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Hamilton–Jacobi Approach to Berezinian Singular Systems
Annals of Physics PH5813 Annals of Physics 267, 7596 (1998) Article No. PH985813
HamiltonJacobi Approach to Berezinian Singular Systems B. M. Pimentel and R. G. Teixeira Instituto de F@ sica Teorica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 Sa~o Paulo, S.P., Brazil
and J. L. Tomazelli Departamento de F@ sica e Qu@ micaFaculdade de Engenharia, Universidade Estadual Paulista Campus de Guaratingueta , Av. Dr. Ariberto Pereira da Cunha, 333, 12500-000 Guaratinqueta , S.P., Brazil E-mail: pimentelaxp.ift.unesp.br Received January 13, 1998
In this work we present a formal generalization of the HamiltonJacobi formalism, recently developed for singular systems, to include the case of Lagrangians containing variables which are elements of Berezin algebra. We derive the HamiltonJacobi equation for such systems, analyzing the singular case in order to obtain the equations of motion as total differential equations and study the integrability conditions for such equations. An example is solved using both HamiltonJacobi and Dirac's Hamiltonian formalisms and the results are compared. 1998 Academic Press
1. INTRODUCTION In this work we intend to study singular systems with Lagrangians containing elements of Berezin algebra from the point of view of the HamiltonJacobi formalism recently developed [1, 2]. The study of such systems through Dirac's generalized Hamiltonian formalism has already been extensively developed in literature [35] and will be used for comparative purposes. Despite the success of Dirac's approach in studying singular systems, which is demonstrated by the wide number of physical systems to which this formalism has been applied, it is always instructive to study singular systems through other formalisms, since different procedures will provide different views for the same problems, even for nonsingular systems. The HamiltonJacobi formalism that we study in this work has been already generalized to higher order singular systems [6, 7] and applied only to a few number of physical examples as the electromagnetic field [8], relativistic particle in an external electromagnetic field [9] and Podolsky's electrodynamics [6]. But a better understanding of this approach utility 75 0003-491698 25.00 Copyright 1998 by Academic Press All rights of reproduction in any form reserved.
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PIMENTEL, TEIXEIRA, AND TOMAZELLI
in the studying singular systems is still lacking, and such understanding can only be achieved through its application to other interesting physical systems. Besides that, Berezin algebra is a useful way to deal simultaneously with bosonic and fermionic variables in a unique and compact notation, which justifies the interest in studying systems composed of its elements using new formalisms. The aim of this work is not only to generalize the HamiltonJacobi approach for singular systems to the case of Lagrangians containing Berezinian variables but also to present an example of its application to a well-known physical system, comparing the results to those obtained through Dirac's method. We will start in Section 2 with some basic definitions and next, in Section 3, we will introduce the HamiltonJacobi formalism to Berezinian systems using Caratheodory's equivalent Lagrangians method. In Section 4 the singular case is considered and the equations of motion are obtained as a system of total differential equations whose integrability conditions are analyzed in Section 5. The equivalence among these integrability conditions and Dirac's consistency conditions will be discussed separately in the Appendix. In Section 6 we present, as an example, the electromagnetic field coupled to a spinor, which is studied using both the formalism presented in this work and Dirac's Hamiltonian one. Finally, the conclusions are presented in Section 7.
2. BASIC DEFINITIONS We will start from a Lagrangian L(q, q* ) that must be an even function of a set of N variables q i that are elements of Berezin algebra. For a basic introduction to such an algebra we suggest the reader see Ref. [3, Appendix D], from which we took the definitions used in this paper. A more complete treatment can be found in Ref. [10]. The Lagrangian equations of motion can be obtained through variational principles from the action S= L dt, $r S r L d r L = & =0, $q i q i dt q* i
(1)
where we must call attention to the use of right derivatives. The passage to Hamiltonian formalism is made, as usual, by defining the momenta variables through right derivatives as pi #
r L q* i
(2)
and introducing the Hamiltonian function as (summing over repeated indexes) H= p i q* i &L,
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BEREZINIAN H-J SINGULAR FORMALISM
77
where the ordering of momenta to the left of velocities shall be observed since they were defined as right derivatives. This ordering will be, of course, irrelevant when we deal with even elements of the Berezin algebra. The Hamiltonian equations of motion will be given by q* i =
r H l H =(&1) P(i) , p i p i
l H r H p* i =& i =&(&1) P(i) . q q i
(4)
If we use the Poisson bracket in the Berezin algebra, given by [F, G] B =
r F l G G F &(&1) P(F) P(G) r i l , q i p i q p i
(5)
we get the known expressions q* i =[q i, H] B ,
p* i =[ p i , H] B .
(6)
For simplicity and clarity we will refer to these brackets as Berezin brackets. These brackets have properties similar to the usual Poisson brackets, [F, G] B =&(&1) P(F) P(G) [G, F] B ,
(7)
[F, GK] B =[F, G] B K+(&1) P(F) P(G) G[F, K] B , (&1)
P(F) P(K)
[F, [G, K] B ] B +(&1)
P(G) P(F)
(8)
[G, [K, F] B ] B
+(&1) P(K) P(G) [K, [F, G] B ] B =0,
(9)
where the last expression is the analogue of Jacobi's identity. Similarly to the usual case, the transition to phase space is only possible if the momenta variables, given by Eq. (2), are independent variables among themselves so that we can express all velocities q* i as functions of canonical variables (q i, p i ). Such necessity implies that the Hessian supermatrix H ij #
r pi 2r L = q* j q* j q* i
(10)
must be nonsingular. Otherwise, if the Hessian has a rank P=N&R, there will be R relations among the momenta variables and coordinates q i that are primary constraints (which we suppose to have definite parity), while R velocities q* i will remain arbitrary variables in the theory. The development of Dirac's generalized Hamiltonian formalism is straightforward: the primary constraints have to be added to the Hamiltonian. We have to work out the consistency conditions, separate the constraints in first- and second-class ones, and define the Dirac brackets using the supermatrix whose elements are the Berezin brackets among the second-class constraints [3].
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3. HAMILTONJACOBI FORMALISM From Caratheodory's equivalent Lagrangians method [11] we can obtain the HamiltonJacobi equation for the even Lagrangian L(q, q* ). The procedure is similar to the one applied to usual variables: given a Lagrangian L, we can obtain a completely equivalent one given by L$=L(q i, q* i )&
dS(q i, t) , dt
where S(q i, t) is an even function in order to keep the equivalent Lagrangian even. These Lagrangians are equivalent because their respective action integrals have simultaneous extremum. Then we choose the function S(q i, t) in such a way that we get an extremum of L$ and, consequently, an extremum of the Lagrangian L. For this purpose, it is enough to find a set of functions ; i (q j, t) and S(q i, t) such that L$(q i, q* i =; i (q j, t), t)=0
(11)
and for all neighborhoods of q* i =; i (q j, t), L$(q i, q* i )>0.
(12)
With these conditions satisfied, the Lagrangian L$ (and consequently L) will have a minimum in q* i =; i (q j, t), so that the solutions of the differential equations given by q* i =; i (q j, t) will correspond to an extremum of the action integral. From the definition of L$ we have L$=L(q j, q* j )&
S(q j, t) r S(q j, t) dq i & , t q i dt
where again we must call attention to the use of the right derivative. Using condition (11) we have
_
S(q j, t) r S(q j, t) i q* & t q i
&} S S(q , t) = L(q , q* )& q* _ &} t } q
L(q j, q* j )&
=0,
q* i =; i
(13)
j
j
q* i =; i
j
r
i
i
. q* i =; i
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BEREZINIAN H-J SINGULAR FORMALISM
79
In addition, since L$ has a minimum in q* i =; i, we must have r L$ q* i
}
r L
q* i =; i
dS
r
_ q* & q* \ dt +&} L S(q , t) _ q* & q &}
=0 O
i
i
=0, q* i =; i
j
r
r
i
i
=0, q* i =; i
or r S(q j, t) q i
}
= q* i =; i
r L q* i
}
.
(14)
q* i =; i
Now, using the definitions for the conjugated momenta given by Eq. (2), we get pi =
r S(q j, t) . q i
(15)
We can see from this result and from Eq. (13) that, in order to obtain an extremum of the action, we must get a function S(q j, t) such that S =&H 0 , t
(16)
where H 0 is the canonical Hamiltonian H 0 = p i q* i &L(q j, q* j )
(17)
and the momenta p i are given by Eq. (15). Besides that, Eq. (16) is the Hamilton Jacobi partial differential equation (HJPDE).
4. THE SINGULAR CASE We now consider the case of a system with a singular Lagrangian. When the Hessian supermatrix is singular with a rank P=N&R we can define the variables q i in such order that the P_P supermatrix in the right bottom corner of the Hessian supermatrix be nonsingular, i.e., det |H ab | {0,
H ab =
2r L , q* b q* a
a, b=R+1, ..., N.
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PIMENTEL, TEIXEIRA, AND TOMAZELLI
This allows us to solve the velocities q* a as functions of coordinates q$s and momenta p a ; i.e., q* a = f a (q i, p b ). There will remain R momenta variables p : dependent upon the other canonical variables, and we can always [3, 4, 12] write expressions like p : =&H :(q i ; p a ),
:=1, ..., R,
(19)
that correspond to the Dirac primary constraints 8 : #p : +H :(q i ; p a )r0. The Hamiltonian H 0 , given by Eq. (17), becomes H 0 = p a f a +p : | p; =&H; } q* : &L(q i, q* : , q* a = f a ),
(20)
where :, ;=1, ..., R; a=R+1, ..., N. On the other hand, we have r L r L r f a r f a r H0 = p +p & & =0, a : q* : q* : q* : q* a q* : so the Hamiltonian H 0 does not depend explicitly upon the velocities q* : . Now we will adopt the following notation: the time parameter t will be called t0 #q 0 ; the coordinates q : will be called t : ; the momenta p : will be called P : ; and the momentum p 0 #P 0 will be defined as
P0 #
S . t
(21)
Then, to get an extremum of the action integral, we must find a function S(t : ; q a, t) that satisfies the set of HJPDE, H$0 #P 0 +H 0 t, t ; ; q a; p a =
\
r S =0, q a
+
(22)
\
r S =0, q a
(23)
H$: #P : +H : t, t ; ; q a ; p a =
+
where :, ;=1, ..., R. If we let the indexes : and ; run from 0 to R we can write both equations as
\
H$: #P : +H : t ; ; q a ; p a =
r S =0. q a
+
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BEREZINIAN H-J SINGULAR FORMALISM
81
From the above definition and Eq. (20) we have r H$0 r H: : r f a r L r f a =& a &(&1) P(b) P(:) q* + p a +(&1) P(b) q* b, p b q* p b p b p b r H$0 r H: : =(&1) P(b) q* b &(&1) P(b) P(:) q* , p b p b where we came back to :=1, ..., R. Multiplying this equation by dt=dt 0 and (&1) P(b), we have dq b =(&1) P(b)
r H$0 0 H$ dt +(&1) P(b) +P(b) P(:) r : dq :. p b p b
Using t : #q :, letting the index : run again from 0 to R, and considering P (0) =P (t 0 ) =0, we have dq b =(&1) P(b) +P(b) P(:)
r H$: : dt . p b
(25)
Noticing that we have the expressions dq ; =(&1) P(;) +P(;) P(:)
r H$: : dt =(&1) P(;) +P(;) P(:) $ :; dt : #dt ; p ;
identically satisfied for :, ;=0, 1, ..., R, we can write Eq. (25) as dq i =(&1) P(i) +P(i) P(:)
r H$: : dt , p i
i=0, 1, ..., N.
