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An alternative derivation of the Faddeev-Popov path integral
Volume 258, number 1,2
PHYSICS LETTERS B
4 April 1991
An alternative derivation of the Faddeev-Popov path integral A. C a b o ", M. C h a i c h i a n b,~, D . L o u i s M a r t i n e z ~ a n d P. Pre~najder ~ Grupo de l"isica Te&ica, ICIMAF; Academia de Ciencich~, Vedado, lIavana 4, Cuba b Theory Division, CERN, CI1-1211 Geneva 23, Switzerland ¢ Department of Theoretical Physics, Comenius University, Mlynskdt dolina k2, CS-84215 Bratislava, Czechoslovakia Received 19 December 1990
A new derivation of the Faddeev-Popov path integral is presented. The use of gauge invariant transformations and gauge fixing conditions in the phase space allows to introduce straightforwardly Lorentz invariant gauge conditions into the path integral, thus avoiding the necessity of going first through a Coulomb-like gauge as it is usually done. The case of systems with finite degrees of freedom and the abelian (QED) one are also presented for illustration.
In this letter we present a new derivation of the F a d d e c v - P o p o v path integral method for quantization of non-abelian gauge theories with Lorentz invariant gauge conditions. Let us start our discussion by considering the case of systems with finite dcgrees of freedom. Following the Dirac formalism [ 1 ] for systems with singular lagrangians, wc can obtain the constraints in the phase space, 0~(q,p)=0,
~x=l .... , m .
(1)
We shall consider systems with first-class constraints, for which the following involution relations hold [2,3]:
{0,~, ep} = c~,/,¢,,,, {¢,,,
It}=D~.01,.
(2)
{ ~ ' ) , ¢~,.1) } = 0 ,
We shall assume a situation which is typical for gauge systems, i.e. that the constraints ( 1 ) are divided into two classes: (i) p r i m a r y constraints. ¢ ( t ) , a = 1, ..., m~, which are just the direct consequence o f the quence o f the equations p,=OL/O(t,, i = l, ..., n, (n is the degree o f freedom of the system); we shall assume that the constraints ~ ) commute: Permanent address: Department of High Energy Physics, University of Helsinki, Siltavuorenpenger 20 C, SF-00170 Helsinki, Finland.
(4)
(ii) secondary constraints'. ¢(2) = { ¢ ( , ) , ii},
a = l .... , m 2 ,
(5)
are supposed to satisfy the conditions ;'*(')~.~, ¢ b ( 2 ) } = 0 ,
(6)
(7)
.,(2) ~ II}=D~¢},2)
tp' a
,.4(2)
~,~
, e~2) } = C ~ b ~ 2 ~
,
(8)
where C ~ are somc constants. We have m~ = m2 and m=m~ +m2. We can choose canonical variables q,, p,, i = 1, ..., n, in such a way that ~(l)
(3)
a,a'=l,...,m,,
,
=p~, a = 1..... m, .
(9)
According to eqs. ( 5 ) - (9), the hamiltonian has the structure 11= tto +q~¢~2) ,
(10)
where Ho and 0~ 2) are functions only of physical variables q,, p~, r = m x + 1, ..., n. The secondary constraints ~ 2 ) , b = 1, ..., m2 are generators of time-independent gauge transformations, i.e. for constant vb, b = 1, ..., m2,
¢[v] = v~,¢~~)
(11 )
is a generator o f gauge transformation. The infinites-
0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
183
Volume 258, number 1,2
PHYSICS LETTERS B
imal change of any physical variable F is given by the formula
6F= {~; 0[vl},
(12)
which means that 0Iv] is a generator of canonical transformation. To fix the gauge freedom, we impose the gauge conditions Z~2) =0,
a=l,...,m2,
(13)
(14)
One starts usually the path integral approach to the quantum theory from the path integral formula ~ cxp(i f
In this letter we shall obtain the same result more directly. Physically, the essence of our approach is the following. The hamiltonian formalism violates from the very beginning the explicit Lorentz invariance e.g. the path integral (16) has only time-independent explicit gauge invariance. To restore the full gauge invariance, we extend ( 11 ) to time-dependent gauge transformations by putting
(p,dL-H)dt)
where vb, b= 1..... m2 can be time dependent. The change of any physical variable is again given by eq. (12), i.e. we realize time-dependent gauge transformations as canonical ones. Now we have to fix this gauge freedom. In order to recover the Lorentz invariance, we shall impose more general gauges of the type
Z -~'(l)-v(z)=O.a=l, a
Xd(z)A(O, Z)O(q~)Dq, Dp,,
(15)
where the functional 5-function 5(0)=6(0(~ ~)) reduces the integration only to configurations satisfying primary constraints and 5(Z)=~(;((2)) guarantees the gauge condition (13). One can show in a standard way, that the path integral (16) does not depend on a choice of gauge fixing conditions, see e.g. [2,31. The integration over unphysical momenta p~= ,0~t ) is trivial, and according to eq. ( 10 ), the integration over unphysical coordinates gives the 6-function ~(0(2)). In this way the path integral ( 15 ) is reduced to the integral containing only integration over physical variables q,; p,, r = m~ + 1, ..., n: ~ exp(i f (p~gl,.-llo)dt)
XiJ(Z(~z))A(O,Z)6(,O(,2)) Dqr Dp~.
