Local symmetries and phase space dynamics of finite-dimensional systems

ANNALS

OF PHYSICS

152, 418-450

(1984)

Local Symmetries and Phase Space Dynamics of Finite-Dimensional Systems KURT Freie

UnicersitCt Berlin, Amitnallee

SUNDERMEYER

Institut fir Theorie der Elementarteilchen. 14, 1000 Berlin 33. German)

Received

May

20, 1983

A method to implement reparametrization covariance and additional local symmetries for finite-dimensional systems in a phase space is presented. The approach is a physically and group-theoretically motivated alternative to the Dirac-Bergman algorithm for theories with Hamiltonian constraints. The method is applied to bosonic and fermionic systems with both first and second class constaints.

1. INTRODUCTION Lagrangians for theories with local symmetries are necessarily singular due to a theorem by Noether and Bessel-Hagen. For the same reason do these theories belong to the class of constrained systems. One usually employs the Dirac-Bergmann algorithm in order to arrive at a consistent phase space formulation [l-3]. At the end are the local symmetries hidden in so-called first class constraints. In two publications [4, 51 we presented a method aimed for realizing local symmetries directly in a phase space. The starting point is not a Lagrangian but instead the symmetries themselves. The idea is to map a subgroup of the local symmetry group to a symplectic group, that is to map certain local transformations to canonical transformations. The program has been explained in 141 for generally covariant theories, exemplified on a two-dimensional theory with scalar fields (under some assumptions leading to the relativistic string), and applied to a four-dimensional theory with rank-2 tensor fields (uniquely leading to the Hamiltonian form of Einstein’s theory of gravitation). In 151 the method was extended to systems possessing besides coordinate symmetries a Yang-Mills gauge symmetry. Although we formulated and applied our program so far only to field theories, it can be used also for finite-dimensional theories (theories with finitely many degrees of freedom). Of course many features arising in the treatment of field theories get lost for finite-dimensional theories (for instance the Poisson bracket algebra of the superHamiltonian and supermomentum constraints). Nevertheless do we find it worthwile to explain the method again for finite-dimensional systems since up to now it only 418 0003-49161’84 Copyright All rights

$7.50

t 1984 by Academic Press. Inc. of reproduction in any form reserved.

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has been applied to theories with bosonic fields and first class constraints. We want to demonstrate on finite-dimensional systems how it can be enlarged to fermionic and/or second class systems. There is another point which we would like to discuss more carefully: The canonical analysis of the Nambu-Goto action for the relativistic string and the Einstein-Hilbert action for gravitation differ in one important point. In the former one only primary constraints arise (which are first class). whereas in the latter case one obtains both primary and secondary first class constraints. Primary constraints originate from the definition of momenta from a Lagrangian, secondary ones may come out because of consistency (time conservation of primary constraints). In our approach, no Lagrangian being present, we introduce “pseudo-momenta.” These are D-invariant functions, whose explicit form is restricted by internal consistency of the program. The difference in the Hamiltonian analysis of the Nambu-Goto and the Einstein-Hilbert theory is reflected in that for the former the pseudo-momenta are functionally dependent, whereas in the latter they are independent. In this article is this difference emphasized on finite-dimensional theories (referred to as “theories with/without p-constraints”). The most simple physical example for “constrained Hamiltonian dynamics” is the relativistic particle. If formulated with reparametrization scalars (what correspondsto the square-root action) p-constraints are present. No p-constraints must occur in the first-order (or einbein) formulation. Since this case is easier to deal with in our approach, we demonstrate how the program works on reparametrization covariant scalars and scalar densities in Section 2. For theories without p-constraints the only restrictions on the functional form of the pseudo-momenta are due to integrability conditions of first order differential equations. For theories with p-constraints (Section 3) there are “fewer” differential equations compensatedby “more” conditions on the pseudo-momenta.Here it also may happen that for consistency second class constraints must be present although they are completely unrelated to local symmetries. Systems with scalars are treated in Section 4. We demonstrate how, by extra physically motivated requirements out of all dynamical systems allowed by consistent choices of pseudo-momenta, specific ones are selected. By triggering on these requirements one may for instance derive either nonrelativistic Lagrangian/ Hamiltonian dynamics, the free relativistic particle, or the interaction of a relativistic particle with external scalar, vector and tensor fields. A further example is a choice of pseudo-momenta leading to the dynamics of a relativistic particle in an external Yang-Mills field. We discussedin some length in Ref. [5 ] on the example of the Einstein-YangMills theory that the additional local gauge symmetry cannot be realized in a phasespace separately from the coordinate symmetry. Instead only a mixed symmetry is implementable. The central role of a mixed symmetry is also observed in the Einstein-Dirac theory [6]. This mixing phenomenon can already be studied for finitedimensional theories. In Section 5 it shall be explained on an example which is both reparametrization and scale covariant. The presence of two symmetries leads to a

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Poisson bracket algebra among constraints, giving essential information on possible consistent choices for the pseudo-momenta. Finally in Section 6 we investigate finite-dimensional theories which besides reparametrization covariant bosonic scalars and scalar densities also contain fermionic quantities, and which are invariant under an additional local supertransformation. For one choice of pseudo-momenta do we arrive at the phase-space formulation of the massless spinning relativistic particle. This section may be considered an exercise ground for the extension of our program to locally supersymmetric field theories. In the concluding remarks (Section 7) we give an outlook to the treatment of these systems together with any other attempts in the arena of “Local Symmetries and Dynamics.”

2. THE IDEA OF THE LOCAL SYMMETRY

REALIZATION

PROGRAM

Consider an example with ‘Yields” q”(r) (a = O,..., N- 1) and e(r) which are scalars and a scalar density, respectively, under reparametrizations (general coordinate transformations in one dimension). That is, under infinitesimal transformations r’=r+

Y[r,q.e]

(2.1)

the q” and e transform as (2.2) where 8Q f Q’(r) - Q( t ), and Q = de/&. The parameter r shall be called time. As indicated in (2.1) the descriptors Y depend functionally on the (q”, e). The transformations (2.2) form a group since the commutator of two transformations Ji (with descriptors !?i) turns out to be --(S,6,-6,6,)=6;[~~=(~~w,-~,~~)-(lH2)1. (2.3) Our goal is to realize “as much” of this local symmetry group in a phase space as possible, that is to implement a maximal subgroup. As demonstrated below realization means that the commutator (2.3) is mapped onto the Poisson bracket of canonical transformations. One observes that the commutator in general contains time derivatives of descriptors and therefore refers to different times, whereas a Poisson bracket refers to one instant of time. Demanding that in (2.3) no time derivatives of descriptors occur leads to a restriction on their functional dependence. We shall impose a further restriction in that the descriptors are allowed to depend on at most first-order time derivatives of the qa, e. As shall be obvious below this restriction is enforced by the quest for second-order equations of motion. The maximal subgroup implementable in a phase-spaceis characterized by descriptors Y(t, q, 4, e, i) such that the commutator of two transformations does not contain time

