Dynamically constraining deconstrained dynamics

PHYSICAL REVIEW D VOLUME 45, NUMBER 4 15 FEBRUARY 1992

Dynamically constraining deconstrained dynamics

Jemal Guven Znstituto de Ciencias Nucleares, Universidad Nacional Autonoma de Mixico, Apartado Postal 70-543,

04510 Mexico, Distrito Federal, Mexico (Received 19 September 1991)

We examine the possibility of modifying the equations of motion for a reparametrization-invariant system in such a way that, no matter what initial data one provides, the system will always relax onto the constraint hypersurface.

PACS numberk): 04.20.Me, 03.20. + i, 03.65.Ca

I. INTRODUCTION

In any reparametrization-invariant theory, the constraints carry a particularly heavy burden, serving not only as one might expect to constrain the dynamical variables but also as the generator of both the dynamical evo- lution and the reparametrization of these variables.

If the constraints are satisfied initially, the Hamiltoni- an equations will guarantee that they will continue to be satisfied (Bianchi identities). Suppose, however, that the constraints are not initially satisfied. It is a remarkable fact that by the addition of some appropriate linear combination of the constraints to the dynamical equations, it is possible to arrange that the constraints will be satisfied exponentially rapidly [I].

I t is also true that we can relax the gauge conditions which freeze-out the reparametrization invariance of the theory. Thus it is possible to start with arbitrary initial data which satisfy neither the constraints nor the gauge condition which will nonetheless evolve towards the gauge-fixed constraint hypersurface.

One can examine the quantum mechanics of the modified system. To do this canonically involves recast- ing the system in Hamiltonian terms. While we can demonstrate that this is straightforward in principle, the Hamiltonian which results is not simple.

11. RELAXING THE CONSTRAINT

To illustrate what is involved, we will examine the simple reparametrization-invariant system described by the action

The Lagrange multiplier N enforces the constraint H =O where

while the dynamics is described by Hamilton's equations

The reparametrization invariance of the theory is broken by fixing the multiplier. The canonical way to do this is to implement a gauge condition of the form @( P, Q, t)=O,

where ( @, H ) f 0 and a, @# 0 . Then N is uniquely fixed by the requirement that the gauge be preserved:

An alternative procedure is to fix the multipliers directly. However, while the description of the dynamics so described is simplified, the identification of the physical de- grees of freedom becomes more difficult.

Clearly, the dynamics is unaltered by the addition of a multiple of the constraint to the right-hand side of these equations. Let us then examine the modified equations

where F and G are two functions of P A and Q A. Howev- er, let us also suppose that H f O . What is remarkable is that it is possible to choose F and G such that [ l ]

where a is some positive constant. Then, even if the constraint is not satisfied at some initial time, it will be exponentially rapidly:

The parameter a can presumably be chosen to be large enough so that the damping time scale is more rapid than any characteristic time scale in the problem. This provides us with a mechanism for solving the constraint for free. Any unconstrained data we choose will be driven onto the constraint hypersurface by the damping term.

An obvious application of this mechanism is to the nu- merical solution of the equations of motion and a similar device was proposed in Ref. [2] within the context of gen- eral relativity. As one propagates the initial data, numer- ical errors will accumulate, driving the propagated data off the constraint hypersurface. The addition of the damping term is a device to nudge the data back.

In the case of the model problem, the most simple choice implementing the requirement ( 6 ) is ( V f O )

For exponential damping, the choice appears to be

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unique. A less judicious choice of F and G, for example,

yields a slower damping onto the constraint hypersurface:

If V=O, then the exponential damping is inconsistent because then the correction is no longer proportional to the constraint so that PA = - a / 2 P A . The solution PA ( t ) would then tend asymptotically to zero which clearly is not the asymptotic behavior we are looking for. The best one can do is to implement the slower damping corresponding to (9a) with V=O.

