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Flip to Helicity from Spin

Song He, Shuxin Shao, Hongbao Zhang

Published Jan 30, 2007• 0 citations
OpenAlex

Abstract

Lorentz transformation of the reduced helicity matrix for a massive spin 1/2 particle is first investigated in the framwork of relativistic quantum information theory. The corresponding entangled helicity entropy is calculated and has no invariant meaning as that of spin. The variation of the entropy as the relative velocity of motion, however, differs significantly from that of spin due to their distinct transformation behaviors under the action of Lorentz group. The underlying physical explanations are also discussed.

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Lorentz transformation of the reduced helicity matrix for a massive spin 1/2 particle is first investigated in the framwork of relativistic quantum information theory. The corresponding entangled helicity entropy is calculated and has no invariant meaning as that of spin. The variation of the entropy as the relative velocity of motion, however, differs significantly from that of spin due to their distinct transformation behaviors under the action of Lorentz group. The underlying physical explanations are also discussed.
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.
For a massive spin 1/2 field, we present the reduced spin and helicity density matrix, respectively, for the same pure one particle state. Their relation has also been developed. Furthermore, we calculate and compare the corresponding entanglement entropy for spin and helicity within the same inertial reference frame. Due to the distinct dependence on momentum degree of freedom between spin and helicity states, the resultant helicity entropy is different from that of spin in general. In particular, we find that both helicity entanglement for a spin eigenstate and spin entanglement for a right handed or left handed helicity state do not vanish, and their Von Neumann entropy has no dependence on the specific form of momentum distribution, as long as it is isotropic.
We compute, for massive particles, the explicit Wigner rotations of one-particle states for arbitrary Lorentz transformations; and the explicit Hermitian generators of the infinite-dimensional unitary representation. For a pair of spin 1/2 particles, Einstein–Podolsky–Rosen–ell entangled states and their behaviour under the Lorentz group are analyzed in the context of quantum field theory. Group theoretical considerations suggest a convenient definition of the Bell states which is slightly different from the conventional assignment. The behaviour of Bell states under arbitrary Lorentz transformations can then be described succinctly. Reduced density matrices applicable to systems of identical particles are defined through Yang's prescription. The von Neumann entropy of each of the reduced density matrix is Lorentz invariant; and its relevance as a measure of entanglement is discussed, and illustrated with an explicit example. A regularization of the entropy in terms of generalized zeta functions is also suggested.
We study the transformation of maximally entangled states under the action of Lorentz transformations in a fully relativistic setting. By explicit calculation of the Wigner rotation, we describe the relativistic analog of the Bell states as viewed from two inertial frames moving with constant velocity with respect to each other. Though the finite dimensional matrices describing the Lorentz transformations are non-unitary, each single particle state of the entangled pair undergoes an effective, momentum dependent, local unitary rotation, thereby preserving the entanglement fidelity of the bipartite state. The details of how these unitary transformations are manifested are explicitly worked out for the Bell states comprised of massive spin 1/21/21/2 particles and massless photon polarizations. The relevance of this work to non-inertial frames is briefly discussed.