(26)
If we consider that we have a solution S(q j, t) of the HJPDE given by Eq. (24) then, differentiating that equation with respect to q i, we obtain r H$: 2r S r H$: r H$: 2r S + + =0 q i P ; q i t ; p a q i q a
(27)
for :, ;=0, 1, ..., R. From the momenta definitions we can obtain dp i =
2r S 2 S dt ; + ar i dq a, ; i t q q q
i=0, 1, ..., N.
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(28)
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PIMENTEL, TEIXEIRA, AND TOMAZELLI
Now, contracting Eq. (27) with dt : (from the right), multiplying by (&1) P(i) P(:), and adding the result to Eq. (28), we get dp i +( &1) P(i) P(:) =
_
2r S r H$: : dq a &(&1) P(i) P(:) (&1) (P(:) +P(a) )(P(i) +P(a) ) (&1) P(i) P(a) dt q a q i p a
\ S + dt &(&1) t q \ 2 r ;
;
i
dp i +(&1) P(i) P(:) =
r H$: : dt q i
_
P(i) P(:)
(&1) (P(:) +P(;) )(P(i) +P(;)) (&1) P(i) P(;)
+ H$ dt +& , P r
:
:
;
r H$: : dt q i
2r S r H$: : dq a &(&1) P(:) P(a) +P(a) dt q a q i p a
\ S + dt &(&1) t q \ 2 r ;
;
i
P(:) P(;) +P(;)
+ H$ dt +& , P r
:
:
(29)
;
where we used the fact that 2r S 2r S P(i) P( j ) j i =(&1) q q q i q j and that we have the parities P
\
2r S =P (i) +P ( j ) , q i q j
+
P
r H$:
\ p + =P
(:)
+P ( j ) .
j
If the total differential equation given by Eq. (26) applies, the above equation becomes dp i =&(&1) P(i) P(:)
r H$: : dt , q i
i=0, 1, ..., N.
(30)
Making Z#S(t : ; q a ) and using the momenta definitions, together with Eq. (26) we have dZ=
r S ; r S a dt + a dq , t ; q
\
dZ=&H ; dt ; + p a (&1) P(a) +P(a) P(:)
r H$: : dt . p a
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+
BEREZINIAN H-J SINGULAR FORMALISM
83
With a little change of indexes we get
\
dZ= &H ; +(&1) P(a) +P(a) P(;) p a
r H$; dt ;. p a
+
(31)
This equation, together with Eq. (26) and Eq. (30), is the total differential equation for the characteristic curves of the HJPDE given by Eq. (24) and, if Eqs. (26), (30), and (31) form a completely integrable set, their simultaneous solutions determine S(t : ; q a ) uniquely from the initial conditions. Besides that, Eq. (26) and Eq. (30) are the equations of motion of the system written as total differential equations. It is important to observe that, in the nonsingular case, we have only H$0 {0 and no others H$: , so that these equations of motion will reduce naturally to the usual expressions given by Eq. (4).
5. INTEGRABILITY CONDITIONS The analysis of integrability conditions of the total differential equations (26), (30), and (31) can be carried out using standard techniques. This has already been made [2, 13, 14] for systems with the usual variables, and here we will present an analysis of the integrability conditions for Berezinian singular systems. To a given set of total differential equations, dq i =4 : i(t ;, q j ) dt :
(32)
(i, j=0, 1, ..., N; and :, ;=0, 1, ..., R
r f i 4 : =0, q i
(33)
where X : are linear operators. Given any twice-differentiable solution of the set (33), it should also satisfy the equation [X : , X ; ] f =0,
(34)
[X : , X ; ] f =(X : X ; &(&1) P(:) P(;) X ; X : ) f
(35)
where
is the Berezin bracket among the operators X : . This implies that we should have [X : , X ; ] f =C :; # X # f.
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(36)
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PIMENTEL, TEIXEIRA, AND TOMAZELLI
So, the commutation relations (35) will give the maximal number of linearly independent equations. Any bracket that results in a expression that cannot be written as Eq. (36) must be written as a new operator X and be joined to the original set (33), having all commutators with the other X's calculated. The process is repeated until all operators X satisfy Eq. (36). If all operators X : satisfy the commutation relations given by Eq. (36) the system of partial differential equations (33) is said to be complete and the corresponding system of total differential equations (32) is integrable if, and only if, the system (33) is complete. Now, we consider the system of differential equations obtained in the previous section. First we shall observe that if the total differential equations (26) and (30) are integrable the solutions of Eq. (31) can be obtained by a quadrature, so we only need to analyze the integrability conditions for the former ones, since the last will be integrable as a consequence. The operators X : corresponding to the system of total differential equations formed by Eq. (26) and Eq. (30) are given by X : f (t ;, q a, p a )=
r H$: r f r H$: r f (&1) P(i) +P(i) P(:) & (&1) P(i) P(:) , i q p i p i q i
X : f (t ;, q a, p a )=
r f l H$: &(&1) P(i) +P(i) P( f ) q i p i _(&1) P(i) P(:) (&1) (P(:) +P(i) )(P( f ) +P(i))
X : f (t ;, q a, p a )=
r H$: l f , q i p i
(37)
r H$: l f r f l H$: &(&1) P(:) P( f ) , i q p i q i p i
X : f (t ;, q a, p a )=[ f, H$: ] B ; where i=0, 1, ..., N; :, ;=0, 1, ..., R; a=R+1, ..., N, and we have used Eq. (32) and Eq. (33) together with the result r A l A =(&1) P(i) (&1) P(i) P(A) . q i q i
(38)
It is important to notice that the Berezin bracket in Eq. (37) is defined in a (2N+2)-dimensional phase space, since we are including q 0 =t as a ``coordinate.'' Now, the integrability condition will be [X : , X ; ] f =(X : X ; &(&1) P(:) P(;) X ; X : ) f =X :[ f, H$; ] B &(&1) P(:) P(;) X ;[ f, H$: ] B =0, [X : , X ; ] f =[[ f, H$; ] B , H$: ] B &(&1)
P(:) P(;)
[[ f, H$: ] B , H$; ] B =0,
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85
BEREZINIAN H-J SINGULAR FORMALISM
that will reduce to [X : , X ; ] f =&(&1) (P( f )+P(;) ) P(:) [H$: , [ f, H$; ] B ] B &(&1) P(:) P(;) (&1) (P( f ) +P(:)) P(;) +P( f ) P(:) [H$; , [H$: , f ] B ] B =0, [X : , X ; ] f =&(&1) P( f ) P(:) (&1) P(;) P(:) [H$: , [ f, H$; ] B ] B &(&1) P(:) P(;) (&1) P( f ) P(;) +P(:) P(;) +P( f) P(:) [H$; , [H$: , f ] B ] B =0, [X : , X ; ] f =&(&1) P( f ) P(:) [(&1) P(;) P(:) [H$: , [ f, H$; ] B ] B +(&1) P( f ) P(;) [H$; , [H$: , f ] B ] B ]=0, [X : , X ; ] f =&(&1) P( f ) P(:) [&(&1) P( f ) P(:) [ f, [H$; , H$: ] B ] B ], [X : , X ; ] f =[ f, [H$; , H$: ] B ] B =0,
(40)
when using the Jacobi relations for Berezin brackets given by Eq. (9) and the fact that P([A, B] B )=P(A)+P(B).
(41)
So, the integrability condition will be [H$; , H$: ] B =0
\:, ;.
(42)
It is important to notice that the above condition can be shown to be equivalent to the consistency conditions in Dirac's Hamiltonian formalism but, to keep the continuity of the presentation, we will postpone the demonstration of this equivalence to the Appendix. Now, the total differential for any function F(t ;, q a, p a ) can be written as dF=
r F a r F a r F : dq + a dp + : dt , q a p t
dF=
F F r F H$ H$ (&1) P(a) +P(a) P(:) r : dt : & r a (&1) P(a) P(:) r : dt : + r : dt :, q a p a p q a t r F l H$:
\ q p &(&1) F H$ dF= \ q p &(&1)
dF=
a
P(a)
(&1) P(a) P(F)
a
r
l
:
a
a
P(:) P(F)
(43)
l F r H$: r F (&1) P(a) P(:) + : dt :, p a q a t
+
r H$: l F r F + dt :, q a p a t :
+
dF=[F, H$: ] B dt : ;
(44)
where the Berezin bracket above is the one defined in the 2N+2 phase space used in Eq. (37). Using this result we have dH$; =[H$; , H$: ] B dt :
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PIMENTEL, TEIXEIRA, AND TOMAZELLI
and, consequently, the integrability condition (42) reduces to dH$: =0
(46)
\:.
If the above conditions are not identically satisfied we will have one of two different cases. First, we may have a new H$=0, which has to satisfy a condition dH$=0 and must be used in all equations. Otherwise, we will have relations among the differentials dt : which also must be used in the remaining equations of the formalism.