111. RELAXING THE GAUGE

If there is no constraint, there is no gauge freedom to fix and it is no longer obvious what role the multiplier plays in the modified equat i~ns of motion, Eq. (5). Sim- ply eliminating N using the @=O criterion [with the obvious modifications implied by Eq. (5)] would reconstrain the system. One possibility is to relax the gauge condition just as we did the constraint. The most obvious way to do this is to introduce a damping term into Eq. (4) so that any (consistent) canonical gauge function @ we choose will relax to @=O in an analogous manner to the relaxation of the constraint. Let us therefore suppose that we have made a particular choice of @# 0. To implement the damping, we replace the condition 6 =O by

where y is some other positive constant. Then, N is determined by the equation

A nice feature is that the asymptotic boundary condition @-to is now automatically satisfied by @. A similar con- struction was exploited in Ref. [3]. If we now substitute for N from Eq. (9) into the modified Hamilton's equations, we are left with equations of motion which will evolve arbitrary (unconstrained) initial data ( eiA, P,, J onto the gauge-fixed constraint hypersurface. This is, of course, no surprise-it is after all no more than we put in by hand. It does, however, provide us with a mechanism for generating solutions to the initial-value problem. No matter what initial data (QiA, P,, j we specify, they will rapidly relax to values ( e l A * , P ; ~ J that are consistent with H=O and @=0. What we have done is trade the initial-value problem for another problem. The ( QiA,pA,) must be tuned appropriately if the data

(QiA*,P;,] are to lie in a physically interesting region of phase space. While we have not gotten anything for free, it does appear that we can learn something about the constrained system by examining the corresponding deconstrained svstem.

There are features of this deconstrained system which invite comparison with the Hamiltonian Becchi-Rouet- Stora-Tyutin (BRST) formalism introduced by Batalin, Fradkin, and Vilkovisky [4]. By introducing additional multiplier and ghost variables, the classical equations of motion can be made to correspond to an unconstrained dynamical system. It is only when appropriate (BRST- invariant) boundary conditions are imposed on the additional variables that we project onto the gauge-fixed constrained equations of motion.

IV. HAMILTONIAN REFORMULATION OF THE PROBLEM

The modification of the dynamics we have explored here suggests an approach to attacking the quantization of the system. Upon first reflection, this route does not appear particularly hopeful because we have completely mutilated the original Hamiltonian formalism. However, this does not exclude the possibility of reconstructing another (new) Hamiltonian formulation of the problem. One way is to first obtain the (unconstrained) equations of motion

which we obtain from the modified Hamiltonian equations upon eliminating PA and N. The key step is to determine if these equations correspond to the stationary point of some action functional, i.e., if

where G A L represents the Euler-Lagrange derivative of a Lagrangian function, L ( Q A, Q A, t ) and WAB is some (nonsingular) matrix. This is a problem which has been studied (the inverse problem in classical mechanics) [5]. The question of the existence can be recast in terms of a set of conditions (Helmholz) on W,,:

and

where we define

a A A B = ~ A B - ( ~ 2 ) A B - - ~ A d t

, and E""=* aQB '

Two of the conditions are algebraic, and two are

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differential. W A B might not exist. If it does, it will not be unique. For a given W A B , the corresponding Lagrang- ian is given by

where P A ( Q A , Q A , Q A , t ) = w ~ ~ E ~ . The boundary term

serves to cancel the second derivatives which would oth- erwise appear.

To put the formalism to the test, let us examine a mas- sive relativistic free particle with v ( Q ) = m 2/2. We need to ensure that the second-order equations of motion (12) do not possess any spurious solutions inconsistent with Eq. (11) prescribing how fast the combination of the dynamical variables @ relaxes to @=O. Suppose, for example, we attempt to relax onto a canonical gauge condition such as Qo- t =o. Whereas solutions to Eq. (1 1) automatically satisfy the boundary condition a - 0 as t - a , the differentiated equation & = - y& possesses spurious solutions which do not. The necessity to specify boundary conditions so as to eliminate these solutions is clearly a problem. How, for example, would such boundary conditions survive the quantization process?

The extreme simplicity of the relativistic free particle model has the drawback that it severely limits the possible gauge-fixing conditions we can implement. In particular, the extrinsic time gauge @ = P o - t = O is not consistent with ( @ , H ] #O and so breaks down. In minisu- perspace models for gravity, however, H will typically de- pend explicitly on Q A and such gauges become viable. The important point is that there will now be no need to supplement the second-order equations of motion with any boundary conditions to reproduce the correct asymptotic behavior of the relaxed @. They therefore are free of the shortcoming we encountered earlier characteristic of (intrinsic time) gauges depending only on Q A and t .

