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Measure in the functional integral over the configuration space
Vol. 25 (1987)
REPORTS
ON
MATHEMATICAL
MEASURE
IN THE FUNCTIONAL CONFIGURATION
Institute
Physics,
of Theoretical
University
No. 1
PHl3ICS
INTEGRAL SPACE+
of Wroclaw,
OVER THE
Wroclaw,
Poland
(Received January 26, 1987)
A general formula is derived for the measure in the functional integral over configuration space, for singular systems. It extends Fradkin’s result given for regular, essentially non-linear systems. Particular cases, up to the most general one of a singular bosonic-fermionic system with both kinds of constraints, are considered. Justification of the Faddeev-Popov ansatz for quantization of gauge theories in non-covariant gauges is thus given. Pure electrodynamics in the radiation gauge is treated in detail.
Introduction In the paper we shall discuss a problem of a measure in path integral representation of generating functional of Green’s functions. In classical papers of Dirac and Feynman [l-4] it was assumed that the transition amplitude has the form
(1.1) The integration
goes over continuous 4’ = 4 (0
paths and
connecting
points
q” = q(t”),
t” S [q;
t”, t’] =
S dtL(q, 4) is a classical action functional of a system. The relevant 1’ measure is thus a continuous product of Lebesgue measures dq(t). It does not contain any non-trivial weight factor which would depend on coordinates of integration q. In other words, all paths contribute to the integral over the phase
while the modulus
* Work supported
by the Polish
is the same N- ’ and does not depend
Ministry
of Higher Education c731
and Science.
on
q.
It is
W. GARCZYr;lSKI
74
known that the above expression mechanical problems characterized
works well for a wide class of quantumby Hamiltonian functions of the form V(q).
H = g+
(1.2)
In systems of this type one does not encounter neither the factor-ordering problem [S-S] nor any non-trivial measure in the integral, like the above one. The point is that the above, let us say, coordinate-space path integral, should be regarded as a result of integration over momenta performed within a corresponding phase-space path integral. As long as one considers Hamiltonians quadratic in momenta, with coefficients not depending on q’s, the integration over momenta yields some rather trivial constants which may be included into an overall normalization factor. When the coefficients do depend on coordinates q both above mentioned problems arise, and a non-trivial measure in coordinate path integral may appear as it was shown by Fradkin [9]. In this paper we would like to analyse this phenomenon for singular systems. One knows from Faddeev’s and Popov’s trick of quantizing gauge theories that non-trivial weight factors appear in generating functionals for Green’s functions of a system [l&12]. Thus, we would like to provide a basis for the Faddeev-Popov ansatz, deriving in a systematic way relevant measures in coordinate-space functional integrals. 2. Measure in a functional integral for generating functional, the case of a regular bosonic system For pedagogical reasons we shall start our considerations with a regular system for which the corresponding measure is known (cf. [9]). Later we will extend this result of Fradkin. Let a system with n degrees of freedom be described by coordinates 4r 3 . . . . q”, Pl, f.., p.. Its Hamiltonian H(q, p) is connected with a classic Lagrange function L(q, 4) in the usual way
H(qv d = i
k= 1
where dk are expressed
by canonical _
= Pk,
(3-4”
4” = dk(q, p) -
solution
Pk dk-
L(q>4)~
coordinates
q, p using the equations
k = 1, . . . . n,
of these equations.
(2.1)
(2.2)
MEASURE
IN THE FUNCTIONAL
This is possible for regular cases
75
INTEGRAL
does not vanish in such
systems since the Hessian
(2.3)
Quantization of the system expression for the generating
z[Jl =
by the path integral method leads to the following functional of the full, time ordered, Green’s functions:
N-lSnndqk(t)dpk(t)exp
~Sdt(Cpk(t)4k(t)-HJ[4(1),
k
&)I) ?
(2.4)
k
where
HJ(q, p) = H(q,
(2.5)
d--CJkqk, k
and
dk
factor
are just time derivatives of variables N is determined by the condition
of integrations
qk. The normalization
z [O] = 1.
(2.6)
For the sake of definiteness we apply here the so-called Weyl’s quantization procedure whenever products of non-commuting operators $, fik enter [13]. Using (2.1) and (2.2) one can rewrite the above formula for Z[J] in the following way:
z[J]
= N-’
SndqkdPkdUkg[Uk-~k(q,p)]eXp
V)+J’q]
,
f,k
(2.7) where we have used, for brevity,
the notations
(2.8) and similarly According
for p. u, and J. q. to the property of the Dirac
rpUkb
[ok-Ljk(q,
II)] = prn(q,
d-function
u)l~dvkd k
k
where
x,,(q,
4) is the
Hessian
of a system,
we may
>
(2.9)
write for the generating
76
W. GARCZYT;ISKI
functional z[J]
as follows :
[aL;; ‘)-pk]
= N-’ Jndqkdp,dukd
x
f,k
G%&L Wv The integrations over momenta functions, and we get
i’&H~.(4-4+L(4, I.
can now be performed
z [J] = IV-l J-ndqkdvk(A?n(q,
‘) (4-u)+L(q,
i Jdt aLt i. [ by a fixed function xexp
the uk variables
using the presence
u)+J.q
4”
I> .
vk = pk+‘jk, we obtain
from the last formula
nrn(% t
4) =
measure
the final expression
d+J.41 ,
m(q, 4) is given by the following
j-ndpkb%t(6?>
6i+P)l
(2.13)
path integral:
x
t,k
L(q, 4+~)-L(a The infinite continuous form
product
of S”(t)
= exp
(2.14) ~+PJ]}.
CP$GL
factors can be written in the exponential
6”‘(O) Jdt Tr In
a2Lh 4 + PI aPk
where 6”‘(O) stands for (dt)- l. Hence, the measure takes on the form vm(G
(2.11)
(2.12)
ZCJI = N-l Jn&‘m(q, 4)exp t,k where the resulting
of 6-
?I)[x
t,k
Shifting
(2.10)
u)+Jd
4) = j-ndP,exP
Lb,
t,k
aL
-p%(q,
aPl
(2.15) ’
[14],
L~+P)--L(q,
@-
i+p)-iI#)(O)Trln /I
*
(2.16)
MEASURE
Instead Lagrangian
IN THE FUNCTIONAL
77
INTEGRAL
of the measure m(q, 4) one can define, LQ” by the definition
let us say, the “quantum
5 exp {i JdtLQ (q, $1
*
xTrln In this case the generating Z[J] A deviation non-triviality
functional
1.
iJdt[&(q,
of the quantum Lagrangian of the measure m(q, 4).
EXAMPLE1. Local scalar field theory.
The canonical
momentum 6
conjugate
iF(t,
2, y) =
= Lagrange
of the scalar
a2
MC Wdk Y)
i.e.
(2.20)
is given by
aLC4k a Q#4, a a&t, x‘)
(221)
.
.
field are equal
p:w@, 3, Qm s(g_g)
the Hessian,
is therefore
31 (2.22)
wt,
of this matrix,
L signals
function.
to the field 4(x)
= ~2m#4~~ a>Qm~ m 3 @(t, FJ The determinant
(2.19)
.
In this case L stands for a Lagrangian,
I^
of the Hess matrix
d)+J.q]
LQ from its “classic ancestor”
~d(~,x‘) . &M(~, n9 Q?w~ m =
n(t, 2) = Elements
8,4(x)]
(2.18)
takes on the form
= N-‘Jndq’exp t.k
JdiL[$(x),
(2.17)
equal
78
W. GARCZYrjSKI
The generating functional of complete outlined above, and reads
can be derived in the way
~S~X[~(X)~(X)-H[~(X), n(x)]+
ZCJI = N-‘JJJd+(x)dn(x)exp i
x
+
=
Green’s functions
~-1JW#WmCN4
1
J (4 $ (41
d(x)lexp i
x
jC~~x[Ll~W, Q+U+J(4W)]
I,
(2.24) where the measure
is given by the following
functional
integral:
The appearance of the additional Dirac d-function in the last formula is due to the time product of the Hessians. Let us consider in detail the self-interacting scalar field 4’ whose Lagrangian is quadratic in time derivatives, and contains an arbitrary polynomial in the field L[$, It is field into with
a,41
easy to check that for any 4 (or its time derivative), the normalization factor. the classical one, up to nm[~$(x), X
The generating
d(x)]
functional Z[J]
(2.26)
=~a,~a~-+l2~2-V[~].
V[C$] the measure m(4, 4) does not depend on the and is a trivial numerical factor, which we include The quantum Lagrangian LQ coincides in this case a constant
= Jfl&I(x)exp x
-&Jdxf12(x) i
.
= const.
takes on in this case the customary =
N-l Jn@(x)exp x
i
(2.27)
I
iJdx(LfJ.4) I
form .
(2.28)
One sees that the above particular form of the Lagrangian obscures the possible appearance of the non-trivial measure in the functional integral over the configuration space of the field. Extension of this form of functional integral to more
MEASURE
IN THE
FUNCTIONAL
79
INTEGRAL
complicated Lagrangians, with coefficients depending on fields, has no grounds. One should check in every case whether the measure m(q, 4) does or does not depend on field variables. In the sequel we shall omit altogether the normalization factor N. It will be restored at the end of calculations. 3. The measure for a singular bosonic svstem with second_clasQ constraints It is evident from formula (2.14) that the measure m(q, 4) vanishes for singular systems since the Hessian vanishes in these cases. One should then reconsider the whole derivation of the resulting measure when constraints are present in a system [15-173. Let us assume now that a system is a singular one
yi”,(q, 4) =
a*wL 4) n=
I
a$ a$
I 1
o
(3.1)
’
-
and that the rank of the Hess matrix is I < n. We assume that a principal dimension r is different from zero
minor of
(3.2) This does not restrict the generality of considerations since by renumeration coordinates it can always be achieved. In this case the tist r equations
am 4) = Pkr
k
aq
=
1, . . . . n,
of the
(3.3)
can be solved for the velocities 4” =fa(ql,
. . . . 9”, Pl, .“? P,, cY+l, ...> 47,
c1 = 1, . ..) 1.
