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New results on systems with second-class constraints
ANNALS
OF PHYSICS
203, 137-156 (1990)
New Results on Systems with Second-Class Constraints P. MITRA Theoretical
Nuclear
Physics Division, Saha Institute Calcutta 700009, India
of Nuclear
Physics,
R. RAJARAMAN Centre for
Theoretical Studies, Indian Institute Bangalore 560012, India
of Science,
Received January 30, 1990
We show how, for large classes of systems with purely second-class constraints, further information can be obtained about the constraint algebra. In particular, a subset consisting of half the full set of constraints is shown to have vanishing mutual brackets. Some other constraint brackets are also shown to be zero. The class of systems for which our results hold includes examples from non-relativistic particle mechanics as well as relativistic field theory. The results are derived at the classical level for Poisson brackets, but in the absence of commutator anomalies the same results will hold for the commutators of the constraint operators in the corresponding quantised theories. 0 1990 Academic Press, Inc.
1. INTRODUCTION The theory of constrained dynamical systems was given its first comprehensive Hamiltonian formulation in the seminal work of Dirac [l]. His classification of constraints into primary versus secondary, and first-class versus second-class, not only enabled a systematic Hamiltonian analysis of constrained systems but, as subsequent reviews of his formulation have elucidated [2, 31, it also threw light on the physical significance of different types of constraints. In particular it is now well understood that first-class constraints correspond to gauge invariance. Of course, as pointed out by Faddeev and Jackiw [4], one can now use elegant techniques of differential geometry to bypass some of Dirac’s comparatively laboured procedure. Nevertheless, the Dirac procedure continues to be correct, clear, and useful. Subsequent to Dirac’s overall formulation there has been a steady stream of work on constrained systems, some exploring general properties and others in the form of applications. In this paper, we derive several new general results on the constraint algebra of 137 0003-4916/90 $7.50 Copyright 0 1990 by Academic Press. Inc. All rights of reproduction m any form reserved.
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large classes of systems with solely second-class constraints. The fact that the constraints of a system are all of the second class requires only that the matrix of Poisson brackets of the constraints have a non-zero determinant. There is, in the definition, no further requirement on the brackets between individual pairs of constraints. We show in this paper, for large classes of such systems of increasing complexity, that many of those constraint brackets vanish. In particular, the first half of the sequence of constraints obtained by the usual time-consistency requirement in Dirac’s procedure have zero brackets with one another. Now the following result is already known [3, 51. Consider a system which has only second-class constraints,
where i = 1, .... 2n and pn, qn (a= 1, .... N > n) are phase space variables. Then it is possible to replace locally in phase space these 2n constraint equations by an equivalent set Pi = Qj = 0, j = 1, .... n, where P, and Qj are some other functions of phase space which obey canonical Poisson brackets with one another. Once we have this set (P,, Qj) there clearly exist many n-member subsets, such as the Pj (j= 1, .... n), or the Qj (j= 1, .... n), where mutual brackets are zero. Obviously such a subset can have at most n members. Now, these Pi and Qj will be local combinations of the original constraints xi, i = 1, .... 2n. In practice, however, finding these combinations explicitly in a given model can be cumbersome and difficult. There is no simple rule for doing so. What our analysis shows, among other things, is that for a large class of systems half of the original constraints form, as they stand, such a maximal subset with zero mutual brackets. In the first instance it is shown that these brackets vanish on the constrained surface, that is, modulo all the constraints x1 =x2= . . . = xZn= 0. Later, it is shown that they vanish identically-a stronger statement-provided one extra assumption is made. One important motivation behind obtaining this result is that it enables us to construct (in a succeeding paper) a gauge-invariant reformulation of such classes of theories with second-class constraints. In Section 2, we present our analysis for “one-chain systems,” by which we mean cases where all the (second-class) constraints flow from a single primary constraint. The remaining 2n - 1 constraints are secondary, forced by consistency under time evolution as per Dirac’s procedure. In Section 3, we extend the results to “twochain systems” which have two primary and the remaining secondary constraints. In Section 4 we comment on the extent to which we can generalise our results to “many-chain systems” and to local field theories (which correspond to infinite-chain systems). Finally, in Section 5, we offer several examples, from particle mechanics as well as from relativistic field theory, which illustrate the different types of situations where our results hold. Our discussion is given entirely at the classical level. However, for systems with no quantum anomalies, corresponding results for commutators will continue to hold upon quantization, modulo the usual ordering problems and the requirement
SECOND-CLASS
CONSTRAINTS,
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139
of the positivity of the quantum Hamiltonian. Our method cannot be used directly in theories with genuine commutator anomalies, by which we mean alterations occurring in the constraint algebra upon quantization. However, anomalous gauge theories in two dimensions do come under our purview; although the usual (fermionic) classical theory has a constraint algebra that is different from the quantized one, bosonization provides us with an alternative classical action carrying the same constraint algebra as the anomalous quantum theory. To such cases, our method can be applied and will be illustrated by an example in Section 5.
2. 2.1. Some General Properties
ONE-CHAIN
of One-Chain
SYSTEMS
Systems
Let us begin by studying the class of dynamical systems whose Lagrangian foruq,, 4a), a = 1, .... N, is singular and yields, in the passage to the Hamiltonian mulation, only one primary constraint and altogether a globally defined set of some 2n constraints (n < N) which are all of the second class. Clearly these systems are not gauge invariant since they have no first-class constraints. Let the primary constraint be denoted by Xl(Pn, 4,)-O.
(2.1)
The remaining (2n - 1) constraints (x2, .... x2,) must then be forced by the requirement that the constrained subspace be time invariant. That is, X*(P0,4J = {Xl, H) = 0, . . . . . . . . . .. .. .. . . .. . . .. . XZn(Pa, 9a) = {X*n-
17 4
(2.2)
= 0,
where { , } stands for the Poisson brackets and H is the full Hamiltonian associated with the Lagrangian L. As per standard constraint theory, the full Hamiltonian H must include, in addition to the canonical Hamiltonian H, (where H,=C, p,Q,- L with x1 = 0 inserted), a term proportional to the primary constraint x1, H=H,+x,u,,
(2.3)
where the “velocity” v0 is, to start with, unspecified. Our hypothesis, that the system has 2n constraints, no more, no less, and all of the second-class, implies the following. (i) (ii)
None of the xi defined as {xi-, , H} in (2.2) vanishes either identically or modulo the earlier constraints xjGi. {xi,xI}=O (onZi) for i<2n. (2.4)
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Here (on Zi) denotes weak equality modulo the constraints x1 = xZ = . . . = xi = 0, which define a surface Zi in phase space. If (2.4) were not true for some i < 2n, then the requirement (Xi7
H)
=
(Xi3
Hcl
+
uO{Xi,
Xl>
could be satisfied by choosing u. appropriately constraints. (iii) ~x2n~~~=~~2n~~c~+~o~~2n~~1~=0
=O
(on
zi)
and we would end up with i < 2n (onz2J.
(2.5)
If this were not true, then we would have more than 2n constraints. Equation (2.5) could be satisfied in two ways: Either
IX2mXl >f0,
with u. = - hZnyK)/{x~~, xl>
or
{x2n9xl> = {x2nyf&l =O
bQ2J.
In the latter case, given (2.4), x1 would remain first class, violating our hypothesis that all 2n constraints are of the second class. Therefore we must have
(iv)
{x2n, x1>= a’(~,, 4J + 0 uo= -(W’
(on zZn),
{XZn, K}.
(2.6a) (2.6b)
Here a’ could be a non-zero constant, or any non-trivial function on phase space which does not vanish anywhere on the constrained surface Cln (defined by x1=x2= . . . =xZn =O). Note that in general the bracket between a pair of constraints could vanish at some points in phase space and not at others. However, the hypothesis that our system has some globally defined set of 2n constraints forces a’(pa, q.) to be non-zero everywhere on Z,,. Correspondingly the function u0 occurring in the Hamiltonian is fully determined on CZ, by Eq. (2.6b). The Hamiltonian is unambiguously fixed on ZZ,,, as expected for a system with only second-class constraints. The above properties (2.4) to (2.6) flowed trivially from the definition of our class of systems. But further analysis yields some more interesting features about the constraint algebra of such a system. Let xr+ i be the first member of the sequence h x2, ..*3x2,,} which has a non-zero bracket (on Z,,) with some earlier member, i.e.,
{xi,xk}=O
(onZZn), i,kGr
(2.7a)
and {Xr+r,
xj}
#O
(onz,,)
for some j
(2.7b)
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I
Clearly there will always exist such an integer r 2 1, since at the very least (xl, x1 > = 0. We now show that (2.7b) will be non-zero only if j= r. To see this, note that by using the Jacobi identity we have for any j and i > 1,
Apply this to the case i = r + 1, and any j < r. We have ix,+ 1, Irj>
=
lr Xr> + ( (Xr3 Xj>?
{Xj+
H).
(2.9)
From (2.7a), ix,, Xj} vanishes modulo all constraints, which themselves have zero brackets with H on .Z2,,, because of (2.2) and (2.5). Thus the second term in (2.9) vanishes on CZn for all j < r. The first term also vanishes on C,, for all j < r - 1, by (2.7a). The only case for which (2.9) can be non-zero is the case j = r. Hence, (2.7b) can be rewritten as IXr+l,
x,} =O
(onz2J
for
j
1
(2.10)
while (2.11) where tl(p,, qu) must be some function not identically zero on C,,. Next we show that r = n. To see this, note that by repeating the steps in (2.9), we have (2.12)
{~r+2,~jj={~j+l,~r+l}+({~r+l,~j},H). Using (2.11) and the same arguments as those above, we see that {X,+23
Xjl
=O
(on
LJ
for jdr-2.