(16)
Here (f(Z(~2) ) guarantees the gauge condition (13), whereas (i(O(~2)) restricts the physical variables to configurations satisfying the secondary constraints (Gauss law). Such a procedure is typical to the quantum theory of gauge fields: first one imposes non-covariant (e.g. Coulomb) gauge and then, using the procedure described in ref. [4], goes over to Lorentz invariant gauge and finally one obtains the Faddeev-Popov form for the path integral representing amplitude. 184
( 17 )
0 [ U] =/)t)0 ~ 1) "~ Ub0})2) ,
depending on the variables Pb, qb, b= 1.... , m2. They should satis~ the requirement
A(O, Z) =det ({0~(z) , Z~2)}) 4:0.
4 April 1991
--
t ~ c,l
f~a
,
-.-.~
m2,
(18)
as partners of time-dependent gauge transformations ( 17 ). We shall understand the condition (18) in the following sense: for arbitrary functions Ua, a= 1, ..., m2 (with compact supports) we shall require that
fZ[u] dt=0,
(19a)
where
Z[u] =h,g~ ~) +u~z~ 2) •
(19b)
The key points of our proposal are the use of timedependent gauge transformations (17) and (timedependent) gauge fixing conditions (19) in the framework of the canonical formalism. We define the quantum theory, by the path integral ( 15 ) with the following generalizations: (i) we put ~(Z) = a ( ~
)=~(2~'> -Z~2)),
(20)
(ii) we define the functional determinant A(0, Z) as
A(O, Z) = d e t ( M , b ) ,
(21)
where "14ahis an operator defined as
6 f{O[vl,z[u]}dt 6v~(t~ ) ~ut,(t2) = Mah6( t, --t2) ,
(22)
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PHYSICS LETTERSB
and we shall require that d(¢, Z) ¢ 0. Similar formulas are well known, but the calculations of Faddeev-Popov determinants is restricted to the case of time-independent ~ [ v], Z[ u ] [ and is given by eq. (14) ]. More general cases are usually calculated indirectly using the Faddeev-Popov trick (see refs. [ 2,3 ] ) instead of the rather direct formula (21 ), (22). Performing finally in the path integral (15) the integration over all momenta p,, i= 1, ..., n, one obtains the path integral in Faddeev-Popov form. Quantum Electrodynamics. We shall consider now QED for illustration. We start from the lagrangian density of the free electromagnetic field
ze= - ¼F..F~" = ½(f4,+ O,Ao) 2_ ~F,jF,:.
4 April 1991
For the transformations of coordinates and momenta generated by @[v] we obtain the following expressions:
A'/'=AU + O/'v(x) ,
(30)
~u---~u.
(31)
Let us now impose the Lorentz gauge condition
O/~A'(x) = 0 ,
(32)
which is of the form (18). The corresponding function Z [ u ] [ defined in ( 19 ) ] is
Z[U] =
f d3x
(OUu)Au.
Using the canonical brackets (23)
{A,(x, t), g"(y,
t)} =6~6(x-y)
(34)
we obtain
The momenta are defined as usual:
{q>[v],Z[w]}= ~ d~x ~Zo- 0,~o = 0 ,
(33)
(O"u)(Ouv).
(35)
(24) The definition (22) for the operator Mgives
n,-
0~
~i
= A ' + a,A ° .
(25)
From this definition we obtain the primary, constraints of the theory as 0(I)=no.
(26)
The hamiltonian for the electromagnetic field is expressed in the following form:
It=
f
[½(n~)2-ze,O,A°+](F,:)2].
