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arXiv:1404.5576 · Physica Scripta

Relativistic dynamics compels a thermalized fermi gas to a unique intrinsic parity eigenstate

Alex E. Bernardini, S S Mizrahi

Published Jun 12, 2014• 22 citations
arXivHTMLPDF

Abstract

Dirac equation describes the dynamics of a relativistic spin-1/2 particle regarding its spatial motion and intrinsic degrees of freedom. Here we adopt the point of view that the spinors describe the state of a massive particle carrying two qubits of information: helicity and intrinsic parity. We show that the density matrix for a gas of free fermions, in thermal equilibrium, correlates helicity and intrinsic parity. Our results introduce the basic elements for discussing the spin-parity correlation for a Fermi gas: (1) at the ultra-relativistic domains, when the temperature is quite high, $T > 10^{10}\ K$, the fermions have no definite intrinsic parity (50% : 50%), which is maximally correlated with the helicity; (2) at very low temperature, $T \approx 3 \ K$, a unique parity dominates (conventionally chosen positive), by $10^{20}$ to $1$, while the helicity goes into a mixed state for spin up and down, and the quantum correlation decoheres. For the anti-fermions we get the opposite behavior. In the framework of quantum information, our result could be considered as a plausible explanation of why we do accept, as a fact (consistent with the experimental observation), that fermions (and anti-fermions), in our present epoch of a cool universe, have a unique intrinsic parity. The framework for constructing spin-parity entangled states is established.

DOI
10.1088/0031-8949/89/7/075105

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Dirac equation describes the dynamics of a relativistic spin-1/2 particle regarding its spatial motion and intrinsic degrees of freedom. Here we adopt the point of view that the spinors describe the state of a massive particle carrying two qubits of information: helicity and intrinsic parity. We show that the density matrix for a gas of free fermions, in thermal equilibrium, correlates helicity and intrinsic parity. Our results introduce the basic elements for discussing the spin-parity correlation for a Fermi gas: (1) at the ultra-relativistic domains, when the temperature is quite high, T>1010 KT > 10^{10}\ KT>1010 K, the fermions have no definite intrinsic parity (50% : 50%), which is maximally correlated with the helicity; (2) at very low temperature, T≈3 KT \approx 3 \ KT≈3 K, a unique parity dominates (conventionally chosen positive), by 102010^{20}1020 to 111, while the helicity goes into a mixed state for spin up and down, and the quantum correlation decoheres. For the anti-fermions we get the opposite behavior. In the framework of quantum information, our result could be considered as a plausible explanation of why we do accept, as a fact (consistent with the experimental observation), that fermions (and anti-fermions), in our present epoch of a cool universe, have a unique intrinsic parity. The framework for constructing spin-parity entangled states is established.
Abstract In this paper we study the single particle entanglement induced by a Lorentz boost. We described the particle as a solution of the free Dirac equation, a Dirac bispinor, and compare the induced momentum-spin entanglement with the result obtained in the widely considered framework of relativistic spin 1/2 states. The spin linear entropy for both approaches agree in the ultra relativistic limit. We also verify that the spin-momentum entanglement differers from the spin entropy for the bispinorial case, indicating a true multipartite entanglement involving the degrees of freedom of a Dirac bispinor state: momentum, spin and intrinsic parity. The fact that Dirac bispinors belongs to an irreducible representation of the complete Lorentz group, which also included parity as a symmetry, is the ultimate reason behind such non-trivial structure.
Lorentz transformation of the reduced helicity density matrix for a massive spin- particle is investigated in the framework of relativistic quantum information theory for the first time. The corresponding helicity entropy is calculated, which shows no invariant meaning as that of spin. The variation of the helicity entropy with the relative speed of motion of inertial observers, however, differs significantly from that of spin due to their distinct transformation behaviors under the action of the Lorentz group. This novel and odd behavior unique to the helicity may be readily detected by high-energy physics experiments. The underlying physical explanations are also discussed.
Dirac's equation with static external fields is considered as an equation of relativistic quantum theory, which describes the quantum dynamics of an electron in four-dimensional space-time. It is shown that the Dirac electron has only one intrinsic degree of freedom -- spin. The quantum ensemble of an electron with a given energy consists of two inseparable from each other subensembles of heavy and light spin-1/2 quasiparticles to have opposite (relative) intrinsic parities. In the case of electrical scalar potential nonuniform in the direction of a particle motion, these subensembles behave as analogs of the electrical and magnetic components of the plane monochromatic electromagnetic wave propagating in a dispersive medium, nonuniform in the direction of a wave motion. The role of states with negative energies and the properties of the velocity operator are explained.
Dirac's equation with static external fields is considered as an equation of relativistic quantum theory, which describes the quantum dynamics of an electron in four-dimensional space-time. It is shown that the Dirac electron has only one intrinsic degree of freedom -- spin. The quantum ensemble of an electron with a given energy consists of two inseparable from each other subensembles of heavy and light spin-1/2 quasiparticles to have opposite (relative) intrinsic parities. In the case of electrical scalar potential nonuniform in the direction of a particle motion, these subensembles behave as analogs of the electrical and magnetic components of the plane monochromatic electromagnetic wave propagating in a dispersive medium, nonuniform in the direction of a wave motion. The role of states with negative energies and the properties of the velocity operator are explained.
Following the information-based approach to Dirac spinors under a constant magnetic field, the phase-space representation of symmetric and anti-symmetric localized Dirac cat states is obtained. The intrinsic entanglement profile implied by the Dirac Hamiltonian is then investigated so as to shed a light on quantum states as carriers of qubits correlated by phase-space variables. Corresponding to the superposition of Gaussian states, cat states exhibit non-trivial elementary information dynamics which include the interplay between intrinsic entanglement and quantum superposition as reported by the corresponding Dirac archetypes. Despite the involved time-evolution as non-stationary states, the Wigner function constrains the elementary information quantifiers according to a robust framework which can be consistently used for quantifying the time-dependent SU(2)⊗SU(2)SU(2) \otimes SU(2)SU(2)⊗SU(2) (spin projection and intrinsic parity) correlation profile of phase-space localized Dirac spinor states. Our results show that the Dirac Wigner functions for cat states -- described in terms of generalized Laguerre polynomials -- exhibit an almost maximized timely persistent mutual information profile which is engendered by either classical- or quantum-like spin-parity correlations, depending on the magnetic field intensity.
The Weyl-Wigner formalism for evaluating the intrinsic information of Dirac bispinors as correlated qubits (localized) in a magnetic field is investigated in the extension to statistical ensembles. The confining external field quantizes the quantum correlation measures implied by the spin-parity qubit structure of the Dirac equation in 3+13+13+1 dimensions, which simplifies the computation of the entanglement quantifier for mixed states in relativistic Landau levels. This allows for the evaluation of quantum and classical correlations in terms of entropy measures for Dirac structures that are eventually mixed. Our results are twofold. First, a family of mixed Gaussian states is obtained in phase space, and its intrinsic correlation structure is computed in closed form. Second, the partition function for the low-dimensional Dirac equation in a magnetic field is derived through complex integration techniques. It describes the low-temperature regime in terms of analytically continued zeta functions and the high-temperature limit as a polynomial on the temperature variable. The connection with lower dimensional systems is further elicited by mapping the spin-parity qubits to valley-sublattice bispinors of the low-energy effective Hamiltonian of graphene.