6. EXAMPLE As an example we analyze the case of the electromagnetic field coupled to a spinor, whose Hamiltonian formalism was analyzed in Ref. [3, 4]. We will consider the Lagrangian density written as L=& 14 F +& F +& +i # +( + +ieA + ) &m,
(47)
where A + are even variables while and are odd ones. The electromagnetic tensor is defined as F +& = +A & & &A + and we are adopting the Minkowski metric ' +& =diag(+1,&1,&1,&1). 6.1. Hamiltonian Formalism Let us first review Dirac's Hamiltonian formalism. The momenta variables conjugated, respectively, to A + , , and , are p+=
r L =&F 0+ , A4 +
p=
r L =i# 0, 4
(48)
p =
r L =0, 4
(49)
where we must call attention to the necessity of being careful with the spinor indexes. Considering, as usual, as a column vector and as a row vector implies that p will be a row vector while p will be a column vector. From the momenta expressions we have the primary constraints , 1 = p 0 r0, , 2 = p &i# 0,
(50) , 3 = p r0.
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BEREZINIAN H-J SINGULAR FORMALISM
The canonical Hamiltonian is given by H C = HC d 3 x= ( p + A4 + +( p ) : (4 ) : +( p ) : (4 ) : &L) d 3x,
|
|
|
|
(52)
H C = HC d 3x= [ 14 F ij F ij & 12 p i p i &A 0( i p i &e# 0) &i# j ( j +ieA j ) +m] d 3x.
(53)
The primary Hamiltonian is
|
H P = (HC +* 1 , 1 +, 2 * 2 +* 3 , 3 ) d 3x,
(54)
where * 1 is an even variable and * 2 , * 3 are odd variables, * 2 being a column vector and * 3 a row vector. The fundamental nonvanishing Berezin brackets (here the brackets are the ones defined by Eq. (5) in the 2N phase space) are [A +(x), p &( y)] B =$ &+ $ 3(x& y),
(55)
3
[(x), p ( y)] B =$ (x& y),
(56)
[( x), p ( y)] B =$ 3(x& y).
(57)
The consistency conditions are [, 1 , H P ] B = i p i &e# 0r0,
(58)
[, 2 , H P ] B =&i( j &ieA j ) # j &e# 0A 0 &m +i* 3 # 0 r0,
(59)
[, 3 , H P ] B =&i( j +ieA j ) # j +e# 0 A 0 +m&i# 0* 2 r0.
(60)
The last two will determine * 2 and * 3 , while the first will give rise to the secondary constraint, /= i p i &e# 0r0,
(61)
for which the consistency condition will be identically satisfied with the use of the expressions for * 2 and * 3 given by Eq. (59) and Eq. (60). Taking the Berezin brackets among the constraints we have as nonvanishing results [(, 2 ) : , (, 3 ) ; ] B =&i[(# 0 ) : , ( p ) ; ] B =&i(# 0 ) ;: , [(, 2 ) : , /] B =&e[( p ) : , # 0] B =e(# 0 ) : , 0
0
[(, 3 ) : , /] B =&e[( p ) : , # ] B =&e(# ) : ,
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(62) (63) (64)
88
PIMENTEL, TEIXEIRA, AND TOMAZELLI
where we explicitly wrote the spinor indexes. Obviously the , 1 constraint is first-class, but we have another first-class constraint. This can be seen from the supermatrix 2 formed by the Berezin brackets among the second-class constraints /, , 2 , and , 3 . Numbering the constraints as 8 1 =/, 8 2 =, 2 , and 8 3 =, 3 , we have this supermatrix in normal form (see Ref. [3, Appendix D]) given, with spinor indexes indicated, by [8 1 , 8 1 ] B 2= [(8 2 ) $ , 8 1 ] B
[8 1 , (8 2 ) : ] B [(8 2 ) $ , (8 2 ) : ] B
[8 1 , (8 3 ) ; ] B [(8 2 ) $ , (8 3 ) ; ] B
[(8 3 ) + , 8 1 ] B
[(8 3 ) + , (8 2 ) : ] B
[(8 3 ) + , (8 3 ) ; ] B
2=
\ \
0 0
e(# ) $ &e(# 0) +
&e(# 0 ) :
e(# 0) ;
0 &i(# 0 ) +:
&i(# 0 ) ;$ 0
+
+
,
(65)
.
(66)
This supermatrix has one eingevector with null eingevalue, that is, 1
\ + ie() : &ie( ) ;
,
(67)
so there is another first-class constraint given by .=/+ie(8 2 ) : () : &ie(8 3 ) ; ( ) ; ,
(68)
.= i p i +ie( p +p )
(69)
that we will substitute for /. So, we have the first-class constraints , 1 and ., and the second-class ones , 2 and , 3 . The supermatrix 2 now reduces to the Berezin brackets among the second-class constraints , 2 and , 3 and is given by 2=
\
[(8 2 ) $ , (8 2 ) : ] B [(8 2 ) $ , (8 3 ) ; ] B 0 &i(# 0 ) ;$ = , &i (# 0 ) +: [(8 3 ) + , (8 2 ) : ] B [(8 3 ) + , (8 3 ) ; ] B 0
+ \
+
(70)
having as the inverse, 2 &1 =
\
0 i (# 0 ) $#
i(# 0 ) :* . 0
+
(71)
With these results, the Dirac brackets among any variables F and G are [F(x), G( y)] D =[F(x), G( y)] B &i
|d
3
z([F(x), (8 2 ) :(z)] B (# 0 ) :;
_[(8 3 ) ; (z), G( y)] B +[F(x), (8 3 ) : (z)] B (# 0 ) ;: _[(8 2 ) ; (z), G( y)] B ).
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(72)
89
BEREZINIAN H-J SINGULAR FORMALISM
The nonvanishing fundamental brackets now will be [A +(x), p &( y)] D =$ &+ $ 3(x& y), 0
(73)
3
[() * (x), ( ) : ( y)] D =&i(# ) *: $ (x& y),
(74)
[() * (x), ( p ) : ( y)] D =$ *: $ 3(x& y).
(75)
Now we can make the second-class constraints as strong equalities and write the equations of motion in terms of the Dirac brackets and the extended Hamiltonian given by
|
|
H E = HE d 3x= (HC +* 1 , 1 +:.) d 3x.
(76)
We must remember that, when making , 2 =, 3 #0 the constraint . becomes identical to the original secondary constraint /. Then, the equations of motion will be A4 i r[A i, H E ] D =&p i + i A 0 & i:,
(77)
A4 0 r[A 0, H E ] D =* 1 ,
(78)
p* i r[ p i, H E ] D = j F ji &e# i ,
(79)
4 r[, H E ] D =&ieA 0 &# 0# j ( j +ieA j ) &im# 0,
(80)
4 r[, H E ] D =ieA 0 &( j &ieA j ) # # +im# .
(81)
j 0
0
Multiplying Eq. (80) from the left by i# 0 we get i# 04 =eA 0 # 0&i# j ( j +ieA j ) +m,
(82)
i( + +ieA + ) # +&m=0,
(83)
while multiplying Eq. (81) from the right by i# 0 we get i4 # 0 =&e# 0A 0 &i( j +ieA j ) # j &m,
(84)
i# +( 0 + &ieA + )+m =0.
(85)
These are the equations of motions with full gauge freedom. It can be seen, from Eq. (78), that A 0 is an arbitrary (gauge dependent) variable since its time derivative is arbitrary. Besides that, Eq. (77) shows the gauge dependence of A i and, taking the curl of its vector form, leads to the known Maxwell equation, A9 B9 =&E9 &{9 (A 0 &:) O =&{9 _E9 . t t
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(86)
90
PIMENTEL, TEIXEIRA, AND TOMAZELLI
Writing j + =e# + we get, from Eq. (79), the inhomogeneous Maxwell equation E9 ={9 _B9 &} , t
(87)
while the other inhomogeneous equation {9 } E9 = j 0,
(88)
follows from the secondary constraint (61). Expressions (83) and (85) are the known Dirac's equations for the spinor fields and . 6.2. HamiltonJacobi Formalism Now we apply the formalism presented in the previous sections. From the momenta definition we have the ``Hamiltonians''
|
(89)
|
(90)
|
(91)
|
(92)
H$0 = (P 0 +HC ) d 3x, H$1 = p 0 d 3x, H$2 = ( p &i# 0 ) d 3x, H$3 = p d 3x,
which are associated, respectively, to t=t 0 (remember that P 0 is the momentum conjugated to t), A 0 , , and . The first two H$ are even variables, while the last two are odd. Then, using Eq. (26), we have dA i =
$H$0 dt=(&p i + iA 0 ) dt. $p i
(93)
From Eq. (30) we have dp i =&
$H$0 dt=( j F ji &e# i) dt, $A i
(94)
dp 0 =&
$H$0 dt=( j p j &e# 0) dt, $A 0
(95)
dp =&
$ r H$0 dt=(&i j # j &e# +A + &m ) dt, $
(96)
dp =&
$ r H$0 $
dt+
$ r H$2 $
d=(&i# j j +eA + # ++m) dt&i# 0 d.
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(97)
BEREZINIAN H-J SINGULAR FORMALISM
91
The integrability conditions require dH$=0, which implies for H$1 dH$1 =d
|p
0
d 3x=0 O dp 0 = j p j &e# 0=0,
(98)
where we made use of Eq. (95). This expression is equivalent to the secondary constraint (61) and has to satisfy
|
H$4 = ( j p j &e# 0) d 3x,
dH$4 =0,
(99)
which is indeed identically satisfied. For H$2 we have dH$2 =d
| (p
&i# 0 ) d 3x=0 O dp =i(d ) # 0,
(100)
which cannot be written as an expression like H$=0 due to the presence of two differentials (dp and d ) but, substituting in Eq. (96), we get i(d ) # 0 =(&i j # j &e# +A + &m ) dt;
(101)
&i4 # 0 &i j # j &e# +A + &m =0,
(102)
i.e. +
i# ( 0 + &ieA + )+m =0.
(103)
For H$3 we have
|
dH$3 =d p d 3x=0 O dp =0,
(104)
that, similarly to the case above, can be used in Eq. (97), giving (&i# j j +eA + # ++m) dt&i# 0 d=0, +
i( + +ieA + ) # &m=0.
(105) (106)
Finally, we can verify, using the above results, that dH$0 =0 is identically satisfied. Equations (103) and (106) are identical, respectively, to Eqs. (85) and (83) obtained in Hamiltonian formalism. Besides that, Eqs. (94) and (79) are equivalent, while Eq. (93) corresponds to Eq. (77), except for an arbitrary gauge factor.