(3.4)
The remaining velocities q+‘, . . . , 4” are arbitrary functions of time. It is convenient to denote the first r components of q’s and p’s as q’ and p’, respectively, while the remaining n-r components as q” and p”, 4 = (4’5 q”),
p = (p’, p”),
etc.
(3.5) If X, is not a principal minor, we shall call the primed variables those which enter the minor and the complementing variables will bear double primes. Using this short notation we may write solution (3.4) as follows: 4’” = “PM, P’, 4’7, Substituting these solutions II -r primary constraints
cl=1 7 . . . . r.
back into equations
C&9(4,PI = P;;-s/&L
P’) z 0,
(3.6)
(3.3), one gets r identities, j? = r+l,
. . . . n.
and
(3.7)
80
W. GARCZYr;JSKI ’
The wavy equality sign means that it holds on the surface of constraints. We consider here the case when all primary and possible secondary constraints are of the second class. In such the case their number is even. We shall renumber them for convenience , . . . . 2m”,
a=1
cb’(q, P) = 0,
(3-g)
and I(Cb’, CL}1 = C’(q, P) # 0.
(3.9)
The double prime over C, stands for the second class character of the constraints. The canonical Hamiltonian of a system with constraints is connected with the singular Lagrangian in the following way: H(q, P’) = i P:_P(4, P’Y 4”)+ a= 1
lf
s&L
P’)4”s-uq,
4hjf=f(q,p’,q”).
(3.10)
p=r+1
The constraints Ci ought to be conserved in time, and this can be achieved due to a choice of unspecified so far functions q“M 5 p* for which one gets the following system of linear equations: Ci z {Ci, H)+p*
a, b = 1, . . . . 2m”.
{Ci, CC} = 0,
(3.11)
A solution exists since, by assumption, the matrix C” is not singular. The generating functional of Green’s functions of the system has the form Z[Jl
where the measure
= Jfl444, t
(3.12)
P)exp
dp is given by, [18, 193,
dPcL(4, P)= s(c”)(q, P) fi dqk‘hr
(3.13)
k= 1
where S(C”) denotes
the Senjanovic
weight (3.14)
Substituting ZCJI
H into
the Hamiltonian
= J~44q~ t
; jd~(&
Pbxp
formula
(3.12), one gets
P; Cd’“-f”(q>
P’, $‘)I +
n
+8=z+,
,
CP;;-s&9
P’)wB+QL
4’9 4’xj~=,+J*q
(3.15) 1
MEASURE IN THE FUNCTIONAL
=
INTEGRAL
81
~l--pwL P~~d~“~C~“-f”(4,P’, 4”)l x f
a
xexp
kJdt[ph(Q’“-o’)+L(q,
U, $‘)+.J.q]
1
.
(3.16)
I
We drop the term with the sum over j in the exponent because of the presence of Dirac d-functions in the measure ,u. Now, by applying formula (2.9) which in this case reads l+kJ”6 [VU-f”(q, a we obtain,
p’, $‘)I = jP,(q,
after performing
integrations
0, $‘)I n&*6 a
p;-
RL(qiyY ““)I,
(3.17)
[
over p:,
(3.18) dji is given
where the new measure
dii(q,
0,
P”) = IZ,(q,
0,
i”)lS(C’?
by the formula 4, aL(q;’
4”), p” n dqkNol duae=fi+ 1 dp;; I > k
(3.19) Changing
the variables
of integration ua = p; +g=,
where $’ is considered
4)exp
t,k
vm(q, 4) =
m(q, 4) is determined
J;dp,l%,(q,
$+p’,
L(q, $+A
This is the be reduced the above which we
basic formula for the to the above case of formula holds. Some shall consider at the
(3.20)
for Z [J]
fixed, we finally obtain
Z [J] = Sndqkm(qy Here the measure
a = 1, . . . . r,
:SdGUq,
1.
4)+Jd
by the following
$‘)ls(c”)
4”)-L(q,
c
OL
(3.21)
path integral:
q, aL(q’ ;p--p”
4-p$(q,
.
“‘), j+
d’+p’,
$)I}.
(3.22)
path integral measure. All other cases can, in fact, a system with second type constraints, for which complications will arise in the case of fermions, end.
W. GARCZYr;rSKI
82
4. The measure for a singular hosonic system with first-class constraints only We consider
a singular
dynamical
system subject to the first-class constraints
Ci(q, P) 2 0, satisfying
a = 1, . . . . m’
(4.1)
the conditions ‘,c;, cg) = U@Yc; E 0, (4.2)
[CL, H) = VaPC;l = 0. According to a general partner G, such that
theory,
one introduces
G, = G,(q, P),
for every constraint
a=1
(complete nonco\ar~ant
det
II&,
CL its gauge
, ..., m’,
(4.3)
gauges)
C~lll~c
f
(4.4)
0.
The last condition means that the G,‘s break the gauge symmetry generated the first-class constraints (CA). In totality, one gets a set of constraints (4,) =
(2)
a =
OL
On the surface
of constraints
the following
by
1, . . . , 2m’. equality
holds: (4.6)
This means that (4,) form the set of second-class the results of the previous section [20]. According may write
constraints, to formulae
and we may apply (3.21) and (3.22) we
where
vmG(q, 4) = Jn4+IX,(q, t,k
x
exp i
=
$+P’, 4”)lS(4) 4,
$s dt Uq, 4’+p’, 4”)-Uq, [
j~bM,(q~ t.lL
+~&‘(q,
4’+~‘, 4”)I[IWlIG,,
d’+p’, 4”)
8P;
C~}ll~~6(C~)6(G,)] 3
11
(4.8)
x
MEASURE
-WI,
IN THE
FUNCTIONAL
83
INTEGRAL
4hzxL 4’+p’, 4”) ‘I ZPL 1 _I
(4.9)
‘.
Here, as before, Xr stands for a maximal non-vanishing principal minor of order r of the Hess matrix of the system. It was shown by Faddeev (cf. [lo]) that matrix elements of the scattering operator determined by Z, [J], do not depend in fact on the choice of gauge constraints (G,), within the class of functions connected by canonical transformations generated by the constraints (CL)
(G, C”E~ 8J
6G 01 =
The gauge constraints
9
’
a, p =
do not need to commute
(4.10)
1, . . . , m’. between
themselves
(cf. [ZO]).
5. The measure for a singular hosonic system with both types of constraints As before, we consider freedom with phase space
a bosonic
system
with a finite number
of degrees
r = {qk, Pk; k = 1, . . . . n), subject
to constraints
(5.1)
of the first-class Ci(q, P) = 0,
and of the second-class,
a=1
, . . . . m',
(5.2)
as well CS(q,
We assume that the number hence the determinant
(5.3)
a” = 1, . . . . 2m”.
P) = 0,
m’ of the first-class det ()(Cy,,, c;q,,
does not vanish
of
constraints
# 0
is a maximal
one, (5.4)
on the surface of constraints c E (C LJ C”) % 0.
(5.5)
The number of second-ciass constraints is even in this case: 2m”. Poisson brackets of the first-class constraints with all other constraints vanish on the constraints surface. As in the previous section we introduce gauge constraints G,; z = 1, . ..’ m’ such that
6 = G,(q, P) detll{G,,
(non-covariant
Cblllt, f 0,
gauges),
a, /? = 1, . . . , m’.
(5.6)
W. GARCZYhSKI
84
As before.
we introduce
the constraints (,..)=(Z),
a’=1
)...)
2m’.
(5.7)
Gdll)1”, f 0.
(5.8)
OL They are of the second-class
since
det IIkL The generating
functional
&)ll), = (det
Ilh
can be written in the form given by Senjanovic
f
3
(cf. [18])
(5.9)
where the measure dpG is given by the product of two characteristic pieces, each of them corresponding to the case of a system with the second-class constraints only &c (4, P) = S (4) S (C”) fl dqk dP, = ldetll(G,,
Cjillln6(G,)6(Cb),detll(C;,, a
C~,,}ll11’2x
x fl6 (Cit.) n dq’ dP,. k 0”
(5.10)
As before, the S-matrix related to Z, does not depend on the choice of gauge constraints (G,). Within the class of functions connected by the canonical transformations (4.10), generated by the first-class constraints, the following equalities hold : z,
= ZG,dC = z.
(5.11)
It is possible to write the whole measure dpG in the form of single piece of Senjanovic type. For this we introduce an equivalent set of constrainsts of the second-class c;,. = La”b”c;. + lalf, CL. The matrix llLa~~b~~~~ is an arbitrary non-singular one, while the matrix subject to the conditions which follow from the requirements: {c”,,,, G,) z 0,
(5.12) Ill,,,,ll is
c1= 1, . . . . m’, u” = 1, . . . . 2m”, (5.13)
or Lawb,,{C$,
G,} + I,,,,, {C;,, G,} z 0.
of the matrix Solution for 1,,,,, exists due to the assumed non-singularity ll{CL,, G,)ll, see (5.6). The characteristic Senjanovic piece of the measure is invari-
MEASURE
ant under
the above
S(@,,
We denote
IN THE
transformations
FUNCTIONAL
INTEGRAL
[21]
= ldet l)(Z_.,.,c,.C;., Lb,,dj, C&:S)11)1/2 n S(L,..,.. Ci,,),, = S(C”)j,. (1”
by (3) the total
They are of the second-class,
2(m’+m”)
,rl,...,
= 2m.
