’-E)&p’-p) 2mE_p2+ie
(3.5)
i@,, @,l=i.
d(Pb--Pa)
Using the equations of motion associated nonical action (2.2) we get
(2.13)
Notice that the Feynman kernel is independent of the unphysical parameter (rb- 7,); this result is due to the nontrivial contribution of the ghost part in eq. (2.10). 310
structure
>
x exp[ -&(2mE,-pa’)]
IPE) = -
(3.2)
and three first-class ones
{@,, @,}=iQ2, (P&b;
@,=17V-fir.
With respect the graded symplectic
dzl(2mE-p*)
The Lagrange multiplier 1 is not restricted at the endpoints, then S(i) selects the constant paths A(7) =&, while the integration on E, p has fixed values at the endpoints; we get
(P’E’
particle
m m
=-
particle
The appearance of the first-class constraints is associated to the reparametrization invariance of the action (3.1). The spurious degrees of freedom 17Vand II, are eliminated using the Dirac bracket associated to the second class constraints @Vand @,+ {a rl>n =i ,
(3.6)
while the other fundamental brackets remain unchanged. We could also have eliminated the variable IZ. However, in the quantum theory we cannot compute the propagator having e fixed at both endpoints. In order to avoid this problem we could still have performed the change of variables (e) --t (8, e, e3 ):
e=;
Jz (e’+ie*),
&+
Jz
(EL-ie*)
(3.7)
Volume 207, number
PHYSICS
3
and followed the procedure described in ref. [ 7 1, having 8 and 0 fixed at ra and ri,, respectively. However this would have broken explicit invariance under rotations because in such a case the variable a3 has to be singled out. Then 17 will be carried throughout the calculation. Since we have used the Dirac bracket for q and ?j the constraint @, is written as 6, =Eq-pe+
mfj+- ip@ .
(3.8)
LETTERS G=
B
23 June 1988
(i’&bnbqb;
=
7b 1
[dpl exp(i&f) ,
(3.13)
where the measure [ d,u] is the Liouville measure of the extended phase space. Performing the integration over the variables x, p, t, E, I&, 1, LT,, x and over the ghosts we get -Ed&P,
G= -d(Et, x
sZ=P,I&-iP,II,,+-C,6,
X exp[-&(7b-7a)
-Pd
I Gdxo [ml [de1[WI [drll (2mEa-~:)l
(3.9)
where P, , P, , C, , C, and Pz, P2, C,, C, are even and odd ghost variables, respectively. The gauge fixing function lJ/=P,x+P*n
7, >
s
Now the effective action is obtained in the same way as in the spinless case. The BRST charge is
+C,@,ti(C’)*P,,
~aEae,va;
)
(3.10)
leads to the relativistic gauge [ 8 ] x =i= If we consider the BRST-invariant conditions I&=I+C”=Ca=O,
0. boundary
r=r,,rb
(3.11a)
and
x
exp[flbe(7b)
x
cd-i
1
(3.14)
[rfbv(7b)+~(7ahl),
where A0 and x0 are the zero modes of the variables A and x. Let us explicitly compute the ghost contribution to the path integral
s
[@aI [fl,l Idcal IdCal m
~(7,)
=~a,
p(7b)
=pb,
q(b)
=&
E(7a)
=&,
E(7b)
e(7a)
=Eb,
q(7bb)=qb
=ea
n(7b)
U
3
x exp i
=17,
dr(P,C2+C2P2-iP
2
P2
7a
,
(3.11b)
>
-P,C’+C’P,
the effective action is [ 3,4] rb
=
s
-PIP,
[dCa]
-ix0C’P2) >
[dCa] m
xexp +,i17,+j17,-x& +C’P, +flbe(rb)
-M+P,C’+P*?
+C2P2-P,P, +
-iP2P2-ixC’P1]
ii[~bv(%)+4(c?hl
=,
(3.12)
where the terms at the endpoints follow from the fact that e, U, q, q satisfy a first order differential equation of motion. The Feynman kernel in momentum space is given
(I -i
1.
dr [iC2(C2+ix0CL)+C’C’] SC
> (3.15)
Notice that in the spinning case the contribution of the ghosts is a constant as in the relativistic case [ 5 1. After integration over q, q, IZand e we are left with
by
311
PHYSICS
Volume 207. number
3
G= -6 t&-E,)
d(~, -A)
x exp( inbea)exp( -
23 June 1988
B
{E’, &‘}D = -id”,
{q, q}b =i , (3.21)
{E’, rl>D = i&l, em =o .
vbI],)
Then
”
X
LETTERS
s
~oexp{-i(zb_T,)~o(2mE,-p~)
e’=i
s&K,
rj=bK,
$=Kb,
(3.22)
where b and b verify [b, 6]+ = - 1 and K is a Klein
x
(3.16)
[E~I],-iip,(~b-~ie~)+m~bl)a
which is clearly independent of the unphysical parameter ( rb - ~,a), as it must be. Now we would like to express (3.16 ) as the matrix element of some operator between the states 1.s~~) and 1nb rfb) . Using the scalar products (ZZle> = exp(iZZe); (4lv> we rewrite the Feynman (PbEbnb)?b
I~aEa~ava
= ew(-rlvl)
,
factor given by K= exp ( - iI7bb). The physical states can be written as
Ix>=@Io>+wm
(3.23)
>
where 4 and v are bispinors. (3.20) as
Then we can rewrite eq.
(p’E’ !P’ IpEY)
(3.17)
kernel as
= - 4 &iS(E’
-E)6(p’
xp/‘ys If $
(YO-l)E
-vp+t
>
Jz
-p)
(Yo+l)ml~‘,
(2mE-p’+ic)-’
(3.24)
where Y is a four-spinor containing both @and I+V. Therefore we have obtained the Levy-Leblond [ 91 propagator for a non-relativistic spin 4 particle. (3.18) Imposition of the second-class constraints @as weak equations on the states defines the physical subspace (3.19)
Ct’ I@lX> =0 and finally gives
-p)
tx’ I (m-Pe^+G)
Ix> (3.20)
In the physical subspace (3.19) the anticommutation relations are no longer those we used in the whole space. We will implement 4, 4 and 6 on the physical subspace fulfilling the anticommutation relations which follow from the Dirac brackets:
312
[ 11R. Casalbuoni, [2] [ 31
CP’E’Y IPEx) =i6(E’ -E)d(p’ 2mE-p’+it
References
[4] [ 51 [6] [ 71
[ 81 [9]
D. Dominici, R. Gatto and J. Gomis, Phys. Lett. B 198 (1987) 165. J. Gomis and M. Novell, Phys. Rev. D 33 (1986) 2212. ES. Fradkin and G.A. Vilkovisky, Phys. Lett. B 55 ( 1975 ) 224. M. Henneaux, Phys. Rep. 126 (1985) I. C. Bathe, J. Gomis and J. Rota, Barcelona preprint UBECMPF-2188. R. Casalbuoni, Nuovo Cimento A 33 (1976) I1 5. A. Barducci, F. Bordi and R. Casalbuoni, Nuovo Cimento B 64 (1981) 287. C. Teitelboim, Phys. Rev. D 25 ( 1982) 3 159. J.M. Levy-Leblond, Commun. Math. Phys. 6 (1967) 286.