Reparametrization invariant dynamics on a sphere, being parameterized by angular momentum coordinates, represents an example of noncommutative theory.It can be quantized according to Berezin-Marinov prescription, replacing the coordinates by Pauli matrices.Following the scheme, we present two semiclassical models for description of spin without use of Grassman variables.The first model implies Pauli equation upon the canonical quantization.The second model produces nonrelativistic limit of the Dirac equation implying correct value for the electron spin magnetic moment.
Reparametrization invariant dynamics on a sphere, being parameterized by angular momentum coordinates, represents an example of noncommutative theory.It can be quantized according to Berezin-Marinov prescription, replacing the coordinates by Pauli matrices.Following the scheme, we present two semiclassical models for description of spin without use of Grassman variables.The first model implies Pauli equation upon the canonical quantization.The second model produces nonrelativistic limit of the Dirac equation implying correct value for the electron spin magnetic moment.
Reparametrization invariant dynamics on a sphere, being parametrized by angular momentum coordinates, represents an appropriate framework for semiclassical description of nonrelativistic spin. The space can be quantized according to Berezin–Marinov prescription, replacing the coordinates by Pauli matrices. Following the scheme, we present two semiclassical models for describing spin without the use of Grassmann variables. The first model implies Pauli equation upon the canonical quantization. The second model produces nonrelativistic limit of the Dirac equation implying correct value for the electron spin magnetic moment.
We construct a nonrelativistic wave equation for spinning particles in the noncommutative space (in a sense, a $\ensuremath{\theta}$ modification of the Pauli equation). To this end, we consider the nonrelativistic limit of the $\ensuremath{\theta}$-modified Dirac equation. To complete the consideration, we present a pseudoclassical model (\`a la Berezin-Marinov) for the corresponding nonrelativistic particle in the noncommutative space. To justify the latter model, we demonstrate that its quantization leads to the $\ensuremath{\theta}$-modified Pauli equation. We extract $\ensuremath{\theta}$-modified interaction between a nonrelativistic spin and a magnetic field from such a Pauli equation and construct a $\ensuremath{\theta}$ modification of the Heisenberg model for two coupled spins placed in an external magnetic field. In the framework of such a model, we calculate the probability transition between two orthogonal Einstein-Podolsky-Rosen states for a pair of spins in an oscillatory magnetic field and show that some of such transitions, which are forbidden in the commutative space, are possible due to the space noncommutativity. This allows us to estimate an upper bound on the noncommutativity parameter.
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]).
A covariant, symplectic, classical dynamical system is presented whose quantization, by replacement of the Poisson brackets with commutators, gives precisely the Dirac electron theory. For the classical system the velocity and momentum are independent dynamical variables, as in Dirac electron; it undergoes a real Zitterbewegung so that the spin is the orbital angular momentum of the Zitterbewegung. Thus the longstanding problem of representing the configuration space of quantum spin by a classical model is solved. Three different forms of the dynamical equations are given.
Introduction The claim that it is impossible to construct a realistic physical model of quantum matter is often presented as definitively and finally proved in the case of systems with spin ½, or internal angular momentum of magnitude ħ/2, which exhibit a characteristic ‘nonclassical two-valuedness’. Nevertheless, the extension of the causal interpretation to embrace nonrelativistic spin ½ systems was successfully carried through in the 1950s (Bohm, Schiller & Tiomno, 1955; Bohm & Schiller, 1955; Takabayasi, 1955; Takabayasi & Vigier, 1957) and developed more recently (Dewdney, Holland & Kyprianidis, 1986, 1987a, b; Dewdney, Holland, Kyprianidis & Vigier, 1988). The idea is to assume that the spinor wave represents a new type of physical field propagating in spacetime that exerts an influence on a corpuscle moving within it. The additional information conveyed by the spinor beyond that contained in the spinless Schrödinger wave implies that the state of the particle is not exhausted by specifying its position and momentum, but that it is endowed with further properties such as an internal angular momentum. The equation governing the dynamics of the spinor wavefunction is the Pauli equation: where the Hamiltonian operator is given by Here e, m and μ are respectively the charge, mass and magnetic moment of the particle, A 0 and A are the electromagnetic potentials, B = ∇ × A is the magnetic field, V is any other external scalar potential, and σ is a vector whose components are the 2 × 2 Pauli matrices. The Hamiltonian (9.1.2) istherefore a 2 × 2 matrix operator and the wavefunction ψ( x , t ) has two complex components.
In this second paper in a series, we show that the the general statistical approach to nonrelativistic quantum mechanics developed in the first paper yields a representation of quantum spin and magnetic moments based on classical nonrelativistic spinning top models, using Euler angle coordinates. The models allow half-odd-integer spin and predict supraluminal speeds only for electrons and other leptons, which must be treated relativistically. The spin operators in the space-fixed frame satisfy the usual commutation rules, while those in the rotating body-fixed frame satisfy "left-handed" rules. The commutation rules are independent of the structure of the top, so all nonrelativistic rigidly rotating objects must have integer or odd-half-integer spin. Physical boundary conditions restrict all mixed spin states to involve only half-odd-integer or only integer spin eigenstates. For spin 1/2, the theory automatically yields a modified Pauli-Schrödinger equation. The Hamiltonian operator in this equation contains a rigid rotator term and a term involving the square of the magmetic field, as well as an interaction term having the usual form in spherically symmetric and some cylindrically symmetric models, valid for any magnetogyric ratio.