Rather than explore extrinsic time gauges in a less trivial model let us explore the of applying a noncanonical gauge to the relativistic particle model. Suppose that N = l whether or not we are on the constraint hypersurface. Off the surface there is no constraint on the Q A and P A implied by this condition. In addition, no boundary conditions will be necessary on the dynamical variables as t + a . With this choice, one can easily demonstrate that a Lagrangian can be constructed using the algorithm we presented earlier, which is reassuring. However, even though the Lagrangian can be constructed, the boundary term (15) which serves to eliminate second derivatives is not entirely trivial. In practice, this makes it difficult to proceed. In a nontrivial model, H will involve Q A explicitly and the situation is even worse.

Let us suppose that we can always construct a La- grangian. We can then proceed with the definition of new momenta FA (no relation to the old ones) conjugate to Q " ,

and construct a Hamiltonian H through the Legendre transformation

H ( H , , Q * , ~ ) = P , Q A - ~ . (17)

The existence of the invertible matrix W A B will also guarantee that the canonical formalism can be imple- mented. This is because

so that Eq. (16) can be inverted for the velocities in terms of the momenta.

The canonical quantization of the system is now, in principle, straightforward. T o begin with, there are no constraints, and no gauge conditions. We let the system evolve using the Schrodinger equation corresponding to fi. Modulo the possible necessity to tune initial conditions, we need only to focus on the asymptotic behavior corresponding to the gauge-fixed constrained classical theory. In this limit, we would expect the constraint to hold only in the mean on physical states, ( H ) =O, with nonvanishing fluctuations ( H ~ ) off the constraint hypersurface. This is very different from the Dirac quantized theory in which physical states are annihilated by the constraints. In addition, there is the distinction that no gauge condition is ever invoked in the Dirac theory.

We have already noted that difficulties arise (associated with the boundary conditions which are necessary to eliminate spurious solutions) when one attempts to implement the program using a canonical gauge. The noncanonical choice we therefore pursued does not appear to allow a simple reduced phase-space interpretation which always involves the specification of a canonical gauge.

V. CONCLUSIONS

We have examined a modification of the equations of motion of a reparametrization-invariant system (involv- ing the addition of an appropriate linear combination of the constraints to Hamilton's equations) in such a way that even if the constraints are not initially satisfied, the system will relax automatically onto the constraint hypersurface. We have discussed the possibility of setting up classical initial data by exploiting the resulting dissi- pation in the equations of motion. We have addressed the problem of canonically quantizing such a theory. While this appears to be straightforward in principle, it turns out to be difficult to implement in practice. What appears to be true is that the asymptotic limit of the modified theory does not correspond to either of the currently favored approaches to quantization. Faced as we are with the shortcomings of these candidates, this is not necessarily a bad thing.

ACKNOWLEDGMENTS

I would like to thank N i a l l 0 Murchadha who suggest- ed the idea and Francisco Pardo for describing to me the solution of the inverse problem in classical mechanics. I would also like to thank Michael Ryan for helpful com- ments.

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[I] The possibiilty appears to have a long history dating back at least to the early 1970s when it was pointed out by Charles Misner.

[2] S. Detweiler, Phys. Rev. D 35, 1095 (1987). Another possibility we do not pursue is the addition of a term quadra- tic in H to the Hamiltonian.

[3] N. 0 Murchadha, in the proceedings of GR9, Jena, 1980, edited by E. Schmutzer (unpublished). A global obstruc- tion to the implementation of the maximal slicing condition which was exploited here was later pointed out by

Dieter Brill, in Relativity and Gravitation, Proceedings of the 10th International Conference, Padua, Italy, 1983, edited by B. Bertotti et al. (Reidel, Dordrecht, Nether- lands, 1989).

[4] For a review, see M. Henneaux, Phys. Rep. 126, 1 (1985). [5] This is classical if not very widely known. See, for exam-

ple, R. M. Santilli, Foundations of Theoretical Mechanics I (Springer-Verlag, New York, 1978). A useful introduction to the subject can be found in F. Pardo, Ph.D. thesis, Uni- versity of Texas, 1987,

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Dynamically constraining deconstrained dynamics