9. CONCLUSIONS In this work we presented a formal generalization of the HamiltonJacobi formalism for singular systems with Berezinian variables, obtaining the equations of
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PIMENTEL, TEIXEIRA, AND TOMAZELLI
motion as total differential equations (26) and (30). In these equations, the coordinates q : =t : (:=1, ..., R), whose momenta are constrained, play the role of evolution parameters of the system, together with the time parameter t=t 0. So, the system's evolution is described by contact transformations generated by the ``Hamiltonians'' H$: and parametrized by t : (with :=0, 1, ..., R), where H$0 is related to the canonical Hamiltonian by Eq. (22) and the other H$: (:=1, ..., R) are the constraints given by Eq. (23). This evolution is considered as being always restricted to the constraint surface in phase space. There is no complete phase space treatment that is later reduced to the constraint surface, as in Dirac's formalism with the use of weak equalities. We should observe that, in the case of systems composed exclusively by even variables, all parities are equal to zero and Eqs. (26), (30), (31) reduce to the results obtained in Ref. [1]. Furthermore, if the system is nonsingular, we have H$: #0, except for :=0, so the total differential equations (26) and (30) will be reduced to the expressions given by Eq. (4). The integrability conditions (which relation to the consistency conditions in Dirac's formalism is discussed in the Appendix) were shown to be equivalent to the necessity of the vanishing of the variation of each H$: (:=0, ..., R); i.e., dH$: =0. The example presented was chosen for its completeness; it is a singular system with even and odd variables and its Hamiltonian treatment contains all kinds of constraints (primary and secondary, first- and second-class ones). This example is very illustrative, since it allows a comparison between all features of Dirac and HamiltonJacobi formalisms. For example, the fact that the integrability conditions dH$2 =0 and dH$3 =0 give expressions involving some differentials dt : is related to the fact that the corresponding Hamiltonian constraints , 2 and , 3 are secondclass constraints and determine some of the arbitrary parameters in the primary Hamiltonian (54). Similarly, the fact that the condition dH$1 =0 generated an expression like H$4 =0 is related to the fact that the corresponding Hamiltonian constraint , 1 is a first class one (see the Appendix). Finally, we must call attention to the presence of arbitrary variables in some of the Hamiltonian equations of motion due to the fact that we have gauge-dependent variables and we have not made any gauge fixing. This does not occur in Hamilton Jacobi formalism since it provides a gauge-independent description of the system's evolution, due to the fact that the HamiltonJacobi function S contains all the solutions that are related by gauge transformations.
APPENDIX: EQUIVALENCE AMONG CONSISTENCY AND INTEGRABILITY CONDITIONS In this appendix we will show the equivalence among the integrability conditions of the formalism shown above and the consistency conditions in Dirac's Hamiltonian formalism, in a way similar to what was made for the usual variables [16]. In the
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BEREZINIAN H-J SINGULAR FORMALISM
93
notation used in this paper the Dirac primary constraints are written, from Eq. (23), as H$: #P : +H :(q i; p a )r0,
(107)
where :=1, ..., R; i=1, ..., N. The canonical Hamiltonian is given by H 0 in Eq. (20), so the primary Hamiltonian H P is H P #H 0 +H$: v :,
(108)
where the v : are unknown coefficients related to the undetermined velocities q* : [4]. The ordering of the v : with respect to the H$: is a matter of choice, since it will simply produce a change of sign, but the natural procedure that identifies v : and q* : suggests the ordering above as a consequence of the ordering adopted in the Hamiltonian (3). This ordering is also the most natural choice to our purpose but is, of course, irrelevant for systems containing only the usual variables. The consistency conditions, which demand that the constraints are preserved by time evolution, are written as H4 $+ r[H$+ , H P ] B =[H$+ , H 0 ] B +[H$+ , H$: ] B q* : =0,
(109)
where :, +=1, ..., R and the Berezin brackets here are those given in Eq. (5) defined in the usual 2N-dimensional phase space and we have made the explicit identification q* : #v :. Multipling the above equation by dt we get dH$+ r[H$+ , H 0 ] B dt+[H$+ , H$: ] B dt : =0,
(110)
where, as before, q : =t :, but we are still making :=1, ..., R. At this point we can already see that, when Dirac's consistency conditions are satisfied we have dH$+ =0 satisfied. We must see now that we have dH$0 =0 when the Dirac consistency conditions are satisfied. The Hamiltonian equation of motion for H$0 is H4 $0 r[H$0 , H P ] B =[H$0 , H 0 ] B +[H$0 , H$: ] B q* :,
(111)
which multiplied by dt becomes dH$0 r[H$0 , H 0 ] B dt+[H$0 , H$: ] B dt :.
(112)
Remembering that the ``momentum'' P 0 in H$0 is independent of the canonical variables q i and p i , we have dH$0 r[H 0 , H 0 ] B dt+[H 0 , H$: ] B dt : =[H 0 , H$: ] B dt :.
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(113)
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PIMENTEL, TEIXEIRA, AND TOMAZELLI
But, if Dirac's consistency conditions are satisfied, we must have only primary first-class constraints, otherwise we would have conditions imposed on the unknown velocities q* :. So, the preservation of constraints in time will reduce to [H$+ , H 0 ] B r0,
(114)
and the right side of Eq. (113) will be zero. This is simply a consequence of the fact that, once all Dirac's conditions are satisfied, the Hamiltonian is preserved. So the condition dH$0 =0 is satisfied when the Dirac consistency conditions are satisfied. This shows that the integrability conditions in HamiltonJacobi formulation will be satisfied when Dirac's consistency conditions are satisfied. Similarly, we can consider that we have the integrability conditions satisfied so that dH$0 =dH$+ =0 and then Eq. (110), which is equivalent to Eq. (109), implies that Dirac's conditions are satisfied. So, both conditions are equivalent. Now, we will consider that these conditions are not initially satisfied. When we have only first class constraints in Hamiltonian formalism we will simply get a new constraint from some of the conditions (109). From Eq. (110) we see that this will imply an expression like dH$+ =H$dt (H$r0 is the secondary Hamiltonian constraint) which means that there will be a new H$ in HamiltonJacobi formalism that has to satisfy dH$=0. If we have some second-class Hamiltonian constraints the consistency conditions (109) will imply a condition over some of the velocities q* :. From Eq. (110) we see that, in the HamiltonJacobi approach there will be conditions imposed on some differentials dt :. Such correspondence among the formalisms can be clearly seen in the example presented in this paper. Besides that, Eq. (110) and Eq. (112) can be written as dH$+ r[H$+ , H$: ] B dt :,
(115)
where now :, +=0, 1, ..., R and the Berezin bracket is again defined in the (2N+2)dimensional phase space containing t 0 and P 0 . This equation is obviously identical to Eq. (45) that leads to the integrability condition dH$+ =0, and its right- hand side was shown to correspond to Dirac's consistency conditions. Consequently, this expression shows directly the relation among consistency and integrability conditions. It is important to notice that here we are not considering any explicit dependence on time, neither of the constraints nor of the canonical Hamiltonian, because it is a usual procedure in the Hamiltonian approach. But the equations of Hamilton Jacobi formalism were obtained without considering this condition and, consequently, remain valid if we consider systems with Lagrangians that are explicitly timedependent. But the Hamiltonian approach is also applicable to such systems (see Ref. [3, p. 229]) and in this case we can follow a procedure similar to the one shown here and demonstrate the correspondence between Dirac's consistency conditions and integrability conditions.
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BEREZINIAN H-J SINGULAR FORMALISM
95
Finally, some words about the simpletic structure. Using Eq. (38), we can write Eq. (26) and Eq. (30) in terms of left derivatives as dq i =
l H$: : dt , p i
dp i =&(&1) P(i)
(116) l H$: : dt , q i
(117)
where, as before, i=0, 1, ..., N ; :=0, 1, ..., R. These expressions can be compactly written as d' I =E IJ
l H$: : dt , ' J
(118)
where we used the notation ' 2i = p i O ' I =(q i, p i ),
' 1i =q i, IJ
i j
` 1
2 _
E =$ ($ $ &(&1)
P(i)
1 _
I=(`=1, 2; i=0, 1, ..., N ),
(119)
I=(`; i),
(120)
` 2
$ $ ),
J=(_; j ),
which was introduced in page 76 of Ref. [3]. The Berezin brackets defined in Eq. (5) can be written as [F, G] B =
r F IJ l F E . ' I ' J
(121)
This simpletic notation allows us to obtain the expression for the total differential for any function F(t ;, q a, p a ) in a more direct way. Using it in Eq. (43) we get r F I d' , ' I
(122)
r F IJ l H$: : E dt =[F, H$: ] B dt :, ' I ' J
(123)
dF= where the use of Eq. (118) gives dF= in agreement with Eq. (44).
ACKNOWLEDGMENTS B.M.P. is partially supported by CNPq and R.G.T. is supported by CAPES.
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REFERENCES 1. Y. Guler, Il Nuovo Cimento B 107 (1992), 1389. 2. Y. Guler, Il Nuovo Cimento B 107 (1992), 1143. 3. D. M. Gitman and I. V. Tyutin, ``Quantization of Fields with Constraints,'' Springer-Verlag, New YorkBerlin, 1990. 4. K. Sundermeyer, ``Lecture Notes in Physics 169Constrained Dynamics,'' Springer-Verlag, New YorkBerlin, 1982. 5. A. Hanson, T. Regge, and C. Teitelboim, ``Constrained Hamiltonian Systems,'' Accad. Naz. Lincei, Rome, 1976. 6. B. M. Pimentel and R. G. Teixeira, Il Nuovo Cimento B 111 (1996), 841. 7. B. M. Pimentel and R. G. Teixeira, Preprint hep-th9704088, Il Nuovo Cimento B, to appear. 8. Y. Guler, Il Nuovo Cimento B 109 (1994), 341. 9. Y. Guler, Il Nuovo Cimento B 111 (1996), 513. 10. F. A. Berezin, ``Introduction to Superanalysis,'' Reidel, Dordrecht, 1987. 11. C. Caratheodory, ``Calculus of Variations and Partial Differential Equations of the First Order,'' Part II, p. 205, HoldenDay, Oakland, CA, 1967. 12. E. C. G. Sudarshan and N. Mukunda, ``Classical Dynamics: A Modern Perspective,'' Wiley, New York, 1974. 13. E. M. Rabei and Y. Guler, Phys. Rev. A 46 (1992), 3513. 14. Y. Guler, Il Nuovo Cimento B 110 (1995), 307. 15. E. T. Whittaker, ``A Treatise on the Analytical Dynamics of Particles and Rigid Bodies,'' 4 th ed., p. 52, Dover, New York, 1944. 16. E. M. Rabei, Hadronic J. 19 (1996), 597.