LOCAL

derivatives of descriptors. the ansatz

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421

The most general dependence of !P may be obtained from

Yi = n(q, 4, e, 6)
(2.4)

and by translating the conditions on Yi into conditions on n and c$. Then

and with

the previous expression becomes

In order to guarantee that the commutator (2.3) does not contain time derivatives of descriptors, IZ cannot depend on P and it must obey F-3) Furthermore c$&must not contain time derivatives of ‘u, . This leads to the concept of D-invariants \7,8]: A quantity A is called D-invariant if &4 does not contain time derivatives of a descriptor. Demanding that 6; are D-invariant meansthat they depend on those independent D-invariants which can be built from(q”. 4”, e, P). From Eq. (2.2) one observes that the q” are D-invariants (called zero-order D-invariants), however the 4” are not since

Since both 4” and e transform as scalar densities, their quotient p

1 =-Q4” e

(2.6)

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transforms like a scalar and is therefore D-invariant. In fact are the q” and rrr the only independent D-invariants constructable out of (q, 4, e. 6): Any function f(q, 4, e, P) can be written asf(q, r, e, P), and

(here - meansmodulo terms containing no time derivatives of the descriptor). Hence 7 is D-invariant if both L$/&! = 0 and af/ae = 0. The most general solution to (2.5) is easily written down since it is a homogeneity relation for ?z.It turns out that the specific solution one choosesis irrelevant for the final result. Namely, if n and n’ are two different solutions one finds from

that if < is D-invariant so will be r’ = (en/en’)l since both en and en’ are D-invariant. A convenient solution to work with is n = l/e. Then the most general dependenceof the descriptors !P is given by Ul(r, q”, Q”, e, 6) = f

t (i. qn,rn = -$).

In terms of r the variations of qm and P are 8q4”= - r”&

(2.7)

&.a = - L p< e

(2.7’)

(s, 6, - 6, s,) q” = - ryi& <, - 6, Cz).

(2.8)

and the commutator is

We next introduce functions P, = %Aq, r)

(2.9)

called “pseudo-momenta.” The further program depends on the rank of the matrix M,, = au, /dr4. The pseudo-momenta are functionally independent if det M # 0. However if M,, is singular, there exist relations of the form @,(p, q) = 0. These relations shall be called p-constraints. In this section we want to investigate theories without p-constraints. For these is it possible to solve (at least locally) the pseudomomenta as r” = r”(p, q) such that the descriptors ((4. r) may be considered as functions of q” and p, : &q,p) = t(q, r(p, q)). N ow everything is prepared to be mapped on a phase space.

LOCAL

Let there be a symplectic The generator

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manifold

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DYNAMICS

(phase-space)

with local coordinates

cl<1 = GQ, PI .#(Q, P)

423 Qe, P, (2.10)

(5 being infinitesimal) generates infinitesimal canonical transformations with commutators

where ( , } is the Poisson bracket in the phase-space.Implementation of the transformations (2.7) with commutators (2.8) in a phase-spaceis now simply achieved by identifying the pairs (q,p) with (Q. P) and the descriptor [(p, q) with ?(P, Q), that is, we demand

This condition may be considered the soul of our approach! The explicit form of (2.11) and (2.12) is {<*,&)2P+

Ir,{.~,r,}-r,{.lyl,r,}].~~(~~z,-8,~~),~.

The first term on the left-hand side contains products (af,/aQR) 1(at,/aP,), etc., whereas similar products do not appear on the right-hand side. From this we conclude that .d vanishes; it is a constraint in phase space (in Dirac’s terminology .#(Q, P) z 0). The other terms to be compared are

or (dropping the indices on f) -8.R 5-

=

!

-Jqs”

=

-$Q”,

@cl

(2.14) Subsequently the (q,p) and (Q, P) (and the r^and t) are no longer distinguished since they are identified by a mapping. Conceptually however are they different. The explicit form of (2.13) is due to (2.7) (2.13’)

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and from

we obtain for (2.14) (2.14’) These equations containing time derivatives of momenta are nothing but half of Hamilton’s equations of motion. Observe that there was no Lagrangian to begin with. Therefore no momenta can be defined, and no way whatsoever towards a Hamiltonian exists. Instead we identified the “fields” q” and the pseudo-momenta p, with coordinates (Qa. P,) in an a priori given symplectic manifold. But after identifying the descriptors of infinitesimal coordinate transformations with descriptors of symplectic transformations we do indeed have a Hamiltonian at our disposal. It is by definition just that generator which translates in time. By (2.1) time translations are characterized by Y= c = const., that is, by [= ce = <, and therefore the Hamiltonian is H=e.$.

(2.15)

From (2.14’) one finds i3H -= w

3.9 eF=e

and Eqs. (2.13’) are compatible with the other half of Hamilton’s equations

We know how H and .R are related, but still is the constraint .R not known Equations (2.13’) however contain genuine information on this constraint. Since the right-hand side can be expressedin terms of q” and p,, Eqs. (2.13’) constitute a set of differential equations for 9:

a.9 - = rrr(4,p). ap, These are integrable if (3r”/apD = ar”/ap, that is, if the pseudo-momentaare chosen such that M,, = au,/&-” is symmetric. We conclude: Every choice of pseudo-

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momenta for which M,, is symmetric and nonsingular leads to a Hamiltonian system with H = e9, where 9 is obtained from integrating Eqs. (2.13 “). The constraint .9 generates the originally envisaged local symmetries for the D-invariants. In order to deal with a specific example a set of physical restrictions shall be imposed: EXAMPLE

1.

R,: The q” and p, are Lorentz four-vectors. R,:

q” has dimension length, p, dimension mass.

R,: Only one-dimensional

parameter (mass) is present.

R,: The theory is translational

invariant.

With these restrictions is p, = tyaDrb (r~,~ being the Minkowski metric) the only possible choice for the pseudo-momenta. Integration of (2.13”) yields R = 4 p, p” + const. The constant may be expressed in terms of a mass, such that the Hamiltonian becomes H=fe(p2 An inverse Legendre transformation

-m2).