(27)
Secondary constraints appear as a result of the consistency condition on the primary constraints: ¢(2~= - O:z~.
d
i.e. M=VI.
(36)
Thus, the functional dcterminant is field independent and can be dropped out. The quantum theory is according to eq. (15) defined by the path integral
f ~(n°)~(O"A.) × exp(i ; d4x(ztUA,,-.~{:))DA~,D~zu ,
(37)
where 1 :2 ~ = ~n% + n / 0iAo + ~J t ,j
(38)
is the hamiltonian density of the electromagnetic field. The simple integration over momenta leads straightforwardly to the Faddeev-Popov path integral
d3x[b(x)rto+v(x)'O,~z,]
= f d3xOub'(X)'Tru .
= [264(x~-x2) ,
(28)
The infinitesimal canonical transformations mapping solutions of the total hamiltonian equations of motion into each other are generated by a function q>[v] which has been obtained in ref. [5] in the following form: ~[V] = f
~v(x,) ~u(x2) &x [O"u(x)] [Our(x)]
(29)
fr(OUA,)exp(ifd4x,~)DA,.
(39) 185
Volume 258, number 1,2
PHYSICS LETTERS B
4 April 1991
•Von-abelian gauge field. For the non-abelian case, as we did for the abelian one (QED), we first write the Yang-Mills lagrangian [ 2,3,6 ] :
Using the canonical brackets
c~.=
we obtain
! p a Fu~
I[r'~aAb = ~ ~'--b,.~o -
a 2
1
,a
OoAi ) - ~ ( F o )
2
{Aau(x, t), Try(y, t) } = 6 g 6 ~ 6 ( x - y ) ,
{ ~ [ v ] , Z [ U ] } = ~ d3x (OUua)(Dg~,v t') •
,
(52)
(53)
(40)
where
The definition (22) gives
,'a
a
a
1 u~ = O u A . - - O~A ~, + f
a
b
a~A uA ~,
(41 )
~Va(Xl) ~'~Ub(X2)
a ~a Dbu =Oh 0u - - f at,cA cu,
d4x [0'ubta(X) ] [Dguuh(X) ]
(42)
and a = 1, ..., m~, is the symmetry index. The momenta are defined as usual: u - / ' ~"~,o. ~z~
(43)
The primary constraints of the theory may be written
:Mg~4(xl
-x2) ,
where
Mi, - DOt, + f t,cA uO
(54)
The functional determinant
as
~b~~) = ~ r ° .
(44)
The hamiltonian tbr the Yang-Mills fields has the following form: H=
f
! , r r l q r a . 4- _. ~~i , e l,~,a~Aob + ~ k 1i j k"a, ) d3x (2'~'~,
"U
•
(45)
The secondary constraints appear as a result of the consistency conditions on the primary ones: 0~(2) - {0~ ~), The
II}
generating
= - Dg, n~~ . function
qS[v]
(46) can
be w r i t t e n
as
follows:
q~[v] = f d 3 x (Dhj, ~ V~,)n~.
A(O,z)=det(Mg)
(55)
is just the Faddeev-Popov determinant corresponding to the Lorcntz gauge (51), see e.g. refs. [2,3]. The quantum theory is (according to (15)) defined by the path integral
f
6(zc°)fi(OUA'/,)A(o,Z)
X e x p ( i f d4 x (pz G' Aau - ), ~ )a
DAuDrr~,
where 1 i a t a b II,'api j J { = iTr. 7r, +rCaDmAo + ~" O--a
(47)
For the infinitesimal canonical transformations of coordinates and momenta generated by q)[ v], we obtain the following expressions: (48)
~'a'u ~ 7 ~ a,,a_Ce I d b a ~ c X *UAt, "
(49)
Let us now impose the Lorentz gaugc conditions
O"A".,,= 0 ,
(50)
which has the form (18). The corresponding function Z[u] is Z[U] = f d3x (0",, u,~)A ua . J
186
(51)
(57)
is the hamiltonian density of the Yang-Mills field. The simple integration over momenta leads to the well-known formula [2,3 ]
f 6(&'A~,)A(O,z)exp(i ~ d4xCdU)DA~. A ,,'~=AT,+Dg,,A ~ ,
(56)
(58)
The Faddeev-Popov determinant can be expressed as usual in terms of a path integral over the ghost fields. Thus the Faddeev-Popov path integral [4] for the Yang-Mills gauge fields is obtained straightforwardly, using gauge invariant transformations in the phase space. Extension to systems of more general character. The above formulation, as we have seen above, was indeed sufficient to derive the Faddeev-Popov path in-
Volume 258, number 1,2
PHYSICS LETTERSB
tegral for the non-abelian gauge theories. This is thc case since the primary constraints of Yang-Mills gauge theories salis~ the commutativity relations assumed in eq. (4). This proccdurc, however, can be generalized to cases when the primary constraints ~ ) , a = 1, ..., m~, do not commute but arc still in involution,
f,~t),~(~) ~-6"~",-~)
"t "," a "
"~" a "
J --
~
aa'
"Y a "
can be proven, provided that the quantities C~'~a in (2) arc constant. In cq. (64) Hphys(q*, P * ) - l l ( q , P) le=o dcnotes the physical hamiltonian. Formally, the proof goes as follows: f exp(i J"
= I cxp(i
+D.,O~
,
q.,.=qs(q,P), P.~=P~(q,P), s= 1, ..., n - m , Q,~=Q~(q,p), P,,=l'(q,p), a=l ..... m (m=m,+me).