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HamiltonJacobi Approach to Berezinian Singular Systems B. M. Pimentel and R. G. Teixeira Instituto de F@ sica Teorica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 Sa~o Paulo, S.P., Brazil
and J. L. Tomazelli Departamento de F@ sica e Qu@ micaFaculdade de Engenharia, Universidade Estadual Paulista Campus de Guaratingueta , Av. Dr. Ariberto Pereira da Cunha, 333, 12500-000 Guaratinqueta , S.P., Brazil E-mail: pimentelaxp.ift.unesp.br Received January 13, 1998
In this work we present a formal generalization of the HamiltonJacobi formalism, recently developed for singular systems, to include the case of Lagrangians containing variables which are elements of Berezin algebra. We derive the HamiltonJacobi equation for such systems, analyzing the singular case in order to obtain the equations of motion as total differential equations and study the integrability conditions for such equations. An example is solved using both HamiltonJacobi and Dirac's Hamiltonian formalisms and the results are compared. 1998 Academic Press
1. INTRODUCTION In this work we intend to study singular systems with Lagrangians containing elements of Berezin algebra from the point of view of the HamiltonJacobi formalism recently developed [1, 2]. The study of such systems through Dirac's generalized Hamiltonian formalism has already been extensively developed in literature [35] and will be used for comparative purposes. Despite the success of Dirac's approach in studying singular systems, which is demonstrated by the wide number of physical systems to which this formalism has been applied, it is always instructive to study singular systems through other formalisms, since different procedures will provide different views for the same problems, even for nonsingular systems. The HamiltonJacobi formalism that we study in this work has been already generalized to higher order singular systems [6, 7] and applied only to a few number of physical examples as the electromagnetic field [8], relativistic particle in an external electromagnetic field [9] and Podolsky's electrodynamics [6]. But a better understanding of this approach utility 75 0003-491698 25.00 Copyright 1998 by Academic Press All rights of reproduction in any form reserved.
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PIMENTEL, TEIXEIRA, AND TOMAZELLI
in the studying singular systems is still lacking, and such understanding can only be achieved through its application to other interesting physical systems. Besides that, Berezin algebra is a useful way to deal simultaneously with bosonic and fermionic variables in a unique and compact notation, which justifies the interest in studying systems composed of its elements using new formalisms. The aim of this work is not only to generalize the HamiltonJacobi approach for singular systems to the case of Lagrangians containing Berezinian variables but also to present an example of its application to a well-known physical system, comparing the results to those obtained through Dirac's method. We will start in Section 2 with some basic definitions and next, in Section 3, we will introduce the HamiltonJacobi formalism to Berezinian systems using Caratheodory's equivalent Lagrangians method. In Section 4 the singular case is considered and the equations of motion are obtained as a system of total differential equations whose integrability conditions are analyzed in Section 5. The equivalence among these integrability conditions and Dirac's consistency conditions will be discussed separately in the Appendix. In Section 6 we present, as an example, the electromagnetic field coupled to a spinor, which is studied using both the formalism presented in this work and Dirac's Hamiltonian one. Finally, the conclusions are presented in Section 7.
2. BASIC DEFINITIONS We will start from a Lagrangian L(q, q* ) that must be an even function of a set of N variables q i that are elements of Berezin algebra. For a basic introduction to such an algebra we suggest the reader see Ref. [3, Appendix D], from which we took the definitions used in this paper. A more complete treatment can be found in Ref. [10]. The Lagrangian equations of motion can be obtained through variational principles from the action S= L dt, $r S r L d r L = & =0, $q i q i dt q* i
(1)
where we must call attention to the use of right derivatives. The passage to Hamiltonian formalism is made, as usual, by defining the momenta variables through right derivatives as pi #
r L q* i
(2)
and introducing the Hamiltonian function as (summing over repeated indexes) H= p i q* i &L,
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(3)
BEREZINIAN H-J SINGULAR FORMALISM
77
where the ordering of momenta to the left of velocities shall be observed since they were defined as right derivatives. This ordering will be, of course, irrelevant when we deal with even elements of the Berezin algebra. The Hamiltonian equations of motion will be given by q* i =
r H l H =(&1) P(i) , p i p i
l H r H p* i =& i =&(&1) P(i) . q q i
(4)
If we use the Poisson bracket in the Berezin algebra, given by [F, G] B =
r F l G G F &(&1) P(F) P(G) r i l , q i p i q p i
(5)
we get the known expressions q* i =[q i, H] B ,
p* i =[ p i , H] B .
(6)
For simplicity and clarity we will refer to these brackets as Berezin brackets. These brackets have properties similar to the usual Poisson brackets, [F, G] B =&(&1) P(F) P(G) [G, F] B ,
(7)
[F, GK] B =[F, G] B K+(&1) P(F) P(G) G[F, K] B , (&1)
P(F) P(K)
[F, [G, K] B ] B +(&1)
P(G) P(F)
(8)
[G, [K, F] B ] B
+(&1) P(K) P(G) [K, [F, G] B ] B =0,
(9)
where the last expression is the analogue of Jacobi's identity. Similarly to the usual case, the transition to phase space is only possible if the momenta variables, given by Eq. (2), are independent variables among themselves so that we can express all velocities q* i as functions of canonical variables (q i, p i ). Such necessity implies that the Hessian supermatrix H ij #
r pi 2r L = q* j q* j q* i
(10)
must be nonsingular. Otherwise, if the Hessian has a rank P=N&R, there will be R relations among the momenta variables and coordinates q i that are primary constraints (which we suppose to have definite parity), while R velocities q* i will remain arbitrary variables in the theory. The development of Dirac's generalized Hamiltonian formalism is straightforward: the primary constraints have to be added to the Hamiltonian. We have to work out the consistency conditions, separate the constraints in first- and second-class ones, and define the Dirac brackets using the supermatrix whose elements are the Berezin brackets among the second-class constraints [3].
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PIMENTEL, TEIXEIRA, AND TOMAZELLI
3. HAMILTONJACOBI FORMALISM From Caratheodory's equivalent Lagrangians method [11] we can obtain the HamiltonJacobi equation for the even Lagrangian L(q, q* ). The procedure is similar to the one applied to usual variables: given a Lagrangian L, we can obtain a completely equivalent one given by L$=L(q i, q* i )&
dS(q i, t) , dt
where S(q i, t) is an even function in order to keep the equivalent Lagrangian even. These Lagrangians are equivalent because their respective action integrals have simultaneous extremum. Then we choose the function S(q i, t) in such a way that we get an extremum of L$ and, consequently, an extremum of the Lagrangian L. For this purpose, it is enough to find a set of functions ; i (q j, t) and S(q i, t) such that L$(q i, q* i =; i (q j, t), t)=0
(11)
and for all neighborhoods of q* i =; i (q j, t), L$(q i, q* i )>0.
(12)
With these conditions satisfied, the Lagrangian L$ (and consequently L) will have a minimum in q* i =; i (q j, t), so that the solutions of the differential equations given by q* i =; i (q j, t) will correspond to an extremum of the action integral. From the definition of L$ we have L$=L(q j, q* j )&
S(q j, t) r S(q j, t) dq i & , t q i dt
where again we must call attention to the use of the right derivative. Using condition (11) we have
_
S(q j, t) r S(q j, t) i q* & t q i
&} S S(q , t) = L(q , q* )& q* _ &} t } q
L(q j, q* j )&
=0,
q* i =; i
(13)
j
j
q* i =; i
j
r
i
i
. q* i =; i
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BEREZINIAN H-J SINGULAR FORMALISM
79
In addition, since L$ has a minimum in q* i =; i, we must have r L$ q* i
}
r L
q* i =; i
dS
r
_ q* & q* \ dt +&} L S(q , t) _ q* & q &}
=0 O
i
i
=0, q* i =; i
j
r
r
i
i
=0, q* i =; i
or r S(q j, t) q i
}
= q* i =; i
r L q* i
}
.
(14)
q* i =; i
Now, using the definitions for the conjugated momenta given by Eq. (2), we get pi =
r S(q j, t) . q i
(15)
We can see from this result and from Eq. (13) that, in order to obtain an extremum of the action, we must get a function S(q j, t) such that S =&H 0 , t
(16)
where H 0 is the canonical Hamiltonian H 0 = p i q* i &L(q j, q* j )
(17)
and the momenta p i are given by Eq. (15). Besides that, Eq. (16) is the Hamilton Jacobi partial differential equation (HJPDE).
4. THE SINGULAR CASE We now consider the case of a system with a singular Lagrangian. When the Hessian supermatrix is singular with a rank P=N&R we can define the variables q i in such order that the P_P supermatrix in the right bottom corner of the Hessian supermatrix be nonsingular, i.e., det |H ab | {0,
H ab =
2r L , q* b q* a
a, b=R+1, ..., N.
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This allows us to solve the velocities q* a as functions of coordinates q$s and momenta p a ; i.e., q* a = f a (q i, p b ). There will remain R momenta variables p : dependent upon the other canonical variables, and we can always [3, 4, 12] write expressions like p : =&H :(q i ; p a ),
:=1, ..., R,
(19)
that correspond to the Dirac primary constraints 8 : #p : +H :(q i ; p a )r0. The Hamiltonian H 0 , given by Eq. (17), becomes H 0 = p a f a +p : | p; =&H; } q* : &L(q i, q* : , q* a = f a ),
(20)
where :, ;=1, ..., R; a=R+1, ..., N. On the other hand, we have r L r L r f a r f a r H0 = p +p & & =0, a : q* : q* : q* : q* a q* : so the Hamiltonian H 0 does not depend explicitly upon the velocities q* : . Now we will adopt the following notation: the time parameter t will be called t0 #q 0 ; the coordinates q : will be called t : ; the momenta p : will be called P : ; and the momentum p 0 #P 0 will be defined as
P0 #
S . t
(21)
Then, to get an extremum of the action integral, we must find a function S(t : ; q a, t) that satisfies the set of HJPDE, H$0 #P 0 +H 0 t, t ; ; q a; p a =
\
r S =0, q a
+
(22)
\
r S =0, q a
(23)
H$: #P : +H : t, t ; ; q a ; p a =
+
where :, ;=1, ..., R. If we let the indexes : and ; run from 0 to R we can write both equations as
\
H$: #P : +H : t ; ; q a ; p a =
r S =0. q a
+
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(24)
BEREZINIAN H-J SINGULAR FORMALISM
81
From the above definition and Eq. (20) we have r H$0 r H: : r f a r L r f a =& a &(&1) P(b) P(:) q* + p a +(&1) P(b) q* b, p b q* p b p b p b r H$0 r H: : =(&1) P(b) q* b &(&1) P(b) P(:) q* , p b p b where we came back to :=1, ..., R. Multiplying this equation by dt=dt 0 and (&1) P(b), we have dq b =(&1) P(b)
r H$0 0 H$ dt +(&1) P(b) +P(b) P(:) r : dq :. p b p b
Using t : #q :, letting the index : run again from 0 to R, and considering P (0) =P (t 0 ) =0, we have dq b =(&1) P(b) +P(b) P(:)
r H$: : dt . p b
(25)
Noticing that we have the expressions dq ; =(&1) P(;) +P(;) P(:)
r H$: : dt =(&1) P(;) +P(;) P(:) $ :; dt : #dt ; p ;
identically satisfied for :, ;=0, 1, ..., R, we can write Eq. (25) as dq i =(&1) P(i) +P(i) P(:)
r H$: : dt , p i
i=0, 1, ..., N.