(2.16)

leads to the Lagrangian (2.17)

which is the Lagrangian for the relativistic particle in first-order (or einbein) formulation. Let us compare the Hamiltonian (2.16) with the one resulting from the canonical analysis. The Dirac algorithm of the Lagrangian (2.17) yields a primary constraint pe ;- BL/BP z 0 and a secondary contraint p2 - m2 z 0. The Dirac Hamiltonian is a linear combination of these constraints ND = z&J* - m’) + up,, where u and v are multipliers. Their arbitrariness reflects the “gauge” symmetry of this system. Both constraints are first class and generate the symmetry (2.2). The Hamiltonian H (2.16) may be obtained from H, by reduction, that is, a gauge fixing for the constraint p, z 0. The other gauge degrees of freedom are still present in H since the multiplier Se in front of the constraint p2 - m* z 0 is not fixed by any dynamical equation. This is related to the fact that time translations, which the Hamiltonian is supposed to generate, are not representable as canonical transformations. (A constant !P is not of the form !P = (l/e) ((r, q, r).)

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3. THEORIES

WITH P-CONSTRAINTS

p-constraints (PC’s) are present if det M,, = det 1au, /iir, 1= 0. In this case is it no longer possible to express the first-order D-invariants rn in terms of (q”,p,) alone. Instead if M is the rank of M,,, only A4 of the ra can locally be expressedby the q’s, M of the pseudo-momentaand N - M of the other r’s: A = l,..., M, I = M + 1)..., N.

+ = p4(PB7r’, q),

(3.1)

Here we assumedthat the index set on the pseudo-momentais the sameas the one on the “solved-for” r’s. This is no loss of generality since due to the arbitrariness of n in Y= nc one may switch from the pairs (p, r) to other pairs (6. Q where r^^ are N independent first-order D-invariants constructed from (r, q). By a suitable choice of the variables r^ one may always arrange that (3.1) holds. Substituting the pA in p, = ua(rr q) one obtains p, = u,(q, rd. r’) = u,@“(pHr r’. 41, r’. 9) Y U,(p$. r’. 9).

(3.2)

The U, can no longer depend on r’ since p, = u,(r. q) has been solved for the maximal number of r’s. Especially p,c = u,~, and P, = MP., 9s>

@,7p, - U,(p,.,, q) = 0.

or

(3.3)

The @,(p, q) are the N - M p-constraints. Define the descriptor &7, p) = &7, PA(9, PR3IJh r’) + E’(q, p, r-7 @,b 4h where the multipliers s’ in front of the PC’s are chosen such that the right-hand side does not depend on the r’ any longer. The “realization of local symmetries in a phase-space” program is now similar to the one for theories without p-constraints. Everything written in the previous section in the paragraphs containing Eqs. (2.10)(2.15) indeed holds true. Especially 9 is a constraint in phase-space.However no longer are all of Eqs. (2.13’) genuine differential equations for .,R. This does not mean that for theories with PC’s the constraint .Y? cannot be determined. Let us separate (2.13’) a.R

- pA(pB, r’, 91, 3P.4

a.9 -= 3P,

r’,

(3.5)

Since .R does not depend on the r’ one may consider these to be arbitrary. A solution to the set of Eqs. (3.5) is then .w = r’(P, - uI(P.4’4)) + aPA 94)

(3.6)

LOCAL

since the factors multiplying fulfilled we demand

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the J are constraints.

427

In order that Eqs. (3.4) are

-

-r’

3

+ g A

= p” (pB ,

rTt

q),

A

which only makes sense if p* is of the form P”(PB,IJ,q)=P;I(PB,q)P+~*(PB,q)

(3.7)

--=au,

a@,

ap,4

-ig=p:.

(3.8

such that

a.3 -- -p”. ap,

(3.9 1

The form (3.7) for p* is a restriction on the choice of pseudo-momenta since not every set of functions p, = u,(q, r) leads to p.’ at most linear in the Y’. Equations (3.9) are differential equations for .2, integrable if ;ip’“/ap, = ap8/ap,4. Equations (3.8) are a further restriction on the pseudo-momenta since they imply apf lap, = apF/ap,. There is another argument leading to (3.6). Define a function R depending on (4, r> by

This function

R (q, r) = #(q, u(q, r)).

(3.10)

aR -a~au,Irb%u, at-=' ar* ap, ai-=

(3.11)

must obey

Integrability demands that &,/W = au,,/&-“. Assume this condition to be fulfilled. Then Eqs. (3.11) determine R up to a function of q’s. The original function .$ is obtained from R only up to PC’s, that is, .a(q,p> = R(q,p.4Co,, r’, 4). #I + A’@,. In going back to the equations for ,Y? one finds i,’ = r’, and comparing this with (3.6) one concludes that R = -2. This equality is only possible if R does no longer depend on the r’. Again is this an implicit restriction on the choice of pseudo-momenta. The Hamiltonian is as before H = e.W, that is, H = er’@, + e.2 = cj’@, + e.2.

(3.12)

But we are not ready yet since further consistency conditions must be checked: Since there are several constraints, namely, @, z 0 and ,g z 0, it is not automatically true

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that they are conserved in time. Let us discuss the cases of one and two PC’s separately, and then return to the general situation. I. One p-Constraint Choose arbitrarily r’ = r”. The Hamiltonian is H = era@, + e.2. The situation becomesvery simple if .? E 0. This is possibleonly if p”’ = 0, implying conditions on the pseudo-momenta.From (3.6) one concludes that thep, can depend only on ratios yA/yo. The Hamiltonian is just H = cj”Qo, and the constraint Q. generates the reparametrizations. If, however, .I # 0, there must exist a condition which intrinsically determines Y’. This condition originates from the quest of time conservation of the constraints Qo, .#. Time conservation means

where r, indicates weak equality on the surface in phase space defined by Q. z 0, .2 z 0. If ( Qo, ./? } z 0 both r” and e would be undetermined, and both Q. and ti would be first class. They would independently generate local symmetry transformations. If we want to realize reparametrizations only, a choice of pseudo-momenta leading to ( QO, .2 } z 0 is not allowed. Instead must the pseudo-momentabe chosen such that

and x is a new constraint, corresponding to a secondary constraint in the BergmannDirac terminology. We demand

where the weak equality now refers to the surface defined by the vanishing of .2, Qo,x. The coefficient r” can only be determined if (x, Qo} Ii-, # 0. The resulting theory is finally characterized by one first-class (FC) constraint (,,9 z 0), and two second class (SC) constraints (Q. =: 0, x z 0). Conclusion: A choice of pseudomomenta is consistent either if (i) .2 = 0 or (ii) ( ao, .2) Ir, =x. (x, Go} I,., # 0. In order to demonstrate that the arguments above indeed restrict the functional form of the pseudo-momentaconsider the following examples. EXAMPLE

2. P.4 =$

P; =p.4

i

/\ p. = u. 7,q r’4 ( ii

$10.