X ~q* ~p* ~Q ~P ~2 = f exp(i
~(p'~l'+PO-II(q',p', Q,P)dt)
× ~ [ P ] det- ~11~/~(q*, P *, Q, P)II × ~ q * ~p* .~Q ~ e .
(65)
Performing the integration over the momenta 1~, we obtain
(p~l-ll-2~O~)dt)~q~p~2
=fexp(if(p*it*-llp,:)dt)~q*~p* (61)
(62)
where 0a,/3= 1, ..., rn, stands for the full system of constraints. The existence of new variables (62) is discussed in refs. [ 6,8 ]. Inverting (63) we obtain
0/,= U/,,~(q*,p*, Q, P)P,,
-;.~ U,~a(q*, p*, Q, P)Pa]dt)
j" exp(i f
The nev¢ momenta P,, are chosen in such a way that det IIV~,ll # 0 ,
~[p*?l*+l'O_-ll(q*,p*, Q,P)
(60)
but there are no ternary constraints, i.e. the system of constraints containing 0~ ') , 0},2), a = 1, ..., rnl, b = l, .... mz, satisfies the involution conditions (2), (3). In this more general situation it is useful to use new canonical variables,
P,~=V./~(q,p)O~(q,p),
(pdl-H-2~)dt):2q~p~2
(59)
•
The main key is now to have a generating function (independent of Lagrange multipliers) which implements the time-dependent gauge transformations as the canonical ones. The existence of such a generating function T has becn proven [7] for the systems having in addition to primary constraints (57), the secondary constraints ~t) ~, b= 1, ..., m:, defined by •~ ,
4 April 1991
X f det- '110,~p(Q)II @Q,
(66)
where wc havc assumed that C]~/j(Q)=C~p(q*, p*, Q, 0) do not depend on q* and p*. This approach being of a more general nature, could be of interest for more complicated gauge theories such as quantum gravity.
(63)
where U,a are elements of the inversc matrix to !1V,j. One can demonstrate that, if the functions 0 ~ Ua~(q*, p*, Q, 0) do not depend on "physical variables" q*, p*, then the relation
~ exp(if (p,~-ll-)..o.)dt).:.~q.p.)~ (64)
References
[1] P.A.M. Dirac, Lectures in quantum mechanics, Belfer Graduate Schoolof Science,YeshivaUniversity(New York, 1964). [2] L.D. Faddcev and A.A. Slavnov,Gauge fields: Introduction to quantum theory (Benjamin/Cummings, Reading, MA, 1983). [3] M. Chaichian and N.F. Nclipa, Introduction to gauge field theories (Springer,Berlin, 1984). 187
Volume 258, number 1,2
PHYSICS LETTERS B
[4] L.D. Faddeev and V.N. Popov, Phys. Lett. B 25 (1967) 30; L.D. Faddecv, Theor. Math. Phys. 1 (1970) 1. [5] R. Sugano, Prog. Theor. Phys. 68 (1982) 1377. [6] D.M. Gitman and I.V. Tyutin, Canonical quantization of constrained fields (Nauka, Moscow, 1989) lin Russian].
188
4 April 1991
[7] A. Cabo, M. Chaichian and D. Louis Martinez, Gauge invariance of systems with first-class constraints, Helsinki University preprint HU-TFI'-90-6. [8] M. Henneaux, Phys. Rep. 126 (1985) 1.
