Lorentz Operations

Mark Linne

Andrés J. Kálnay, P. L. Torres

In parts II and III of the present series of papers, the localization problem was studied for the cases of relativistic nonzero mass (spin 0, $\frac{1}{2}$) and zero mass (spin 0, $\frac{1}{2}$, and 1). With a set of postulates, of which the basic one is the imposition of Lorentz invariance of localization (LIL) (which is a self-consistency requirement), it was possible to define the position operator uniquely (except for a constant in the spin-$\frac{1}{2}$, nonzero-mass case). For the determination of the position operators it was enough to consider only very special cases of homogeneous Lorentz transformations. One of the purposes of the present paper is to consider the effect of arbitrary inhomogeneous continuous Lorentz transformations in order to obtain general Poincar\'e-invariant localization (GPIL). Our main result will be that localization is LIL if and only if it is GPIL. In II and III, LIL was obtained in spite of the fact that the position operators were not 4-vector operators but only 3-vector operators. In this paper, following Fleming, we construct formally covariant position operators from the 3-vector operators. We prove that their eigenfunctions are just the localized states defined in the GPIL sense.

R.A. Frick

In this paper we extend our previous result on the description of the partcle motion in a generalized Heisenberg picture to a relativistic fermion. The operators of the Lorentz algebra in this picture may be regarded as field operators. In this approach the transition amplitudes for the particle are constructed in terms of two-component functions in the unitary representations of the Lorentz group.

Mark Linne

Stefano Olivares

Hoshang Heydari

In this paper we establish a relation between quantum relative phase, $m$-tangle, and multi-local Lorentz-group invariant or $SL(2,\mathbb{C})^{\times m}$-invariant $S^{2}_{(m)}$. Our construction is based on the orthogonal complement of a positive operator valued measure on quantum phase. In particular, we propose a quantity based on the quantum relative phase of a multi-qubit operator that coincides with $m$-tangle, and multi-local Lorentz-group invariant.