Abstract

We present a new covariant approach to the quantum mechanics of a charged 1/2-spin particle in given electromagnetic and gravitational fields. The background space is assumed to be a curved Galileian spacetime, that is a curved spacetime with absolute time. This setting is intended both as a suitable approximation for the case of low speeds and feeble gravitational fields, and as a guide for eventual extension to fully Einstenian space-time. Moreover, in the flat spacetime case one completely recovers standard non-relativistic quantum mechanics.

This work is a generalization of previous work by Jadczyk and Modugno, where the quantum mechanics of scalar particles was formulated with a similar approach.

1991 MSC: 15A66, 53A50, 53C07, 81R20, 17B66, 53C15, 58A20, 58F06

Keywords: spin, Galileian spacetime, quantum mechanics on a curved background, jets, connections.

Introduction

Recently Jadczyk and Modugno have proposed a new geometric formulation of the quantum mechanics of a scalar charged particle, with given gravitational and electromagnetic classical fields, in the framework of a general relativistic Galileian space-time. In this paper we extend that formulation to the quantum mechanics of a particle with spin 1/2.

Our work is related to a wide literature on classical and quantum Galileian theory, starting from E. Cartan. Moreover our theory has evident relations, but also important differences, with geometric quantization. Our touchstone is standard quantum mechanic.

Our research is intended as a step toward a covariant formulation of quantum mechanics in an Einstein general relativistic background. In fact, such a full goal would demand the solutions of too many problems at the same time; so, it is worth splitting the research into steps by separating different kinds of difficulties.

We found that the Galileian general relativistic spacetime provides a suitable background for a start. Thus our current setting stands in between a non-relativistic and a fully relativistic formulation of quantum mechanics. It is mathematically self-consistent, while from the physical point of view it is intended both as a suitable approximation for the case of low speeds and feeble gravitational fields, and as a guide for eventual extension to fully Einstenian spacetime. Actually, the assumptions of a classical spacetime with absolute time and a Euclidean spacelike metric allows us to skip (temporarirly) some difficulties related to the Lorentz metric, but we pay a price for that. Namely, we are forced to consider a weaker version of the Maxwell and Einstein equations. Nevertheless, what we learn in this weakened context seems to preserve its interest in view of future developments. Moreover, in the flat spacetime case one completely recovers standard non-relativistic quantum mechanics along with new understanding of known objects.

The mathematical language of the paper is that of the geometry of fibred manifolds, jets and non-linear connections. We do not deal explicitly with theoretical group representations: rather we directly obtain physical objects from our starting structures via functorial methods; of course, the resulting objects are automatically equivariant with respect to the action of the groups of automorphisms of the starting structures. The reader who is not completely acquainted with this language will find, besides intrinsic formulations, a full coordinate description of all results.

The main points of our theory can be summarized as follows.

First, we sketch the essential features of our background classical spacetime. Namely, we assume a 4-dimensional spacetime fibred over time and equipped with a spacelike Euclidean metric, a time preserving linear connection (the gravitational field) and a 2-form (the electromagnetic field). We can couple the gravitational and electromagnetic fields into a unique spacetime connection; this yields a number of 'total' geometric objects, including a cosymplectic 2-form which will play a key role. We postulate the closure of this form thus obtaining a link between the above geometrical structures and the first Maxwell equation; moreover, we postulate a kind of 'reduced' Einstein and second Maxwell equations expressing the interaction of the above fields with their matter sources. The cosymplectic form yields a distinguished Lie algebra of functions, which are called 'quantizable' in view of their role in the theory of quantum operators.