(2.13)
While the j= r - 1 case yields
cG+2, xr-*I = {x,7xr+J= -a,> qa), where a is the same function as that in (2.11). Proceeding similarly this result so that for any s = 0, 1, 2, . ... r, {Xr+s,Xj}=O
(onC,,)
for
(2.14) we can extend
jbr-s
(2.15)
{Xr+l,Xr)=(-l)S-la(p~,q~).
(2.16)
while {Xr+srXr--s+l}=(-l)S-l
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when s = r, (2.16) becomes (2.17)
IX2n Xl I= (- 1 I’- l a(Pa, 4J.
Now, recall that xr+ I denotes the first constraint in the sequence (2.2) which fails to have zero brackets (on C,,) with all earlier constraints. As a logical corollary, the requirement on the function c((p,, qa) in (2.11) was only that it be non-zero somewhere on CZn. Then in those regions of Zc,, where a is non-zero, the constraint sequence can be terminated at x2r, since the requirement o=
fX2n
HI
=
{X2rl
f&l
+
{X2rY
=
1~2~~
ff,)
+(-I)‘-’
Xl
> uo abay
9J
u.
(2.18)
can be satisfied by inverting (2.18) for the velocity zio. But if a vanished at some other points on ZZn, (2.18) would yield further constraints there. This would violate the hypothesis that our system has some globally fixed number (2~2) of constraints. Therefore, for such systems, a(pa, q,) must be non-zero everywhere on Z,, . The constraint sequence must terminate at xzr. Hence we can set r =n and identify (- l)“- ’ a(pa, q,) with the function a’(pa, qu) in (2.6a). Upon setting r = n, we can summarize our results conveniently by giving the matrix of constraint brackets M, = {xi, xi}, which has the form 0
0
0 0
0 0
M= (j 0 a
0
-a
0 .O a 0.:“--a,? : : : : : : 0 . . a Q”() t
t t
-a t
0 ...
t” t ...
0
t t
0 t
on
ZPn.
(2.19)
i t 0
Here the crosses stand for unknown elements. The following features of the matrix M may be pointed out: (1)
Each element on the reverse diagonal (i + j = 2n + 1) is f a(pa, qa), where which is non-zero everywhere on the fully constrained surface. This follows from Eq. (2.16). (2) All elements above this line, in the upper left triangle, are zero on Zcz, by virtue of Eq. (2.15), and the fact that r = n. (3) No information is available on elements in the lower right triangle except for diagonal elements Mii = {xi, xi>, which clearly vanish. (4) Although all elements of M, are not known, its determinant is known in terms of a. Clearly a is some function
det M= a2”
on
C,,.
(2.20)
SECOND-CLASS
CONSTRAINTS.
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143
(5) Since Y= n, the first n constraints xi, .... xn have zero brackets with one another on .?YZn,by virtue of Eq. (2.7a). This however is not the only such subset. For instance, the n-member subset (xi, .... xnP i, x,,+ ,) also has this property, by virtue of (2.10) and (2.7). (6) Finally, for the class of systems studied in this section, these results have been proved without any further assumptions. Therefore they hold for any system with some fixed set of second-class constraints, of which only one is primary and the remaining are secondary. 2.2. Constraint Brackets Which Vanish Identically In Section 2.1, the vanishing of all mutual brackets between the first half of the constraints (x, with i < n) has been demonstrated only on the submanifold L’,,. In general, these brackets may not vanish identically everywhere in phase space, or in any larger submanifold containing C,, . Recall that this limitation entered our proof in the very first step (Eq. 2.7a)), where r was defined as the largest integer for which Lt i g ;zi’,e;~~~zon C2,. As a corollary, the function c~(p~, qa) in (2.11) had Subsequently (2.18) could be used to terminate the 2n. constraint chain at xlr, thereby proving 2r = 2n. Now, let us see in what circumstances the brackets (xi<,,, x~~,,} will vanish identically everywhere on phase space. To answer this, we proceed just as in Section 2.1, but with a modified starting point. Let r” be the largest integer such that {xi, xk} =O,
(2.21)
i, k
The difference between (2.21) and (2.7a) is that for i, k < r” the brackets vanish identically in (2.21), whereas for i, k < r, the corresponding brackets (2.7a) need vanish only on ,ZZn. Note that there will always exist some r” for which (2.21) holds, since at the very least {xi, x1 } = 0. Also, clearly 7 < r. Given that r” is the largest integer for which (2.21) holds, it follows that {Xi+l,
Xj>
#O
for some j < ?,
(2.22)
where in the absence of further assumptions we can only require that the left-hand side of (2.22) not vanish identically in the full phase space. Now, let us repeat the type of argumentation used in Section 2.1. Using the Jacobi identity, we have, just as in (2.9),
(2.23) Clearly, use of (2.21) in (2.23) yields (x~+~, xi} =O,
j
1.
(2.24)
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Then (2.22) implies (2.25) where a” is some phase-space function which is not identically such steps, we have, analogous to (2.15k(2.16), {Xi+s9
Xj>=
{Xj+l, Xi+s-1) + {{Xi+s-13 = 0, j d r”- s, s = 1, ...) ?,
Xj>,
zero. On repeating
H) (2.26)
while
{xi+s~xi~s+l}=(-l)s-l In particular,
(xi+1,xi}=(-l)“-‘B(p,,q,).
(2.27)
for s = ?,
{xmXl)= (- 1Ii- ’ a”(P,, 4d
(2.28)
Remember that ti(p,, qa) is known only to be some function which is not identically zero on the full phase space. Consider its behaviour on the surface ZZi. There are three possibilities. (a) &(p,, qa) # 0 on some parts of CZi but vanishes on other parts. In that case, the condition O=
{X7?,
H,
=
{X*i,
Hc)
+
uO{XZi3
Xl>
(2.29)
is satisfied on some parts of CZi by choosing u0 suitably, whereas on the remaining portion it will force further constraints, violating our hypothesis that the system has some fixed number (2n) of constraints. We therefore discard this possibility. (b) Although d(p,, qp) is not identically zero, it happens to vanish everywhere on Czi. Then (2.29) forces more constraints, and hence 2r”< 2n. (c) a(~,, qJ is non-zero everywhere on CZi. In that case, (2.29) can be inverted for u0 everywhere on ZZi. There will be no further constraints, and hence 2?= 2n. Equations (2.21) to (2.29) then hold for ?= n. In summary, for those systems for which possibility (b) occurs, r” will be less than n. The possibility (a) has been ruled out by hypothesis. On the other hand, for those systems for which possibility (c) occurs, we see that r’= n, and hence the first half of the constraint sequence (namely, xi, .... x,) will have identically vanishing pairwise brackets. Note that the requirement of possibility (c), that Z(p,, qa) be non-zero everywhere on ZZi, is satisfied when Ial > 0 on Ci with any i< 2R For instance, if 5 is some non-zero constant, we can immediately conclude that (c) is realised.
SECOND-CLASS CONSTRAINTS, I
3. TWO-CHAIN
145
SYSTEMS
Let us again consider systems with some fixed set of 2n second-class constraints, but which now descend from two primary constraints bl(p,, qJ and 1+9~(p,, qa). We outline the derivation of results similar to those obtained for the one-chain systems, by using the same type of arguments. Those aspects of the argumentation which are common to Section 2.1 will not be repeated; only the additional ,features that arise in the generalisation to the two-chain case will be stressed. If ff, denotes the canonical Hamiltonian, the full Hamiltonian now involves two velocities, uO, wO, and has the form H=H,+uo(b,+w,$,. The simplest possibility the requirements
(3.1)
is that { 4 i, II/, } # 0 on the surface 4 1 = II/i = 0. In that case
{h,~}=(ICIl,~}=O
moduloh,ICI,
(3.2)
can be satisfied by suitably fixing u0 and wO, and there will be no more constraints (n = 1). This is a trivial case from our point of view, since no other constraint brackets exist apart from (b,, tc/, >. A general case (n > 1) will involve more constraints coming from the double sequence of time consistency conditions h-
(41, ff) =o,
43s (42, H} =o, . . . .. . . . .. . . . . . .. (j-{(jl,H}=O,
$*={h,~)=o~ ICI3= {h
fq = 0,
(3.3)
““.““..“.“..{l/GM’-,, H} =o, ljG,&f’E
with A4 + M’ = 2n. We must emphasize at this point that although the total number 2n of constraints is lixed, their separation into Q-s and tj-s is not necessarily unique. Thus, M and M’ may vary, although their sum is fixed. To understand how this may happen, consider a situation where we have a 4, and a tik such that both have non-zero brackets with the same primary constraint, say dl. It is possible that the sequences terminate with the pair (d,, tik): this happens if u0 and w0 can be found so as to satisfy the consistency conditions (3.4a) (3.4b)
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What happens if this determinant vanishes? The velocities u,,, w,, can fix either (3.4a) or (3.4b), but not both. If (3.4a) is chosen, the b-chain terminates at 4,, but (3.4b) becomes (3.6)
which, being independent of u,, and wO, is a new constraint $k + I . Alternatively, (3.4b) may be chosen, in which case $k is the last constraint in its chain, but (3.4a) becomes the new constraint (3.7)
which should be called 41+, . Clearly, (3.6) is the same as (3.7), so that the new constraint is unique, but one has the option of putting it in the $-chain or in the &chain, closing the other chain at the same time. The single remaining chain must now be pursued until a second relation can be found between uO and wO, so that the two can be determined. In general, the termination of the two sequences in (3.3) at #M, tjM, requires that the 2 x 2 matrix h4,4,> 01’= ( (h4~~ hl
&4> 4h> {h? 4h> )
have a non-zero determinant on CM,M,, which is the surface #1 = 4~ = . . . = 4~ = $1=$2= . . . z$~,=O. Th is is the generalization of (2.6a). To obtain further information on the constraint algebra, we have to use the Jacobi identity (2.8), where the xi and xi are to be replaced by members of the & or $-chain. First we consider the generalization of the results given in Section 2.1, i.e., pertaining to the vanishing of Poisson brackets on ZM,M,. In the next few paragraphs, we use the symbol x to denote equality modulo 419 42, .... bM, $r, ij2, +,,,,,. Let r and r’ stand for the largest integers such that, in the ordered sequence of constraints listed in Eq. (3.3),
(3.9) As corollary, inclusion of d,+ , or $rV+ 1 will spoil the above property; i.e., each of these will have a non-zero bracket on CM,,. with at least one member of the above subset di<,, $i’
and
W
r’+l,~i
(3.10)
SECOND-CLASS
CONSTRAINTS,
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I
Hence the only members of the earlier set allowed to have non-zero brackets with are #,, tirZ. Hence the 2 x 2 matrix 4 r+l or hfl w+1m I* r’+lrhl
bf%+dr’l _ all bh+1, tic> )=(a2,
I::)=
(3.11)
must have at least one non-zero element in each row. The Jacobi identity also equates the non-diagonal elements {d,, , , $,,} = {$r.+ i, dr}. Given all this, the possible forms of the matrix CIfall into three categories, which need to be analysed separately. Category (i). clll $0, a,,$O, anda,,zaZI z 0. In this case, the d-chain and the $-chain can be treated separately, each in the manner of Section 2.1. The results (2.15), (2.16), and (2.17) hold for each chain separately. That is,
In particular
From (3.13) we can expect that the constraint sequence can be terminated at bzr and tiZrS, with {&,,H}=O and {tiZr,, H} z 0 satisfied by fixing the velocities u0 and wO, respectively. Therefore 2r=M
and
2r’ = M’,
(3.14)
provided the matrix N’ in Eq. (3.8), with this identification, has a non-zero determinant. This is ensured by the following. In addition to the results (3.12) we can see that starting with {drfl, tirZ> x {$,,+i, d,} z 0, repeated Jacobi identity arguments show that for s= 1,2, .... (3.15)
Clearly, if r
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Category (ii). aI2 x clZ, ~0 with aZ2 ~0 (the case aI1 ~0 is treated the same way). Then the sequence of Jacobi identity relations starting from {4,+ i, $+ f
shows that {4r+r,, $r} ~0. Similarly (til, r,Gr+r,} ~$0. Hence c$,+,, and $r+r, will be final constraints of their respective chains, with {#r + ,, , H} z 0 and ( tir + r,, H} z 0 satisfied fixing w0 and uO, respectively. Hence, M=M’=r+r’=n.