(5.15)
since we have 10
{C’, C’> {C’, G} {C’, ,,‘} $p}ll =
{G, C’} {G, G} (G, c”} {c”l’, C’> {c,‘, G} {c”,‘, ,“}
=
{Cl, G) 0
{G, C’> {G, G) 0 0
0 {El!, ,,,} (5.16)
= (det ll{G,, C;P}ll)’det Il{c”$, e;,,}ll,c # 0. Therefore,
using the invariance S($) =
(5.14)
set of new constraints
CL G, , -II 0 C a”
($,)=
detll@,,
85
property
(5.14) we obtain
the formula
[detll{G,,cb>lllns(C~)s(G,)S(C”).
The last step follows by noting that one obtains from the original set of constraints
0
(5.17)
a
exactly
the same result starting
G
($,) =
G,
,
y = 1, . . . . 2(m’+m”)
= 2m.
(5.18)
C-11 0” L= 1 In other words, using properties the equalities det
of the Dirac b-functions,
and determinants,
II{&,, $Jll = W U2 *detll~~,~hJll
we get (5.19)
and (5.20)
S(G) = S($). Finally,
we obtain
the results &G (4, P) = s ($) (4, P) n dqk dPk
(5.21)
k
and
z[Jl =
j~dqkdPkldetlI{tiy9 ‘k,)III”2n~(‘&)exP t,k
Y
. (5.22)
86
W. GARCZYhSKI
Thus we have succeeded in casting the measure into the Senjanovic form, which is characteristic of a system with the second-class constraints only. Hence, we may apply the results of Section 3 and obtain immediately in this way the formulae z
CJI= Jn dqkMq, 4bxp fi$U4, i. t,k
where the measure
mG is given by
4)+Jd
,
(5.23)
d
~m,(% 4) = ~~h’ki%( q, f,k
L(q, ~‘+p’, d”)-L(q,
4)-p;$(q, d’+p’, 4j}. 01
(5.24)
Here, as before, X, stands for rth order minor of the Hess matrix of the system, and r is the rank of this matrix. 2. The pure electrodynamics in Minkowski space-time, in the secondorder formulation, with t = x0 as the dynamical parameter. The Lagrangian of the system is EXAMPLE
L(;’ - J_ = -$FrVFCY,
(5.25)
where J’pYzz ?“A”-PA’ , / We consider, tials A’,
as independent
p,v=O,1,2,3.
field variables,
the four components
At every fixed point 2 one has four degrees of treedom. conjugate to A are =
aL a
of the poten-
k = (p, 2).
qk - A”(T),
I$
(5.26)
=
(5.27) The canonical
momenta
-Fop.
The Hess matrix of the system, at a fixed point X, has the remarkably
simple form
0000
a2L II~~VII =
I/
a(#)
a(aOAv)
0100 II
= IISJLO~YO-CJl(YCIOOlI =
0010’
(5.29)
0001 Notice that it does not depend on field variables, so it remains unchanged throughout the phase space of the system. One sees that the determinant of this
MEASURE IN THE FUNCTIONAL
matrix,
the Hessian,
vanishes,
INTEGRAL
87
and that its rank is r = 3,
z3
(5.30)
= 1.
To be exact one should rename the zero-component A0 as A4 in order that X3 become the principal minor of the Hess matrix. Therefore, the system is singular, and contains (5.31)
n-r=4-3=1 primary constraints, since F,, vanishes,
at every fixed point Z. This constraint and we get from (5.28)
can easily be identified
c1 = no z 0. Using this constraint one may construct according to rule (3.10) z The Latin Since
(5.32)
the spatial
indicies k, 1, etc. run from
we have for the Hamiltonian
that the behaviour
?/IO,
for the canonical H = j%% Hamiltonian
(5.35)
$A’.
of II, A0 at spatial
infinity
is such that (5.36)
A’) = 0,
Hamiltonian
of the electromagnetic
field the expres-
= ~d::(~nknk+tFk,Fk,-Ao~nk).
(5.37)
is thus equal
&I = H+~d%)&(X) where t’ is an arbitrary Poisson brackets
(5.34)
density
jd3 x8(n,
The primary
(5.33)
1 to 3.
% = +n,n,+@,;&,+n,
we obtain sion
of the Hanultonian
= n, kP - L(c,Ja=fP = nkkk-tnkn,+aF,,F,,ldP=rC.
AL = FOk+ $A0 = n,+
Assuming
density
Lagrange
= Sd~(tn,n,+tF,,Fkl-A’$n,+vn,), multiplier.
{A”(x), K(Y))
Using
the fundamental
(5.38) equal time
=&8(X-Y), (5.39)
x0 = yo, {A’(x), A”(Y)} = (n,(x),
K(Y))
= 0
88
W. GARCZYhSKI
we may calculate
the time derivative
Cl = Ii, =
of the constraint
C1,
{no,HP} = a%,= 0.
(5.40)
The last equality is imposed as a consistency condition of the formalism. therefore, another constraint, called the secondary one c2 = &7, There
are no further
constraints,
(5.41)
= 0.
since C2 is already
conserved
Cz z {c,, HP) = +(A%-~~)F,, It is straightforward
C,(Y)) = 0,
{Cl (4,
C,(Y)}
both constraints
equalities:
W,(x)9Cl (Y)) = 09
= 0,
G(x),
(5.43)
H(xO)) = 0.
are of the first-class
c, = c; = no,
c2 =
c; = a%,,
and we are in the situation described in Section generate canonical transformations of the form 61 A”(x) = J&A’(x), 61 n,(X)
= jd; {n,(x),
=
4. These
constraints
Ci, C; (5.45)
no(Y))%(Y) = 07
(5.46)
E2
(Y)= -sE 8
3WY)}&2(Y)
#w,(X),
(544)
=&&l(X),
no(Y)
62 AM(x) = Id; {A” (XL 8 n,(Y)} ~24(4
(5.42)
W,(x)9 G(Y)) = 0,
{C, (x), H(xO)} = 0, Hence,
in time,
ZE0.
to check, using (5.39), the following G(x),
We get,
=o.
E2 (4,
(5.47) (5.48)
We see that the constraint no z 0 generates changes of A0 only, while the secondary constraint ak nk z 0 changes spatial components of the potentials A” adding derivatives of s2. The momenta n,, do not change under these transformations. Similarly, the spatial components of the strength tensor do not transform 6, F&.1= 82Fkl = 0. Putting
(5.49)
in particular s1 (x) = aoE(x) (5.50)
and E2 (4
=
-E
(4,
MEASURE
where E(X) is some arbitrary transformation (4.10)
IN THE
FUNCTIONAL
differentiable
89
INTEGRAL
function,
we get for a full canonical
&4’(x) = jd; {A’(x), no(y)dOE(yO)-~n,(Y,E(~)}
= a&(X).
(5.51)
Hence, the Abelian gauge transformations of the above particular form follows from the above canonical transformations generated by the first-class constraints. Every first-class constraint should be supplemented by a gauge constraint. We shall choose the traditional radiation gauge
This gauge is admissible
0,
G1 =
A0 z
G2 =
dkAk z 0.
since the determinant
(5.52)
(4.4) differs from zero in this case
{A07“0) {AO,$n,} WI@,, C;}ll = (akAk, Ho) (akAk, d’l7,) =
S(Z--g)
0
0
AS(x‘-8)
II
# 0. (5.53)
Moreover, the determinant does not depend on field variables. Both gauge constraints are conserved in time, on the expense of choice of the Lagrange multiplier u, and due to the other constraints. We have Gr = k0 = {AO, HP} = u = 0
(5.54)
and & = akkk = {&A’, HP} = -AA’-&& Thus the primary
Hamiltonian
H, and canonical
z 0.
one H coincide,
(5.55) in the radiation
gauge H PIAO=,=O for onto primed and Hessian let us say, A4. In this case the principal one. So primed variables the double-primed ones are +
measure one needs the division One sees from form (5.29) the the third-order are with qrfk
time components
the and II,, of these vectors
W. GARCZYfiSKI
90
The next important
of formula
(4.9) is
(4,“(”;;“’ “‘),p”)=fs(n,)s($n,)s(~o)6(a,~k)
~(C;)6(G,)
fi a=
ingredient
1
Therefore
we get for the measure, ‘,’
since the following
mG
(4,
continual
4)
dropping =
F
integral
an insignificant
6 CA0
@)I
6
[&
is a constant
Ak
(5.59)
constant, (5.61)
(x)lj
not depending
ndn,(.~)~[n,(X)]~[C?kn,(x)]exp
unon
=COnSt.
Thus we end up wit1 the following formula for the generating functional in the radiation gauge: Green’s functions for pure electrodynamics, Z[J](Rad
= N-’
fldA’(x)G
[A’(x)]6
[tZLAk(s)]
fields: (5.62) of full
x
X.P
(5.63) = exp
(564)
This gives a full justification of various procedures applied earlier in quantization 6. The measure
for a general
singular
rather formal and not quite correct of the pure electrodynamics [22, 23).
bosonic-fermionic
system
with both kinds
of constraints
When
fermions
are present
in a system one should consider the variables of a Grassmann algebra, identifying even elements with bosonic degrees of freedom, and odd elements with fermionic ones [24]. It is convenient to identify the parities of coordinates with parities of their indices
qk, pk; k = 1, . . . . n as elements
II
= n(p,) = n(k) =
0
1
for bosonic for fermionic
k, k.
