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Journal of Physics A Mathematical and General

A physical interpretation of the quantum group Uq(SU(2))

Miguel Angel Martin-Delgado

Published Aug 7, 1991• 4 citations
Publisher

Abstract

The quantum statistics of a system of free spins in the presence of a constant magnetic field H is interpreted in terms of the representation theory of the simplest quantum group U q (SU(2)) with q=e ( mu 0 mu B kBT)/ being the deformation parameter which has a definite physical meaning. The classical limit q to 1 corresponds to weak magnetic field regime H to 0.

DOI
10.1088/0305-4470/24/15/004

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The quantum statistics of a system of free spins in the presence of a constant magnetic field H is interpreted in terms of the representation theory of the simplest quantum group U q (SU(2)) with q=e ( mu 0 mu B kBT)/ being the deformation parameter which has a definite physical meaning. The classical limit q to 1 corresponds to weak magnetic field regime H to 0.
An approach to study a generalization of the classical-quantum transition for general systems is proposed. In order to develop the idea, a deformation of the ladder operators algebra is proposed that contains a realization of the quantum group SU(2)qSU(2{)}_{q}SU(2)q​ as a particular case. In this deformation Planck's constant becomes an operator whose eigenvalues approach $\ensuremath{\hbar}$ for small values of nnn (the eigenvalue of the number operator) and zero for large values of nnn (the system is classicalized).
Abstract A consistent theory of quantum entanglement requires that the constituent single-particle states belong to the same Hilbert space, the coherent eigenstates of a complete set of operators in a given representation, defined with respect to a shared continuous parameterization. A covariant theory of relativistic spin and entanglement has been given by Horwitz and Arshansky, in which representations of O(3, 1) are induced with respect to an arbitrarily selected timelike unit vector nµ . In this paper, we construct an induced relativistic representation of spin on an extended phase space {(xµ , pµ ), (ζµ , πµ )}, in which we associate the unit vector nµ with the momentum πµ , thus providing a dynamical interpretation for this new quantity. Studying the unitary representations of the Poincaré group on the extended phase space allows us to define basis quantities for quantum states and develop the gauge invariant electromagnetic Hamiltonian in classical and quantum mechanics. We write plane wave solutions for free particles and construct stable singlet states. As in non-relativistic quantum theory the magnetic field couples to the rotation generators of SU(2) ⊂ SL(2, C), however the electric field couples to the non-compact and anti-Hermitian boost generators. We show that to first order in perturbation theory, a constant external magnetic field will produce a perturbative contribution to the total mass (the eigenvalue of the Lorentz scalar Hamiltonian) or a transition between states in the same representation of SL(2, C), and so will not disrupt a singlet state. However, an electric field normal to πµ can produce a perturbative transition between states in inequivalent representations of SL(2, C), and therefore potentially cause decoherence of the relativistic entanglement.
We derive the representation theory of SU(2)SU(2)SU(2) from the expository theory of Lie groups and Lie algebras. Based on this, the mathematics of non-relativistic quantum mechanics of a spin 12\frac{1}{2}21​ particle are described from a representation-theoretic perspective, and are extended to many particle systems.
The Stern–Gerlach experiment is notoriously counter-intuitive. The official theory is that the spin of a fermion remains always aligned with the magnetic field. Its directions are thus quantized: It can only be spin-up or spin-down. However, that theory is based on mathematical errors in the way it (mis)treats spinors and group theory. We present here a mathematically rigorous theory for a fermion in a magnetic field, which is no longer counter-intuitive. It is based on an understanding of spinors in SU(2) which is only Euclidean geometry. Contrary to what Pauli has been reading into the Stern–Gerlach experiment, the spin directions are not quantized. The new corrected paradigm, which solves all conceptual problems, is that the fermions precess around the magnetic-field just as Einstein and Ehrenfest had conjectured. Surprisingly, this leads to only two energy states, which should be qualified as precession-up and precession-down rather than spin-up and spin-down. Indeed, despite the presence of the many different possible angles θ between the spin axis s and the magnetic field B, the fermions can only have two possible energies m0c2±μB. The values ±μB thus do not correspond to the continuum of values −μ·B Einstein and Ehrenfest had conjectured. The energy term V=−μ·B is a macroscopic quantity. It is a statistical average over a large ensemble of fermions distributed over the two microscopic states with energies ±μB, and as such not valid for individual fermions. The two fermion states with energy ±μB are not potential-energy states. We also explain the mathematically rigorous meaning of the up and down spinors. They represent left-handed and right-handed reference frames, such that now everything is intuitively clear and understandable in simple geometrical terms. The paradigm shift does not affect the Pauli principle.