(3.16)
We must ensure that the determinant of a’ in Eq. (3.8) is non-zero with this identification of M and M’. Since ~1~~= { tir,+, , I+G~,}z 0, then if r < r’, repeated use of the Jacobi identity will yield {ICI,,+,, $r} ~0. If r > r’, then (c$~+~,, ~,~5i}~0 by virtue of (3.12). Along with the fact that (c$~+~,, It/r} and {ICI,+,., d,} are non-zero, this confirms that det a’ # 0 with the identification (3.16). One can again enumerate a list of vanishing constraint brackets obtained by use of Jacobi identities. We do not list them all but merely note that with r + r’ = n, Eq. (3.9) again gives n constraints whose pairwise brackets are all zero, just as in earlier cases. It is of some interest to note that r and r’ are not unique, but r’ + r = n is. Category (iii). All elements of (3.11) viz., ul,, ctZZ, and c112~ac1z1,are non-zero. This case is like a combination of the two earlier categories, and the closure of the two constraint sequences can be achieved in either way. First consider r # r’, say r < r’. Then Jacobi identities starting from alI $0 and aZ2 # 0 give { &, 4, } $0 and {$2r,r $r}$O, as in (3.13). At the same time {&, It/r} will vanish because ($,, tijlj,,,} x 0. Hence, constraints can be terminated at M= 2r and M’ = 2r’, with det IX’ non-zero. Alternately, c(iz z aZ1 % 0 leads to {d,+ rl, +I } $0 and r+r,, 4,} $0 just as in category (ii), while {1+5~+~,,$r} w0 by virtue of (3.12). W This permits identifying M = M’ = r + r’, with det CC’# 0 as required. These two possible ways of closing the constraint chains are just a reflection of the physically equivalent options discussed at the beginning of this section. For a given choice of some M constraints in the &chain, and M’ constraints in the $-chain, category (iii) may be resolved either as in (i) or as in (ii). Of course, when r = r’, the two possibilities are the same. Either way, the subset of constraints having mutually vanishing brackets, as per (3.9), will be (r + r’), which is half the total number of constraints 2n = M + M’. However, to prove that det a’ # 0, one has to invoke the fact that a’ is now proportional to 01and assume that det LY# 0.
Note that this analysis shows that either M and M’ are both even numbers, so that M = 2r and M’ = 2r’ may be satisfied, or if they are both odd, they must be equal to each other so that M = M’ = r + r’ may hold. The possibility that either M (or M’) is even while the other is odd is ruled out anyway, since, for a second-class system, the total number of constraints M+ M’ = 2n has to be even. But our analysis seems to indicate that the case in which M and M’ are both odd, but unequal, is not realised. Of course we have not elaborated here on non-generic “accidental” possibilities such as the possibility that det a is zero despite the fact that all its elements are non-zero.
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SECOND-CLASS CONSTRAINTS, I
The preceding analysis has shown that half the constraints have vanishing brackets with one another in all the categories, since r + r’ = f(M+ M’) = n in all cases. But this vanishing, as in Section 2a, has been shown to hold only on the fully constrained surface Z,, M,. To see in what circumstances these brackets will vanish identically, we repeat the analysis from a different starting point, just as we did in Section 2.2. We will outline the steps very briefly. Let r” and r”’ be the largest integers such that
{~i,~~}={~i,~i~}={~i~,~,~}=O,
i,j
where, in contrast to (3.9), here the brackets vanish identically space. Correspondingly, the matrix
(3.17) throughout
phase
must have, in each row, at least one element which is not identically zero in phase space. We can again separate all cases into three categories, as before. Then if we make the extra assumption that all elements which are not identically zero in each category are strictly non-zero on the fully ,constrained surface, then the entire analysis done earlier (for all three categories) can be repeated. This will yield either M= 2?, M’ = 27, or M = M’ = ?+ ?‘. Then, by virtue of (3.17), the first r” constraints in the &chain and the first ?’ constraints in the $-chain will have identically zero brackets with one another, with their total number ?+ ?’ half the total number of constraints 2n = M+ M’ in all cases. 4. MANY
CHAINS AND FIELD THEORY:
The extension of our analysis from the one-chain case to the two-chain case indicates that a similar approach can also be extended, at least in principle, to the many-chain case, i.e., a general second-class system with an arbitrary number of primary constraints. Consider a system with K primary constraints, denoted by xi’), Xl(2),...?Xl (U . From time consistency suppose each primary constraint leads to a chain as shown below, Xl x:l)=
Cl)=0
{Xl (1) > H}
(2)=0
XI =o
~~~~{~$~:‘,l,H}=O
xi’)=
{Xl C2) 2 H}=O
&+{&,,H}=O
. .. ...
Xl x2 (W
(K) = 0
=-(xyQ,H}=O
(4.1)
... ~~;={~~;~I,H}=O,
where (4.2)
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Altogether the full set of 2n constraints (where 2n = M, + Mz + ... + MK) is taken to be of the second class. Let us define rl, r2, .... rx- to be the largest integers such that { xy, p>
z 0,
i < r,, j < r,; 1, m arbitrary.
(4.3)
Depending on the results we are seeking, the brackets in (4.3) could be taken to vanish module all constraints, as in Section 2.1, or identically, as in Section 2.2. Then, proceeding as before, we see that the K x K matrix a11 @=
...
alK
:
(. aK1
(4.4) ...
am
-1
where cl,,,,- {xl:‘, i, x:‘J’}, must have at least one non-zero element in each row. The next step, analogous to Section 3, would be to divide all permissible matrices a into different categories. There will be one category that is particularly simple, where only the diagonal elements of c1 are non-zero. Call it category (i). This is the generalisation of category (i) in the two-chain case. As argued there, in this category, each chain can be treated separately and the problem reduces to that of K separate one-chain cases, with 2r, = M,, 2r, = Mz, .... and 2r, = Mk. The set of constraints which form the first half of each chain will have zero brackets with one another. To begin with, their brackets can be expected to vanish only modulo all the constraints. But if, just as in Section 2.2, we take (4.3) to hold everywhere in phase space, and assume that the non-zero elements of a are strictly non-zero on the fully constrained surface, then the top half of the constraints will have identically zero brackets with one another. These results are for category (i) defined above. For a K-chain system with arbitrary K> 2 there will be many other categories. The general principles for attacking the K-chain problem are the same as those in the K= 1 or the K= 2 case, but in practice they can become very tedious. Our analysis of the two-chain case suggests that in the generic situation the constraints {xi”’ 1i 6 rm, m = 1, 2, .... K), which form a first-class subset, are exactly half the total number of constraints. Next let us turn to local field theories. Suppose we have a set of fields $Jx, t) governed by a local Lagrangian L = s & 9(4,(x), d,(x)). Suppose the Lagrangian is singular and yields solely second-class constraints. Now, in field theory each constraint will be some field equation of the form x(d,(xh %2(x)) = x(x) = 0,
(4.5)
where X,(X) are the momenta conjugate to #Jx). Equation (4.5) is actually an infinite number of constraint equations, one at each space point x. In particular, even if a theory has only one primary constraint equation in the field theoretic sense, it corresponds to an infinite number of primary constraints and to an intinitechain system from our point of view. Fortunately, the property of locality simplifies
151
SECOND-CLASS CONSTRAINTS. I
matters greatly. The bracket between any pair of constraints x,(x) and xj(x) must, unless it vanishes identically, be proportional to 6(x - y) or to some derivative of 6(x - y). Now suppose the system has a set of second-class constraints xi(x), x2(x), ...Yx,(x), of which only the first, xl(x), is primary. In general, in field theory, the integer A4 need not be even. But suppose further that the different brackets {x,(x), x,(y)) either vanish or are proportional only to 6(x-y) and not its derivatives. Then although it is an infinite chain system, it clearly belongs to category (i) in the sense defined earlier. Our analysis can be carried out for each chain (that is, at each x) separately, with the results of Section 2 applying at each x. The integer M must then be even (M= 2n), and the arguments of Section 2 show that the matrix MV(x, y) = (x,(x), xj(y)} will have the same form as Eq. (2.19) with an extra factor of 6(x - y) on the right-hand side and with the non-zero entries (such as c( in (2.19)) now functions a(x). In particular the top half of the constraint functions xi(x), x*(x), .... x,(x) will have zero brackets with one another. Without further assumptions, these brackets will vanish only modulo all the M= 2n constraints. But, as in Section 2.2, if in the field theoretic analogue of Eq. (2.26), i.e., {xi+], xi} = G(x) 6(x-y) (where r” is defined as in Section 2.2), the function G(x) is non-zero everywhere on the submanifold satisfying xi(x) = x2(x) = . . = x,(x) =O, with i,<2J, then 2F=2n and the tirst half of the constraints (x,(x), x*(x), “‘> x,,(x)) will have identically zero brackets with one another. We illustrate this class of field theoretic systems with two examples in the next section. When some constraint brackets involve X(x - y), etc., the system does not belong to category (i) in the sense described above. Non-trivial multi-chain analysis will have to be carried out. The well-known model of a free scalar field in (1 + 1) dimensions, briefly recalled in the next section, will offer a simple example of this type.
