Localization of relativistic particles and uncertainty relations

Abstract

Localization of relativistic particles and their position-momentum uncertainty relations are not yet fully understood. We discuss two schemes of photon localization that are based on the energy density. One scheme produces a positive operator-valued measure for localization. It coincides with the number density operator and reproduces an effective 3 × 3 polarization density matrix. Another scheme results in a probability distribution that is conditioned on the detection. In both schemes the uncertainty relations for transversal position and momentum approach the Heisenberg bound ∆p∆z = 1 2 .

The Information Recovery Problem

“…Thus there is no scalar field supporting a wormhole at a moment of time. Although the scalar field is classical, appeal to the quantum violation of the energy conditions is impossible: localization of quantum scalar particles can only be described using the positivity of their energy densities [24,25].…”

Spin and localization of relativistic fermions and uncertainty relations

“…Interesting features of localization of fermions in a cavity are described in [43]. Limitations of the localization POVM built from the field operators in general, and of the use of energy density in particular are discussed in [11,12].…”

“…On the other hand, particle's position is not a dynamical variable in the field picture and thus does not have to be a part of a self-adjoint Hamiltonian. Hence we describe the localized detection events in terms of positive operator-valued measures (POVMs) [9,12]. A POVM constitutes a nonorthogonal decomposition of the identity by means of positive operatorsΠ(x), resulting in detection probabilities P (x) = trρΠ(x) for the set of events {x} [1,2,42].…”

“…To enforce the convexity of the trace formula, i.e., to maintain that the probability of a particular outcome for a weighted mixture of states is a weighted mixture of the corresponding probabilities, normalisation should be performed at the level of operators [12]. Hence the centre of energy POVM is constructed aŝ…”

“…Construction of relativistic position is fraught with technical difficulties and compromises between different reasonable requirements (see, e.g., [3,[6][7][8][9][10][11][12] and references therein). There is no unique way to describe localization of a relativistic particle, even when particles are unambiguously defined.…”