Due to (3.8) is k, = -f~,~p~ + u(q) the only consistent choice for uo.

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3. pA = rA p. = uo(rA, 4) 1

/I

p’i = O p”’ zp,.

The function u0 cannot depend on r4 due to (3.8) Q0 = p0 - u,(q), and .T = 4pA pA + .Y?. We have

u0 = u,(q). The PC is

which vanishes on r, if &/aq* = 0 and a.z/aq’ = 0. So we must impose on no the condition au,/c3qA # 0 and must furthermore demand (x, Qo} /rs # 0. That is.

ix>@Ol=~~+PA a2uo aqA a@ +$f+

must not vanish on I’,. II. Two p-Constraints Without Ioss of generality assumethat ra and Y’ are those D-invariants for which the pseudo-momenta(p,,,p,) cannot be solved. So H = doGo + 4’@, + e.g. If .@ E 0 there are still two constraints. They are conserved in time if @‘{@,, QJ} Ir, = 0. Both constraints would be first class if {Qo, @i ] (r(. = 0, simulating a larger invariance than we started with. Therefore we insist that by choice of pseudo-momenta

for at least one 1. If these conditions are met is the theory characterized by one FC and two SC constraints. If, however, 5%? # 0, we must demand

{~‘J,HJl,r=~‘(~,,~,}l,c+e(~,,.~}l,r,4’~,,-ec,=0.

(3.13)

This system of weak equations does have a solution for the 4’ if d k 0 Ire. There is no need for a secondary constraint in this case. Again is 9 a first class constraint and two independent linear combinations of .G$? and @, are second class. In the presence of K = N - A4 p-constraints is the choice of p-constraints again restricted implicitly by insisting that H (or 9) is the only FC constraint. All other independent constraints linearly built from (@(, 3) must be SC. Two casesare to be distinguished. Case A:

.a E 0. This is already an implicit condition on the pseudo-momenta

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since p, = u,(q, r) must be solvable for M I’S in the form r4 = ,D:r’. Further conditions may follow from (3.13’) This is a (weak) homogeneouslinear system of equations for the 4’. We want it to have only one (nontrivial) fundamental solution. This only exists if &(A,,) = K - 1. Since in this section only bosonic finite-dimensional systems are dealt with is the matrix A, always antisymmetric. Therefore is it singular for an odd number of PC-s. We insist det(A,J) to be zero in such a way that only one linear combination a’@, exists which has zero Poisson brackets with all other p-constraints. This is an implicit condition on the pseudo-momenta. For an even number of PC’s however the condition rk(A,J) = K - 1 cannot be met for any choice of pseudo-momentasince the rank of A,, will always be even. The discussion of the case with two PC’s has shown that this can be cured by secondary constraints. An odd number (
4. THEORIES WITH SCALARS An example with scalars and a scalar density has been treated in Section 1. If instead there are only scalars, p-constraints cannot be avoided. This is becausethe N D-invariants ra = 4”/e are no longer available. Only N - 1 independent first-order D-invariants exist (they shall be called r again). These are ‘Q rQ =-- 4.o ’ 4

LOCAL SYMMETRIES AND DYNAMICS

431

Distinguishing 4” is of course not necessary. Instead one may select any other velocity, or take for instance ra = 4” (Go,44@‘)“’ with G,, # 0. In any case, N - 1 of the r” are independent since Ga4rar4 = 1. Instead of Eq. (2.5), n must obey

As solution one may choose without loss of generality n = l/Q”. Y= (L q”, 4”) = (l/4”) t-(G cl”, r’) the variations of qn and ra are 8qa = -rat,

Then with

r*;_ 1,

where

(4.3)

and - (6,6,-6,6,)qa=-ra(~~z,-g,r2).

The pseudo-momenta p, = u,(q, ra) cannot be independent and obey at least one p-constraint. Realizing the symmetries (4.3) in a phase-space yields the same differential expressions for 9 as in the example of Section 2 (cf. Eqs. (2.13’) and (2.14’)), that is, (4.4) (4.5)

These are compatible with the Hamiltonian H = tj”.%‘. They are however not to be considered as differential equations for 9 in general because of the presence of (a) pconstraint(s). Expressions (4.4) may be analysed essentially according to the arguments in the previous section, with one difference however: The r” in (4.4) is not to be interpreted as an arbitrary phase-space function, it is “one”! This fact allows for an easier discussion of consistency conditions than before. Let us immediately turn to examples. EXAMPLE

4.

For the pseudo-momenta

choose

p, = ra, PO= uo(ral 4).

(4.6)

Independent of the actual form of the PC Q. =po - I,(p,, q) the equations (4.4) can be solved as .9=Po+fPaPo+

0).

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KURTSUNDERMEYER

Since we do not want to have any other FC constraint besides 5%’we do demand that 9 and Q0 be proportional, i.e., ziO= -jp,p, - V(q). The Lagrangian for this theory is calculated to be 1 4”Lj” 2 go

L(q,@=p,4”=u,4*=---

which allows for an interesting interpretation:

4” V(q),

The action

.f’=

V(q)i

becomes by cj”/d’ = dq”/dq’: ,d =

.r

dqO +$$i

v(q*,qO)

1

.

This action describes N- 1 mass points in one dimension moving in a (timedependent) potential V(qa, q’), or for N - 1 = 3K, K particles in three dimensions in the potential V. So the choice of momenta (4.6) simulates homogenized regular nonrelativistic mechanics! Of course, many other choices of pseudo-momenta lead to a consistent Hamiltonian for scalar quantities. In order to select specific ones, we impose the same set of restrictions (R r - R4) on (q,p) as was done in Example 1. EXAMPLE 5. The restriction (R4) excludes functions U, depending on the q’s. From ra one may construct a first-order D-invariant Lorentz vector

The only choice for U, compatible

with (R,) to (R4) is

p, = l&f?) = f?Qse

where m is a constant with dimension mass. The p-constraint is @ =p’ - m2. If one solves Eq. (4.2) by n = (d2)-li2, the equation for .9 corresponding to (4.4) is SF/8p, = r”“. Running through the previous arguments one arrives at H = l(p* - rn’),

A being a multiplier. This is the Hamiltonian If restriction R, is droped, the ansatz

for the free relativistic

Gd? rp

P, = (g&.b)l/z

+ Ax(q)

particle.