(6.1)
Constraints themselves are elements of a Grassmann algebra, and the same holds for a Hamiltonian which is assumed to be an even element. The Poisson brackets are modified to incorporate fermionic degrees of freedom. For two elements F, G of the Grassmann algebra, with given parities n(F) and n(G) respectively, one
MEASURE
defines a generalized
IN THE
FUNCTIONAL
bracket
as follows
Poisson
INTEGRAL
91
[25-281:
(6.2) where the left a’/@ and right sense :
Z’/dq derivatives
are understood
in the following
. We have for F with a definite
parity
8F = -(84 8’F &-
(6.3)
I)‘@~
_F
for q -
odd, (6.4)
t?F izq
With definition (6.2) of the generalized preserves its form
for q -
even.
Poisson
brackets,
the equation
c = a,c+{c,H), where the Hamrltonian
H is defined
of motion
(6.5)
as follows:
H = i L.=
pk‘jk-L,
(6.6)
*
and the order of variables is essential when fermionic degrees of freedom enter. All the formulae derived previously for pure bosonic systems remain valid also in the mixed bosonic-fermionic system, if we replace the determinants with superdeterminants (Berezinians). This comes about because the key property (2.9) of Dirac 6functions, which was used in deriving a measure, is now replaced by = nS(C,),
(6.7)
a
where the superdeterminant s det
of a generalized
det(e, -0, =
matrix is of the following
ei ’ 02) =
det e2
block form:
Ber
.
(6.8)
Such matrix transforms, e.g. constraints, and preserves its arrangement into bosonic and fermionic parts when e,, e,, are even while O,, O2 are odd elements of
W. GARCZYhSKI
92
Grassmann
algebra (6.9)
Clearly, the ez matrix is assumed one needs is the following:
non.-singular
SdetA where the supertrace
is defined
here. Another
useful formula
which (6.10)
= exp(sTrlnA),
as STrA
= i
(6.11)
~~~~~
k=l
and Ek = Berezinians
for k bosonic,
1
also enjoy the fundamental Sdet(A,
(6.12)
for k fermionic.
-1
property
of determinants
AZ) = (Sdet A,)(Sdet
A,).
(6.13)
In the most general situation of a mixed bosonic-fermionic system subject to both kinds of constraints we write for the generating functional (6.14) where
(6.15) Here (II/,) stands
for the totality
of constraints
,
y = 1, . . . . 2(m’+m”)
= 2m.
(6.16)
The number m’ denotes the maximal quantity of the first-class constraints Ci = 0, cl=1 9 **.> m’, while 2m” denotes the number of the remaining second-class constraints C$ Z 0; Q” = 1, . . . , 2m”. They do satisfy the condition Sdet ll{Ci,,, Cb:,}ll,c# 0.
(6.17)
MEASURE IN THE FUNCTIONAL
All the appearing in this section Poisson ized sense (cf. (6.2)). The gauge constraints restriction
INTEGRAL
93
brackets are understood in the generalG, z 0, a = 1, . . . , m’ are subject to the (6.18)
Sdet ll{G,, C;)lllc # 0. The momenta
conjugate
to the coordinates pk =
Since we are dealing
awl> 4) a$
’
qk are
k = 1, . ..) n.
here with a singular
(6.19)
system, we have the inequality (6.20)
r = rank
and Xr(q,
Repeting, generating
mutatis mutandis, functional
the previous
o
z
4) = Sdet
(6.21)
.
considerations
we may
write
for the
(6.22) where the measure
h(q,
4) =
m, is given by the integral
~fldp,(i%f,.(q,d’+p’, 4”) s($)
(q,“L(q’ zp-“” “‘I,f) x
t,k
L(q, $+p’,
4”)-L(q,
4)--p:
‘~3~’
zpTp”
“‘)]I.
(6.23)
OL The resulting generating functional does not depend on the choice of gauge constraints within the gauges belonging to orbits of the gauge transformations generated by the first-class constraints 6G, = {G,, Cb)@,
(6.24)
z,
(6.25)
= ZG,&
= z.
7. Concluding remarks Summarizing, we have found a general form of the measures which appear in functional integrals over the configuration space of singular systems. Considerations were carried out for the non-covariant gauges only. We justify in this way
94
W. GARCZYfiSKI
the insertion of Dirac b-functions of constraints under the functional integral over A% in the case of Maxwell electrodynamics in radiation gauge (cf. [22, 231). The full justification of the Faddeev-Popov ansatz for quantization of gauge fields requires inclusion of covariant gauges also for, both, Abelian and non-Abelian gauge fields, and for Einstein gravity, as well. Covariant gauges require employment of a more powerful formalism developed recently by Fradkin, Batalin and Vilkovisky in a series of papers, which were reviewed by Henneaux [29], and Batalin and Fradkin [30]. The case of Einstein gravity requires special attention in view of the controversy around the shape of the measure appearing in the generating functional of Green’s functions [31, 321. We bention in this connection a recent paper by Unz [33] where the Faddeev-Popov ansatz was rather used for deriving the measure than justified as in the our case, where we derive the measure independently of that ansatz. We postpone these interesting problems connected with covariant gauges and non-Abelian gauge theories to subsequent publications. Acknowledgements The author wishes to thank Professor E. S. Fradkin and Dr. I. A. Batalin for useful col;versations concerning the measure for essentially non-linear field theories. The author wishes to thank Professor J. Rzewuski for his interest in the present work.
REFERENCES [l] Dirac, P.A. M.: Physikalische Zeitschrif der Sowjetunion, 3 (1933), 64. [Z] Feynman, R. P.: Rev. Mod. Phys., 20 (1948), 367. [3] Feynman, R. P.: Phys. Rev., 80 (1950), 440; 84 (1951), 108. [4] Feynman, R. P. and A. R. Hibbs: Quantum Mechanics and Path Integruls, MC Graw-Hill, New York 1965. [S] Berezin, F. A.: Teor. Math. Phys., 6 (1971), 194 (in Russian). [6] Slavnov, A. A. and L. D. Faddeev: Vvedenie v kvantovuyu teoriu kalibrovochnykh polei, Nauka, Moscow 1978 (in Russian). [7] Garczyriski, W.: Bull. Acad. Polon. Sci., CL II!, 21 (1973), 355. [8] Garczytiski, W.: in: Functional Integration. Theory and Applications, J. P. Antoine and E. Tirapegui (Eds.), Plenum Press, New York-London 1980, pp. 1755189. [9] Fradkin, E. S.: in: A Memorial Volume to I. E. Tamm, Nauka, Moscow 1972, pp. 146176. [lo] Faddeev, L. D.: Teor. Math. Phys., I (1969), 3. [l l] Faddeev, L. D. and V. N. Popov: Phys. Lett. B25 (1967), 30. [I21 Faddeev, L. D. and V. N. Popov: Preprint ITP, Acad. Sci. USSR, Kiev 1967. [13] Berezin, F. A. and M. A, Shubin: Schriidinger Equation, Moscow University Press, Moscow 1983 (in Russian). 1143 Fradkin, E. S. and G. A. Vilkovisky: Phys. Rev., D8 (1973), 4241. [15] Dirac, P. A. M.: Canad. T Math., 2 (1950), 129. [16] Dirac, P. A. M.: Lectures on Quantum Mechanics, Belfer Graduate School of Science, Yeshiva University, New York 1964.
MEASURE IN THE FUNCTIONAL
INTEGRAL
95
For a broad review of other references see also: [17] Sundermeyer, K.: Constrained Dynamics, Lecture Notes in Physics No 169, Springer-Verlag, Berlin-Heidelberg-New York 1982. [18] Senjanovic, P.: Ann. of Phys. (US), 100 (1976), 227. [19] Fradkin, E. S.: Acta Universitatis Wratislaviensis, No 207, Proceedings of the X-th Winter School of Theoretical Physics in Karpacz, Poland. 1973. [20] Garczyriski, W.: Ann. of Phys. (US), 174 (1987), 26. [21] Garczynski, W.: Invariance Property of Senjanovic Measure, and Its Role in Quantization of Constrained Systems, Proceedings of the XXII Winter School of Theoretical Physics in Karpacz, Poland, 1986, World Scientific Publishing Co, Singapore 1986. [22] Bialynicki-Birula, I. J.: r Math. Phys., 3 (1962) 1094. 1231 Konopleva. N. P. and V. N. Popov: Gauge Fields. 2nd ed., Moscow 1972 (in Russian). [24] Berezin, F. A.: Introduction to an Algebra and Analysis with Anticommuting Variables, Moscow University Press, Moscow 1983 (in Russian), and references given there. [25] Fradkin, E. S. and G. A. Vilkovisky: Phys. Letr., 55B (1975) 224. [26] Casalbuoni, R.: Nuovo Cim., 33A (1976), 115 and 384. 1271 Batahn, I. A. and G. A. Vilkovisky: Phys. Lett., 61B (1977), 369. [28] Maharana, T.: Fortsch. Phys., 33 (1985), 645. [29] Henneaux, M.: Phys. Rep., 126 (1985) 1. [30] Batalin, I. A. and E. S. Fradkin: Works of the P. Lebedev Physical Institute, Academy of Sciences USSR, 165 (1986) 31. [31] Fradkin, E. S. and G. A. Vilkovisky: CERN preprint TH-2332 (1977), and references @en there. [32] Toms, D. J.: Preprint of the University of Newcastle upon Tyne NCL-86 TP6, and references given there. [333 Unz, R. K.: Nuovo Cim., 92A (1986), 397. [34] Rzewuski, J.: Field Theory, Part II, PWN-Polish Scientific Publishers, Warszawa 1969 (in Polish).