PHYSICS LETTERS B
4 April 1991
An alternative derivation of the Faddeev-Popov path integral A. C a b o ", M. C h a i c h i a n b,~, D . L o u i s M a r t i n e z ~ a n d P. Pre~najder ~ Grupo de l"isica Te&ica, ICIMAF; Academia de Ciencich~, Vedado, lIavana 4, Cuba b Theory Division, CERN, CI1-1211 Geneva 23, Switzerland ¢ Department of Theoretical Physics, Comenius University, Mlynskdt dolina k2, CS-84215 Bratislava, Czechoslovakia Received 19 December 1990
A new derivation of the Faddeev-Popov path integral is presented. The use of gauge invariant transformations and gauge fixing conditions in the phase space allows to introduce straightforwardly Lorentz invariant gauge conditions into the path integral, thus avoiding the necessity of going first through a Coulomb-like gauge as it is usually done. The case of systems with finite degrees of freedom and the abelian (QED) one are also presented for illustration.
In this letter we present a new derivation of the F a d d e c v - P o p o v path integral method for quantization of non-abelian gauge theories with Lorentz invariant gauge conditions. Let us start our discussion by considering the case of systems with finite dcgrees of freedom. Following the Dirac formalism [ 1 ] for systems with singular lagrangians, wc can obtain the constraints in the phase space, 0~(q,p)=0,
~x=l .... , m .
(1)
We shall consider systems with first-class constraints, for which the following involution relations hold [2,3]:
{0,~, ep} = c~,/,¢,,,, {¢,,,
It}=D~.01,.
(2)
{ ~ ' ) , ¢~,.1) } = 0 ,
We shall assume a situation which is typical for gauge systems, i.e. that the constraints ( 1 ) are divided into two classes: (i) p r i m a r y constraints. ¢ ( t ) , a = 1, ..., m~, which are just the direct consequence o f the quence o f the equations p,=OL/O(t,, i = l, ..., n, (n is the degree o f freedom of the system); we shall assume that the constraints ~ ) commute: Permanent address: Department of High Energy Physics, University of Helsinki, Siltavuorenpenger 20 C, SF-00170 Helsinki, Finland.
(4)
(ii) secondary constraints'. ¢(2) = { ¢ ( , ) , ii},
a = l .... , m 2 ,
(5)
are supposed to satisfy the conditions ;'*(')~.~, ¢ b ( 2 ) } = 0 ,
(6)
(7)
.,(2) ~ II}=D~¢},2)
tp' a
,.4(2)
~,~
, e~2) } = C ~ b ~ 2 ~
,
(8)
where C ~ are somc constants. We have m~ = m2 and m=m~ +m2. We can choose canonical variables q,, p,, i = 1, ..., n, in such a way that ~(l)
(3)
a,a'=l,...,m,,
,
=p~, a = 1..... m, .
(9)
According to eqs. ( 5 ) - (9), the hamiltonian has the structure 11= tto +q~¢~2) ,
(10)
where Ho and 0~ 2) are functions only of physical variables q,, p~, r = m x + 1, ..., n. The secondary constraints ~ 2 ) , b = 1, ..., m2 are generators of time-independent gauge transformations, i.e. for constant vb, b = 1, ..., m2,
¢[v] = v~,¢~~)
(11 )
is a generator o f gauge transformation. The infinites-
0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
183
Volume 258, number 1,2
PHYSICS LETTERS B
imal change of any physical variable F is given by the formula
6F= {~; 0[vl},
(12)
which means that 0Iv] is a generator of canonical transformation. To fix the gauge freedom, we impose the gauge conditions Z~2) =0,
a=l,...,m2,
(13)
(14)
One starts usually the path integral approach to the quantum theory from the path integral formula ~ cxp(i f
In this letter we shall obtain the same result more directly. Physically, the essence of our approach is the following. The hamiltonian formalism violates from the very beginning the explicit Lorentz invariance e.g. the path integral (16) has only time-independent explicit gauge invariance. To restore the full gauge invariance, we extend ( 11 ) to time-dependent gauge transformations by putting
(p,dL-H)dt)
where vb, b= 1..... m2 can be time dependent. The change of any physical variable is again given by eq. (12), i.e. we realize time-dependent gauge transformations as canonical ones. Now we have to fix this gauge freedom. In order to recover the Lorentz invariance, we shall impose more general gauges of the type
Z -~'(l)-v(z)=O.a=l, a
Xd(z)A(O, Z)O(q~)Dq, Dp,,
(15)
where the functional 5-function 5(0)=6(0(~ ~)) reduces the integration only to configurations satisfying primary constraints and 5(Z)=~(;((2)) guarantees the gauge condition (13). One can show in a standard way, that the path integral (16) does not depend on a choice of gauge fixing conditions, see e.g. [2,31. The integration over unphysical momenta p~= ,0~t ) is trivial, and according to eq. ( 10 ), the integration over unphysical coordinates gives the 6-function ~(0(2)). In this way the path integral ( 15 ) is reduced to the integral containing only integration over physical variables q,; p,, r = m~ + 1, ..., n: ~ exp(i f (p~gl,.-llo)dt)
XiJ(Z(~z))A(O,Z)6(,O(,2)) Dqr Dp~.
