# A hidden constraint on the Hamiltonian formulation of relativistic worldlines

## Abstract

Gauge theories with general covariance are particularly reluctant to quantization. We discuss the example of the Hamiltonian formulation of the relativistic point particle that, despite its apparent simplicity, is of crucial importance since a number of point particle systems can be cast into this form on a higher dimensional Rindler background, as recently pointed out by Hojman. It is shown that this system can be equipped with a hidden local, symmetry generating, constraint which on the one hand does not bother the classical evolution and on the other hand simplifies the realization of the path integral quantization. Even though the positive impact of the hidden symmetry is more evident in the Lagrangian version of the theory, it is still present through the suggested Hamiltonian constraint.

## 1 Introduction

Symmetries have been the guiding principle of theoretical physics throughout centuries. In particular local symmetries have shown to be particularly useful for the description of fundamental interactions. The description of all four known forces of nature: electromagnetism, weak interaction, strong interaction, and gravitation have been cast in this language. However, the last member of this illustrious list poses serious problems, when it comes to a quantum formulation of gravity. Numerous attempts have been made to solve the problem but, up to now, no conclusion could be reached (see [1, 2] for a review). Clearly, one needs to better understand the quantization of this theory, based on a particularly beautiful and complicated symmetry called general covariance. Since the rich structure of the full gravitational covariant system appears to be too complicated to tackle the problem directly, it seems instead that a wiser strategy is to learn more about general covariance in simpler systems.

- (a)
declare the Lagrangian action (1) to be wrong or at least inadequate for the purpose of quantization, and stick to the Hamiltonian version. This Hamiltonian version of the action was first formulated in [6, 9, 10];

- (b)
stick to the straight forward PI quantization of (1) and try to tackle the arising problems by a re-definition of the usual interpretation of probability, the super-probability [11, 12];

- (c)
- (d)
realize that the action has a hidden symmetry, which, when factored out of the PI solves the inconsistencies and the quantization works just fine, giving the expected results. The factorization was shown to work in a formal Fadeev Popov construction [14] and in a purely geometrical approach [15].

Is there a corresponding additional symmetry in the Hamiltonian system? If it exists, can this additional symmetry be written in terms of a (local) constraint? If this can be done, how does this constrain affect the PI construction of the Hamiltonian system?

On the following pages, those questions will be answered.

The paper is organized as follows: We first start with a thorough discussion of the different criteria that a Hamiltonian formulation of the path integral for the RPP must meet, in order to capture the symmetries involved. Then, we discuss the consequences of treating a Hamiltonian formulation that does not take into account the hidden symmetry. Finally, we propose an ansatz to involve this symmetry explicitly in the action, via a suitable constraint and a corresponding Lagrange multiplier. Moreover, by explicit calculation of the path integral, we show that this constraint does indeed allow us to recover the correct result for the RPP propagator. At last, we present the conclusions and possible extensions of our formulation to other physical systems of current interest.

## 2 Hamiltonian PI for the RPP

The aim is to formulate a Hamiltionian theory for the relativistic point particle that meets several criteria.

Summary of different criteria and corresponding motivation for symmetry-related constraints in the Hamiltonian formulation

Criterium | Motivation | |
---|---|---|

(1) | The Hamiltonian action includes an additional constraint \(\phi ^\mu \) that reflects the local velocity rotations symmetry discussed in the Lagrangian formulation | This is imposed because the original motivation is to find the meaning of the local velocity rotations in the Hamiltonian picture |

(2) | The constraint \(\phi ^\mu \) generates local symmetry transformation of the action | A local symmetry is needed in order to justify a later factorization from the PI, similar to the known redundant gauge configurations |

(3) | The equations of motion are in agreement with the classical equations of the RPP | Only if the systems are classically equivalent, one is still solving the same problem one was up for in the first place, namely the quantization of the RPP |

(4) | The PI of the additional constraint can be done and the result does not modify the expected Klein Gordon propagator | The factorization of the new symmetry is meant to act as improvement of the naive PI approach and thus should not introduce modifications where this naive approach already works |

### 2.1 Hamiltonian without hidden symmetry: summary

*n*is the Lagrange multiplier imposing the Hamiltonian constraint

*F*-intervals, such that \(\epsilon = (t_f - t_1)/F\) is the size of each time-slice. Hence, for \(t_j = t_1 + (j-1)\epsilon \), (\(1 \le j \le F\)), the discrete coordinates \(x(t_j)\rightarrow x_j\), and momenta \(p(t_j)\rightarrow p_j\), with fixed coordinates at the ends \(x(t_1) \rightarrow x_1\) and \(x(t_f) \rightarrow x_f\), respectively.

