Marco Modugno

Quantum mechanics of a spin particle in a curved spacetime with absolute time

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 nonrelativistic quantum mechanics. This work is a generalization of [JM93], where the quantum mechanics of scalar particles was formulated with a similar approach.

Covariant Schrödinger operator

We analyse the Schrodinger operator for a quantum scalar particle in a curved spacetime which is fibred over absolute time and is equipped with given spacelike metric, gravitational field and electromagnetic field. We approach the Schrodinger operator in three independent ways: in terms of covariant differentials induced by the quantum connection, via a quantum Lagrangian and directly by the only requirement of general covariance. In particular, in the flat case, our Schrodinger operator coincides with the standard one.

Torsion and Ricci tensor for non-linear connections

Abstract We study a natural generalization of the concepts of torsion and Ricci tensor for a non- linear connection on a fibred manifold, with respect to a given fibred soldering form. Our results are achieved by means of the differentials and codifferentials induced by the Frolicher-Nijenhuis graded Lie algebra of tangent valued forms.

Some variations on the notion of connection

Distributions on manifolds are studied in terms of jets of submanifolds and are interpreted as «pre-connections» or «almost-fibrings»; the associated differential calculus is developed in detail. A comparison with connections on fibred manifolds is analysed. Moreover, «higher order pre-connections», defined as pre-connections dependent on jets of arbitrary order, are introduced and studied. It is shown that infinite jets play an essential role in the associated differential calculus.

On the geometric structure of gauge theories

In the framework of the adjoint forms over the jet spaces of connections and using a canonical jet shift differential, we give a geometrical interpretation of the Yang–Mills equations both in a direct and Lagrangian formulation.

Classification of infinitesimal symmetries in covariant classical mechanics

In the framework of general relativistic classical mechanics on a spacetime with absolute time, we classify the infinitesimal symmetries of the classical structure by means of distinguished Lie subalgebras of the Lie algebra of “special phase function.” These subalgebras are crucial also for the classification of infinitesimal quantum symmetries, which will be analyzed in a forthcoming paper.

Hermitian vector fields and special phase functions

We start by analyzing the Lie algebra of Hermitian vector fields of a Hermitian line bundle. Then, we specify the base space of the above bundle by considering a Galilei, or an Einstein spacetime. Namely, in the first case, we consider, a fibred manifold over absolute time equipped with a spacelike Riemannian metric, a spacetime connection (preserving the time fibring and the spacelike metric) and an electromagnetic field. In the second case, we consider a spacetime equipped with a Lorentzian metric and an electromagnetic field. In both cases, we exhibit a natural Lie algebra of special phase functions and show that the Lie algebra of Hermitian vector fields turns out to be naturally isomorphic to the Lie algebra of special phase functions. Eventually, we compare the Galilei and Einstein cases.

GEOMETRIC STRUCTURES OF THE CLASSICAL GENERAL RELATIVISTIC PHASE SPACE

This paper is concerned with basic geometric properties of the phase space of a classical general relativistic particle, regarded as the 1st jet space of motions, i.e. as the 1st jet space of timelike 1-dimensional submanifolds of spacetime. This setting allows us to skip constraints. Our main goal is to determine the geometric conditions by which the Lorentz metric and a connection of the phase space yield contact and Jacobi structures. In particular, we specialize these conditions to the cases when the connection of the phase space is generated by the metric and an additional tensor. Indeed, the case generated by the metric and the electromagnetic field is included, as well.

Covariant Quantum Mechanics and Quantum Symmetries

We sketch the basic ideas and results on the covariant formulation of quantum mechanics on a curved spacetime with absolute time equipped with given gravitational and electromagnetic fields. Moreover, we analyse the classical and quantum symmetries and show their relations.

Natural maps on the iterated jet prolongation of a fibred manifold

SummaryUsing an analytic procedure by the first author [5, 6], we first determine all natural transformations of the second iterated jet prolongation J1J1Y of a fibred manifold Y→X into itself depending on a linear connection Λ on the base manifold X. We obtain two 3-parameter families and we interpret them geometrically. Our results clarify the distinguished role of the involution on J1 J1 Y depending on Λ introduced by the second author [11]. Then we discuss the role of our transformations in the theory of the natural operators transforming a connection Γ on a fibred manifold Y→X and a linear connection Λ on X into a connection on the first jet prolongation J1Y→X of Y. In the final remark we determine all natural transformations of the second sesquiholonomic and holomonic jet prolongations of Y into themselves. Our attention to second order jet spaces is due to the role they play in fundamental geometric and physical theories (cf. curvature of connections and lagrangian theories).