(16)
Here (f(Z(~2) ) guarantees the gauge condition (13), whereas (i(O(~2)) restricts the physical variables to configurations satisfying the secondary constraints (Gauss law). Such a procedure is typical to the quantum theory of gauge fields: first one imposes non-covariant (e.g. Coulomb) gauge and then, using the procedure described in ref. [4], goes over to Lorentz invariant gauge and finally one obtains the Faddeev-Popov form for the path integral representing amplitude. 184
( 17 )
0 [ U] =/)t)0 ~ 1) "~ Ub0})2) ,
depending on the variables Pb, qb, b= 1.... , m2. They should satis~ the requirement
A(O, Z) =det ({0~(z) , Z~2)}) 4:0.
4 April 1991
--
t ~ c,l
f~a
,
-.-.~
m2,
(18)
as partners of time-dependent gauge transformations ( 17 ). We shall understand the condition (18) in the following sense: for arbitrary functions Ua, a= 1, ..., m2 (with compact supports) we shall require that
fZ[u] dt=0,
(19a)
where
Z[u] =h,g~ ~) +u~z~ 2) •
(19b)
The key points of our proposal are the use of timedependent gauge transformations (17) and (timedependent) gauge fixing conditions (19) in the framework of the canonical formalism. We define the quantum theory, by the path integral ( 15 ) with the following generalizations: (i) we put ~(Z) = a ( ~
)=~(2~'> -Z~2)),
(20)
(ii) we define the functional determinant A(0, Z) as
A(O, Z) = d e t ( M , b ) ,
(21)
where "14ahis an operator defined as
6 f{O[vl,z[u]}dt 6v~(t~ ) ~ut,(t2) = Mah6( t, --t2) ,
(22)
Volume 258, number 1,2
PHYSICS LETTERSB
and we shall require that d(¢, Z) ¢ 0. Similar formulas are well known, but the calculations of Faddeev-Popov determinants is restricted to the case of time-independent ~ [ v], Z[ u ] [ and is given by eq. (14) ]. More general cases are usually calculated indirectly using the Faddeev-Popov trick (see refs. [ 2,3 ] ) instead of the rather direct formula (21 ), (22). Performing finally in the path integral (15) the integration over all momenta p,, i= 1, ..., n, one obtains the path integral in Faddeev-Popov form. Quantum Electrodynamics. We shall consider now QED for illustration. We start from the lagrangian density of the free electromagnetic field
ze= - ¼F..F~" = ½(f4,+ O,Ao) 2_ ~F,jF,:.
4 April 1991
For the transformations of coordinates and momenta generated by @[v] we obtain the following expressions:
A'/'=AU + O/'v(x) ,
(30)
~u---~u.
(31)
Let us now impose the Lorentz gauge condition
O/~A'(x) = 0 ,
(32)
which is of the form (18). The corresponding function Z [ u ] [ defined in ( 19 ) ] is
Z[U] =
f d3x
(OUu)Au.
Using the canonical brackets (23)
{A,(x, t), g"(y,
t)} =6~6(x-y)
(34)
we obtain
The momenta are defined as usual:
{q>[v],Z[w]}= ~ d~x ~Zo- 0,~o = 0 ,
(33)
(O"u)(Ouv).
(35)
(24) The definition (22) for the operator Mgives
n,-
0~
~i
= A ' + a,A ° .
(25)
From this definition we obtain the primary, constraints of the theory as 0(I)=no.
(26)
The hamiltonian for the electromagnetic field is expressed in the following form:
It=
f
[½(n~)2-ze,O,A°+](F,:)2].