5. EXAMPLES The analysis in Sections 2 and 3 makes predictions about the constraint algebra of any system with purely second-class constraints which flow from either one or two primary constraints. For many-primary-constraint cases and local field theory, Section 4 makes some predictions for certain categories. In this section we offer a few examples chosen to illustrate the different types of situations in which our analysis applies, both in non-relativistic mechanics and in relativistic field theory. a. Consider the family of models described by the Lagrangian 2
N-l +qN
c
i=l
where qi (i= 1, .... N) are coordinates, 595/203/l.
11
biqi,
(5.1)
and bi, di, and cii are some real constants.
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Although for special sets of values of the constants b,, di, and cii this Lagrangian could yield first-class constraints, for generic values it gives a theory with purely second-class constraints of the type we are interested in. As a specific illustration let us take, say, N = 5 and
The canonical Hamiltonian
is (5.3)
Following Dirac’s procedure, one can see that this system carries the following six second-class constraints:
(5.4)
Of these, only the first constraint, x1, is primary; the rest are secondary. Thus, this is an example of the one-chain case with 2n = 6 and we can see that the results of Section 2 hold. The matrix M, = {xi, xi}, given below, has indeed the form predicted in Eq. (2.19).
(5.5)
(We have not bothered to calculate the elements denoted by crosses in (5.5) since our theory has no predictions for them.) Note that this model is an example of
SECOND-CLASS
CONSTRAINTS.
153
I
possibility (c) discussed in Section 2.2. The bracket c(z {x4, x3} = - & is clearly non-zero everywhere. Correspondingly, as per the analysis in Section 2.2, we see that the top half of the constraints, viz., x1, xZ, and x3, have mutual brackets which vanish identically. b. Our next example illustrates the two-chain case. Consider the Lagrangian L = fcot + 45Y + 242 + 4612 + a: +
@6(@1
+
v2(@3,
+
43-q&2
+ Vl(@,, 941)
(5.6)
94)) - v3(@3>@4),
where VI, V,, and V, are any functions of their arguments. Then
This system has six second-class constraints, but two of them-#, primary, and the rest secondary as shown below.
and til--are
(5.8)
This system illustrates category (ii) of the two-chain case analysed in Section 3. As predicted, the upper half subset of the constraint set has zero mutual brackets. This subset can be taken to consist of #1, I++~,and either #2 or ti2. c. The preceding examples involved a finite number of degrees of freedom. The chiral Schwinger model in (1 + 1) dimensions offers a nice field theoretic example. The bosonized Lagrangian for the system (in the regularization a = 1; see Ref. [6]) is, using Lorentz covariant notation,
L=Jd+(a,4
+ Ap)(apd+ AF)- +F~,P- 8~”a,+u,].
Let 7c, no, E stand for momenta
(5.9)
conjugate respectively to 4, A,, A,. Then
K= I ~xC~(nz+(a,~)2+E2)+Ea,~o+(~+a,~+~I)(~1-~0)].
(5.10)
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The sequence of constraints is Xl(X) E no(x) z50 ~*(X)~a,(E+(d)+7c+A~MO
(5.11)
x3(x) = EwO x4(x)-
-7c-~a,4-2A,+A,~o.
It can be verified that these constraints form a second-class set, with only x1(x) primary. This is an infinite-chain case in our terminology, since (5.11) gives a chain of constraints at each x. However, all the brackets MU(x, y) = {x,(x), xi(r)} are proportional to 6(x - y). Thus, the example belongs to category (i) as described in Section 4. The chains at different values of x are not coupled by brackets and all single-chain results of Section 2 should hold at each x. This is indeed so, and the M,(x, y) can be verified to be
(5.12)
The chiral Schwinger model in the 01> 1 regularisation can be treated similarly, but it is less interesting from our point of view, since it has only two constraint equations. d. Our next example is the non-linear sigma model which, unlike the chiral Schwinger model, is a non-trivially interacting theory. Let G - (#(x, t), a = 1, 2, ..*, N} be a multiplet of N real scalar fields in (1 + 1) dimensions and 2(x, t) another scalar field. Let the Lagrangian be L =
s
dx[-a,a.
Pa + lf((a)],
(5.13)
where f(a) is any function of the fields CP such that the equations f(o) = 0 and V,f(a) =0 have no common roots. The well-known O(N) sigma model corresponds to f(a) = 0’ - 1 and is one such example. (At the quantum level, these models are renormalisable only in (1 + 1) dimensions, but at the classical level, our analysis will apply in any spatial dimensions.) If zl(x) and n(x) stand for fields conjugate to 1(x) and a(x), respectively, then
H,=jdX[tr2+f(axa)2-nf(u)].
(5.14)
SECOND-CLASS
CONSTRAINTS,
I
155
There are four constraint equations, all of the second class, given below. Of these only xl(x) is primary. The constraints are Xl(X) c ?rj,(X) z 0 x*(x) =f(4x))
= 0
x3(x) E 11:.V,fz x4(x) = A(V,f)’
(5.15)
0 + a+.
v,
f+ (II .V,)’ fx0.
We see that
~xl(x)ALYH= By assumption,
~x3w&(YH=
-P,f)'W-Y).
these brackets are non-vanishing
on the constrained
(5.16) surface since
V,f does not vanish where f does. The constraint sequence can be terminated with x4(x) since the requirement i4 = {x4, H} = 0 can be satisfied by including in term 1 dx xl(x) uO(x) with Q,(X) chosen suitably. We also see that the upper half of the constraint sequence, namely xl(x) and x2(y), will have identically vanishing brackets among themselves for all x and y. All these results are in accordance with our analysis in Section 4. H a permissible
e. Our last example is described by the Lagrangian
L=[dx{a,~a,+
+2*&j.
(5.17)
This is a trivial and well-known model-it is just the free Klein-Gordon theory in (1 + 1) dimensions in light cone coordinates. We mention it briefly since it illustrates an infinite-chain case which is not in category (i) in the terminology of Section 4. There is only one constraint equation,
X,(x)-~(X)--.~~cx)~oo,
(5.18)
with PAX), ~ha= The Hamiltonian
4
6(x - Y).
(5.19)
is H=
s
dx [$m’cj’+
u,,(x) x,(x)].
(5.20)
The requirement 0 = {x,(x),
H} = -rn’c$ - a,uO
(5.21)
is satisfied by choosing the arbitrary function Q,(X) appropriately. Hence there are no further constraints beyond (5.18). This is an infinite-chain case since (5.18) gives one primary constraint for each x, but each chain consists of only one member.
156
MITRAANDRAJARAMAN
Consider the Fourier transform (5.22)
Clearly { iI(
jgk’)}
= 2ik 6(k + k’).
(5.23)
Thus, the subset of iI with k 2 0 (which is, roughly speaking, half the set of constraints) has vanishing mutual brackets.
ACKNOWLEDGMENTS One of us (R. R.) thanks Dr. N. Mukunda for a discussion. P. M. acknowledges Centre for Theoretical Studies, Indian Institute of Science, Bangalore, where part
the hospitality of the of this work was done.
REFERENCES 1. P. A. M. DIRAC, “Lectures on Quantum Mechanics,” Academic Press, New York, 1964. 2. E. C. G. SUDAIUHAN AND N. MUKUNDA, “Classical Dynamics,” Wiley, New York 1972; A. HANSON, T. REGGE, AND C. TEITELBOIM, Acad. Naz. Lincei (Rome) (1976). Lecture Notes in Physics, Vol. 169, Springer-Verlag, 3. K. SUNDERMEYER, “Constrained Dynamics,” New York/Berlin, 1982. 4. L. D. FADDEEV AND R. JACKIW, Phys. Rev. Let?. 60 (1988), 1692. 5. J. A. SHOUTEN AND W. VAN DER KLUK, “Pfatl’s Problem and Its Generalization,” Clarendon Press, Oxford, 1949; L. P. EISENHART, “Continuous Groups of Transformations,” Dover, New York, 1969. 6. R. JACKIW AND R. RAJARAMAN, Phys. Rev. Left. 54 (1985), 1219; J. LOTT AND R. RAJARAMAN, Phys. La. B 165 (1985), 321.