(26)
If we consider that we have a solution S(q j, t) of the HJPDE given by Eq. (24) then, differentiating that equation with respect to q i, we obtain r H$: 2r S r H$: r H$: 2r S + + =0 q i P ; q i t ; p a q i q a
(27)
for :, ;=0, 1, ..., R. From the momenta definitions we can obtain dp i =
2r S 2 S dt ; + ar i dq a, ; i t q q q
i=0, 1, ..., N.
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PIMENTEL, TEIXEIRA, AND TOMAZELLI
Now, contracting Eq. (27) with dt : (from the right), multiplying by (&1) P(i) P(:), and adding the result to Eq. (28), we get dp i +( &1) P(i) P(:) =
_
2r S r H$: : dq a &(&1) P(i) P(:) (&1) (P(:) +P(a) )(P(i) +P(a) ) (&1) P(i) P(a) dt q a q i p a
\ S + dt &(&1) t q \ 2 r ;
;
i
dp i +(&1) P(i) P(:) =
r H$: : dt q i
_
P(i) P(:)
(&1) (P(:) +P(;) )(P(i) +P(;)) (&1) P(i) P(;)
+ H$ dt +& , P r
:
:
;
r H$: : dt q i
2r S r H$: : dq a &(&1) P(:) P(a) +P(a) dt q a q i p a
\ S + dt &(&1) t q \ 2 r ;
;
i
P(:) P(;) +P(;)
+ H$ dt +& , P r
:
:
(29)
;
where we used the fact that 2r S 2r S P(i) P( j ) j i =(&1) q q q i q j and that we have the parities P
\
2r S =P (i) +P ( j ) , q i q j
+
P
r H$:
\ p + =P
(:)
+P ( j ) .
j
If the total differential equation given by Eq. (26) applies, the above equation becomes dp i =&(&1) P(i) P(:)
r H$: : dt , q i
i=0, 1, ..., N.
(30)
Making Z#S(t : ; q a ) and using the momenta definitions, together with Eq. (26) we have dZ=
r S ; r S a dt + a dq , t ; q
\
dZ=&H ; dt ; + p a (&1) P(a) +P(a) P(:)
r H$: : dt . p a
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+
BEREZINIAN H-J SINGULAR FORMALISM
83
With a little change of indexes we get
\
dZ= &H ; +(&1) P(a) +P(a) P(;) p a
r H$; dt ;. p a
+
(31)
This equation, together with Eq. (26) and Eq. (30), is the total differential equation for the characteristic curves of the HJPDE given by Eq. (24) and, if Eqs. (26), (30), and (31) form a completely integrable set, their simultaneous solutions determine S(t : ; q a ) uniquely from the initial conditions. Besides that, Eq. (26) and Eq. (30) are the equations of motion of the system written as total differential equations. It is important to observe that, in the nonsingular case, we have only H$0 {0 and no others H$: , so that these equations of motion will reduce naturally to the usual expressions given by Eq. (4).
5. INTEGRABILITY CONDITIONS The analysis of integrability conditions of the total differential equations (26), (30), and (31) can be carried out using standard techniques. This has already been made [2, 13, 14] for systems with the usual variables, and here we will present an analysis of the integrability conditions for Berezinian singular systems. To a given set of total differential equations, dq i =4 : i(t ;, q j ) dt :
(32)
(i, j=0, 1, ..., N; and :, ;=0, 1, ..., R
r f i 4 : =0, q i
(33)
where X : are linear operators. Given any twice-differentiable solution of the set (33), it should also satisfy the equation [X : , X ; ] f =0,
(34)
[X : , X ; ] f =(X : X ; &(&1) P(:) P(;) X ; X : ) f
(35)
where
is the Berezin bracket among the operators X : . This implies that we should have [X : , X ; ] f =C :; # X # f.
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(36)
84
PIMENTEL, TEIXEIRA, AND TOMAZELLI
So, the commutation relations (35) will give the maximal number of linearly independent equations. Any bracket that results in a expression that cannot be written as Eq. (36) must be written as a new operator X and be joined to the original set (33), having all commutators with the other X's calculated. The process is repeated until all operators X satisfy Eq. (36). If all operators X : satisfy the commutation relations given by Eq. (36) the system of partial differential equations (33) is said to be complete and the corresponding system of total differential equations (32) is integrable if, and only if, the system (33) is complete. Now, we consider the system of differential equations obtained in the previous section. First we shall observe that if the total differential equations (26) and (30) are integrable the solutions of Eq. (31) can be obtained by a quadrature, so we only need to analyze the integrability conditions for the former ones, since the last will be integrable as a consequence. The operators X : corresponding to the system of total differential equations formed by Eq. (26) and Eq. (30) are given by X : f (t ;, q a, p a )=
r H$: r f r H$: r f (&1) P(i) +P(i) P(:) & (&1) P(i) P(:) , i q p i p i q i
X : f (t ;, q a, p a )=
r f l H$: &(&1) P(i) +P(i) P( f ) q i p i _(&1) P(i) P(:) (&1) (P(:) +P(i) )(P( f ) +P(i))
X : f (t ;, q a, p a )=
r H$: l f , q i p i
(37)
r H$: l f r f l H$: &(&1) P(:) P( f ) , i q p i q i p i
X : f (t ;, q a, p a )=[ f, H$: ] B ; where i=0, 1, ..., N; :, ;=0, 1, ..., R; a=R+1, ..., N, and we have used Eq. (32) and Eq. (33) together with the result r A l A =(&1) P(i) (&1) P(i) P(A) . q i q i
(38)
It is important to notice that the Berezin bracket in Eq. (37) is defined in a (2N+2)-dimensional phase space, since we are including q 0 =t as a ``coordinate.'' Now, the integrability condition will be [X : , X ; ] f =(X : X ; &(&1) P(:) P(;) X ; X : ) f =X :[ f, H$; ] B &(&1) P(:) P(;) X ;[ f, H$: ] B =0, [X : , X ; ] f =[[ f, H$; ] B , H$: ] B &(&1)
P(:) P(;)
[[ f, H$: ] B , H$; ] B =0,
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(39)
85
BEREZINIAN H-J SINGULAR FORMALISM
that will reduce to [X : , X ; ] f =&(&1) (P( f )+P(;) ) P(:) [H$: , [ f, H$; ] B ] B &(&1) P(:) P(;) (&1) (P( f ) +P(:)) P(;) +P( f ) P(:) [H$; , [H$: , f ] B ] B =0, [X : , X ; ] f =&(&1) P( f ) P(:) (&1) P(;) P(:) [H$: , [ f, H$; ] B ] B &(&1) P(:) P(;) (&1) P( f ) P(;) +P(:) P(;) +P( f) P(:) [H$; , [H$: , f ] B ] B =0, [X : , X ; ] f =&(&1) P( f ) P(:) [(&1) P(;) P(:) [H$: , [ f, H$; ] B ] B +(&1) P( f ) P(;) [H$; , [H$: , f ] B ] B ]=0, [X : , X ; ] f =&(&1) P( f ) P(:) [&(&1) P( f ) P(:) [ f, [H$; , H$: ] B ] B ], [X : , X ; ] f =[ f, [H$; , H$: ] B ] B =0,
(40)
when using the Jacobi relations for Berezin brackets given by Eq. (9) and the fact that P([A, B] B )=P(A)+P(B).
(41)
So, the integrability condition will be [H$; , H$: ] B =0
\:, ;.
(42)
It is important to notice that the above condition can be shown to be equivalent to the consistency conditions in Dirac's Hamiltonian formalism but, to keep the continuity of the presentation, we will postpone the demonstration of this equivalence to the Appendix. Now, the total differential for any function F(t ;, q a, p a ) can be written as dF=
r F a r F a r F : dq + a dp + : dt , q a p t
dF=
F F r F H$ H$ (&1) P(a) +P(a) P(:) r : dt : & r a (&1) P(a) P(:) r : dt : + r : dt :, q a p a p q a t r F l H$:
\ q p &(&1) F H$ dF= \ q p &(&1)
dF=
a
P(a)
(&1) P(a) P(F)
a
r
l
:
a
a
P(:) P(F)
(43)
l F r H$: r F (&1) P(a) P(:) + : dt :, p a q a t
+
r H$: l F r F + dt :, q a p a t :
+
dF=[F, H$: ] B dt : ;
(44)
where the Berezin bracket above is the one defined in the 2N+2 phase space used in Eq. (37). Using this result we have dH$; =[H$; , H$: ] B dt :
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(45)
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PIMENTEL, TEIXEIRA, AND TOMAZELLI
and, consequently, the integrability condition (42) reduces to dH$: =0
(46)
\:.
If the above conditions are not identically satisfied we will have one of two different cases. First, we may have a new H$=0, which has to satisfy a condition dH$=0 and must be used in all equations. Otherwise, we will have relations among the differentials dt : which also must be used in the remaining equations of the formalism.
