Symmetries in covariant classical mechanics

Abstract

In the framework of covariant classical mechanics (i.e. , generally relativistic classical mechanics on a spacetime with absolute time) developed by Jadczyk and Modugno, we analyse systematically the relations between symmetries of geometric objects. We show that the (holonomic) infinitesimal symmetries of the cosymplectic structure and of its horizontal potentials are also symmetries of spacelike metric, gravitational and electromagnetic fields, Euler-Lagrange morphism and Lagrangians. Then, we provide a definition for a covariant momentum map associated with a group of cosymplectic symmetries using a covariant lift of functions of phase space. In the case when the cosymplectic symmetries projects on spacetime we see that the components of this momentum map are quantisable functions in the sense of Jadczyk and Modugno. Finally, we illustrate the results by some examples.At the beginning of the 90's M. Modugno and A. Jadczyk proposed a new geometric framework for a generally covariant classical and quantum mechanics on a curved spacetime with absolute time [15,16,31], based on jets, connections and cosymplectic forms. This approach was later developed in [4,14,30,32,38,39,41]. The theory will be referred to as 'covariant classical Galilei theory' (CCG) and 'covariant quantum Galilei theory' (CQG).The theory was partially inspired by the wide literature on geometric formulations of classical and quantum mechanics. Main sources were symplectic and cosymplectic classical mechanics [1,3,6,13,24,26,27,28], Newton or Galilei classical mechanics as general relativistic theories [2,7,8,9,10,21,22,36,37], quantum theories of mechanics within a symplectic or cosymplectic framework (such as geometric quantisation) [5,9,13,19,23,34,35,37,42].The models of CCG and CQG share nice ideas with the above literature, trying at the same time to avoid some typical problems. For instance, both theories are explicitly covariant with respect to changes of coordinates, even time-dependent ones. This feature partially comes from the cosymplectic structure of the phase space in the classical theory, which is also the classical background for the quantum theory, and overcomes the problem of explicit time independence of symplectic mechanics. On the other hand, the cosymplectic structure of CCG is of a special, physically reasonable, type. Therefore, it excludes those problems of the general cosymplectic formalism, which have no physical meaning. Additionally, both theories are covariant with respect to the choice

An Algebraic Approach to Physical Scales

“…Finally, we show how unit spaces can be employed in physical theories by the language of differential geometry. The formalism discussed in this section has been widely used in several papers dealing with physical theories (see, for instance, [8,17,18,25,31,34,35]). In the present paper we analyse the mathematical foundations of this formalism for the first time.…”

A covariant approach to the quantization of a rigid body

“…We start with a brief sketch of the basic notions of "covariant quantum mechanics", paying attention just to the facts that are strictly needed in the present paper. We follow [29,31,32,33,35,53,61]. For further details and discussions the reader should refer to the above literature and references therein.…”

“…A covariant formulation of classical and quantum mechanics on a curved spacetime with absolute time based on fibred manifolds, jets, non linear connections, cosymplectic forms and Frölicher smooth spaces has been proposed by A. Jadczyk and M. Modugno [28,29] and further developed by several authors (see, for instance, [8,30,31,34,53,61,48]). We shall briefly call this approach "covariant quantum mechanics".…”

A differential equations approach to Hamiltonian systems

“…Here, the symbol denotes objects derived from the gravitational field. The pair (Ω , dt) defines a scaled cosymplectic structure on the phase space [16].…”

“…The total 2-form Ω encodes the full structure of spacetime (metric, gravitational and electromagnetic field) and hence plays a central role in the theory. Moreover, (J 1 E, Ω, dt) is a scaled cosymplectic manifold [16].…”