Canonical group quantization and rotation generators for indistinguishable particles

H. D. Doebner, Pavel Šťovíček, J. Tolar

Carlos Benavides, Andrés Fernando Reyes-Lega

Using the method of canonical group quantization, we construct the angular momentum operators associated to configuration spaces with the topology of (i) a sphere and (ii) a projective plane. In the first case, the angular momentum operators derived this way are the quantum version of Poincaré's vector, i.e., the physically correct angular momentum operators for an electron coupled to the field of a magnetic monopole. In the second case, the operators one obtains represent the angular momentum operators of a system of two indistinguishable spin zero quantum particles in three spatial dimensions. The relevance of the proposed formalism for the progress in our understanding of the spin–statistics connection in nonrelativistic quantum mechanics is discussed. Introduction The connection between the spin of quantum particles and the statistics they obey is a remarkable example of a simply stated physical fact without the recognition of which many physical phenomena (ranging from the stability of matter and the electronic configuration of atoms to Bose–Einstein condensation and superconductivity) would not have an explanation. Nevertheless, the simplicity of the assertion “integer spin particles obey Bose statistics, and half-integer spin particles obey Fermi statistics” stands in bold contrast to its intricate physical origin. Indeed, Pauli's proof of the spin–statistics theorem [Pau40] (improving on earlier work by Fierz [Fie39]), showed that the spin–statistics connection was deeply rooted in relativistic quantum field theory. The path to a rigorous proof of this theorem (from the mathematical point of view) was a long one and involved the efforts of many people (see, e.g., the book by Duck and Sudarshan [DS98b]).

George Sterman

Yoshio Ohnuki, S. Kamefuchi

Ahamad Sumadi, Hishamuddin B Zainuddin

Using Isham's group-theoretic quantization scheme, we construct the canonical groups of the systems on the two-dimensional sphere and one-dimensional complex projective space, which are homeomorphic. In the first case, we take SO(3) as the natural canonical Lie group of rotations of the two-sphere and find all the possible Hamiltonian vector fields, and followed by verifying the commutator and Poisson bracket algebra correspondences with the Lie algebra of the group. In the second case, the same technique is resumed to define the Lie group, in this case SU (2), of CP'.We show that one can simply use a coordinate transformation from S 2 to CP 1 to obtain all the Hamiltonian vector fields of CP 1 . We explicitly show that the Lie algebra structures of both canonical groups are locally homomorphic. On the other hand, globally their corresponding canonical groups are acting on different geometries, the latter of which is almost complex. Thus the canonical group for CP 1 is the double-covering group of SO(3), namely SU(2). The relevance of the proposed formalism is to understand the idea of CP 1 as a space of where the qubit lives which is known as a Bloch sphere.

Aleksandra Pȩdrak, ANDRZEJ GÓŹDŹ, Włodzimierz Piechocki +2 more

Abstract We develop a relativistic framework of integral quantization applied to the motion of spinless particles in the four-dimensional Minkowski spacetime. The proposed scheme is based on coherent states generated by the action of the Heisenberg–Weyl group and has been motivated by the Hamiltonian description of the geodesic motion in General Relativity. We believe that this formulation should also allow for a generalization to the motion of test particles in curved spacetimes. A key element in our construction is the use of suitably defined positive operator-valued measures. We show that this approach can be used to quantize the one-dimensional nonrelativistic harmonic oscillator, recovering the standard Hamiltonian as obtained by the canonical quantization. A direct application of our model, including a computation of transition amplitudes between states characterized by fixed positions and momenta, is postponed to a forthcoming article.