6. EXAMPLE As an example we analyze the case of the electromagnetic field coupled to a spinor, whose Hamiltonian formalism was analyzed in Ref. [3, 4]. We will consider the Lagrangian density written as L=& 14 F +& F +& +i # +( + +ieA + ) &m,
(47)
where A + are even variables while and are odd ones. The electromagnetic tensor is defined as F +& = +A & & &A + and we are adopting the Minkowski metric ' +& =diag(+1,&1,&1,&1). 6.1. Hamiltonian Formalism Let us first review Dirac's Hamiltonian formalism. The momenta variables conjugated, respectively, to A + , , and , are p+=
r L =&F 0+ , A4 +
p=
r L =i# 0, 4
(48)
p =
r L =0, 4
(49)
where we must call attention to the necessity of being careful with the spinor indexes. Considering, as usual, as a column vector and as a row vector implies that p will be a row vector while p will be a column vector. From the momenta expressions we have the primary constraints , 1 = p 0 r0, , 2 = p &i# 0,
(50) , 3 = p r0.
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87
BEREZINIAN H-J SINGULAR FORMALISM
The canonical Hamiltonian is given by H C = HC d 3 x= ( p + A4 + +( p ) : (4 ) : +( p ) : (4 ) : &L) d 3x,
|
|
|
|
(52)
H C = HC d 3x= [ 14 F ij F ij & 12 p i p i &A 0( i p i &e# 0) &i# j ( j +ieA j ) +m] d 3x.
(53)
The primary Hamiltonian is
|
H P = (HC +* 1 , 1 +, 2 * 2 +* 3 , 3 ) d 3x,
(54)
where * 1 is an even variable and * 2 , * 3 are odd variables, * 2 being a column vector and * 3 a row vector. The fundamental nonvanishing Berezin brackets (here the brackets are the ones defined by Eq. (5) in the 2N phase space) are [A +(x), p &( y)] B =$ &+ $ 3(x& y),
(55)
3
[(x), p ( y)] B =$ (x& y),
(56)
[( x), p ( y)] B =$ 3(x& y).
(57)
The consistency conditions are [, 1 , H P ] B = i p i &e# 0r0,
(58)
[, 2 , H P ] B =&i( j &ieA j ) # j &e# 0A 0 &m +i* 3 # 0 r0,
(59)
[, 3 , H P ] B =&i( j +ieA j ) # j +e# 0 A 0 +m&i# 0* 2 r0.
(60)
The last two will determine * 2 and * 3 , while the first will give rise to the secondary constraint, /= i p i &e# 0r0,
(61)
for which the consistency condition will be identically satisfied with the use of the expressions for * 2 and * 3 given by Eq. (59) and Eq. (60). Taking the Berezin brackets among the constraints we have as nonvanishing results [(, 2 ) : , (, 3 ) ; ] B =&i[(# 0 ) : , ( p ) ; ] B =&i(# 0 ) ;: , [(, 2 ) : , /] B =&e[( p ) : , # 0] B =e(# 0 ) : , 0
0
[(, 3 ) : , /] B =&e[( p ) : , # ] B =&e(# ) : ,
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(62) (63) (64)
88
PIMENTEL, TEIXEIRA, AND TOMAZELLI
where we explicitly wrote the spinor indexes. Obviously the , 1 constraint is first-class, but we have another first-class constraint. This can be seen from the supermatrix 2 formed by the Berezin brackets among the second-class constraints /, , 2 , and , 3 . Numbering the constraints as 8 1 =/, 8 2 =, 2 , and 8 3 =, 3 , we have this supermatrix in normal form (see Ref. [3, Appendix D]) given, with spinor indexes indicated, by [8 1 , 8 1 ] B 2= [(8 2 ) $ , 8 1 ] B
[8 1 , (8 2 ) : ] B [(8 2 ) $ , (8 2 ) : ] B
[8 1 , (8 3 ) ; ] B [(8 2 ) $ , (8 3 ) ; ] B
[(8 3 ) + , 8 1 ] B
[(8 3 ) + , (8 2 ) : ] B
[(8 3 ) + , (8 3 ) ; ] B
2=
\ \
0 0
e(# ) $ &e(# 0) +
&e(# 0 ) :
e(# 0) ;
0 &i(# 0 ) +:
&i(# 0 ) ;$ 0
+
+
,
(65)
.
(66)
This supermatrix has one eingevector with null eingevalue, that is, 1
\ + ie() : &ie( ) ;
,
(67)
so there is another first-class constraint given by .=/+ie(8 2 ) : () : &ie(8 3 ) ; ( ) ; ,
(68)
.= i p i +ie( p +p )
(69)
that we will substitute for /. So, we have the first-class constraints , 1 and ., and the second-class ones , 2 and , 3 . The supermatrix 2 now reduces to the Berezin brackets among the second-class constraints , 2 and , 3 and is given by 2=
\
[(8 2 ) $ , (8 2 ) : ] B [(8 2 ) $ , (8 3 ) ; ] B 0 &i(# 0 ) ;$ = , &i (# 0 ) +: [(8 3 ) + , (8 2 ) : ] B [(8 3 ) + , (8 3 ) ; ] B 0
+ \
+
(70)
having as the inverse, 2 &1 =
\
0 i (# 0 ) $#
i(# 0 ) :* . 0
+
(71)
With these results, the Dirac brackets among any variables F and G are [F(x), G( y)] D =[F(x), G( y)] B &i
|d
3
z([F(x), (8 2 ) :(z)] B (# 0 ) :;
_[(8 3 ) ; (z), G( y)] B +[F(x), (8 3 ) : (z)] B (# 0 ) ;: _[(8 2 ) ; (z), G( y)] B ).
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BEREZINIAN H-J SINGULAR FORMALISM
The nonvanishing fundamental brackets now will be [A +(x), p &( y)] D =$ &+ $ 3(x& y), 0
(73)
3
[() * (x), ( ) : ( y)] D =&i(# ) *: $ (x& y),
(74)
[() * (x), ( p ) : ( y)] D =$ *: $ 3(x& y).
(75)
Now we can make the second-class constraints as strong equalities and write the equations of motion in terms of the Dirac brackets and the extended Hamiltonian given by
|
|
H E = HE d 3x= (HC +* 1 , 1 +:.) d 3x.
(76)
We must remember that, when making , 2 =, 3 #0 the constraint . becomes identical to the original secondary constraint /. Then, the equations of motion will be A4 i r[A i, H E ] D =&p i + i A 0 & i:,
(77)
A4 0 r[A 0, H E ] D =* 1 ,
(78)
p* i r[ p i, H E ] D = j F ji &e# i ,
(79)
4 r[, H E ] D =&ieA 0 &# 0# j ( j +ieA j ) &im# 0,
(80)
4 r[, H E ] D =ieA 0 &( j &ieA j ) # # +im# .
(81)
j 0
0
Multiplying Eq. (80) from the left by i# 0 we get i# 04 =eA 0 # 0&i# j ( j +ieA j ) +m,
(82)
i( + +ieA + ) # +&m=0,
(83)
while multiplying Eq. (81) from the right by i# 0 we get i4 # 0 =&e# 0A 0 &i( j +ieA j ) # j &m,
(84)
i# +( 0 + &ieA + )+m =0.
(85)
These are the equations of motions with full gauge freedom. It can be seen, from Eq. (78), that A 0 is an arbitrary (gauge dependent) variable since its time derivative is arbitrary. Besides that, Eq. (77) shows the gauge dependence of A i and, taking the curl of its vector form, leads to the known Maxwell equation, A9 B9 =&E9 &{9 (A 0 &:) O =&{9 _E9 . t t
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(86)
90
PIMENTEL, TEIXEIRA, AND TOMAZELLI
Writing j + =e# + we get, from Eq. (79), the inhomogeneous Maxwell equation E9 ={9 _B9 &} , t
(87)
while the other inhomogeneous equation {9 } E9 = j 0,
(88)
follows from the secondary constraint (61). Expressions (83) and (85) are the known Dirac's equations for the spinor fields and . 6.2. HamiltonJacobi Formalism Now we apply the formalism presented in the previous sections. From the momenta definition we have the ``Hamiltonians''
|
(89)
|
(90)
|
(91)
|
(92)
H$0 = (P 0 +HC ) d 3x, H$1 = p 0 d 3x, H$2 = ( p &i# 0 ) d 3x, H$3 = p d 3x,
which are associated, respectively, to t=t 0 (remember that P 0 is the momentum conjugated to t), A 0 , , and . The first two H$ are even variables, while the last two are odd. Then, using Eq. (26), we have dA i =
$H$0 dt=(&p i + iA 0 ) dt. $p i
(93)
From Eq. (30) we have dp i =&
$H$0 dt=( j F ji &e# i) dt, $A i
(94)
dp 0 =&
$H$0 dt=( j p j &e# 0) dt, $A 0
(95)
dp =&
$ r H$0 dt=(&i j # j &e# +A + &m ) dt, $
(96)
dp =&
$ r H$0 $
dt+
$ r H$2 $
d=(&i# j j +eA + # ++m) dt&i# 0 d.
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(97)
BEREZINIAN H-J SINGULAR FORMALISM
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The integrability conditions require dH$=0, which implies for H$1 dH$1 =d
|p
0
d 3x=0 O dp 0 = j p j &e# 0=0,
(98)
where we made use of Eq. (95). This expression is equivalent to the secondary constraint (61) and has to satisfy
|
H$4 = ( j p j &e# 0) d 3x,
dH$4 =0,
(99)
which is indeed identically satisfied. For H$2 we have dH$2 =d
| (p
&i# 0 ) d 3x=0 O dp =i(d ) # 0,
(100)
which cannot be written as an expression like H$=0 due to the presence of two differentials (dp and d ) but, substituting in Eq. (96), we get i(d ) # 0 =(&i j # j &e# +A + &m ) dt;
(101)
&i4 # 0 &i j # j &e# +A + &m =0,
(102)
i.e. +
i# ( 0 + &ieA + )+m =0.
(103)
For H$3 we have
|
dH$3 =d p d 3x=0 O dp =0,
(104)
that, similarly to the case above, can be used in Eq. (97), giving (&i# j j +eA + # ++m) dt&i# 0 d=0, +
i( + +ieA + ) # &m=0.
(105) (106)
Finally, we can verify, using the above results, that dH$0 =0 is identically satisfied. Equations (103) and (106) are identical, respectively, to Eqs. (85) and (83) obtained in Hamiltonian formalism. Besides that, Eqs. (94) and (79) are equivalent, while Eq. (93) corresponds to Eq. (77), except for an arbitrary gauge factor.