(27)
Secondary constraints appear as a result of the consistency condition on the primary constraints: ¢(2~= - O:z~.
d
i.e. M=VI.
(36)
Thus, the functional dcterminant is field independent and can be dropped out. The quantum theory is according to eq. (15) defined by the path integral
f ~(n°)~(O"A.) × exp(i ; d4x(ztUA,,-.~{:))DA~,D~zu ,
(37)
where 1 :2 ~ = ~n% + n / 0iAo + ~J t ,j
(38)
is the hamiltonian density of the electromagnetic field. The simple integration over momenta leads straightforwardly to the Faddeev-Popov path integral
d3x[b(x)rto+v(x)'O,~z,]
= f d3xOub'(X)'Tru .
= [264(x~-x2) ,
(28)
The infinitesimal canonical transformations mapping solutions of the total hamiltonian equations of motion into each other are generated by a function q>[v] which has been obtained in ref. [5] in the following form: ~[V] = f
~v(x,) ~u(x2) &x [O"u(x)] [Our(x)]
(29)
fr(OUA,)exp(ifd4x,~)DA,.
(39) 185
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4 April 1991
•Von-abelian gauge field. For the non-abelian case, as we did for the abelian one (QED), we first write the Yang-Mills lagrangian [ 2,3,6 ] :
Using the canonical brackets
c~.=
we obtain
! p a Fu~
I[r'~aAb = ~ ~'--b,.~o -
a 2
1
,a
OoAi ) - ~ ( F o )
2
{Aau(x, t), Try(y, t) } = 6 g 6 ~ 6 ( x - y ) ,
{ ~ [ v ] , Z [ U ] } = ~ d3x (OUua)(Dg~,v t') •
,
(52)
(53)
(40)
where
The definition (22) gives
,'a
a
a
1 u~ = O u A . - - O~A ~, + f
a
b
a~A uA ~,
(41 )
~Va(Xl) ~'~Ub(X2)
a ~a Dbu =Oh 0u - - f at,cA cu,
d4x [0'ubta(X) ] [Dguuh(X) ]
(42)
and a = 1, ..., m~, is the symmetry index. The momenta are defined as usual: u - / ' ~"~,o. ~z~
(43)
The primary constraints of the theory may be written
:Mg~4(xl
-x2) ,
where
Mi, - DOt, + f t,cA uO
(54)
The functional determinant
as
~b~~) = ~ r ° .
(44)
The hamiltonian tbr the Yang-Mills fields has the following form: H=
f
! , r r l q r a . 4- _. ~~i , e l,~,a~Aob + ~ k 1i j k"a, ) d3x (2'~'~,
"U
•
(45)
The secondary constraints appear as a result of the consistency conditions on the primary ones: 0~(2) - {0~ ~), The
II}
generating
= - Dg, n~~ . function
qS[v]
(46) can
be w r i t t e n
as
follows:
q~[v] = f d 3 x (Dhj, ~ V~,)n~.
A(O,z)=det(Mg)
(55)
is just the Faddeev-Popov determinant corresponding to the Lorcntz gauge (51), see e.g. refs. [2,3]. The quantum theory is (according to (15)) defined by the path integral
f
6(zc°)fi(OUA'/,)A(o,Z)
X e x p ( i f d4 x (pz G' Aau - ), ~ )a
DAuDrr~,
where 1 i a t a b II,'api j J { = iTr. 7r, +rCaDmAo + ~" O--a
(47)
For the infinitesimal canonical transformations of coordinates and momenta generated by q)[ v], we obtain the following expressions: (48)
~'a'u ~ 7 ~ a,,a_Ce I d b a ~ c X *UAt, "
(49)
Let us now impose the Lorentz gaugc conditions
O"A".,,= 0 ,
(50)
which has the form (18). The corresponding function Z[u] is Z[U] = f d3x (0",, u,~)A ua . J
186
(51)
(57)
is the hamiltonian density of the Yang-Mills field. The simple integration over momenta leads to the well-known formula [2,3 ]
f 6(&'A~,)A(O,z)exp(i ~ d4xCdU)DA~. A ,,'~=AT,+Dg,,A ~ ,
(56)
(58)
The Faddeev-Popov determinant can be expressed as usual in terms of a path integral over the ghost fields. Thus the Faddeev-Popov path integral [4] for the Yang-Mills gauge fields is obtained straightforwardly, using gauge invariant transformations in the phase space. Extension to systems of more general character. The above formulation, as we have seen above, was indeed sufficient to derive the Faddeev-Popov path in-
Volume 258, number 1,2
PHYSICS LETTERSB
tegral for the non-abelian gauge theories. This is thc case since the primary constraints of Yang-Mills gauge theories salis~ the commutativity relations assumed in eq. (4). This proccdurc, however, can be generalized to cases when the primary constraints ~ ) , a = 1, ..., m~, do not commute but arc still in involution,
f,~t),~(~) ~-6"~",-~)
"t "," a "
"~" a "
J --
~
aa'
"Y a "
can be proven, provided that the quantities C~'~a in (2) arc constant. In cq. (64) Hphys(q*, P * ) - l l ( q , P) le=o dcnotes the physical hamiltonian. Formally, the proof goes as follows: f exp(i J"
= I cxp(i
+D.,O~
,
q.,.=qs(q,P), P.~=P~(q,P), s= 1, ..., n - m , Q,~=Q~(q,p), P,,=l'(q,p), a=l ..... m (m=m,+me).