LOCAL SYMMETRIES

AND DYNAMICS

433

with G,,(q), g,,(q) symmetric and regular matrices is admissible. It implies the PC @ = g,, GayG”*(py - A ,)(pa - A6) - 1 = 0, where Gay is the inverse to G, (GaYGY5= SF). With F* = ran, n = (ga4rffr4)-“’ Eqs. (4.4) equivalent to a.9 - = G”‘(p,

-A,),

.S? = fGa5(p, -A,)@,

-A&

3P,

are

which upon integration yield + ego(q).

Since we want 9 to be proportional to @, .,A! = cp@ say, the matrix G,, must be proportional to g,, , G,, = (o(q) g,, . So the Hamiltonian for this theory becomes

~=~[g”4(~,-~,)(~~-~~~-cp21. The Lagrangian L = cj”u,(r(d), q) is L=qd;x+A,cj*.

It subsumes the following situations: (a> For v = m, g,, = va4’ A, = O-free relativistic particle in N dimensions. particle in N dimensions (b) For v = m + @(qh g,, = va4, A, = O-relativistic interacting with a scalar field (p(q). particle in N dimensions interacting (cl For Y = m, ga5= rlao-relativistic with an external vector field. If restriction R, is dropped, this could be an external electromagnetic field. (d) For (D= m, A, = O-relativistic particle in N dimensions interacting with a tensor field gaB, i.e., relativistic particle in curved space-time. (e) For cp= m, A, = 0, ga4

qa,,

+

v,

‘b

a = O,..., N - 2,

= v,’

relativistic particle in N - 1 dimensions Kaluza type theory). (f) . . * EXAMPLE

6.

interacting

with a vector field V, (Klein-

N even, N = 2K, N > 6. Split the complete set of scalars into 4” = (q”, d, 3’x

,a = o,..., 3, I= 4,..., K + 1.

434

KURT

SUNDERMEYER

Assume that the qw are Lorentz vectors, and as before with r” = $‘/4” pseudo-momenta P, = my,,

choose

IL’ (vap rarp)1’2 - D,Wh

where the functions D,, C,, CI are subject to restrictions specified below. Because of = 3 there are N - 3 p-constraints Oi. They are easily seen to be

rk 1apa/&‘/

@= (p, + D,J2 - m2 = 0, @,=p,-c,=o, @I=J71-c,=o.

For the Hamiltonian

one finds (we do not repeat all arguments again) H = A@ + A’@, + ii’&,

with certain multipliers. We want only one linear combination of constraints to be first class. That is, neither shall { Qi, H} z 0 be identically fulfilled, nor shall it have the trivial solution 2’ = 0 only. The Poisson brackets are

There are choices of pseudo-momenta such that no secondary constraints are necessiated. This is the case if the rank of the matrix defmed by (Qi, cPj} is N - 4. For instance we may impose on the pseudo-momenta the conditions

LOCAL SYMMETRIES

AND DYNAMICS

435

Then fp= @ + jJ;,@J - +f,&J is first class, whereas Q1 and &I are second class. The example incorporates the interaction of a relativistic particle with an external Yang-Mills field A,* by the following explicit functional dependence of D, C,, c,. With the shorthand notation 0’ F qr + iq”, @ F q1 - i$ choose D,(q) = g@A,“(q”) T; OJ,

g = coupling constant,

C, = iO’, I;, z -0’. Here TFJ are numbers (elements of the representation of the algebra of the gauge vector fields A;). Indeed for the Lagrangian L(q”, 4”) one finds

where D, is the covariant derivative D,JOJ = 6, + ig$‘AETFJOJ. This is the Lagrangian investigated by Balachandran

5. MIXED

et al. 191.

SYMMETRIES

In field theory examples one often meets situations where the fields are not invariant under general coordinate transformations only, but also under additional local transformations. We observed that both symmetry groups cannot be realized separately in a phase-space (41. Instead a mixed symmetry has to be implemented. Now we shall give an example of this mixing for a finite-dimensional example. Consider a system of “fields” q”(r) (a = O,..., 3), V(T), W(s) which under reparametrizations behave as scalars and scalar densities, respectively:

(5.1)

the index R standing for reparametrizations. symmetry (scale transformation)

Assume that there is an additional

local

436

KURT

SUNDERMEYER

6,qU= -q”A, 6, v= - 2VA,

(5.2)

6, w =*A with descriptor II. These transformations form a group: Define 6 = JR + a,, then the commutator of two 6 transformations, 6,6, - 6,6, is another 6 transformation with

How much of this local symmetry can be implemented in a phase-space? In search for an answer we need the D-invariants. Only the “fields” q” are both JR and 6, Dinvariant; these are zero-order D-invariants. There also exist exactly four independent first-order D-invariants

That these are zR D-invariant

is easily recognized by observing that Indeed

q”,

W/V and

da/V are separately 6R D-invariant.

(5-4)

The ra are the only first-order D-invariants by the following argument: Any Iirstorder JR D-invariant f may be written as f=f(q”, x, P, y), where x = W/V and y=i/V.

Now

Since 6,i contains a term proportional

to /i’ we must demand

such that every D-invariant is a function of q”, ra. Guided by the reasoning of previous sections one may try to find a “normal” such that

n

LOCAL

SYMMETRIES

AND

431

DYNAMICS

is sR D-invariant if $ and g = (V . n) are C?~D-invariant. However we also want that must not depend on A. We may &;Rqa is 6, D-invariant. Therefore S,((da/V)g) assume that g = g(q”, x, P, y) and find

Hence g must obey

which besides an unwanted solution g = 0 do not have a solution at all. From this simple calculation we conclude that the JR and 6, transformations cannot be realized in a phase-space separately. However there exists a mixed symmetry which together with the scale transformations may be realized in a phase-space. This mixed transformation can indeed be derived from the observation that P is D-invariant. Then obviously also JMqa F --rat is D-invariant for a D-invariant descriptor <. Now

=-pY/-q=wYI

with

<= Vu/,

so 6, is a mixture of a reparametrization (Y) and a scale transformation descriptor A = WY. So instead of JR, 6, we realize

with

JMqa = -rat,

(5.6)

6 s qa = -q”A.

For subsequent calculations one also needs the transformation &re

=--J-(P

dSra

= PA.

of P:

- WY)& (5.7)

Next we determine the commutators of these transformations. transformations (with descriptors <, and &) we obtain

which is again a JM transformation. scale transformation (descriptor A): (S,&

For a 8, transformation

- SMS,) qa = -2r”A<

- r”6,< - q”&A.

