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Geometric aspects of the quantization of a rigid body

MODUGNO M., TEJERO PRIETO C., Raffaele Vitolo

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

In this paper we review our results on the quantization of a rigid body. The fact that the configuration space is not simply connected yields two inequivalent quantizations. One of the quantizations allows us to recover classically double-valued wave functions as single valued sections of a non-trivial complex line bundle. This reopens the problem of a physical interpretation of these wave functions.

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In this paper we review our results on the quantization of a rigid body. The fact that the configuration space is not simply connected yields two inequivalent quantizations. One of the quantizations allows us to recover classically double-valued wave functions as single valued sections of a non-trivial complex line bundle. This reopens the problem of a physical interpretation of these wave functions.
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.
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.
Classical mechanics has a natural mathematical setting in symplectic geometry and it may be asked if the same is true for quantum mechanics. More precisely, is it possible to capture certain quantum idiosyncrasies within the symplectic framework of classical mechanics? Beginning with Dirac, the idea of quantizing a classical system involved associating the phase space variables with Hermitian operators which act on some Hilbert space, as well as associating the Poisson bracket on the phase space with the commutator for the operators. Mathematically the phase space is associated with some symplectic manifold MMM and the non-degenerate 2-form ωωω, which comes with MMM. Geometric prequantization is a process by which one does this in a mathematically "rigorous" manner and we shall attempt to just introduce the methods here. We do this by exploring this contruction for (R2n,∑i=1ndpi∧dqi\mathbb{R}^{2n}, \sum_{i=1}^n dp_i \wedge dq_iR2n,∑i=1n​dpi​∧dqi​) which avoids using complex line bundles in any non-trivial way. One should note however that the Hilbert space one obtains is in fact too "big", in the sense that it has too many functions in order to correspond with actual physically significant Hilbert spaces. Geometric quantization remedies this situation but it should be noted that not all manifolds are prequantizable. We shall not discuss either of these issues however.
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.:
Classical dynamics of rigid bodies The difficulties in formulating the theory of spin ½ systems outlined at the end of the last chapter may be avoided if we start from a classical theory of rotation in which the translational and orientational coordinates enter on a symmetrical footing, and pass to the quantum theory via the standard canonical quantization procedure. The result is not simply the Pauli theory written in a different matrix representation but a more general physical theory which explicitly incorporates in the wave equation the extended character of the bodies considered, through moment-of-inertia-dependent terms. As expected, the spin ½ character of the wave is manifested through the dependence of the body motion on the total quantum state via quantum potential and torque terms. Spin is not usually approached from this angle in basic quantum mechanics texts so it is worthwhile to build up the theory from scratch (for a more rigorous treatment see Normand (1980) and for further details: Bopp & Haag (1950), Rosen (1951, 1984), van Winter (1954), Dankel (1970), Holland (1988a, b)). We begin by reviewing some basic results in the classical canonical theory of rigid body motion. Our treatment is confined to a rigid homogeneous sphere with moment of inertia I about any axis passing through its centre, where the centre of mass is located. The sphere carries a rigid charge and a magnetic moment. By Chasle's theorem an arbitrary displacement of a rigid body is composed of a translation together with a rotation about a line fixed in the body.