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Physical Review A

Spin densities in pseudoclassical kinetic theory

F. M. C. Witte

Published Jun 1, 1997• 2 citations
Publisher

Abstract

It is shown that pseudoclassical many-particle systems allow for observable consequences of the spin degrees of freedom, in contrast to the one-particle system, where a consistent formulation of spin space density is impossible. After quantization, this density can then be directly compared to the decomposition of the Wigner operator in terms of spin matrices, in both the nonrelativistic and the relativistic context. From an expansion of the Dirac spinors in terms of large and small components it is seen that in the nonrelativistic limit the pseudoscalar, the timelike component of the axial-vector current, and the spatial components of the vector current vanish. The spatial components of the axial-vector current vanish in the pseudoclassical limit. The pseudoclassical appearance of spin is due to the spin-tensor contribution.

DOI
10.1103/physreva.55.4093

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It is shown that pseudoclassical many-particle systems allow for observable consequences of the spin degrees of freedom, in contrast to the one-particle system, where a consistent formulation of spin space density is impossible. After quantization, this density can then be directly compared to the decomposition of the Wigner operator in terms of spin matrices, in both the nonrelativistic and the relativistic context. From an expansion of the Dirac spinors in terms of large and small components it is seen that in the nonrelativistic limit the pseudoscalar, the timelike component of the axial-vector current, and the spatial components of the vector current vanish. The spatial components of the axial-vector current vanish in the pseudoclassical limit. The pseudoclassical appearance of spin is due to the spin-tensor contribution.
We examine the quantization of pseudoclassical dynamical systems, models that have classically anticommuting variables, in the Schrödinger picture. We quantize these systems, which can be viewed as classical models of particle spin, using the generalized Gupta–Bleuler method as well as the reduced phase space method in even dimensions. With minimal modifications, the standard constructions of Schrödinger quantum mechanics of constrained systems work for pseudoclassical systems. We generalize the standard Schrödinger norm and implement the correct adjointness properties of observables and constraints. We construct the state space corresponding to spinors as physical wave functions of anticommuting variables, finding that there are superselection sectors in both the physical and ghost subspaces. The physical states are isomorphic to those of the Dirac–Kähler formulation of fermions, though the inner product in Dirac–Kähler theory is not equivalent to ours.
Spin of elementary particles is the only kinematic degree of freedom not having classical corre- spondence. It arises when seeking for the finite-dimensional representations of the Lorentz group, which is the only symmetry group of relativistic quantum field theory acting on multiple-component quantum fields non-unitarily. We study linear transformations, acting on the space of spatial and proper-time velocities rather than on coordinates. While ensuring the relativistic in- variance, they avoid these two exceptions: they describe the spin degree of freedom of a pointlike particle yet at a classical level and form a compact group hence with unitary finite-dimensional rep- resentations. Within this approach changes of the velocity modulus and direction can be accounted for by rotations of two independent unit vectors. Dirac spinors just provide the quantum description of these rotations.
Recently, in the context of Relativistic Quantum Information Theory (RQI) of massive spin-1/2 particles, it has been suggested that it is impossible to perform a momentum-independent spin measurement, showing the inadequacy of the spin reduced density matrix as a legitimate information resource. This is because there is an unavoidable ambiguity in the definition of the spin of a moving particle. In this paper, by introducing a vector bundle theoretic way to view the single-particle state space, we try to rule out this ambiguity. The discrete degree of freedom of the resulting representation space contains information about the Pauli–Lubanski four-vector of the particle instead of the ambiguous spin. Comparing this representation with the standard one used in the RQI literature, we show that the discrete degree of freedom of the standard representation space attains the meaning of the Newton–Wigner spin. In addition, using this viewpoint, we give a mathematical proof of why the spin reduced density matrix is meaningless, which is stronger than the previous claims in that it asserts that the matrix is void of any meaning at all, not just in terms of the impossibility of measurement or Lorentz non-covariance. We give a way (which turns out to be the only way) to modify it to obtain the Pauli–Lubanski reduced density matrix, which is covariant under Lorentz transformations.
This thesis provides a detailed derivation of dissipative spin hydrodynamics from quantum field theory for systems composed of spin-0, spin-1/2, or spin-1 particles. The Wigner function formalism is introduced for quantum fields in the respective representations of the Poincaré group, and the conserved currents, i.e., the energy-momentum tensor and the total angular momentum tensor, in various so-called pseudogauges are derived. An expansion around the semiclassical limit in powers of the Planck constant is performed. Subsequently, kinetic equations are obtained for binary elastic scattering, using both the de Groot-van Leeuwen-van Weert and Kadanoff-Baym method, with the latter retaining the effect of quantum statistics. The resulting collision term features both local and nonlocal contributions, with the latter providing a relaxation mechanism for the spin degrees of freedom of the quasiparticles. The local-equilibrium distribution function is derived from the requirement that the local part of the collision term vanishes. From quantum kinetic theory, dissipative spin hydrodynamics is then constructed via the method of moments, extended to particles with spin. The system of moment equations is closed via the Inverse-Reynolds Dominance (IReD) approach, resulting in a set of equations of motion describing the evolution of both ideal and dissipative degrees of freedom. The application to polarization phenomena relevant to heavy-ion collisions is discussed.
The connection between the (orthodox) single-particle Newton-Wigner position and the Bohm/de Broglie hidden-particle position in the relativistic regime is investigated. The effect of non-positive excursions of the ensemble density for extreme cases of positive-energy waves is easily computed using an integral of the equations of motion developed here and is interpreted in terms of virtual-like pair creation and annihilation beneath the Compton wavelength. A Bohm-theoretic description of the acausal explosion of Newton-Wigner-localized states is presented in detail for spin-0 particles in attempt to relate Bohm/de Broglie and orthodox positions. Spin-1/2 particles are considered as well with the assumption that the convective current component describes ensemble dynamics as with the spin-0 theory. For spin-1/2 particles, a specific second-rank tensor is defined in terms of Dirac spinors for generalizing from simply a quantum potential to a stress tensor required to account for the force of pilot wave on particle. A simple dependence of the stress tensor on a two-component spin pseudovector field is determined for approaching a non-relativistic limit. Consistency is found with an earlier non-relativistic theory of objects with spin.
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.