REPORTS
ON
MATHEMATICAL
MEASURE
IN THE FUNCTIONAL CONFIGURATION
Institute
Physics,
of Theoretical
University
No. 1
PHl3ICS
INTEGRAL SPACE+
of Wroclaw,
OVER THE
Wroclaw,
Poland
(Received January 26, 1987)
A general formula is derived for the measure in the functional integral over configuration space, for singular systems. It extends Fradkin’s result given for regular, essentially non-linear systems. Particular cases, up to the most general one of a singular bosonic-fermionic system with both kinds of constraints, are considered. Justification of the Faddeev-Popov ansatz for quantization of gauge theories in non-covariant gauges is thus given. Pure electrodynamics in the radiation gauge is treated in detail.
Introduction In the paper we shall discuss a problem of a measure in path integral representation of generating functional of Green’s functions. In classical papers of Dirac and Feynman [l-4] it was assumed that the transition amplitude has the form
(1.1) The integration
goes over continuous 4’ = 4 (0
paths and
connecting
points
q” = q(t”),
t” S [q;
t”, t’] =
S dtL(q, 4) is a classical action functional of a system. The relevant 1’ measure is thus a continuous product of Lebesgue measures dq(t). It does not contain any non-trivial weight factor which would depend on coordinates of integration q. In other words, all paths contribute to the integral over the phase
while the modulus
* Work supported
by the Polish
is the same N- ’ and does not depend
Ministry
of Higher Education c731
and Science.
on
q.
It is
W. GARCZYr;lSKI
74
known that the above expression mechanical problems characterized
works well for a wide class of quantumby Hamiltonian functions of the form V(q).
H = g+
(1.2)
In systems of this type one does not encounter neither the factor-ordering problem [S-S] nor any non-trivial measure in the integral, like the above one. The point is that the above, let us say, coordinate-space path integral, should be regarded as a result of integration over momenta performed within a corresponding phase-space path integral. As long as one considers Hamiltonians quadratic in momenta, with coefficients not depending on q’s, the integration over momenta yields some rather trivial constants which may be included into an overall normalization factor. When the coefficients do depend on coordinates q both above mentioned problems arise, and a non-trivial measure in coordinate path integral may appear as it was shown by Fradkin [9]. In this paper we would like to analyse this phenomenon for singular systems. One knows from Faddeev’s and Popov’s trick of quantizing gauge theories that non-trivial weight factors appear in generating functionals for Green’s functions of a system [l&12]. Thus, we would like to provide a basis for the Faddeev-Popov ansatz, deriving in a systematic way relevant measures in coordinate-space functional integrals. 2. Measure in a functional integral for generating functional, the case of a regular bosonic system For pedagogical reasons we shall start our considerations with a regular system for which the corresponding measure is known (cf. [9]). Later we will extend this result of Fradkin. Let a system with n degrees of freedom be described by coordinates 4r 3 . . . . q”, Pl, f.., p.. Its Hamiltonian H(q, p) is connected with a classic Lagrange function L(q, 4) in the usual way
H(qv d = i
k= 1
where dk are expressed
by canonical _
= Pk,
(3-4”
4” = dk(q, p) -
solution
Pk dk-
L(q>4)~
coordinates
q, p using the equations
k = 1, . . . . n,
of these equations.
(2.1)
(2.2)
MEASURE
IN THE FUNCTIONAL
This is possible for regular cases
75
INTEGRAL
does not vanish in such
systems since the Hessian
(2.3)
Quantization of the system expression for the generating
z[Jl =
by the path integral method leads to the following functional of the full, time ordered, Green’s functions:
N-lSnndqk(t)dpk(t)exp
~Sdt(Cpk(t)4k(t)-HJ[4(1),
k
&)I) ?
(2.4)
k
where
HJ(q, p) = H(q,
(2.5)
d--CJkqk, k
and
dk
factor
are just time derivatives of variables N is determined by the condition
of integrations
qk. The normalization
z [O] = 1.
(2.6)
For the sake of definiteness we apply here the so-called Weyl’s quantization procedure whenever products of non-commuting operators $, fik enter [13]. Using (2.1) and (2.2) one can rewrite the above formula for Z[J] in the following way:
z[J]
= N-’
SndqkdPkdUkg[Uk-~k(q,p)]eXp
V)+J’q]
,
f,k
(2.7) where we have used, for brevity,
the notations
(2.8) and similarly According
for p. u, and J. q. to the property of the Dirac
rpUkb
[ok-Ljk(q,
II)] = prn(q,
d-function
u)l~dvkd k
k
where
x,,(q,
4) is the
Hessian
of a system,
we may
>
(2.9)
write for the generating
76
W. GARCZYT;ISKI
functional z[J]
as follows :
[aL;; ‘)-pk]
= N-’ Jndqkdp,dukd
x
f,k
G%&L Wv The integrations over momenta functions, and we get
i’&H~.(4-4+L(4, I.
can now be performed
z [J] = IV-l J-ndqkdvk(A?n(q,
‘) (4-u)+L(q,
i Jdt aLt i. [ by a fixed function xexp
the uk variables
using the presence
u)+J.q
4”
I> .
vk = pk+‘jk, we obtain
from the last formula
nrn(% t
4) =
measure
the final expression
d+J.41 ,
m(q, 4) is given by the following
j-ndpkb%t(6?>
6i+P)l
(2.13)
path integral:
x
t,k
L(q, 4+~)-L(a The infinite continuous form
product
of S”(t)
= exp
(2.14) ~+PJ]}.
CP$GL
factors can be written in the exponential
6”‘(O) Jdt Tr In
a2Lh 4 + PI aPk
where 6”‘(O) stands for (dt)- l. Hence, the measure takes on the form vm(G
(2.11)
(2.12)
ZCJI = N-l Jn&‘m(q, 4)exp t,k where the resulting
of 6-
?I)[x
t,k
Shifting
(2.10)
u)+Jd
4) = j-ndP,exP
Lb,
t,k
aL
-p%(q,
aPl
(2.15) ’
[14],
L~+P)--L(q,
@-
i+p)-iI#)(O)Trln /I
*
(2.16)
MEASURE
Instead Lagrangian
IN THE FUNCTIONAL
77
INTEGRAL
of the measure m(q, 4) one can define, LQ” by the definition
let us say, the “quantum
5 exp {i JdtLQ (q, $1
*
xTrln In this case the generating Z[J] A deviation non-triviality
functional
1.
iJdt[&(q,
of the quantum Lagrangian of the measure m(q, 4).
EXAMPLE1. Local scalar field theory.
The canonical
momentum 6
conjugate
iF(t,
2, y) =
= Lagrange
of the scalar
a2
MC Wdk Y)
i.e.
(2.20)
is given by
aLC4k a Q#4, a a&t, x‘)
(221)
.
.
field are equal
p:w@, 3, Qm s(g_g)
the Hessian,
is therefore
31 (2.22)
wt,
of this matrix,
L signals
function.
to the field 4(x)
= ~2m#4~~ a>Qm~ m 3 @(t, FJ The determinant
(2.19)
.
In this case L stands for a Lagrangian,
I^
of the Hess matrix
d)+J.q]
LQ from its “classic ancestor”
~d(~,x‘) . &M(~, n9 Q?w~ m =
n(t, 2) = Elements
8,4(x)]
(2.18)
takes on the form
= N-‘Jndq’exp t.k
JdiL[$(x),
(2.17)
equal
78
W. GARCZYrjSKI
The generating functional of complete outlined above, and reads
can be derived in the way
~S~X[~(X)~(X)-H[~(X), n(x)]+
ZCJI = N-‘JJJd+(x)dn(x)exp i
x
+
=
Green’s functions
~-1JW#WmCN4
1
J (4 $ (41
d(x)lexp i
x
jC~~x[Ll~W, Q+U+J(4W)]
I,
(2.24) where the measure
is given by the following
functional
integral:
The appearance of the additional Dirac d-function in the last formula is due to the time product of the Hessians. Let us consider in detail the self-interacting scalar field 4’ whose Lagrangian is quadratic in time derivatives, and contains an arbitrary polynomial in the field L[$, It is field into with
a,41
easy to check that for any 4 (or its time derivative), the normalization factor. the classical one, up to nm[~$(x), X
The generating
d(x)]
functional Z[J]
(2.26)
=~a,~a~-+l2~2-V[~].
V[C$] the measure m(4, 4) does not depend on the and is a trivial numerical factor, which we include The quantum Lagrangian LQ coincides in this case a constant
= Jfl&I(x)exp x
-&Jdxf12(x) i
.
= const.
takes on in this case the customary =
N-l Jn@(x)exp x
i
(2.27)
I
iJdx(LfJ.4) I
form .
(2.28)
One sees that the above particular form of the Lagrangian obscures the possible appearance of the non-trivial measure in the functional integral over the configuration space of the field. Extension of this form of functional integral to more
MEASURE
IN THE
FUNCTIONAL
79
INTEGRAL
complicated Lagrangians, with coefficients depending on fields, has no grounds. One should check in every case whether the measure m(q, 4) does or does not depend on field variables. In the sequel we shall omit altogether the normalization factor N. It will be restored at the end of calculations. 3. The measure for a singular bosonic svstem with second_clasQ constraints It is evident from formula (2.14) that the measure m(q, 4) vanishes for singular systems since the Hessian vanishes in these cases. One should then reconsider the whole derivation of the resulting measure when constraints are present in a system [15-173. Let us assume now that a system is a singular one
yi”,(q, 4) =
a*wL 4) n=
I
a$ a$
I 1
o
(3.1)
’
-
and that the rank of the Hess matrix is I < n. We assume that a principal dimension r is different from zero
minor of
(3.2) This does not restrict the generality of considerations since by renumeration coordinates it can always be achieved. In this case the tist r equations
am 4) = Pkr
k
aq
=
1, . . . . n,
of the
(3.3)
can be solved for the velocities 4” =fa(ql,
. . . . 9”, Pl, .“? P,, cY+l, ...> 47,
c1 = 1, . ..) 1.
