A covariant approach to classical and quantum mechanics of a rigid body (original) (raw)

A covariant approach to the quantisation of a rigid body

This paper concerns the quantisation of a rigid body in the framework of ``covariant quantum mechanics'' on a curved spacetime with absolute time. The basic idea is to consider the multi-configuration space, i.e. the configuration space for nnn particles, as the nnn-fold product of the configuration space for one particle. Then we impose a rigid constraint on the multi-configuration space. The resulting space is then dealt with as a configuration space of a single abstract `particle'. The same idea is applied to all geometric and dynamical structures. We show that the above configuration space fits into the general framework of ``covariant quantum mechanics''. Hence, the methods of this theory can be applied to the rigid body. Accordingly, we find exactly two inequivalent choices of quantum structures for the rigid body. Then, we evaluate the quantum energy and momentum operators and the `rotational part' of their spectra. We provide a new mathematical interp...

Quantum mechanics of a generalised rigid body

2015

We consider the quantum version of Arnold's generalisation of a rigid body in classical mechanics. Thus, we quantise the motion on an arbitrary Lie group manifold of a particle whose classical trajectories correspond to the geodesics of any one-sided-invariant metric. We show how the derivation of the spectrum of energy eigenstates can be simplified by making use of automorphisms of the Lie algebra and (for groups of Type I) by methods of harmonic analysis. We show how the method can be extended to cosets, generalising the linear rigid rotor. As examples, we consider all connected and simply-connected Lie groups up to dimension 3. This includes the universal cover of the archetypical rigid body, along with a number of new exactly-solvable models. We also discuss a possible application to the topical problem of quantising a perfect fluid.

Generally covariant quantum theory

Journal of Physics: Conference Series

This paper represents a concise introduction to the quantum theory of point particles in a time orientable curved spacetime part of which was presented in the DICE conference in Castiglioncello, Italy.

Covariant quantum mechanics and quantum symmetries

2002

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.

Spacetime states and covariant quantum theory

In it's usual presentation, classical mechanics appears to give time a very special role. But it is well known that mechanics can be formulated so as to treat the time variable on the same footing as the other variables in the extended configuration space. Such covariant formulations are natural for relativistic gravitational systems, where general covariance conflicts with the notion of a preferred physical-time variable. The standard presentation of quantum mechanics, in turns, gives again time a very special role, raising well known difficulties for quantum gravity. Is there a covariant form of (canonical) quantum mechanics?

On obtaining classical mechanics from quantum mechanics

Classical and Quantum Gravity, 2007

Constructing a classical mechanical system associated with a given quantum mechanical one, entails construction of a classical phase space and a corresponding Hamiltonian function from the available quantum structures and a notion of coarser observations. The Hilbert space of any quantum mechanical system naturally has the structure of an infinite dimensional symplectic manifold ('quantum phase space'). There is also a systematic, quotienting procedure which imparts a bundle structure to the quantum phase space and extracts a classical phase space as the base space. This works straight forwardly when the Hilbert space carries weakly continuous representation of the Heisenberg group and one recovers the linear classical phase space R 2N. We report on how the procedure also allows extraction of non-linear classical phase spaces and illustrate it for Hilbert spaces being finite dimensional (spin-j systems), infinite dimensional but separable (particle on a circle) and infinite dimensional but non-separable (Polymer quantization). To construct a corresponding classical dynamics, one needs to choose a suitable section and identify an effective Hamiltonian. The effective dynamics mirrors the quantum dynamics provided the section satisfies conditions of semiclassicality and tangentiality.

A Novel Approach to Quantum Gravity in the Presence of Matter without the Problem of Time

International Journal of .Modern Physics A 35 (2020) 04, 2050015, 2020

An approach to the quantization of gravity in the presence matter is examined which starts from the classical Einstein-Hilbert action and matter approximated by "point" particles minimally coupled to the metric. Upon quantization, the Hamil-ton constraint assumes the form of the Schrödinger equation: it contains the usual Wheeler-DeWitt term and the term with the time derivative of the wave function. In addition, the wave function also satisfies the Klein-Gordon equation, which arises as the quantum counterpart of the constraint among particles' momenta. Comparison of the novel approach with the usual one in which matter is represented by scalar fields is performed, and shown that those approaches do not exclude, but complement each other. In final discussion it is pointed out that the classical matter could consist of superparticles or spinning particles, described by the commuting and anticommuting Grassmann coordinates, in which case spinor fields would occur after quantization. .

Classical and quantum dynamics for an extended free rigid body

Differential Geometry and its Applications, 2010

In this paper, a free rigid body of dimension three is extended and analysed both in classical and quantum mechanics. The extension is performed by bringing the inverse inertia tensor, which is a positive-definite symmetric matrix for the ordinary rigid body, into an arbitrary real symmetric one. With an arbitrary real symmetric matrix chosen, associated is a Lie-Poisson structure on the Euclidean space of dimension three, through which the classical dynamics for an extended free rigid body is defined, and characterized by two first integrals. In parallel to this, the quantum dynamics is formulated as the problem of simultaneous spectral resolution of the two operators which are viewed as the quantization of the two classical first integrals. Intensive use is made of the unitary representation theory for Lie groups concerned. The explicit spectral resolution is obtained, in particular, when the extended free rigid body is an extended free symmetric top.