For two 6,+,

(descriptor <) and a (5.8b)

438

KURT

SUNDERMEYER

So this is a &,, transformation with descriptor U< + S,& plus a 6, transformation with descriptor &,J. Finally the commutator for two scale transformations is

Introduce pseudo-momenta

We restrict ourselves for simplicity to functions U, with det ]a~,/&-~( # 0, that is, to the case without p-constraints. Cases with PC’s go through by the procedure discussed in Section 3. According to the rules of our game do we now switch to a phase-space. In this let (q”,p,) denote the local coordinates. Infinitesimal canonical transformations are generated by

where again we not distinguish c, [ and 4, this being justified by the mapping of local transformations to canonical transformations. The Hamiltonian for the conceived theory may be found by observing that Y = E(< = EV) mediates translations in 7 and one must subtract the scale transformations with /i = WY = E W, hence the Hamiltonian is H=

VS-

From the comparison of the commutators G[t,,-4,1/ we find 9 E 0,

WsY.

(5.9)

(5.8) with Poisson brackets {G[<,, /1*], .4”‘ z 0,

{
(5.10)

(5.1 lb)

Observe that because of the existence of two constraints Z@, ,Y there is a new outcome in this comparison, namely, an algebra of constraints (5.10). Conditions (5.11) are explicitly

LOCAL

SYMMETRIES

AND

DYNAMICS

439 (5.12a)

a3 apa= r=,

They are compatible

(5.12b)

a9= au, 5 +$p_ aq a@ q

(5.12c)

a9 apa=@

(5.12d)

with the Hamiltonian aH -z& aqa

.

-Pa7

equations of motion, that is, the identities aH -= ape - 4”

hold. Since from p, = u,(q, r) we assumed to be able obtaining P = r”(q,p) are Eqs. (5.12bj(5.12d) differential equations for S? and Y. But also (5.10) has the meaning of a differential equation, By the use of (5.12bj(5.12d) it becomes

a9 ap"qa+yn (au, a454Lc?!!q, a@ 1 =-237*

(5.13)

The differential equations (5.12bj(5.12d) and (5.13) are only integrable if the pseudo-momenta obey further conditions besides det jau,/&” 1f 0. In order to obtain these it turns out convenient to define functions R and S, depending on q” and P by R(93 r> F qq,

u(q, r)),

(5.14)

WA r> F ~y’(9, u(q, r)).

Then (5.13) and (5.12b j(5.12d)

translated to R and S become (5.15) (5.16) (5.17) (5.18)

440

KURT SUNDERMEYER

It turns out that this system of differential equations is integrable only for functions u,(q, r) obeying

(5.21) These conditions are rather strong as may be seenfrom the following EXAMPLE

7.

u, = G,,(q) r4 + A,(q)y

det(G,,) # 0

must fulfill Go, = G,, 9 ‘&gY

= 0,

GCZ4.V- %,5~ (Ad? -A,,a),yqY+

W,,,

-Ad

= 0.

(5.19’) (5.20’) (5.21’) (5.21”)

From (5.19’) and (5.20’) we infer that G,, must be derivable from a function G(q) as G,, = G.a,- Condition (5.21’) implies that G,, dependson q only through sy= Qqy

with

Q f (a,gqnq4)-1’2, aa0 = const.

Condition (5.2 1”) enforces Fao F A,,, -A,,, Fao =

to depend on the q’s as

Q'&dO

Let us integrate the differential equations for R and S. For R we have ?-q” &la

+ 2R = rarBGaB,

(5.15’)

CYR -= r4Ga4. L3r”

(5.16’)

so R = fGaBrnril + R,,(q)

(5.22)

LOCAL

with R,(q)

SYMMETRIES

AND

441

DYNAMICS

obeying

(5.23)

R,,?qY+ 2R, = 0, which is R, = Q’I?,(s).

The differential

equations for S are

as Ga5r5 t q'(A,., -k,,), asa as = G,,& -=

(5.18’)

arm

yielding

S = Ga5qar5 + S,(q),

(5.24)

S 0,a = 45(4~a -A,.,)*

(5.25)

with S,,(q) obeying

Of course (5.23) and (5.25) are fulfilled for a consistent choice of pseudo-momenta, i.e., pseudo-momenta fulfilling (5.19’)-(5.21’) since they are integrability conditions themselves. The constraints .P and .F are obtained by “replacing” f’ by p, in (5.22~(5.24), that is, by making use of where G”*G

r* = Gu5(p, -A,),

BY

= 6”Y‘

so .S = fG”‘(p,

- A,)(pB -A,)

.i” = 4”(P, -A,)

+ R,(q),

+ S,(q),

with R,, S, obeying (5.23) and (5.25), respectively. The Lagrangian L(q, 4) = s(q, r(4)) 4” - vR(q, r(4)) t wS(q, r(4)) turns out to be L =&

G,,(f

t Wq”)(4” t Wq4) + A,Q* - P-R,(q) + WS,(q).

One may check that this Lagrangian is (quasi-) invariant under transformations (5.1) and (5.2) by R, and S, fulfilling (5.23) and (5.25). This Lagrangian constitutes a generalization of the one studied in Ref. [lo] for G,, = v,~, A, = 0 (implying So = 0), and R, = 0.

442

KURTSUNDERMEYER

6. THEORIES WITH FERMIONIC QUANTITIES The techniques developed in previous sections shall be extended and applied to situations where the ‘Yields” are objects of a Grassmann algebra. The example discussed is essentially the massless relativistic particle with pseudoclassical spin. Many topics occurring previously actually are present in this example too, among these the appearance of second class constraints and the mixing of symmetries. The input are a set of fields q”, e (bosonic) and @“, x (fermionic), where ,u = o,..., 3. Under reparametrizations (with descriptor !P) the fields shall transform as f&q’ = -4” !P, (6.1)

and under supertransformations

(with odd descriptor A) as asqu = i/i@@, 6,e = i/lx, 6, w = AT”,

(6.2)

t&x = 2i. Here r” is an abbreviation

for pf-

1 p-i e (

. 0’ 2x 1.

(6.3)

These transformations do form a group, and we are aiming at realizing as much of this group in a phase-space as is feasible. One reads off from (6.1) and (6.2) that (q”, 0“) are both JR and 6, D-invariant. However for no choice of the descriptors are the variations JRqu, JR@@ D-invariant. Consequently symmetries (6.1) and (6.2) cannot be realized in a phase-space separately. Similar to the arguments leading to the observation of a transformation 8M in Section 5, we find here a mixture of reparametrization and supertransformations to be realizable in a phase-space. It is given by &p]

= &[I//] + 6,[A = &Yu].

