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Physical Review A

Wigner function of relativistic spin-1/2 particles

Ghi Ryang Shin, Iwo Białynicki‐Birula, Johann Rafelski A

Published Jul 1, 1992• 22 citations
Publisher

Abstract

Using the recently developed relativistic Wigner formulation for the density matrix of spin-1/2 particles, we study the $^{\mathrm{\ensuremath{\pi}}}$=1/${2}^{+}$ Coulomb-like and cavity relativistic states. One of our objectives is to understand the sharing of the total angular momentum of a quantum state between the spin and rotational degrees of freedom, arising due to the spin-orbit coupling. Another is to demonstrate that the 4\ifmmode\times\else\texttimes\fi{}4-matrix Wigner function is the appropriate generalization from the 2\ifmmode\times\else\texttimes\fi{}2 form of the nonrelativistic theory.

DOI
10.1103/physreva.46.645

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Using the recently developed relativistic Wigner formulation for the density matrix of spin-1/2 particles, we study the \ensuremathπ^{\mathrm{\ensuremath{\pi}}}\ensuremathπ=1/2+{2}^{+}2+ Coulomb-like and cavity relativistic states. One of our objectives is to understand the sharing of the total angular momentum of a quantum state between the spin and rotational degrees of freedom, arising due to the spin-orbit coupling. Another is to demonstrate that the 4\ifmmode\times\else\texttimes\fi{}4-matrix Wigner function is the appropriate generalization from the 2\ifmmode\times\else\texttimes\fi{}2 form of the nonrelativistic theory.
Formal connections between the spin density matrix and the Wigner function for spin-1/2 particles forming a relativistic gas are explored to determine their general structures. They suggest that the commonly used form of the local equilibrium Wigner function should be replaced by a new expression. The latter fulfills the necessary condition for the normalization of the mean spin polarization, which the former fails to reproduce. The new definition of the Wigner function leads to generalized thermodynamic relations for perfect spin hydrodynamics, identical to those obtained earlier using the classical concept of spin. Moreover, one can prove that the perfect spin hydrodynamics based on the new equilibrium Wigner function is nonlinearly causal and stable. Finally, the selection rule for the Lagrange multipliers, which is satisfied by real systems, is discussed.
One of the most fundamental phenomena of quantum physics is entanglement. It describes an inseparable connection between quantum systems, and properties thereof. In a quantum mechanical description even systems far apart from each other can share a common state. This entanglement of the subsystems, although arising from mathematical principles, is no mere abstract concept, but can be tested in experiment, and be utilized in modern quantum information theory procedures, such as quantum teleportation. In particular, entangled states play a crucial role in testing our understanding of reality, by violating Bell inequalities. While the role of entanglement is well studied in the realm of nonrelativistic quantum mechanics, its significance in a relativistic quantum theory is a relatively new field of interest. In this work the consequences of a relativistic description of quantum entanglement are discussed. We analyze the representations of the symmetry groups of special relativity, i.e. of the Lorentz group, and the Poincaré group, on the Hilbert space of states. We describe how unitary, irreducible representations of the Poincaré group for massive spin 1/2 particles are constructed from representations of Wigner's little group. We then proceed to investigate the role of the Wigner rotations in the transformation of quantum states under a change of inertial reference frame. Considering different partitions of the Hilbert space of 2 particles, we find that the entanglement of the quantum states appears different in different inertial frames, depending on the form of the states, the chosen inertial frames, and the particular choice of partition. It is explained, how, despite of this, the maximally possible violation of Bell inequalities is frame independent, when using appropriate spin observables, which are related to the Pauli-Ljubanski vector, a Casimir operator of the Poincaré group.
By the use of complete orthonormal sets of nonrelativistic scalar orbitals introduced by the author in previous papers the new complete orthonormal basis sets for two-and four-component spinor wave functions, and Slater spinor orbitals useful in the quantum-mechanical description of the spin - 1/2 particles by the quasirelativistic and relativistic equations are established in position, momentum and four-dimensional spaces. These function sets are expressed through the corresponding nonrelativistic orbitals. The analytical formulas for overlap integrals over four component relativistic Slater spinor orbitals with the same screening constants in position space are also derived. The relations obtained in this study can be useful in the study of different problems arising in the quasirelativistic and relativistic quantum mechanics when the position, momentum and four dimensional spaces are employed.
In this work we represent the 1/21/21/2 Spin particles with complex quaternions using a transformation to 2x2 matrices in order to obtain the Pauli matrices. With this representation we determine the states, rotation operators and the total angular momentum function in the complex quaternion space. Using this representation we work the solution for the relativistic hydrogen atom.
The spin reduced matrix defined by taking partial trace with the momentum variable has been shown not to transform covariantly under Lorentz transformations. In this paper, I suggest another representation space of the Poincare group, the elements of which are in a sense apparent wave functions. In this new Hilbert space, the naive spin observable 12τ\frac{1}{2}\boldsymbol{\tau}21​τ from nonrelativistic quantum mechanics on the old Hilbert space is represented as the Newton-Wigner spin operator. Also on this space, one can naturally associate to each state a Lorentz covariant spin reduced density matrix, which is then given a new interpretation.