Then we develop the quantum theory starting from the quantum bundle, defined to be a Hermitian bundle over spacetime; its fibres are either 1-dimensional (scalar case) or 2-dimensional (spin case). On the scalar quantum bundle we assume a Hermitian connection which, in a sense, is parametrized by all classical observers, and has some natural properties (it is 'universal' and its curvature is proportional to the cosymplectic form). In the spin case we postulate a 'Pauli map', which is an isometry between the bundle of spacelike vectors and the bundle of Hermitian endomorphisms of the quantum spin bundle; this, via a natural link with the scalar case, yields a Hermitian connection on the quantum spin bundle. This is our only primitive quantum structure; all other objects will be derived from it getting free from observers through a 'principle of projectability' which is our implementation of covariance. In particular we obtain a distinguished Lagrangian, which yields the generalized Pauli equation and conserved quantities. Quantum operators are obtained in three steps. First, we exhibit a distinguished algebra of quantum vector fields which preserves the quantum structures, and study its relation with the algebra of quantizable functions. Then, we show the natural action of quantum vector fields, as 'almost-quantum operators', on 'quantum histories' (sections of the quantum bundle). Eventually we introduce the quantum Hilbert bundle over time and show how to obtain quantum operators from almost-quantum operators. To this end we have to eliminate the time derivative; we accomplish this task by a geometric procedure which uses the quantum Euler-Lagrange operator.

The original features of the paper can be summarized as follows.

i) Time, both in the classical and quantum theory, is not merely a parameter, but an essential ingredient which deeply affects all involved structures. Actually we point out - in contrast with an approach usually implicit in geometric quantization - that spacelike structures do not carry sufficient physical information for a covariant theory. Accordingly, we deal with a cosymplectic rather than symplectic form, with a spacetime rather than vertical (spacelike) connection, and so on. Also, jets are required by a manifestly covariant formulation; in particular, the jet space of spacetime plays the role of phase-space and replaces the more standard tangent space.

ii) New connections are introduced and studied. These play a fundamental and unifying role. In particular, the coupling of the electromagnetic and gravitational fields is represented by a spacetime connection which works in classical field theory and mechanics as well as in quantum mechanics; on the other hand, all quantum structures are derived from the quantum connection. With regard to the latter, we observe that the notion of 'universality' of a connection allows us to skip the problem of polarizations, typical of geometric quantization (we do not need to know the constants of motion in order to develop the quantum theory). Furthermore, the quantum Euler-Lagrange operator is interpreted as a connection on the infinite-dimensional Hilbert bundle (whose definition uses the notion of smoothness introduced by A. Frölicher).

iii) We obtain a generalized Pauli equation and quantum operators in the curved case. Actually, a quantization procedure (a way of obtaining quantum operators from classical observables) was not a primary goal of our approach; however, as a matter of fact, we get a quantization just as a free consequence of geometric results arising naturally in our discussion. We obtained natural algebras of quantizable functions and quantum vector fields, which yield quantum operators in two steps: first by considering sections of the quantum bundle over spacetime (almost-quantum operators), and then sections of the Hilbert bundle over time. In particular we are able to skip the problems of ordering, and achieve the quantum operator corresponding to energy. Note also that, differently from other geometrical approaches to quantum mechanics, no new quantum example is required (all non-relativistic examples of standard quantum mechanics hold automatically in our formulation).

iv) By the way, several results are obtained within the covariant approach to classical mechanics on a curved Galileian background. In particular, the study of the first and second order spacetime connections and the cosymplectic form, and a compact formulation of the link between the (non-relativistic) metric and spacetime connection. Moreover we draw conclusions which are not common belief: classical mechanics cannot be covariantly formulated through a Lagrangian or Hamiltonian approach; only an approach based on a non-linear connection is suitable for that (the Hamiltonian stuff, however, has an important role in the correspondence principle for quantum mechanics).

v) Finally, we introduce a new mathematically rigorous treatment of physical quantities, which makes our approach manifestly independent of the choice of measurement units. By the way, these methods may have also pedagogical interest.

Acknowledgements. This research has been supported by Italian MURST (national and local funds), by GNFM of Consiglio Nazionale delle Ricerche and by the EEC contract N. ERB CHRXCT 930096. Thanks are due to Andrzej Trautman for stimulating discussion.