OF PHYSICS
203, 137-156 (1990)
New Results on Systems with Second-Class Constraints P. MITRA Theoretical
Nuclear
Physics Division, Saha Institute Calcutta 700009, India
of Nuclear
Physics,
R. RAJARAMAN Centre for
Theoretical Studies, Indian Institute Bangalore 560012, India
of Science,
Received January 30, 1990
We show how, for large classes of systems with purely second-class constraints, further information can be obtained about the constraint algebra. In particular, a subset consisting of half the full set of constraints is shown to have vanishing mutual brackets. Some other constraint brackets are also shown to be zero. The class of systems for which our results hold includes examples from non-relativistic particle mechanics as well as relativistic field theory. The results are derived at the classical level for Poisson brackets, but in the absence of commutator anomalies the same results will hold for the commutators of the constraint operators in the corresponding quantised theories. 0 1990 Academic Press, Inc.
1. INTRODUCTION The theory of constrained dynamical systems was given its first comprehensive Hamiltonian formulation in the seminal work of Dirac [l]. His classification of constraints into primary versus secondary, and first-class versus second-class, not only enabled a systematic Hamiltonian analysis of constrained systems but, as subsequent reviews of his formulation have elucidated [2, 31, it also threw light on the physical significance of different types of constraints. In particular it is now well understood that first-class constraints correspond to gauge invariance. Of course, as pointed out by Faddeev and Jackiw [4], one can now use elegant techniques of differential geometry to bypass some of Dirac’s comparatively laboured procedure. Nevertheless, the Dirac procedure continues to be correct, clear, and useful. Subsequent to Dirac’s overall formulation there has been a steady stream of work on constrained systems, some exploring general properties and others in the form of applications. In this paper, we derive several new general results on the constraint algebra of 137 0003-4916/90 $7.50 Copyright 0 1990 by Academic Press. Inc. All rights of reproduction m any form reserved.
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large classes of systems with solely second-class constraints. The fact that the constraints of a system are all of the second class requires only that the matrix of Poisson brackets of the constraints have a non-zero determinant. There is, in the definition, no further requirement on the brackets between individual pairs of constraints. We show in this paper, for large classes of such systems of increasing complexity, that many of those constraint brackets vanish. In particular, the first half of the sequence of constraints obtained by the usual time-consistency requirement in Dirac’s procedure have zero brackets with one another. Now the following result is already known [3, 51. Consider a system which has only second-class constraints,
where i = 1, .... 2n and pn, qn (a= 1, .... N > n) are phase space variables. Then it is possible to replace locally in phase space these 2n constraint equations by an equivalent set Pi = Qj = 0, j = 1, .... n, where P, and Qj are some other functions of phase space which obey canonical Poisson brackets with one another. Once we have this set (P,, Qj) there clearly exist many n-member subsets, such as the Pj (j= 1, .... n), or the Qj (j= 1, .... n), where mutual brackets are zero. Obviously such a subset can have at most n members. Now, these Pi and Qj will be local combinations of the original constraints xi, i = 1, .... 2n. In practice, however, finding these combinations explicitly in a given model can be cumbersome and difficult. There is no simple rule for doing so. What our analysis shows, among other things, is that for a large class of systems half of the original constraints form, as they stand, such a maximal subset with zero mutual brackets. In the first instance it is shown that these brackets vanish on the constrained surface, that is, modulo all the constraints x1 =x2= . . . = xZn= 0. Later, it is shown that they vanish identically-a stronger statement-provided one extra assumption is made. One important motivation behind obtaining this result is that it enables us to construct (in a succeeding paper) a gauge-invariant reformulation of such classes of theories with second-class constraints. In Section 2, we present our analysis for “one-chain systems,” by which we mean cases where all the (second-class) constraints flow from a single primary constraint. The remaining 2n - 1 constraints are secondary, forced by consistency under time evolution as per Dirac’s procedure. In Section 3, we extend the results to “twochain systems” which have two primary and the remaining secondary constraints. In Section 4 we comment on the extent to which we can generalise our results to “many-chain systems” and to local field theories (which correspond to infinite-chain systems). Finally, in Section 5, we offer several examples, from particle mechanics as well as from relativistic field theory, which illustrate the different types of situations where our results hold. Our discussion is given entirely at the classical level. However, for systems with no quantum anomalies, corresponding results for commutators will continue to hold upon quantization, modulo the usual ordering problems and the requirement
SECOND-CLASS
CONSTRAINTS,
I
139
of the positivity of the quantum Hamiltonian. Our method cannot be used directly in theories with genuine commutator anomalies, by which we mean alterations occurring in the constraint algebra upon quantization. However, anomalous gauge theories in two dimensions do come under our purview; although the usual (fermionic) classical theory has a constraint algebra that is different from the quantized one, bosonization provides us with an alternative classical action carrying the same constraint algebra as the anomalous quantum theory. To such cases, our method can be applied and will be illustrated by an example in Section 5.
2. 2.1. Some General Properties
ONE-CHAIN
of One-Chain
SYSTEMS
Systems
Let us begin by studying the class of dynamical systems whose Lagrangian foruq,, 4a), a = 1, .... N, is singular and yields, in the passage to the Hamiltonian mulation, only one primary constraint and altogether a globally defined set of some 2n constraints (n < N) which are all of the second class. Clearly these systems are not gauge invariant since they have no first-class constraints. Let the primary constraint be denoted by Xl(Pn, 4,)-O.
(2.1)
The remaining (2n - 1) constraints (x2, .... x2,) must then be forced by the requirement that the constrained subspace be time invariant. That is, X*(P0,4J = {Xl, H) = 0, . . . . . . . . . .. .. .. . . .. . . .. . XZn(Pa, 9a) = {X*n-
17 4
(2.2)
= 0,
where { , } stands for the Poisson brackets and H is the full Hamiltonian associated with the Lagrangian L. As per standard constraint theory, the full Hamiltonian H must include, in addition to the canonical Hamiltonian H, (where H,=C, p,Q,- L with x1 = 0 inserted), a term proportional to the primary constraint x1, H=H,+x,u,,
(2.3)
where the “velocity” v0 is, to start with, unspecified. Our hypothesis, that the system has 2n constraints, no more, no less, and all of the second-class, implies the following. (i) (ii)
None of the xi defined as {xi-, , H} in (2.2) vanishes either identically or modulo the earlier constraints xjGi. {xi,xI}=O (onZi) for i<2n. (2.4)
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MITRA
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RAJARAMAN
Here (on Zi) denotes weak equality modulo the constraints x1 = xZ = . . . = xi = 0, which define a surface Zi in phase space. If (2.4) were not true for some i < 2n, then the requirement (Xi7
H)
=
(Xi3
Hcl
+
uO{Xi,
Xl>
could be satisfied by choosing u. appropriately constraints. (iii) ~x2n~~~=~~2n~~c~+~o~~2n~~1~=0
=O
(on
zi)
and we would end up with i < 2n (onz2J.
(2.5)
If this were not true, then we would have more than 2n constraints. Equation (2.5) could be satisfied in two ways: Either
IX2mXl >f0,
with u. = - hZnyK)/{x~~, xl>
or
{x2n9xl> = {x2nyf&l =O
bQ2J.
In the latter case, given (2.4), x1 would remain first class, violating our hypothesis that all 2n constraints are of the second class. Therefore we must have
(iv)
{x2n, x1>= a’(~,, 4J + 0 uo= -(W’
(on zZn),
{XZn, K}.
(2.6a) (2.6b)
Here a’ could be a non-zero constant, or any non-trivial function on phase space which does not vanish anywhere on the constrained surface Cln (defined by x1=x2= . . . =xZn =O). Note that in general the bracket between a pair of constraints could vanish at some points in phase space and not at others. However, the hypothesis that our system has some globally defined set of 2n constraints forces a’(pa, q.) to be non-zero everywhere on Z,,. Correspondingly the function u0 occurring in the Hamiltonian is fully determined on CZ, by Eq. (2.6b). The Hamiltonian is unambiguously fixed on ZZ,,, as expected for a system with only second-class constraints. The above properties (2.4) to (2.6) flowed trivially from the definition of our class of systems. But further analysis yields some more interesting features about the constraint algebra of such a system. Let xr+ i be the first member of the sequence h x2, ..*3x2,,} which has a non-zero bracket (on Z,,) with some earlier member, i.e.,
{xi,xk}=O
(onZZn), i,kGr
(2.7a)
and {Xr+r,
xj}
#O
(onz,,)
for some j
(2.7b)
SECOND-CLASS
CONSTRAINTS,
141
I
Clearly there will always exist such an integer r 2 1, since at the very least (xl, x1 > = 0. We now show that (2.7b) will be non-zero only if j= r. To see this, note that by using the Jacobi identity we have for any j and i > 1,
Apply this to the case i = r + 1, and any j < r. We have ix,+ 1, Irj>
=
lr Xr> + ( (Xr3 Xj>?
{Xj+
H).
(2.9)
From (2.7a), ix,, Xj} vanishes modulo all constraints, which themselves have zero brackets with H on .Z2,,, because of (2.2) and (2.5). Thus the second term in (2.9) vanishes on CZn for all j < r. The first term also vanishes on C,, for all j < r - 1, by (2.7a). The only case for which (2.9) can be non-zero is the case j = r. Hence, (2.7b) can be rewritten as IXr+l,
x,} =O
(onz2J
for
j
1
(2.10)
while (2.11) where tl(p,, qu) must be some function not identically zero on C,,. Next we show that r = n. To see this, note that by repeating the steps in (2.9), we have (2.12)
{~r+2,~jj={~j+l,~r+l}+({~r+l,~j},H). Using (2.11) and the same arguments as those above, we see that {X,+23
Xjl
=O
(on
LJ
for jdr-2.