(6.4)

With reY

(6.5)

LOCAL

SYMMETRIES

AND

443

DYNAMICS

one obtains (6.6)

where

Since (r“,p“) are D-invariant (see Eqs. (6.8) and (6.9) below) under 8,,,, and 6, transformations, the quantities 8M(6s)(q”, 0”) are D-invariant if the descriptors 5 and A do depend on D-invariants only. One finds

and dsrrc = Mp”, (6.9) 6,p” = - &*kpP

+ 2ii@),

confirming that (Y, p”) are first-order D-invariants. We obtain for the commutator of two q, transformations

- 8;‘&.

(6.10a)

So this is another a,,,, transformation. The commutator of a JM transformation (descriptor 5) and a 6, transformation (descriptor A) turns out to be (6. lob) Finally the commutator

of two supertransformations

is r3 = 2iA,A,, ‘4 3 = p)/! s I

-

$‘)A s

(6.10~) 2’

Now introduce pseudo-momenta (p, , rrP) depending on D-invariants. Because of the Grassman-odd nature of the quantities 0” do we expect first-order equations of

444

KURT

SUNDERMEYER

motion for these. Therefore we demand the pseudo-momenta first-order D-invariants p’; hence P, =p,(4,@

to not depend on the

r),

(6.11)

71, = n,(q, 0, r). These eight functions do only depend on four first-order D-invariants are at least four p-constraints of the form Qi (fields, p,, TC,) = 0. Define the generator of canonical transformations as G[(, A] = ~$53’+ A5 7 c&Pi.

r”. Hence there

(6.12)

Since G shall be even, .A??is an even and .Y is an odd element of the Grassmann algebra. The Hamiltonian for this theory is H=e,5?-$ii.

(6.13)

This comes about since Y = E(( = se) generates translations in time, and we have to adjust for a supertransformation wih A = $xY. The Poisson bracket of two generators G[<, A ] and G[r’. A ’ ] is

Here n, is zero (one) for an even (odd) element. We are using the rules of pseudoclassical mechanics as derived by Casalbuoni [ 111. For “field’‘-independent descriptors the first term on the right-hand-side the Poisson bracket algebra of 9 and 9 yields: ~,‘~i{~i,ov}

= ((‘A -A’~)(.9’,.2P}

and comparing it with the commutators

+A’/l{9,.Y‘}.

(6.15)

(6.10) we obtain (6.16)

Comparison of the last term in (6.14) with (6.10) allows for the conclusion that Xi are constraints in phase-space; .9? z 0, .Y z 0. The other terms give rise to the following conditions on the constraints (6.17) (6.18)

LOCAL

SYMMETRIES

AND

DYNAMICS

445 (6.19) (6.20)

Equations (6.17) and (6.19) read explicitly

a.9 %=I"'

(6.17a) (6.17b) (6.19a) (6.19b)

The explicit form of (6.18) and (6.20) is (6.18a) (6.18b) (6.20a) (6.20b) All of the relations (6.17~(6.20) are compatible with Hamilton’s equations of motion. The analysis of Eqs. (6.17), (6.19), and (6.20) could be done essentially along the line of reasoning in Section 3. That analysis has however been made for bosonic systems, where the matrix A,, = (@,, QJ} is always antisymmetric. For fermionic systems is this no longer true. Furthermore rested the analysis in Section 3 on only one local symmetry (reparametrizations), whereas in the present case there are two symmetries. Instead of adapting the analysis to the present situation it turns out to be convenient to proceed along the following line: In defining (6.21)

446 conditions differential

KURT

SUNDERMEYER

(6.17), (6.19), and (6.20) may be used to set up genuine first-order equations for R and S. For instance an, - p”.

The function R does not depend on p”, however the right-hand side contains these quantities. Similar terms appear in X3/&,, &S/8q”, For consistency we are forced to demand

an vark p” = 0, Taking this into account equations for R and S is

in (6.17),

sp’:

= 0.

(6.22)

(6.19), and (6.20)

the full set of differential

(6.23) (6.24) (6.25) (6.26)

It does not make sense to write relations for aRIaq”, aR/aW since LGP/aqw, d.2/&3” contain time derivatives of r’. However we still have the information on the constraint algebra, that is, the Poisson brackets (6.16). The first one of these is

With

as aR ap"a.9 aR ap, -=----=---r", ap ag aqpap, ag aqfl 8.9 -=-aw

aR aw

apvr. aw

LOCAL SYMMETRIES AND DYNAMICS

447

this is a genuine differential equation for R:

igOy

(6.27)

+$P=O.

The other Poisson bracket condition is

which by (6.20a) and (6.20b) is an algebraic relation for R: (6.28)

This equation for R together with (6.23) and (6.27), and the differential equations (6.24)-(6.26) for S are solvable only for a restricted class of pseudo-momenta. Instead of writing down the integrability conditions in full length, let us resort to an example. EXAMPLE

8. Pg = r II’

(6.29)

7r,=aO,.

The constant a is determined by the consistency of Eqs. (6.23) (6.27), and (6.28) to be a = i/2,

The differential equations (6.24)-(6.26)

yield

S=-i@r

P’

The functions R and S differ from .%?and .Y by the p-constraints @,, F 71, - (i/2) 0,:

9 = -i(Op) + vu@, .

The multipliers u“, V’ are determined by exposing these expressions to the original conditions on the constraints (2, Y), that is, to Eqs. (6.17)-(6.20); the result is uu = -p’ = 0

v” E-p’.

(due to (6.22) and (6.29))

448

KIJRTSUNDERMEYER

So finally the dynamics is described by the Hamiltonian

and the constraints @u =: 0. The constraints (5P,.Y) are first class; Qu are second class constraints. The Lagrangian found by an inverse Legendre transformation (using rc, = (i/2) 0,) turns out to be

This is the Lagrangian for the massless spin l/2 particle, see for instance Ref.

