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arXiv:1405.2688

Quantized Affinely Rigid Bodies

J. J. S lawianowski, Vasyl Kovalchuk, Agnieszka Martens, Ewa Eliza Rożko

Published May 12, 2014• 0 citations
arXivHTMLPDF

Abstract

Developed are the main ideas of the quantized version of affinely-rigid motion. We base on the usual Schrodinger formulation of quantum me- chanics in the configuration manifold. The latter is given, in our case, by the affine group or equivalently by the semi-direct product ofthe linear group GL(n,R) and the space of translations R n , where n equals the di- mension of the physical space. In particular, we discuss the problem of the dynamical invariance of the kinetic energy under the action of affine group, not only under the isometry subgroup. Technically, the treatment is based on the two-polar decomposition of the matrix of the internal con- figuration space and on the Peter-Weyl theory of generalized Fourier series on Lie groups. One can hope that our results may be applied in quantum problems of nuclear dynamics or even in apparently exotic phenomena in vibrating neutron stars. And, of course, some more prosaic applications in molecular dynamics are possible. It is well known that every dynamical scheme, both mechanical and field- theoretical, may be formulated in two alternative versions: classical and quan- tum. This is always formally possible, although there are situations when one of those versions looks artificial, either classical or quantum. Besides, there are still some non-solved problems concerning the relationship between classical and quantum levels of description. Roughly speaking, they have to do with the mea- surement phenomena and decoherence processes. There is a whole spectrum of views, like, e.g.:

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Developed are the main ideas of the quantized version of affinely-rigid motion. We base on the usual Schrodinger formulation of quantum me- chanics in the configuration manifold. The latter is given, in our case, by the affine group or equivalently by the semi-direct product ofthe linear group GL(n,R) and the space of translations R n , where n equals the di- mension of the physical space. In particular, we discuss the problem of the dynamical invariance of the kinetic energy under the action of affine group, not only under the isometry subgroup. Technically, the treatment is based on the two-polar decomposition of the matrix of the internal con- figuration space and on the Peter-Weyl theory of generalized Fourier series on Lie groups. One can hope that our results may be applied in quantum problems of nuclear dynamics or even in apparently exotic phenomena in vibrating neutron stars. And, of course, some more prosaic applications in molecular dynamics are possible. It is well known that every dynamical scheme, both mechanical and field- theoretical, may be formulated in two alternative versions: classical and quan- tum. This is always formally possible, although there are situations when one of those versions looks artificial, either classical or quantum. Besides, there are still some non-solved problems concerning the relationship between classical and quantum levels of description. Roughly speaking, they have to do with the mea- surement phenomena and decoherence processes. There is a whole spectrum of views, like, e.g.:
Abstract 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.
This paper concerns the quantization of a rigid body in the framework of 'covariant quantum mechanics' on a curved spacetime with absolute time. We consider the configuration space of n classical particles as the n -fold product of the configuration space of one particle. Then, we impose a rigid constraint and the resulting space is dealt with as a configuration space of a single abstract 'particle'. This classical framework turns out to be suitable for the formulation of covariant quantum mechanics according to this scheme. Thus, we quantize such a 'particle' accordingly. This scheme can model, e.g., the quantum dynamics of extremely cold molecules. We provide a new mathematical interpretation of two-valued wavefunctions on SO (3) in terms of single-valued sections of a new non-trivial quantum bundle. These results have clear analogies with spin.
It is well known from the work of Berezin [24] (Commun. Math. Phys. 40:153–174, 1975) in 1975 that the quantization problem of a classical mechanical system is closely related with coherent states. In particular coherent states help to understand the limiting behaviour of a quantum system when the Planck constant ℏ\hbar ℏ becomes negligible in macroscopic units. This problem is called the semi-classical limit problem. In this chapter we discuss properties of quantum systems when the configuration space is the Euclidean space Rn\mathbb R^nRn. So that in the Hamiltonian formalism, the phase space is Rn×Rn\mathbb R^n\times \mathbb R^nRn×Rn with its canonical symplectic form σ\sigma σ. The quantization problem has many solutions, so we choose a convient one, introduced by Weyl [259] (The Classical Groups, 1997) and Wigner [261] (Group Theory and Its Applications to Quantum Mechanics of Atomic Spectra, 1959). We study the symmetries of Weyl quantization, the operationnal calculus and applications to propagation of observables. We show that Wick quantization is a natural bridge between Weyl quantization and coherent states. Applications are given to the semi-classical limit after introducing an efficient modern tool: the semi-classical measures. We illustrate the general results proved in this chapter by explicit computations for the harmonic oscillator. More applications will be given in the following chapters, in particular concerning propagators and trace formulas for a large class of quantum systems.
Following previous works by E. Prugovečki [Physica A 91A, 202 (1978) and Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)] on common features of classical and quantum mechanics, we develop a unified mathematical framework for classical and quantum mechanics (based on L2-spaces over classical phase space), in order to investigate to what extent quantum mechanics can be obtained as a simple modification of classical mechanics (on both logical and analytical levels). To obtain this unified framework, we split quantum theory in two parts: (i) general quantum axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoints operators, and so on) and (ii) quantum mechanics proper that specifies the Hilbert space as L2(Rn); the Heisenberg rule [pi,qj]=−iℏδij with p=−iℏ∇, the free Hamiltonian H=−ℏ2Δ/2m and so on. We show that general quantum axiomatics (up to a supplementary “axiom of classicity”) can be used as a nonstandard mathematical ground to formulate physical ideas and equations of ordinary classical statistical mechanics. So, the question of a “true quantization” with “ℏ” must be seen as an independent physical problem not directly related with quantum formalism. At this stage, we show that this nonstandard formulation of classical mechanics exhibits a new kind of operation that has no classical counterpart: this operation is related to the “quantization process,” and we show why quantization physically depends on group theory (the Galilei group). This analytical procedure of quantization replaces the “correspondence principle” (or canonical quantization) and allows us to map classical mechanics into quantum mechanics, giving all operators of quantum dynamics and the Schrödinger equation. The great advantage of this point of view is that quantization is based on concrete physical arguments and not derived from some “pure algebraic rule” (we exhibit also some limit of the correspondence principle). Moreover spins for particles are naturally generated, including an approximation of their interaction with magnetic fields. We also recover by this approach the semi-classical formalism developed by E. Prugovečki [Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)].
The non-relativistic version of the multi-temporal quantization scheme of relativistic particles in a family of non-inertial frames (see hep-th/0502194) is defined. At the classical level the description of a family of non-rigid non-inertial frames, containing the standard rigidly linear accelereted and rotating ones, is given in the framework of parametrized Galilei theories. Then the multi-temporal quantization, in which the gauge variables, describing the non-inertial effects, are not quantized but considered as c-number generalized times, is applied to non relativistic particles. It is shown that with a suitable ordering there is unitary evolution in all times and that, after the separation of center of mass, it is still possible to identify the inertial bound states. The few existing results of quantization in rigid non-inertial frames are recovered as special cases.
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 interpretation of two-valued wavefunctions on SO(3) in terms of single-valued sections of a new non-trivial quantum bundle. These results have clear analogies with spin.