X ~q* ~p* ~Q ~P ~2 = f exp(i
~(p'~l'+PO-II(q',p', Q,P)dt)
× ~ [ P ] det- ~11~/~(q*, P *, Q, P)II × ~ q * ~p* .~Q ~ e .
(65)
Performing the integration over the momenta 1~, we obtain
(p~l-ll-2~O~)dt)~q~p~2
=fexp(if(p*it*-llp,:)dt)~q*~p* (61)
(62)
where 0a,/3= 1, ..., rn, stands for the full system of constraints. The existence of new variables (62) is discussed in refs. [ 6,8 ]. Inverting (63) we obtain
0/,= U/,,~(q*,p*, Q, P)P,,
-;.~ U,~a(q*, p*, Q, P)Pa]dt)
j" exp(i f
The nev¢ momenta P,, are chosen in such a way that det IIV~,ll # 0 ,
~[p*?l*+l'O_-ll(q*,p*, Q,P)
(60)
but there are no ternary constraints, i.e. the system of constraints containing 0~ ') , 0},2), a = 1, ..., rnl, b = l, .... mz, satisfies the involution conditions (2), (3). In this more general situation it is useful to use new canonical variables,
P,~=V./~(q,p)O~(q,p),
(pdl-H-2~)dt):2q~p~2
(59)
•
The main key is now to have a generating function (independent of Lagrange multipliers) which implements the time-dependent gauge transformations as the canonical ones. The existence of such a generating function T has becn proven [7] for the systems having in addition to primary constraints (57), the secondary constraints ~t) ~, b= 1, ..., m:, defined by •~ ,
4 April 1991
X f det- '110,~p(Q)II @Q,
(66)
where wc havc assumed that C]~/j(Q)=C~p(q*, p*, Q, 0) do not depend on q* and p*. This approach being of a more general nature, could be of interest for more complicated gauge theories such as quantum gravity.
(63)
where U,a are elements of the inversc matrix to !1V,j. One can demonstrate that, if the functions 0 ~ Ua~(q*, p*, Q, 0) do not depend on "physical variables" q*, p*, then the relation
~ exp(if (p,~-ll-)..o.)dt).:.~q.p.)~ (64)
References
[1] P.A.M. Dirac, Lectures in quantum mechanics, Belfer Graduate Schoolof Science,YeshivaUniversity(New York, 1964). [2] L.D. Faddcev and A.A. Slavnov,Gauge fields: Introduction to quantum theory (Benjamin/Cummings, Reading, MA, 1983). [3] M. Chaichian and N.F. Nclipa, Introduction to gauge field theories (Springer,Berlin, 1984). 187
Volume 258, number 1,2
PHYSICS LETTERS B
[4] L.D. Faddeev and V.N. Popov, Phys. Lett. B 25 (1967) 30; L.D. Faddecv, Theor. Math. Phys. 1 (1970) 1. [5] R. Sugano, Prog. Theor. Phys. 68 (1982) 1377. [6] D.M. Gitman and I.V. Tyutin, Canonical quantization of constrained fields (Nauka, Moscow, 1989) lin Russian].
188
4 April 1991
[7] A. Cabo, M. Chaichian and D. Louis Martinez, Gauge invariance of systems with first-class constraints, Helsinki University preprint HU-TFI'-90-6. [8] M. Henneaux, Phys. Rep. 126 (1985) 1.