(3.4)
The remaining velocities q+‘, . . . , 4” are arbitrary functions of time. It is convenient to denote the first r components of q’s and p’s as q’ and p’, respectively, while the remaining n-r components as q” and p”, 4 = (4’5 q”),
p = (p’, p”),
etc.
(3.5) If X, is not a principal minor, we shall call the primed variables those which enter the minor and the complementing variables will bear double primes. Using this short notation we may write solution (3.4) as follows: 4’” = “PM, P’, 4’7, Substituting these solutions II -r primary constraints
cl=1 7 . . . . r.
back into equations
C&9(4,PI = P;;-s/&L
P’) z 0,
(3.6)
(3.3), one gets r identities, j? = r+l,
. . . . n.
and
(3.7)
80
W. GARCZYr;JSKI ’
The wavy equality sign means that it holds on the surface of constraints. We consider here the case when all primary and possible secondary constraints are of the second class. In such the case their number is even. We shall renumber them for convenience , . . . . 2m”,
a=1
cb’(q, P) = 0,
(3-g)
and I(Cb’, CL}1 = C’(q, P) # 0.
(3.9)
The double prime over C, stands for the second class character of the constraints. The canonical Hamiltonian of a system with constraints is connected with the singular Lagrangian in the following way: H(q, P’) = i P:_P(4, P’Y 4”)+ a= 1
lf
s&L
P’)4”s-uq,
4hjf=f(q,p’,q”).
(3.10)
p=r+1
The constraints Ci ought to be conserved in time, and this can be achieved due to a choice of unspecified so far functions q“M 5 p* for which one gets the following system of linear equations: Ci z {Ci, H)+p*
a, b = 1, . . . . 2m”.
{Ci, CC} = 0,
(3.11)
A solution exists since, by assumption, the matrix C” is not singular. The generating functional of Green’s functions of the system has the form Z[Jl
where the measure
= Jfl444, t
(3.12)
P)exp
dp is given by, [18, 193,
dPcL(4, P)= s(c”)(q, P) fi dqk‘hr
(3.13)
k= 1
where S(C”) denotes
the Senjanovic
weight (3.14)
Substituting ZCJI
H into
the Hamiltonian
= J~44q~ t
; jd~(&
Pbxp
formula
(3.12), one gets
P; Cd’“-f”(q>
P’, $‘)I +
n
+8=z+,
,
CP;;-s&9
P’)wB+QL
4’9 4’xj~=,+J*q
(3.15) 1
MEASURE IN THE FUNCTIONAL
=
INTEGRAL
81
~l--pwL P~~d~“~C~“-f”(4,P’, 4”)l x f
a
xexp
kJdt[ph(Q’“-o’)+L(q,
U, $‘)+.J.q]
1
.
(3.16)
I
We drop the term with the sum over j in the exponent because of the presence of Dirac d-functions in the measure ,u. Now, by applying formula (2.9) which in this case reads l+kJ”6 [VU-f”(q, a we obtain,
p’, $‘)I = jP,(q,
after performing
integrations
0, $‘)I n&*6 a
p;-
RL(qiyY ““)I,
(3.17)
[
over p:,
(3.18) dji is given
where the new measure
dii(q,
0,
P”) = IZ,(q,
0,
i”)lS(C’?
by the formula 4, aL(q;’
4”), p” n dqkNol duae=fi+ 1 dp;; I > k
(3.19) Changing
the variables
of integration ua = p; +g=,
where $’ is considered
4)exp
t,k
vm(q, 4) =
m(q, 4) is determined
J;dp,l%,(q,
$+p’,
L(q, $+A
This is the be reduced the above which we
basic formula for the to the above case of formula holds. Some shall consider at the
(3.20)
for Z [J]
fixed, we finally obtain
Z [J] = Sndqkm(qy Here the measure
a = 1, . . . . r,
:SdGUq,
1.
4)+Jd
by the following
$‘)ls(c”)
4”)-L(q,
c
OL
(3.21)
path integral:
q, aL(q’ ;p--p”
4-p$(q,
.
“‘), j+
d’+p’,
$)I}.
(3.22)
path integral measure. All other cases can, in fact, a system with second type constraints, for which complications will arise in the case of fermions, end.
W. GARCZYr;rSKI
82
4. The measure for a singular hosonic system with first-class constraints only We consider
a singular
dynamical
system subject to the first-class constraints
Ci(q, P) 2 0, satisfying
a = 1, . . . . m’
(4.1)
the conditions ‘,c;, cg) = U@Yc; E 0, (4.2)
[CL, H) = VaPC;l = 0. According to a general partner G, such that
theory,
one introduces
G, = G,(q, P),
for every constraint
a=1
(complete nonco\ar~ant
det
II&,
CL its gauge
, ..., m’,
(4.3)
gauges)
C~lll~c
f
(4.4)
0.
The last condition means that the G,‘s break the gauge symmetry generated the first-class constraints (CA). In totality, one gets a set of constraints (4,) =
(2)
a =
OL
On the surface
of constraints
the following
by
1, . . . , 2m’. equality
holds: (4.6)
This means that (4,) form the set of second-class the results of the previous section [20]. According may write
constraints, to formulae
and we may apply (3.21) and (3.22) we
where
vmG(q, 4) = Jn4+IX,(q, t,k
x
exp i
=
$+P’, 4”)lS(4) 4,
$s dt Uq, 4’+p’, 4”)-Uq, [
j~bM,(q~ t.lL
+~&‘(q,
4’+~‘, 4”)I[IWlIG,,
d’+p’, 4”)
8P;
C~}ll~~6(C~)6(G,)] 3
11
(4.8)
x
MEASURE
-WI,
IN THE
FUNCTIONAL
83
INTEGRAL
4hzxL 4’+p’, 4”) ‘I ZPL 1 _I
(4.9)
‘.
Here, as before, Xr stands for a maximal non-vanishing principal minor of order r of the Hess matrix of the system. It was shown by Faddeev (cf. [lo]) that matrix elements of the scattering operator determined by Z, [J], do not depend in fact on the choice of gauge constraints (G,), within the class of functions connected by canonical transformations generated by the constraints (CL)
(G, C”E~ 8J
6G 01 =
The gauge constraints
9
’
a, p =
do not need to commute
(4.10)
1, . . . , m’. between
themselves
(cf. [ZO]).
5. The measure for a singular hosonic system with both types of constraints As before, we consider freedom with phase space
a bosonic
system
with a finite number
of degrees
r = {qk, Pk; k = 1, . . . . n), subject
to constraints
(5.1)
of the first-class Ci(q, P) = 0,
and of the second-class,
a=1
, . . . . m',
(5.2)
as well CS(q,
We assume that the number hence the determinant
(5.3)
a” = 1, . . . . 2m”.
P) = 0,
m’ of the first-class det ()(Cy,,, c;q,,
does not vanish
of
constraints
# 0
is a maximal
one, (5.4)
on the surface of constraints c E (C LJ C”) % 0.
(5.5)
The number of second-ciass constraints is even in this case: 2m”. Poisson brackets of the first-class constraints with all other constraints vanish on the constraints surface. As in the previous section we introduce gauge constraints G,; z = 1, . ..’ m’ such that
6 = G,(q, P) detll{G,,
(non-covariant
Cblllt, f 0,
gauges),
a, /? = 1, . . . , m’.
(5.6)
W. GARCZYhSKI
84
As before.
we introduce
the constraints (,..)=(Z),
a’=1
)...)
2m’.
(5.7)
Gdll)1”, f 0.
(5.8)
OL They are of the second-class
since
det IIkL The generating
functional
&)ll), = (det
Ilh
can be written in the form given by Senjanovic
f
3
(cf. [18])
(5.9)
where the measure dpG is given by the product of two characteristic pieces, each of them corresponding to the case of a system with the second-class constraints only &c (4, P) = S (4) S (C”) fl dqk dP, = ldetll(G,,
Cjillln6(G,)6(Cb),detll(C;,, a
C~,,}ll11’2x
x fl6 (Cit.) n dq’ dP,. k 0”
(5.10)
As before, the S-matrix related to Z, does not depend on the choice of gauge constraints (G,). Within the class of functions connected by the canonical transformations (4.10), generated by the first-class constraints, the following equalities hold : z,
= ZG,dC = z.
(5.11)
It is possible to write the whole measure dpG in the form of single piece of Senjanovic type. For this we introduce an equivalent set of constrainsts of the second-class c;,. = La”b”c;. + lalf, CL. The matrix llLa~~b~~~~ is an arbitrary non-singular one, while the matrix subject to the conditions which follow from the requirements: {c”,,,, G,) z 0,
(5.12) Ill,,,,ll is
c1= 1, . . . . m’, u” = 1, . . . . 2m”, (5.13)
or Lawb,,{C$,
G,} + I,,,,, {C;,, G,} z 0.
of the matrix Solution for 1,,,,, exists due to the assumed non-singularity ll{CL,, G,)ll, see (5.6). The characteristic Senjanovic piece of the measure is invari-
MEASURE
ant under
the above
S(@,,
We denote
IN THE
transformations
FUNCTIONAL
INTEGRAL
[21]
= ldet l)(Z_.,.,c,.C;., Lb,,dj, C&:S)11)1/2 n S(L,..,.. Ci,,),, = S(C”)j,. (1”
by (3) the total
They are of the second-class,
2(m’+m”)
,rl,...,
= 2m.