(2.13)
While the j= r - 1 case yields
cG+2, xr-*I = {x,7xr+J= -a,> qa), where a is the same function as that in (2.11). Proceeding similarly this result so that for any s = 0, 1, 2, . ... r, {Xr+s,Xj}=O
(onC,,)
for
(2.14) we can extend
jbr-s
(2.15)
{Xr+l,Xr)=(-l)S-la(p~,q~).
(2.16)
while {Xr+srXr--s+l}=(-l)S-l
142
MITRA
In particular
AND
RAJARAMAN
when s = r, (2.16) becomes (2.17)
IX2n Xl I= (- 1 I’- l a(Pa, 4J.
Now, recall that xr+ I denotes the first constraint in the sequence (2.2) which fails to have zero brackets (on C,,) with all earlier constraints. As a logical corollary, the requirement on the function c((p,, qa) in (2.11) was only that it be non-zero somewhere on CZn. Then in those regions of Zc,, where a is non-zero, the constraint sequence can be terminated at x2r, since the requirement o=
fX2n
HI
=
{X2rl
f&l
+
{X2rY
=
1~2~~
ff,)
+(-I)‘-’
Xl
> uo abay
9J
u.
(2.18)
can be satisfied by inverting (2.18) for the velocity zio. But if a vanished at some other points on ZZn, (2.18) would yield further constraints there. This would violate the hypothesis that our system has some globally fixed number (2~2) of constraints. Therefore, for such systems, a(pa, q,) must be non-zero everywhere on Z,, . The constraint sequence must terminate at xzr. Hence we can set r =n and identify (- l)“- ’ a(pa, q,) with the function a’(pa, qu) in (2.6a). Upon setting r = n, we can summarize our results conveniently by giving the matrix of constraint brackets M, = {xi, xi}, which has the form 0
0
0 0
0 0
M= (j 0 a
0
-a
0 .O a 0.:“--a,? : : : : : : 0 . . a Q”() t
t t
-a t
0 ...
t” t ...
0
t t
0 t
on
ZPn.
(2.19)
i t 0
Here the crosses stand for unknown elements. The following features of the matrix M may be pointed out: (1)
Each element on the reverse diagonal (i + j = 2n + 1) is f a(pa, qa), where which is non-zero everywhere on the fully constrained surface. This follows from Eq. (2.16). (2) All elements above this line, in the upper left triangle, are zero on Zcz, by virtue of Eq. (2.15), and the fact that r = n. (3) No information is available on elements in the lower right triangle except for diagonal elements Mii = {xi, xi>, which clearly vanish. (4) Although all elements of M, are not known, its determinant is known in terms of a. Clearly a is some function
det M= a2”
on
C,,.
(2.20)
SECOND-CLASS
CONSTRAINTS.
I
143
(5) Since Y= n, the first n constraints xi, .... xn have zero brackets with one another on .?YZn,by virtue of Eq. (2.7a). This however is not the only such subset. For instance, the n-member subset (xi, .... xnP i, x,,+ ,) also has this property, by virtue of (2.10) and (2.7). (6) Finally, for the class of systems studied in this section, these results have been proved without any further assumptions. Therefore they hold for any system with some fixed set of second-class constraints, of which only one is primary and the remaining are secondary. 2.2. Constraint Brackets Which Vanish Identically In Section 2.1, the vanishing of all mutual brackets between the first half of the constraints (x, with i < n) has been demonstrated only on the submanifold L’,,. In general, these brackets may not vanish identically everywhere in phase space, or in any larger submanifold containing C,, . Recall that this limitation entered our proof in the very first step (Eq. 2.7a)), where r was defined as the largest integer for which Lt i g ;zi’,e;~~~zon C2,. As a corollary, the function c~(p~, qa) in (2.11) had Subsequently (2.18) could be used to terminate the 2n. constraint chain at xlr, thereby proving 2r = 2n. Now, let us see in what circumstances the brackets (xi<,,, x~~,,} will vanish identically everywhere on phase space. To answer this, we proceed just as in Section 2.1, but with a modified starting point. Let r” be the largest integer such that {xi, xk} =O,
(2.21)
i, k
The difference between (2.21) and (2.7a) is that for i, k < r” the brackets vanish identically in (2.21), whereas for i, k < r, the corresponding brackets (2.7a) need vanish only on ,ZZn. Note that there will always exist some r” for which (2.21) holds, since at the very least {xi, x1 } = 0. Also, clearly 7 < r. Given that r” is the largest integer for which (2.21) holds, it follows that {Xi+l,
Xj>
#O
for some j < ?,
(2.22)
where in the absence of further assumptions we can only require that the left-hand side of (2.22) not vanish identically in the full phase space. Now, let us repeat the type of argumentation used in Section 2.1. Using the Jacobi identity, we have, just as in (2.9),
(2.23) Clearly, use of (2.21) in (2.23) yields (x~+~, xi} =O,
j
1.
(2.24)
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Then (2.22) implies (2.25) where a” is some phase-space function which is not identically such steps, we have, analogous to (2.15k(2.16), {Xi+s9
Xj>=
{Xj+l, Xi+s-1) + {{Xi+s-13 = 0, j d r”- s, s = 1, ...) ?,
Xj>,
zero. On repeating
H) (2.26)
while
{xi+s~xi~s+l}=(-l)s-l In particular,
(xi+1,xi}=(-l)“-‘B(p,,q,).
(2.27)
for s = ?,
{xmXl)= (- 1Ii- ’ a”(P,, 4d
(2.28)
Remember that ti(p,, qa) is known only to be some function which is not identically zero on the full phase space. Consider its behaviour on the surface ZZi. There are three possibilities. (a) &(p,, qa) # 0 on some parts of CZi but vanishes on other parts. In that case, the condition O=
{X7?,
H,
=
{X*i,
Hc)
+
uO{XZi3
Xl>
(2.29)
is satisfied on some parts of CZi by choosing u0 suitably, whereas on the remaining portion it will force further constraints, violating our hypothesis that the system has some fixed number (2n) of constraints. We therefore discard this possibility. (b) Although d(p,, qp) is not identically zero, it happens to vanish everywhere on Czi. Then (2.29) forces more constraints, and hence 2r”< 2n. (c) a(~,, qJ is non-zero everywhere on CZi. In that case, (2.29) can be inverted for u0 everywhere on ZZi. There will be no further constraints, and hence 2?= 2n. Equations (2.21) to (2.29) then hold for ?= n. In summary, for those systems for which possibility (b) occurs, r” will be less than n. The possibility (a) has been ruled out by hypothesis. On the other hand, for those systems for which possibility (c) occurs, we see that r’= n, and hence the first half of the constraint sequence (namely, xi, .... x,) will have identically vanishing pairwise brackets. Note that the requirement of possibility (c), that Z(p,, qa) be non-zero everywhere on ZZi, is satisfied when Ial > 0 on Ci with any i< 2R For instance, if 5 is some non-zero constant, we can immediately conclude that (c) is realised.
SECOND-CLASS CONSTRAINTS, I
3. TWO-CHAIN
145
SYSTEMS
Let us again consider systems with some fixed set of 2n second-class constraints, but which now descend from two primary constraints bl(p,, qJ and 1+9~(p,, qa). We outline the derivation of results similar to those obtained for the one-chain systems, by using the same type of arguments. Those aspects of the argumentation which are common to Section 2.1 will not be repeated; only the additional ,features that arise in the generalisation to the two-chain case will be stressed. If ff, denotes the canonical Hamiltonian, the full Hamiltonian now involves two velocities, uO, wO, and has the form H=H,+uo(b,+w,$,. The simplest possibility the requirements
(3.1)
is that { 4 i, II/, } # 0 on the surface 4 1 = II/i = 0. In that case
{h,~}=(ICIl,~}=O
moduloh,ICI,
(3.2)
can be satisfied by suitably fixing u0 and wO, and there will be no more constraints (n = 1). This is a trivial case from our point of view, since no other constraint brackets exist apart from (b,, tc/, >. A general case (n > 1) will involve more constraints coming from the double sequence of time consistency conditions h-
(41, ff) =o,
43s (42, H} =o, . . . .. . . . .. . . . . . .. (j-{(jl,H}=O,
$*={h,~)=o~ ICI3= {h
fq = 0,
(3.3)
““.““..“.“..{l/GM’-,, H} =o, ljG,&f’E
with A4 + M’ = 2n. We must emphasize at this point that although the total number 2n of constraints is lixed, their separation into Q-s and tj-s is not necessarily unique. Thus, M and M’ may vary, although their sum is fixed. To understand how this may happen, consider a situation where we have a 4, and a tik such that both have non-zero brackets with the same primary constraint, say dl. It is possible that the sequences terminate with the pair (d,, tik): this happens if u0 and w0 can be found so as to satisfy the consistency conditions (3.4a) (3.4b)
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What happens if this determinant vanishes? The velocities u,,, w,, can fix either (3.4a) or (3.4b), but not both. If (3.4a) is chosen, the b-chain terminates at 4,, but (3.4b) becomes (3.6)
which, being independent of u,, and wO, is a new constraint $k + I . Alternatively, (3.4b) may be chosen, in which case $k is the last constraint in its chain, but (3.4a) becomes the new constraint (3.7)
which should be called 41+, . Clearly, (3.6) is the same as (3.7), so that the new constraint is unique, but one has the option of putting it in the $-chain or in the &chain, closing the other chain at the same time. The single remaining chain must now be pursued until a second relation can be found between uO and wO, so that the two can be determined. In general, the termination of the two sequences in (3.3) at #M, tjM, requires that the 2 x 2 matrix h4,4,> 01’= ( (h4~~ hl
&4> 4h> {h? 4h> )
have a non-zero determinant on CM,M,, which is the surface #1 = 4~ = . . . = 4~ = $1=$2= . . . z$~,=O. Th is is the generalization of (2.6a). To obtain further information on the constraint algebra, we have to use the Jacobi identity (2.8), where the xi and xi are to be replaced by members of the & or $-chain. First we consider the generalization of the results given in Section 2.1, i.e., pertaining to the vanishing of Poisson brackets on ZM,M,. In the next few paragraphs, we use the symbol x to denote equality modulo 419 42, .... bM, $r, ij2, +,,,,,. Let r and r’ stand for the largest integers such that, in the ordered sequence of constraints listed in Eq. (3.3),
(3.9) As corollary, inclusion of d,+ , or $rV+ 1 will spoil the above property; i.e., each of these will have a non-zero bracket on CM,,. with at least one member of the above subset di<,, $i’
and
W
r’+l,~i
(3.10)
SECOND-CLASS
CONSTRAINTS,
147
I
Hence the only members of the earlier set allowed to have non-zero brackets with are #,, tirZ. Hence the 2 x 2 matrix 4 r+l or hfl w+1m I* r’+lrhl
bf%+dr’l _ all bh+1, tic> )=(a2,
I::)=
(3.11)
must have at least one non-zero element in each row. The Jacobi identity also equates the non-diagonal elements {d,, , , $,,} = {$r.+ i, dr}. Given all this, the possible forms of the matrix CIfall into three categories, which need to be analysed separately. Category (i). clll $0, a,,$O, anda,,zaZI z 0. In this case, the d-chain and the $-chain can be treated separately, each in the manner of Section 2.1. The results (2.15), (2.16), and (2.17) hold for each chain separately. That is,
In particular
From (3.13) we can expect that the constraint sequence can be terminated at bzr and tiZrS, with {&,,H}=O and {tiZr,, H} z 0 satisfied by fixing the velocities u0 and wO, respectively. Therefore 2r=M
and
2r’ = M’,
(3.14)
provided the matrix N’ in Eq. (3.8), with this identification, has a non-zero determinant. This is ensured by the following. In addition to the results (3.12) we can see that starting with {drfl, tirZ> x {$,,+i, d,} z 0, repeated Jacobi identity arguments show that for s= 1,2, .... (3.15)
Clearly, if r
148
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Category (ii). aI2 x clZ, ~0 with aZ2 ~0 (the case aI1 ~0 is treated the same way). Then the sequence of Jacobi identity relations starting from {4,+ i, $+ f
shows that {4r+r,, $r} ~0. Similarly (til, r,Gr+r,} ~$0. Hence c$,+,, and $r+r, will be final constraints of their respective chains, with {#r + ,, , H} z 0 and ( tir + r,, H} z 0 satisfied fixing w0 and uO, respectively. Hence, M=M’=r+r’=n.