7. CONCLUDING

112

REMARKS

We demonstrated again how general covariance and additional local symmetries can be implemented in a phase space, and to what extent these local symmetries determine the phase-space dynamics. The approach may be summarized in the following form: Start with a set of fields and state their transformations. Among these must be general coordinate transformations, and the transformations must close. Furthermore state the degree of the envisaged field equations. The program then yields (a) Hamiltonian in the form H = Z (first class constraints), plus eventually some second class constraints. There are as many Hamiltonian systems (not necessarily different) as there are “compatible” choices of pseudo-momenta. Compatibility refers to restrictions originating from integrability conditions on differential equations of the constraints and/or the presence of secondary constraints. Although the reasoning in selecting conditions on the pseudo-momenta resemble arguments in the Dirac procedure, our approach is entirely different in the following respect. The Dirac algorithm terminates on the level on which all constraints vi satisfy {pi, H} 2 0. The algorithm may lead from primary to secondary, tertiary, etc., constraints. Finally has the number of first class constraints to be maximized. In our program are secondary constraints necessiated in specific situations (see Section 2). Tertiary constraints can never arise, and the choice of pseudo-momenta has to be done in a manner that the number of FC constraints is minimized. The examples discussed in Section 3 are meant to demonstrate how by selecting compatible pseudo-momenta different Hamiltonian systems arise. Different realizations of the mapping lead to either nonrelativistic mechanics of an unconstrained system, or to the dynamics of a relativistic particle in the presence of external fields. By still other choices of pseudo-momenta one could contribute to the large amount of publications on the problem of “relativistic action at-a-distance”; work on this topic is in progress. Most “physical” seem to be field theories with second-order field equations for

LOCAL

SYMMETRIES

AND

DYNAMICS

449

bosons, and first-order ones for fermions. This is a criterion for the subgroup of the full symmetry group realizable in a phase-space. One also may think of theories with field equations of order higher than two. In these one should look for higher order Dinvariants (implying an enlargement of the subgroup). The approach would then be an alternative to the corresponding Hamiltonian formulation. An extension of our program in this respect is currently investigated. More interesting seems to be an application of the method to locally supersymmetric theories. The treatment of the relativistic massless spinning particle in this article has to be considered a first step. The “fields” and transformations (6.1) and (6.2) look rather artificial and would appear more natural if everything is treated with superfield/space techniques. However we deleted these methods in this paper since it is mainly aimed at demonstrating the extension of our program to theories with fermionic degrees of freedom. The massive spinning particle is treated in Ref. [ 131. The examples give a hint what one may expect for locally supersymmetric field theories with fermions (supergravity). Finally let us discuss similarities and dissimilarities in our approach as far as finite-dimensional theories and field theories are concerned. Typical for a generally covariant field theory in D dimensions is the occurrence of D descriptors Y” with decomposition

where n“ has the meaning of a normal [4]. The last term is absent in one dimension. In the examples discussed in this paper there are expressions (‘v”) = Y = nor where rr” transforms like a normal. We have shown in an appendix in (41 that the actual form of n@is unessential for the results one wants to obtain. (Use of this freedom has been made of in several examples in this article.) The descriptors r” do always depend on D-invariants. However in field theories an infinite number of such objects exists in contrast to finite-dimensional theories. This is because every spatial derivative of a D-invariant is a D-invariant of the same order. Therefore in field theories the descriptors are functionals as opposed to functions in nonfield theories. The pseudo-momenta in field theories are functionals too. This seems to render the treatment of field theories much more difficult. However there is a stringent restriction on the pseudo-momenta: Call Q the fields and D the pseudo-momenta (indices suppressed). The Z7 must transform under hypersurface transformations (to = 0) in such a manner that (D’Q) transforms as a scalar density. For a generally covariant field theory in D dimensions with additional symmetries (characterized by descriptors A*) a phase-space generator is given by

The comparison of commutators with Poisson brackets always yields a Poisson bracket algebra among the constraints &,X0) which in this paper happened only for theories with additional symmetry. Part of that algebra contains the Poisson brackets among the super-Hamiltonian LX, and the supermomenta &. In linite-

450

KURT SUNDERMEYER

dimensional examples one typically finds conditions on the first-order derivatives of the constraints with respect to the phase-space coordinates. In contradistinction in field theories these are replaced by first-order functional derivatives. Integrating these functional differential equations yields the constraints and therefore the Hamilton function, and ultimately the phase-space dynamics. As in finite-dimensional theories integration is only possible for consistent pseudo-momenta. It turns out that the form of the supermomenta is independent of pseudo-momenta actually chosen. The ~q are solely determined by the behaviour of the fields and the pseudo-momenta under Co= 0 transformations, but the latter in turn are determined by the transformation property of the fields [ 141. Therefore the real severe conditions on the pseudomomenta essentially stem from integrability conditions for the super-Hamiltonian & and the constraints 8. This is similar to finite-dimensional theories. But in general it is very hard to evaluate the integrability conditions as conditions on the pseudomomenta because (ZU, PJ depend on infinitely many variables. With so many examples from one dimensional and field theories at hand it should be possible to formulate our approach in geometrical terms (jet extensions seems to be appropriate). Results of this investigation shall be published elsewhere in due time.

REFERENCES 1. P. A. M. DIRAC, “Lectures on Quantum Mechanics,” Belfer Graduate School of Science, Yeshiva University, New York, 1966. 2. A. J. HANSON, T. REGGE, AND C. TEITELBOIM, “Constrained Hamiltonian Systems.” Accademia Nazionale dei Lincei, Rome, 1976. 3. K. SUNDERMEYER. “Constrained Dynamics-with Applications to Yang-Mills Theory, General Relativity, Classical Spin, Dual String Model,” Lecture Notes in Phys. No. 169. Springer-Verlag. Berlin/Heidelberg/New York, 1982. 4. D. SALISEIURYAND K. SUNDERMEYER.Phys. Rev. D 21 (1983). 74C-756. 5. D. SALISBURYAND K. SUNDERMEYER,Phys. Rev. D 27 (1983). 757-763. 6. D. SALISBURYAND K. SUNDERMEYER,in preparation. 7. P. G. BERGMANN. “The General Theory of Gravitation Encyclopedia of Physics, Vol. IV” (S. Fluegge, Ed.), Springer-Verlag, Berlin, 1962. 8. P. G. BERGMANN AND A. KOMAR, Intern. J. Theoret. Phys. 5 (I 972), 15-28. 9. A. P. BALACHANDRAN, P. SALOMONSON,B.-S. SKAGERSTAM, AND J.-O. WINNBERG, Phys. Rev. D 15 (1977), 2308-2317. 10. R. MARNELIUS AND B. NILSSON, Phys. Rev. D 20 (1979), 839-847. Il. R. CASALBUONI, Nuovo Cimento 33A (1976). 115-125, 389-431. 12. L. BRINK, S. DESER, B. ZUMINO, P. DI VECCHIA. AND P. HOWE, Phys. Len. 64B (1976), 435-438. 13. K. SUNDERMEYER.“Phase Space Realization of D = 1 Local Supersymmetry,” Lecture presented at the IXX Winter School in Theoretical Physics, Karpacz. February, 1983. 14. K. KUCHA~. J. Mafh. Phys. 17 (1976). 777-791.