9. CONCLUSIONS In this work we presented a formal generalization of the HamiltonJacobi formalism for singular systems with Berezinian variables, obtaining the equations of
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PIMENTEL, TEIXEIRA, AND TOMAZELLI
motion as total differential equations (26) and (30). In these equations, the coordinates q : =t : (:=1, ..., R), whose momenta are constrained, play the role of evolution parameters of the system, together with the time parameter t=t 0. So, the system's evolution is described by contact transformations generated by the ``Hamiltonians'' H$: and parametrized by t : (with :=0, 1, ..., R), where H$0 is related to the canonical Hamiltonian by Eq. (22) and the other H$: (:=1, ..., R) are the constraints given by Eq. (23). This evolution is considered as being always restricted to the constraint surface in phase space. There is no complete phase space treatment that is later reduced to the constraint surface, as in Dirac's formalism with the use of weak equalities. We should observe that, in the case of systems composed exclusively by even variables, all parities are equal to zero and Eqs. (26), (30), (31) reduce to the results obtained in Ref. [1]. Furthermore, if the system is nonsingular, we have H$: #0, except for :=0, so the total differential equations (26) and (30) will be reduced to the expressions given by Eq. (4). The integrability conditions (which relation to the consistency conditions in Dirac's formalism is discussed in the Appendix) were shown to be equivalent to the necessity of the vanishing of the variation of each H$: (:=0, ..., R); i.e., dH$: =0. The example presented was chosen for its completeness; it is a singular system with even and odd variables and its Hamiltonian treatment contains all kinds of constraints (primary and secondary, first- and second-class ones). This example is very illustrative, since it allows a comparison between all features of Dirac and HamiltonJacobi formalisms. For example, the fact that the integrability conditions dH$2 =0 and dH$3 =0 give expressions involving some differentials dt : is related to the fact that the corresponding Hamiltonian constraints , 2 and , 3 are secondclass constraints and determine some of the arbitrary parameters in the primary Hamiltonian (54). Similarly, the fact that the condition dH$1 =0 generated an expression like H$4 =0 is related to the fact that the corresponding Hamiltonian constraint , 1 is a first class one (see the Appendix). Finally, we must call attention to the presence of arbitrary variables in some of the Hamiltonian equations of motion due to the fact that we have gauge-dependent variables and we have not made any gauge fixing. This does not occur in Hamilton Jacobi formalism since it provides a gauge-independent description of the system's evolution, due to the fact that the HamiltonJacobi function S contains all the solutions that are related by gauge transformations.
APPENDIX: EQUIVALENCE AMONG CONSISTENCY AND INTEGRABILITY CONDITIONS In this appendix we will show the equivalence among the integrability conditions of the formalism shown above and the consistency conditions in Dirac's Hamiltonian formalism, in a way similar to what was made for the usual variables [16]. In the
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BEREZINIAN H-J SINGULAR FORMALISM
93
notation used in this paper the Dirac primary constraints are written, from Eq. (23), as H$: #P : +H :(q i; p a )r0,
(107)
where :=1, ..., R; i=1, ..., N. The canonical Hamiltonian is given by H 0 in Eq. (20), so the primary Hamiltonian H P is H P #H 0 +H$: v :,
(108)
where the v : are unknown coefficients related to the undetermined velocities q* : [4]. The ordering of the v : with respect to the H$: is a matter of choice, since it will simply produce a change of sign, but the natural procedure that identifies v : and q* : suggests the ordering above as a consequence of the ordering adopted in the Hamiltonian (3). This ordering is also the most natural choice to our purpose but is, of course, irrelevant for systems containing only the usual variables. The consistency conditions, which demand that the constraints are preserved by time evolution, are written as H4 $+ r[H$+ , H P ] B =[H$+ , H 0 ] B +[H$+ , H$: ] B q* : =0,
(109)
where :, +=1, ..., R and the Berezin brackets here are those given in Eq. (5) defined in the usual 2N-dimensional phase space and we have made the explicit identification q* : #v :. Multipling the above equation by dt we get dH$+ r[H$+ , H 0 ] B dt+[H$+ , H$: ] B dt : =0,
(110)
where, as before, q : =t :, but we are still making :=1, ..., R. At this point we can already see that, when Dirac's consistency conditions are satisfied we have dH$+ =0 satisfied. We must see now that we have dH$0 =0 when the Dirac consistency conditions are satisfied. The Hamiltonian equation of motion for H$0 is H4 $0 r[H$0 , H P ] B =[H$0 , H 0 ] B +[H$0 , H$: ] B q* :,
(111)
which multiplied by dt becomes dH$0 r[H$0 , H 0 ] B dt+[H$0 , H$: ] B dt :.
(112)
Remembering that the ``momentum'' P 0 in H$0 is independent of the canonical variables q i and p i , we have dH$0 r[H 0 , H 0 ] B dt+[H 0 , H$: ] B dt : =[H 0 , H$: ] B dt :.
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(113)
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PIMENTEL, TEIXEIRA, AND TOMAZELLI
But, if Dirac's consistency conditions are satisfied, we must have only primary first-class constraints, otherwise we would have conditions imposed on the unknown velocities q* :. So, the preservation of constraints in time will reduce to [H$+ , H 0 ] B r0,
(114)
and the right side of Eq. (113) will be zero. This is simply a consequence of the fact that, once all Dirac's conditions are satisfied, the Hamiltonian is preserved. So the condition dH$0 =0 is satisfied when the Dirac consistency conditions are satisfied. This shows that the integrability conditions in HamiltonJacobi formulation will be satisfied when Dirac's consistency conditions are satisfied. Similarly, we can consider that we have the integrability conditions satisfied so that dH$0 =dH$+ =0 and then Eq. (110), which is equivalent to Eq. (109), implies that Dirac's conditions are satisfied. So, both conditions are equivalent. Now, we will consider that these conditions are not initially satisfied. When we have only first class constraints in Hamiltonian formalism we will simply get a new constraint from some of the conditions (109). From Eq. (110) we see that this will imply an expression like dH$+ =H$dt (H$r0 is the secondary Hamiltonian constraint) which means that there will be a new H$ in HamiltonJacobi formalism that has to satisfy dH$=0. If we have some second-class Hamiltonian constraints the consistency conditions (109) will imply a condition over some of the velocities q* :. From Eq. (110) we see that, in the HamiltonJacobi approach there will be conditions imposed on some differentials dt :. Such correspondence among the formalisms can be clearly seen in the example presented in this paper. Besides that, Eq. (110) and Eq. (112) can be written as dH$+ r[H$+ , H$: ] B dt :,
(115)
where now :, +=0, 1, ..., R and the Berezin bracket is again defined in the (2N+2)dimensional phase space containing t 0 and P 0 . This equation is obviously identical to Eq. (45) that leads to the integrability condition dH$+ =0, and its right- hand side was shown to correspond to Dirac's consistency conditions. Consequently, this expression shows directly the relation among consistency and integrability conditions. It is important to notice that here we are not considering any explicit dependence on time, neither of the constraints nor of the canonical Hamiltonian, because it is a usual procedure in the Hamiltonian approach. But the equations of Hamilton Jacobi formalism were obtained without considering this condition and, consequently, remain valid if we consider systems with Lagrangians that are explicitly timedependent. But the Hamiltonian approach is also applicable to such systems (see Ref. [3, p. 229]) and in this case we can follow a procedure similar to the one shown here and demonstrate the correspondence between Dirac's consistency conditions and integrability conditions.
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BEREZINIAN H-J SINGULAR FORMALISM
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Finally, some words about the simpletic structure. Using Eq. (38), we can write Eq. (26) and Eq. (30) in terms of left derivatives as dq i =
l H$: : dt , p i
dp i =&(&1) P(i)
(116) l H$: : dt , q i
(117)
where, as before, i=0, 1, ..., N ; :=0, 1, ..., R. These expressions can be compactly written as d' I =E IJ
l H$: : dt , ' J
(118)
where we used the notation ' 2i = p i O ' I =(q i, p i ),
' 1i =q i, IJ
i j
` 1
2 _
E =$ ($ $ &(&1)
P(i)
1 _
I=(`=1, 2; i=0, 1, ..., N ),
(119)
I=(`; i),
(120)
` 2
$ $ ),
J=(_; j ),
which was introduced in page 76 of Ref. [3]. The Berezin brackets defined in Eq. (5) can be written as [F, G] B =
r F IJ l F E . ' I ' J
(121)
This simpletic notation allows us to obtain the expression for the total differential for any function F(t ;, q a, p a ) in a more direct way. Using it in Eq. (43) we get r F I d' , ' I
(122)
r F IJ l H$: : E dt =[F, H$: ] B dt :, ' I ' J
(123)
dF= where the use of Eq. (118) gives dF= in agreement with Eq. (44).
ACKNOWLEDGMENTS B.M.P. is partially supported by CNPq and R.G.T. is supported by CAPES.
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REFERENCES 1. Y. Guler, Il Nuovo Cimento B 107 (1992), 1389. 2. Y. Guler, Il Nuovo Cimento B 107 (1992), 1143. 3. D. M. Gitman and I. V. Tyutin, ``Quantization of Fields with Constraints,'' Springer-Verlag, New YorkBerlin, 1990. 4. K. Sundermeyer, ``Lecture Notes in Physics 169Constrained Dynamics,'' Springer-Verlag, New YorkBerlin, 1982. 5. A. Hanson, T. Regge, and C. Teitelboim, ``Constrained Hamiltonian Systems,'' Accad. Naz. Lincei, Rome, 1976. 6. B. M. Pimentel and R. G. Teixeira, Il Nuovo Cimento B 111 (1996), 841. 7. B. M. Pimentel and R. G. Teixeira, Preprint hep-th9704088, Il Nuovo Cimento B, to appear. 8. Y. Guler, Il Nuovo Cimento B 109 (1994), 341. 9. Y. Guler, Il Nuovo Cimento B 111 (1996), 513. 10. F. A. Berezin, ``Introduction to Superanalysis,'' Reidel, Dordrecht, 1987. 11. C. Caratheodory, ``Calculus of Variations and Partial Differential Equations of the First Order,'' Part II, p. 205, HoldenDay, Oakland, CA, 1967. 12. E. C. G. Sudarshan and N. Mukunda, ``Classical Dynamics: A Modern Perspective,'' Wiley, New York, 1974. 13. E. M. Rabei and Y. Guler, Phys. Rev. A 46 (1992), 3513. 14. Y. Guler, Il Nuovo Cimento B 110 (1995), 307. 15. E. T. Whittaker, ``A Treatise on the Analytical Dynamics of Particles and Rigid Bodies,'' 4 th ed., p. 52, Dover, New York, 1944. 16. E. M. Rabei, Hadronic J. 19 (1996), 597.
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