(3.16)
We must ensure that the determinant of a’ in Eq. (3.8) is non-zero with this identification of M and M’. Since ~1~~= { tir,+, , I+G~,}z 0, then if r < r’, repeated use of the Jacobi identity will yield {ICI,,+,, $r} ~0. If r > r’, then (c$~+~,, ~,~5i}~0 by virtue of (3.12). Along with the fact that (c$~+~,, It/r} and {ICI,+,., d,} are non-zero, this confirms that det a’ # 0 with the identification (3.16). One can again enumerate a list of vanishing constraint brackets obtained by use of Jacobi identities. We do not list them all but merely note that with r + r’ = n, Eq. (3.9) again gives n constraints whose pairwise brackets are all zero, just as in earlier cases. It is of some interest to note that r and r’ are not unique, but r’ + r = n is. Category (iii). All elements of (3.11) viz., ul,, ctZZ, and c112~ac1z1,are non-zero. This case is like a combination of the two earlier categories, and the closure of the two constraint sequences can be achieved in either way. First consider r # r’, say r < r’. Then Jacobi identities starting from alI $0 and aZ2 # 0 give { &, 4, } $0 and {$2r,r $r}$O, as in (3.13). At the same time {&, It/r} will vanish because ($,, tijlj,,,} x 0. Hence, constraints can be terminated at M= 2r and M’ = 2r’, with det IX’ non-zero. Alternately, c(iz z aZ1 % 0 leads to {d,+ rl, +I } $0 and r+r,, 4,} $0 just as in category (ii), while {1+5~+~,,$r} w0 by virtue of (3.12). W This permits identifying M = M’ = r + r’, with det CC’# 0 as required. These two possible ways of closing the constraint chains are just a reflection of the physically equivalent options discussed at the beginning of this section. For a given choice of some M constraints in the &chain, and M’ constraints in the $-chain, category (iii) may be resolved either as in (i) or as in (ii). Of course, when r = r’, the two possibilities are the same. Either way, the subset of constraints having mutually vanishing brackets, as per (3.9), will be (r + r’), which is half the total number of constraints 2n = M + M’. However, to prove that det a’ # 0, one has to invoke the fact that a’ is now proportional to 01and assume that det LY# 0.
Note that this analysis shows that either M and M’ are both even numbers, so that M = 2r and M’ = 2r’ may be satisfied, or if they are both odd, they must be equal to each other so that M = M’ = r + r’ may hold. The possibility that either M (or M’) is even while the other is odd is ruled out anyway, since, for a second-class system, the total number of constraints M+ M’ = 2n has to be even. But our analysis seems to indicate that the case in which M and M’ are both odd, but unequal, is not realised. Of course we have not elaborated here on non-generic “accidental” possibilities such as the possibility that det a is zero despite the fact that all its elements are non-zero.
149
SECOND-CLASS CONSTRAINTS, I
The preceding analysis has shown that half the constraints have vanishing brackets with one another in all the categories, since r + r’ = f(M+ M’) = n in all cases. But this vanishing, as in Section 2a, has been shown to hold only on the fully constrained surface Z,, M,. To see in what circumstances these brackets will vanish identically, we repeat the analysis from a different starting point, just as we did in Section 2.2. We will outline the steps very briefly. Let r” and r”’ be the largest integers such that
{~i,~~}={~i,~i~}={~i~,~,~}=O,
i,j
where, in contrast to (3.9), here the brackets vanish identically space. Correspondingly, the matrix
(3.17) throughout
phase
must have, in each row, at least one element which is not identically zero in phase space. We can again separate all cases into three categories, as before. Then if we make the extra assumption that all elements which are not identically zero in each category are strictly non-zero on the fully ,constrained surface, then the entire analysis done earlier (for all three categories) can be repeated. This will yield either M= 2?, M’ = 27, or M = M’ = ?+ ?‘. Then, by virtue of (3.17), the first r” constraints in the &chain and the first ?’ constraints in the $-chain will have identically zero brackets with one another, with their total number ?+ ?’ half the total number of constraints 2n = M+ M’ in all cases. 4. MANY
CHAINS AND FIELD THEORY:
The extension of our analysis from the one-chain case to the two-chain case indicates that a similar approach can also be extended, at least in principle, to the many-chain case, i.e., a general second-class system with an arbitrary number of primary constraints. Consider a system with K primary constraints, denoted by xi’), Xl(2),...?Xl (U . From time consistency suppose each primary constraint leads to a chain as shown below, Xl x:l)=
Cl)=0
{Xl (1) > H}
(2)=0
XI =o
~~~~{~$~:‘,l,H}=O
xi’)=
{Xl C2) 2 H}=O
&+{&,,H}=O
. .. ...
Xl x2 (W
(K) = 0
=-(xyQ,H}=O
(4.1)
... ~~;={~~;~I,H}=O,
where (4.2)
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Altogether the full set of 2n constraints (where 2n = M, + Mz + ... + MK) is taken to be of the second class. Let us define rl, r2, .... rx- to be the largest integers such that { xy, p>
z 0,
i < r,, j < r,; 1, m arbitrary.
(4.3)
Depending on the results we are seeking, the brackets in (4.3) could be taken to vanish module all constraints, as in Section 2.1, or identically, as in Section 2.2. Then, proceeding as before, we see that the K x K matrix a11 @=
...
alK
:
(. aK1
(4.4) ...
am
-1
where cl,,,,- {xl:‘, i, x:‘J’}, must have at least one non-zero element in each row. The next step, analogous to Section 3, would be to divide all permissible matrices a into different categories. There will be one category that is particularly simple, where only the diagonal elements of c1 are non-zero. Call it category (i). This is the generalisation of category (i) in the two-chain case. As argued there, in this category, each chain can be treated separately and the problem reduces to that of K separate one-chain cases, with 2r, = M,, 2r, = Mz, .... and 2r, = Mk. The set of constraints which form the first half of each chain will have zero brackets with one another. To begin with, their brackets can be expected to vanish only modulo all the constraints. But if, just as in Section 2.2, we take (4.3) to hold everywhere in phase space, and assume that the non-zero elements of a are strictly non-zero on the fully constrained surface, then the top half of the constraints will have identically zero brackets with one another. These results are for category (i) defined above. For a K-chain system with arbitrary K> 2 there will be many other categories. The general principles for attacking the K-chain problem are the same as those in the K= 1 or the K= 2 case, but in practice they can become very tedious. Our analysis of the two-chain case suggests that in the generic situation the constraints {xi”’ 1i 6 rm, m = 1, 2, .... K), which form a first-class subset, are exactly half the total number of constraints. Next let us turn to local field theories. Suppose we have a set of fields $Jx, t) governed by a local Lagrangian L = s & 9(4,(x), d,(x)). Suppose the Lagrangian is singular and yields solely second-class constraints. Now, in field theory each constraint will be some field equation of the form x(d,(xh %2(x)) = x(x) = 0,
(4.5)
where X,(X) are the momenta conjugate to #Jx). Equation (4.5) is actually an infinite number of constraint equations, one at each space point x. In particular, even if a theory has only one primary constraint equation in the field theoretic sense, it corresponds to an infinite number of primary constraints and to an intinitechain system from our point of view. Fortunately, the property of locality simplifies
151
SECOND-CLASS CONSTRAINTS. I
matters greatly. The bracket between any pair of constraints x,(x) and xj(x) must, unless it vanishes identically, be proportional to 6(x - y) or to some derivative of 6(x - y). Now suppose the system has a set of second-class constraints xi(x), x2(x), ...Yx,(x), of which only the first, xl(x), is primary. In general, in field theory, the integer A4 need not be even. But suppose further that the different brackets {x,(x), x,(y)) either vanish or are proportional only to 6(x-y) and not its derivatives. Then although it is an infinite chain system, it clearly belongs to category (i) in the sense defined earlier. Our analysis can be carried out for each chain (that is, at each x) separately, with the results of Section 2 applying at each x. The integer M must then be even (M= 2n), and the arguments of Section 2 show that the matrix MV(x, y) = (x,(x), xj(y)} will have the same form as Eq. (2.19) with an extra factor of 6(x - y) on the right-hand side and with the non-zero entries (such as c( in (2.19)) now functions a(x). In particular the top half of the constraint functions xi(x), x*(x), .... x,(x) will have zero brackets with one another. Without further assumptions, these brackets will vanish only modulo all the M= 2n constraints. But, as in Section 2.2, if in the field theoretic analogue of Eq. (2.26), i.e., {xi+], xi} = G(x) 6(x-y) (where r” is defined as in Section 2.2), the function G(x) is non-zero everywhere on the submanifold satisfying xi(x) = x2(x) = . . = x,(x) =O, with i,<2J, then 2F=2n and the tirst half of the constraints (x,(x), x*(x), “‘> x,,(x)) will have identically zero brackets with one another. We illustrate this class of field theoretic systems with two examples in the next section. When some constraint brackets involve X(x - y), etc., the system does not belong to category (i) in the sense described above. Non-trivial multi-chain analysis will have to be carried out. The well-known model of a free scalar field in (1 + 1) dimensions, briefly recalled in the next section, will offer a simple example of this type.