### 2.2 Ansatz for the constraint

As listed above, one wants a constraint which reflects the velocity rotations in the Lagrangian picture. Since velocity is a vector, a scalar constraint is insufficient. One needs at least a vector or tensor for this task. Further, in the equations of motion (5–7) the velocity is associated to the momentum, thus one might first attempt to formulate a constraint which transforms the momenta \(p^\mu \). In order to transform the momenta with a Poisson bracket with \(\phi ^\mu \), one needs this constraint to depend on the positions \(\phi ^\mu =\phi ^\mu (x)\). However, when working out the equations of motion and algebra, it becomes clear that such a position dependent constraint would have to be non-local in the position variables. This can be done, but we prefer to avoid the problems that come along with non-locality, and hence we search for a constraint that is local in momentum space but independent of \(x^\nu \), namely \(\phi ^\mu =\phi ^\mu (p)\). Still a change in the momentum can be achieved but in a different way, as will be seen.

### 2.3 Equations of motion

^{1}This can be seen more elegantly if one notes that \(p^\mu \) is not the canonical momentum \(\pi ^\mu \) any more since introducing (14) gives

### 2.4 The algebra of local transformations

*S*.

### 2.5 Path integral

It is very interesting to note, that in this PI construction the integrals of the two constraints canceled each other and thus, we did not even have to fix some gauges explicitly as in the usual case. This non-trivial impact on the PI formulation gives further evidence, that the new constraint, even though “hidden” at the classical level, is not “trivial” at the level of the quantum mechanical path integral formulation.

## 3 Discussion summary and outlook

We have presented a new constraint (12) for the RPP and shown that it reflects all the nice features demanded in table 1. In particular, it generates a non-trivial local symmetry of the action without altering the classical equations of motion of the system. This new PI formulation worked out in a straight forward way. Based on our results, we can conclude that the local symmetry of the Lagrangian version of the RPP action (1), which was discussed in [14, 15] can also be implement in the Hamiltonian version (19).

In the discussion of constraints, it is a useful exercise to count the degrees of freedom. For the symmetry presented in this paper, this discussion has to be differentiated between classical paths \(S=S_{cl}\) and quantum paths \(S>S_{cl}\). For the former it has been shown that the additional constraints are ineffective, leaving the usual degrees of freedom of a relativistic point particle. For the latter, however, the constraints become effective. In this case, three of the four conditions are independent, reducing the degrees of freedom for a given action *S* to zero. As shown in Sect. 2.5, this leaves, after applying the constraints, only one integral corresponding to \(\int _{} dS\) itself for the path integral.

This seems to be a very isolated result, only valid for a very particular system with general covariance. However, as recently shown by Hojman, there is a very large class of point particle systems which can be cast in the form of the free RPP which is living on a higher dimensional Rindler background [19, 20]. Thus, we believe that our results might be applicable to a much larger class of problems. Further, it would be interesting to explore certain similarities of the presented constraint with constraints imposed in delta-theories such as delta gravity [21]. Finally, our result encourages further investigation on more complicated covariant systems such as quantum cosmology [22] or ultimately quantum gravity in the canonical formulation [23] analogous to (2/19).

## Footnotes

- 1.
It is important to realize that this classical reduction to the usual equations of motion fails for the non-relativistic theory, because in this theory the classical relation between \(\dot{\mathbf {x}}\) and \(\mathbf {p}\) is fixed and an arbitary \(\mathbf {N}\) which is just proportional to \(\mathbf {p}\) would break this relation.

## Notes

### Acknowledgements

B. K. was supported by Fondecyt 1161150 and Fondecyt 1181694. E. M. was supported by Fondecyt Regular 1190361.

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