(5.15)
since we have 10
{C’, C’> {C’, G} {C’, ,,‘} $p}ll =
{G, C’} {G, G} (G, c”} {c”l’, C’> {c,‘, G} {c”,‘, ,“}
=
{Cl, G) 0
{G, C’> {G, G) 0 0
0 {El!, ,,,} (5.16)
= (det ll{G,, C;P}ll)’det Il{c”$, e;,,}ll,c # 0. Therefore,
using the invariance S($) =
(5.14)
set of new constraints
CL G, , -II 0 C a”
($,)=
detll@,,
85
property
(5.14) we obtain
the formula
[detll{G,,cb>lllns(C~)s(G,)S(C”).
The last step follows by noting that one obtains from the original set of constraints
0
(5.17)
a
exactly
the same result starting
G
($,) =
G,
,
y = 1, . . . . 2(m’+m”)
= 2m.
(5.18)
C-11 0” L= 1 In other words, using properties the equalities det
of the Dirac b-functions,
and determinants,
II{&,, $Jll = W U2 *detll~~,~hJll
we get (5.19)
and (5.20)
S(G) = S($). Finally,
we obtain
the results &G (4, P) = s ($) (4, P) n dqk dPk
(5.21)
k
and
z[Jl =
j~dqkdPkldetlI{tiy9 ‘k,)III”2n~(‘&)exP t,k
Y
. (5.22)
86
W. GARCZYhSKI
Thus we have succeeded in casting the measure into the Senjanovic form, which is characteristic of a system with the second-class constraints only. Hence, we may apply the results of Section 3 and obtain immediately in this way the formulae z
CJI= Jn dqkMq, 4bxp fi$U4, i. t,k
where the measure
mG is given by
4)+Jd
,
(5.23)
d
~m,(% 4) = ~~h’ki%( q, f,k
L(q, ~‘+p’, d”)-L(q,
4)-p;$(q, d’+p’, 4j}. 01
(5.24)
Here, as before, X, stands for rth order minor of the Hess matrix of the system, and r is the rank of this matrix. 2. The pure electrodynamics in Minkowski space-time, in the secondorder formulation, with t = x0 as the dynamical parameter. The Lagrangian of the system is EXAMPLE
L(;’ - J_ = -$FrVFCY,
(5.25)
where J’pYzz ?“A”-PA’ , / We consider, tials A’,
as independent
p,v=O,1,2,3.
field variables,
the four components
At every fixed point 2 one has four degrees of treedom. conjugate to A are =
aL a
of the poten-
k = (p, 2).
qk - A”(T),
I$
(5.26)
=
(5.27) The canonical
momenta
-Fop.
The Hess matrix of the system, at a fixed point X, has the remarkably
simple form
0000
a2L II~~VII =
I/
a(#)
a(aOAv)
0100 II
= IISJLO~YO-CJl(YCIOOlI =
0010’
(5.29)
0001 Notice that it does not depend on field variables, so it remains unchanged throughout the phase space of the system. One sees that the determinant of this
MEASURE IN THE FUNCTIONAL
matrix,
the Hessian,
vanishes,
INTEGRAL
87
and that its rank is r = 3,
z3
(5.30)
= 1.
To be exact one should rename the zero-component A0 as A4 in order that X3 become the principal minor of the Hess matrix. Therefore, the system is singular, and contains (5.31)
n-r=4-3=1 primary constraints, since F,, vanishes,
at every fixed point Z. This constraint and we get from (5.28)
can easily be identified
c1 = no z 0. Using this constraint one may construct according to rule (3.10) z The Latin Since
(5.32)
the spatial
indicies k, 1, etc. run from
we have for the Hamiltonian
that the behaviour
?/IO,
for the canonical H = j%% Hamiltonian
(5.35)
$A’.
of II, A0 at spatial
infinity
is such that (5.36)
A’) = 0,
Hamiltonian
of the electromagnetic
field the expres-
= ~d::(~nknk+tFk,Fk,-Ao~nk).
(5.37)
is thus equal
&I = H+~d%)&(X) where t’ is an arbitrary Poisson brackets
(5.34)
density
jd3 x8(n,
The primary
(5.33)
1 to 3.
% = +n,n,+@,;&,+n,
we obtain sion
of the Hanultonian
= n, kP - L(c,Ja=fP = nkkk-tnkn,+aF,,F,,ldP=rC.
AL = FOk+ $A0 = n,+
Assuming
density
Lagrange
= Sd~(tn,n,+tF,,Fkl-A’$n,+vn,), multiplier.
{A”(x), K(Y))
Using
the fundamental
(5.38) equal time
=&8(X-Y), (5.39)
x0 = yo, {A’(x), A”(Y)} = (n,(x),
K(Y))
= 0
88
W. GARCZYhSKI
we may calculate
the time derivative
Cl = Ii, =
of the constraint
C1,
{no,HP} = a%,= 0.
(5.40)
The last equality is imposed as a consistency condition of the formalism. therefore, another constraint, called the secondary one c2 = &7, There
are no further
constraints,
(5.41)
= 0.
since C2 is already
conserved
Cz z {c,, HP) = +(A%-~~)F,, It is straightforward
C,(Y)) = 0,
{Cl (4,
C,(Y)}
both constraints
equalities:
W,(x)9Cl (Y)) = 09
= 0,
G(x),
(5.43)
H(xO)) = 0.
are of the first-class
c, = c; = no,
c2 =
c; = a%,,
and we are in the situation described in Section generate canonical transformations of the form 61 A”(x) = J&A’(x), 61 n,(X)
= jd; {n,(x),
=
4. These
constraints
Ci, C; (5.45)
no(Y))%(Y) = 07
(5.46)
E2
(Y)= -sE 8
3WY)}&2(Y)
#w,(X),
(544)
=&&l(X),
no(Y)
62 AM(x) = Id; {A” (XL 8 n,(Y)} ~24(4
(5.42)
W,(x)9 G(Y)) = 0,
{C, (x), H(xO)} = 0, Hence,
in time,
ZE0.
to check, using (5.39), the following G(x),
We get,
=o.
E2 (4,
(5.47) (5.48)
We see that the constraint no z 0 generates changes of A0 only, while the secondary constraint ak nk z 0 changes spatial components of the potentials A” adding derivatives of s2. The momenta n,, do not change under these transformations. Similarly, the spatial components of the strength tensor do not transform 6, F&.1= 82Fkl = 0. Putting
(5.49)
in particular s1 (x) = aoE(x) (5.50)
and E2 (4
=
-E
(4,
MEASURE
where E(X) is some arbitrary transformation (4.10)
IN THE
FUNCTIONAL
differentiable
89
INTEGRAL
function,
we get for a full canonical
&4’(x) = jd; {A’(x), no(y)dOE(yO)-~n,(Y,E(~)}
= a&(X).
(5.51)
Hence, the Abelian gauge transformations of the above particular form follows from the above canonical transformations generated by the first-class constraints. Every first-class constraint should be supplemented by a gauge constraint. We shall choose the traditional radiation gauge
This gauge is admissible
0,
G1 =
A0 z
G2 =
dkAk z 0.
since the determinant
(5.52)
(4.4) differs from zero in this case
{A07“0) {AO,$n,} WI@,, C;}ll = (akAk, Ho) (akAk, d’l7,) =
S(Z--g)
0
0
AS(x‘-8)
II
# 0. (5.53)
Moreover, the determinant does not depend on field variables. Both gauge constraints are conserved in time, on the expense of choice of the Lagrange multiplier u, and due to the other constraints. We have Gr = k0 = {AO, HP} = u = 0
(5.54)
and & = akkk = {&A’, HP} = -AA’-&& Thus the primary
Hamiltonian
H, and canonical
z 0.
one H coincide,
(5.55) in the radiation
gauge H PIAO=,=O for onto primed and Hessian let us say, A4. In this case the principal one. So primed variables the double-primed ones are +
measure one needs the division One sees from form (5.29) the the third-order are with qrfk
time components
the and II,, of these vectors
W. GARCZYfiSKI
90
The next important
of formula
(4.9) is
(4,“(”;;“’ “‘),p”)=fs(n,)s($n,)s(~o)6(a,~k)
~(C;)6(G,)
fi a=
ingredient
1
Therefore
we get for the measure, ‘,’
since the following
mG
(4,
continual
4)
dropping =
F
integral
an insignificant
6 CA0
@)I
6
[&
is a constant
Ak
(5.59)
constant, (5.61)
(x)lj
not depending
ndn,(.~)~[n,(X)]~[C?kn,(x)]exp
unon
=COnSt.
Thus we end up wit1 the following formula for the generating functional in the radiation gauge: Green’s functions for pure electrodynamics, Z[J](Rad
= N-’
fldA’(x)G
[A’(x)]6
[tZLAk(s)]
fields: (5.62) of full
x
X.P
(5.63) = exp
(564)
This gives a full justification of various procedures applied earlier in quantization 6. The measure
for a general
singular
rather formal and not quite correct of the pure electrodynamics [22, 23).
bosonic-fermionic
system
with both kinds
of constraints
When
fermions
are present
in a system one should consider the variables of a Grassmann algebra, identifying even elements with bosonic degrees of freedom, and odd elements with fermionic ones [24]. It is convenient to identify the parities of coordinates with parities of their indices
qk, pk; k = 1, . . . . n as elements
II
= n(p,) = n(k) =
0
1
for bosonic for fermionic
k, k.