5. EXAMPLES The analysis in Sections 2 and 3 makes predictions about the constraint algebra of any system with purely second-class constraints which flow from either one or two primary constraints. For many-primary-constraint cases and local field theory, Section 4 makes some predictions for certain categories. In this section we offer a few examples chosen to illustrate the different types of situations in which our analysis applies, both in non-relativistic mechanics and in relativistic field theory. a. Consider the family of models described by the Lagrangian 2
N-l +qN
c
i=l
where qi (i= 1, .... N) are coordinates, 595/203/l.
11
biqi,
(5.1)
and bi, di, and cii are some real constants.
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MITRA
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Although for special sets of values of the constants b,, di, and cii this Lagrangian could yield first-class constraints, for generic values it gives a theory with purely second-class constraints of the type we are interested in. As a specific illustration let us take, say, N = 5 and
The canonical Hamiltonian
is (5.3)
Following Dirac’s procedure, one can see that this system carries the following six second-class constraints:
(5.4)
Of these, only the first constraint, x1, is primary; the rest are secondary. Thus, this is an example of the one-chain case with 2n = 6 and we can see that the results of Section 2 hold. The matrix M, = {xi, xi}, given below, has indeed the form predicted in Eq. (2.19).
(5.5)
(We have not bothered to calculate the elements denoted by crosses in (5.5) since our theory has no predictions for them.) Note that this model is an example of
SECOND-CLASS
CONSTRAINTS.
153
I
possibility (c) discussed in Section 2.2. The bracket c(z {x4, x3} = - & is clearly non-zero everywhere. Correspondingly, as per the analysis in Section 2.2, we see that the top half of the constraints, viz., x1, xZ, and x3, have mutual brackets which vanish identically. b. Our next example illustrates the two-chain case. Consider the Lagrangian L = fcot + 45Y + 242 + 4612 + a: +
@6(@1
+
v2(@3,
+
43-q&2
+ Vl(@,, 941)
(5.6)
94)) - v3(@3>@4),
where VI, V,, and V, are any functions of their arguments. Then
This system has six second-class constraints, but two of them-#, primary, and the rest secondary as shown below.
and til--are
(5.8)
This system illustrates category (ii) of the two-chain case analysed in Section 3. As predicted, the upper half subset of the constraint set has zero mutual brackets. This subset can be taken to consist of #1, I++~,and either #2 or ti2. c. The preceding examples involved a finite number of degrees of freedom. The chiral Schwinger model in (1 + 1) dimensions offers a nice field theoretic example. The bosonized Lagrangian for the system (in the regularization a = 1; see Ref. [6]) is, using Lorentz covariant notation,
L=Jd+(a,4
+ Ap)(apd+ AF)- +F~,P- 8~”a,+u,].
Let 7c, no, E stand for momenta
(5.9)
conjugate respectively to 4, A,, A,. Then
K= I ~xC~(nz+(a,~)2+E2)+Ea,~o+(~+a,~+~I)(~1-~0)].
(5.10)
154
MITRA
AND
RAJARAMAN
The sequence of constraints is Xl(X) E no(x) z50 ~*(X)~a,(E+(d)+7c+A~MO
(5.11)
x3(x) = EwO x4(x)-
-7c-~a,4-2A,+A,~o.
It can be verified that these constraints form a second-class set, with only x1(x) primary. This is an infinite-chain case in our terminology, since (5.11) gives a chain of constraints at each x. However, all the brackets MU(x, y) = {x,(x), xi(r)} are proportional to 6(x - y). Thus, the example belongs to category (i) as described in Section 4. The chains at different values of x are not coupled by brackets and all single-chain results of Section 2 should hold at each x. This is indeed so, and the M,(x, y) can be verified to be
(5.12)
The chiral Schwinger model in the 01> 1 regularisation can be treated similarly, but it is less interesting from our point of view, since it has only two constraint equations. d. Our next example is the non-linear sigma model which, unlike the chiral Schwinger model, is a non-trivially interacting theory. Let G - (#(x, t), a = 1, 2, ..*, N} be a multiplet of N real scalar fields in (1 + 1) dimensions and 2(x, t) another scalar field. Let the Lagrangian be L =
s
dx[-a,a.
Pa + lf((a)],
(5.13)
where f(a) is any function of the fields CP such that the equations f(o) = 0 and V,f(a) =0 have no common roots. The well-known O(N) sigma model corresponds to f(a) = 0’ - 1 and is one such example. (At the quantum level, these models are renormalisable only in (1 + 1) dimensions, but at the classical level, our analysis will apply in any spatial dimensions.) If zl(x) and n(x) stand for fields conjugate to 1(x) and a(x), respectively, then
H,=jdX[tr2+f(axa)2-nf(u)].
(5.14)
SECOND-CLASS
CONSTRAINTS,
I
155
There are four constraint equations, all of the second class, given below. Of these only xl(x) is primary. The constraints are Xl(X) c ?rj,(X) z 0 x*(x) =f(4x))
= 0
x3(x) E 11:.V,fz x4(x) = A(V,f)’
(5.15)
0 + a+.
v,
f+ (II .V,)’ fx0.
We see that
~xl(x)ALYH= By assumption,
~x3w&(YH=
-P,f)'W-Y).
these brackets are non-vanishing
on the constrained
(5.16) surface since
V,f does not vanish where f does. The constraint sequence can be terminated with x4(x) since the requirement i4 = {x4, H} = 0 can be satisfied by including in term 1 dx xl(x) uO(x) with Q,(X) chosen suitably. We also see that the upper half of the constraint sequence, namely xl(x) and x2(y), will have identically vanishing brackets among themselves for all x and y. All these results are in accordance with our analysis in Section 4. H a permissible
e. Our last example is described by the Lagrangian
L=[dx{a,~a,+
+2*&j.
(5.17)
This is a trivial and well-known model-it is just the free Klein-Gordon theory in (1 + 1) dimensions in light cone coordinates. We mention it briefly since it illustrates an infinite-chain case which is not in category (i) in the terminology of Section 4. There is only one constraint equation,
X,(x)-~(X)--.~~cx)~oo,
(5.18)
with PAX), ~ha= The Hamiltonian
4
6(x - Y).
(5.19)
is H=
s
dx [$m’cj’+
u,,(x) x,(x)].
(5.20)
The requirement 0 = {x,(x),
H} = -rn’c$ - a,uO
(5.21)
is satisfied by choosing the arbitrary function Q,(X) appropriately. Hence there are no further constraints beyond (5.18). This is an infinite-chain case since (5.18) gives one primary constraint for each x, but each chain consists of only one member.
156
MITRAANDRAJARAMAN
Consider the Fourier transform (5.22)
Clearly { iI(
jgk’)}
= 2ik 6(k + k’).
(5.23)
Thus, the subset of iI with k 2 0 (which is, roughly speaking, half the set of constraints) has vanishing mutual brackets.
ACKNOWLEDGMENTS One of us (R. R.) thanks Dr. N. Mukunda for a discussion. P. M. acknowledges Centre for Theoretical Studies, Indian Institute of Science, Bangalore, where part
the hospitality of the of this work was done.
REFERENCES 1. P. A. M. DIRAC, “Lectures on Quantum Mechanics,” Academic Press, New York, 1964. 2. E. C. G. SUDAIUHAN AND N. MUKUNDA, “Classical Dynamics,” Wiley, New York 1972; A. HANSON, T. REGGE, AND C. TEITELBOIM, Acad. Naz. Lincei (Rome) (1976). Lecture Notes in Physics, Vol. 169, Springer-Verlag, 3. K. SUNDERMEYER, “Constrained Dynamics,” New York/Berlin, 1982. 4. L. D. FADDEEV AND R. JACKIW, Phys. Rev. Let?. 60 (1988), 1692. 5. J. A. SHOUTEN AND W. VAN DER KLUK, “Pfatl’s Problem and Its Generalization,” Clarendon Press, Oxford, 1949; L. P. EISENHART, “Continuous Groups of Transformations,” Dover, New York, 1969. 6. R. JACKIW AND R. RAJARAMAN, Phys. Rev. Left. 54 (1985), 1219; J. LOTT AND R. RAJARAMAN, Phys. La. B 165 (1985), 321.