(6.1)
Constraints themselves are elements of a Grassmann algebra, and the same holds for a Hamiltonian which is assumed to be an even element. The Poisson brackets are modified to incorporate fermionic degrees of freedom. For two elements F, G of the Grassmann algebra, with given parities n(F) and n(G) respectively, one
MEASURE
defines a generalized
IN THE
FUNCTIONAL
bracket
as follows
Poisson
INTEGRAL
91
[25-281:
(6.2) where the left a’/@ and right sense :
Z’/dq derivatives
are understood
in the following
. We have for F with a definite
parity
8F = -(84 8’F &-
(6.3)
I)‘@~
_F
for q -
odd, (6.4)
t?F izq
With definition (6.2) of the generalized preserves its form
for q -
even.
Poisson
brackets,
the equation
c = a,c+{c,H), where the Hamrltonian
H is defined
of motion
(6.5)
as follows:
H = i L.=
pk‘jk-L,
(6.6)
*
and the order of variables is essential when fermionic degrees of freedom enter. All the formulae derived previously for pure bosonic systems remain valid also in the mixed bosonic-fermionic system, if we replace the determinants with superdeterminants (Berezinians). This comes about because the key property (2.9) of Dirac 6functions, which was used in deriving a measure, is now replaced by = nS(C,),
(6.7)
a
where the superdeterminant s det
of a generalized
det(e, -0, =
matrix is of the following
ei ’ 02) =
det e2
block form:
Ber
.
(6.8)
Such matrix transforms, e.g. constraints, and preserves its arrangement into bosonic and fermionic parts when e,, e,, are even while O,, O2 are odd elements of
W. GARCZYhSKI
92
Grassmann
algebra (6.9)
Clearly, the ez matrix is assumed one needs is the following:
non.-singular
SdetA where the supertrace
is defined
here. Another
useful formula
which (6.10)
= exp(sTrlnA),
as STrA
= i
(6.11)
~~~~~
k=l
and Ek = Berezinians
for k bosonic,
1
also enjoy the fundamental Sdet(A,
(6.12)
for k fermionic.
-1
property
of determinants
AZ) = (Sdet A,)(Sdet
A,).
(6.13)
In the most general situation of a mixed bosonic-fermionic system subject to both kinds of constraints we write for the generating functional (6.14) where
(6.15) Here (II/,) stands
for the totality
of constraints
,
y = 1, . . . . 2(m’+m”)
= 2m.
(6.16)
The number m’ denotes the maximal quantity of the first-class constraints Ci = 0, cl=1 9 **.> m’, while 2m” denotes the number of the remaining second-class constraints C$ Z 0; Q” = 1, . . . , 2m”. They do satisfy the condition Sdet ll{Ci,,, Cb:,}ll,c# 0.
(6.17)
MEASURE IN THE FUNCTIONAL
All the appearing in this section Poisson ized sense (cf. (6.2)). The gauge constraints restriction
INTEGRAL
93
brackets are understood in the generalG, z 0, a = 1, . . . , m’ are subject to the (6.18)
Sdet ll{G,, C;)lllc # 0. The momenta
conjugate
to the coordinates pk =
Since we are dealing
awl> 4) a$
’
qk are
k = 1, . ..) n.
here with a singular
(6.19)
system, we have the inequality (6.20)
r = rank
and Xr(q,
Repeting, generating
mutatis mutandis, functional
the previous
o
z
4) = Sdet
(6.21)
.
considerations
we may
write
for the
(6.22) where the measure
h(q,
4) =
m, is given by the integral
~fldp,(i%f,.(q,d’+p’, 4”) s($)
(q,“L(q’ zp-“” “‘I,f) x
t,k
L(q, $+p’,
4”)-L(q,
4)--p:
‘~3~’
zpTp”
“‘)]I.
(6.23)
OL The resulting generating functional does not depend on the choice of gauge constraints within the gauges belonging to orbits of the gauge transformations generated by the first-class constraints 6G, = {G,, Cb)@,
(6.24)
z,
(6.25)
= ZG,&
= z.
7. Concluding remarks Summarizing, we have found a general form of the measures which appear in functional integrals over the configuration space of singular systems. Considerations were carried out for the non-covariant gauges only. We justify in this way
94
W. GARCZYfiSKI
the insertion of Dirac b-functions of constraints under the functional integral over A% in the case of Maxwell electrodynamics in radiation gauge (cf. [22, 231). The full justification of the Faddeev-Popov ansatz for quantization of gauge fields requires inclusion of covariant gauges also for, both, Abelian and non-Abelian gauge fields, and for Einstein gravity, as well. Covariant gauges require employment of a more powerful formalism developed recently by Fradkin, Batalin and Vilkovisky in a series of papers, which were reviewed by Henneaux [29], and Batalin and Fradkin [30]. The case of Einstein gravity requires special attention in view of the controversy around the shape of the measure appearing in the generating functional of Green’s functions [31, 321. We bention in this connection a recent paper by Unz [33] where the Faddeev-Popov ansatz was rather used for deriving the measure than justified as in the our case, where we derive the measure independently of that ansatz. We postpone these interesting problems connected with covariant gauges and non-Abelian gauge theories to subsequent publications. Acknowledgements The author wishes to thank Professor E. S. Fradkin and Dr. I. A. Batalin for useful col;versations concerning the measure for essentially non-linear field theories. The author wishes to thank Professor J. Rzewuski for his interest in the present work.
REFERENCES [l] Dirac, P.A. M.: Physikalische Zeitschrif der Sowjetunion, 3 (1933), 64. [Z] Feynman, R. P.: Rev. Mod. Phys., 20 (1948), 367. [3] Feynman, R. P.: Phys. Rev., 80 (1950), 440; 84 (1951), 108. [4] Feynman, R. P. and A. R. Hibbs: Quantum Mechanics and Path Integruls, MC Graw-Hill, New York 1965. [S] Berezin, F. A.: Teor. Math. Phys., 6 (1971), 194 (in Russian). [6] Slavnov, A. A. and L. D. Faddeev: Vvedenie v kvantovuyu teoriu kalibrovochnykh polei, Nauka, Moscow 1978 (in Russian). [7] Garczyriski, W.: Bull. Acad. Polon. Sci., CL II!, 21 (1973), 355. [8] Garczytiski, W.: in: Functional Integration. Theory and Applications, J. P. Antoine and E. Tirapegui (Eds.), Plenum Press, New York-London 1980, pp. 1755189. [9] Fradkin, E. S.: in: A Memorial Volume to I. E. Tamm, Nauka, Moscow 1972, pp. 146176. [lo] Faddeev, L. D.: Teor. Math. Phys., I (1969), 3. [l l] Faddeev, L. D. and V. N. Popov: Phys. Lett. B25 (1967), 30. [I21 Faddeev, L. D. and V. N. Popov: Preprint ITP, Acad. Sci. USSR, Kiev 1967. [13] Berezin, F. A. and M. A, Shubin: Schriidinger Equation, Moscow University Press, Moscow 1983 (in Russian). 1143 Fradkin, E. S. and G. A. Vilkovisky: Phys. Rev., D8 (1973), 4241. [15] Dirac, P. A. M.: Canad. T Math., 2 (1950), 129. [16] Dirac, P. A. M.: Lectures on Quantum Mechanics, Belfer Graduate School of Science, Yeshiva University, New York 1964.
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For a broad review of other references see also: [17] Sundermeyer, K.: Constrained Dynamics, Lecture Notes in Physics No 169, Springer-Verlag, Berlin-Heidelberg-New York 1982. [18] Senjanovic, P.: Ann. of Phys. (US), 100 (1976), 227. [19] Fradkin, E. S.: Acta Universitatis Wratislaviensis, No 207, Proceedings of the X-th Winter School of Theoretical Physics in Karpacz, Poland. 1973. [20] Garczyriski, W.: Ann. of Phys. (US), 174 (1987), 26. [21] Garczynski, W.: Invariance Property of Senjanovic Measure, and Its Role in Quantization of Constrained Systems, Proceedings of the XXII Winter School of Theoretical Physics in Karpacz, Poland, 1986, World Scientific Publishing Co, Singapore 1986. [22] Bialynicki-Birula, I. J.: r Math. Phys., 3 (1962) 1094. 1231 Konopleva. N. P. and V. N. Popov: Gauge Fields. 2nd ed., Moscow 1972 (in Russian). [24] Berezin, F. A.: Introduction to an Algebra and Analysis with Anticommuting Variables, Moscow University Press, Moscow 1983 (in Russian), and references given there. [25] Fradkin, E. S. and G. A. Vilkovisky: Phys. Letr., 55B (1975) 224. [26] Casalbuoni, R.: Nuovo Cim., 33A (1976), 115 and 384. 1271 Batahn, I. A. and G. A. Vilkovisky: Phys. Lett., 61B (1977), 369. [28] Maharana, T.: Fortsch. Phys., 33 (1985), 645. [29] Henneaux, M.: Phys. Rep., 126 (1985) 1. [30] Batalin, I. A. and E. S. Fradkin: Works of the P. Lebedev Physical Institute, Academy of Sciences USSR, 165 (1986) 31. [31] Fradkin, E. S. and G. A. Vilkovisky: CERN preprint TH-2332 (1977), and references @en there. [32] Toms, D. J.: Preprint of the University of Newcastle upon Tyne NCL-86 TP6, and references given there. [333 Unz, R. K.: Nuovo Cim., 92A (1986), 397. [34] Rzewuski, J.: Field Theory, Part II, PWN-Polish Scientific Publishers, Warszawa 1969 (in Polish).