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Journal of Physics A Mathematical and General

Localizability of relativistic particles in fuzzy phase space

Eduard Prugovečki

Published Nov 1, 1976• 17 citations
Publisher

Abstract

The concept of fuzzy phase space Gamma s , recently introduced in non-relativistic quantum mechanics, is extended to the relativistic case. The L 2 ( Gamma s ) representation of the wave packet is used as a basis for discussing localizability in Gamma s for non-zero mass particles as well as for the photon.
DOI
10.1088/0305-4470/9/11/010

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Although the concept of localization in non-relativistic quantum mechanics is mathematically well-defined there is no obvious and unambiguous way to generalize it to relativistic quantum theory. After a brief review of localization in quantum mechanics the Newton-Wigner localization scheme is introduced and it is shown by the example of a massive spinless system how it leads to a position operator with seemingly unphysical properties. This motivated the development of several theorems which claim to rule out the existence of localizable particles. A particularly important one of them is presented and its validity discussed. In the last section however it is shown that if localization is considered with respect to spacelike hyperplanes, the properties of the Newton-Wigner position operators are not necessarily unphysical and localizable particles can exist.
Abstract The well-definedness of particles of any kind depends on the limits, approximations, or other conditions that may or may not be involved, for example, whether there are interactions and whether ostensibly related energy is localizable. In particular, their theoretical status differs between its non-relativistic and relativistic versions: One can properly define interacting elementary particles in single-system non-relativistic quantum mechanics, at least in the case of non-zero mass systems; by contrast, one is severely challenged to define even these properly in the relativistic quantum field theories that now underlie the study of particle physics. Here, the impact of localizability on this status is reviewed in relation to the work of Paul Busch on positive operator valued measures that significantly probes the relevance of quantum unsharpness to it.
The starting point of this research is a representation of a quantum particle according to the Schrodinger equation of the conventional quantum mechanics. In this representation, a quantum particle is described by a wave packet in the coordinate space and the conjugated wave packet in the momentum space. The problem is that while in the coordinate space the group velocity is in agreement with one of the Hamilton equations, the group velocity in the momentum space is in contradiction with the other Hamilton equation - a minus sign is missing. Group velocities in agreement with the Hamilton equations are obtained only when in the time dependent phase of a quantum particle wave packet, instead of the Hamiltonian coming from the conventional Schrodinger equation, the Lagrangian is considered. This suggests us to consider the relativistic Lagrangian in the time dependent phase. In this way, the conventional relativistic principle of invariance of the time-space interval gets the more physically understandable form of the invariance of the time dependent phase of a quantum particle - the time dependent phase of a quantum particle is the same in any system of coordinates. Based on the relativistic time dependent phase invariance of a quantum particle, from the group velocities of this particle the relativistic kinematics and dynamics are obtained. The interaction with an electromagnetic field is described by a modification of the time dependent phase with a scalar potential conjugated to time and a vector potential conjugated to the space coordinates. In this paper, we find that, according to the formalism of the general relativity, any matter element in a field of forces is accelerated only perpendicularly to its velocity. This means that the matter propagation of a quantum particle can be conceived in planes perpendicular to velocity, while the mater distribution can be considered in a Fourier representation - quantum waves. We study the quantum particle dynamics in gravitational waves and derive the graviton spin. We consider a quantum particle in electromagnetic field and obtain a Schrödinger-Dirac-type equation, with additional terms describing the velocity dependence of the particle dynamics.
The concepts of configuration and momentum representation space for state vectors are generalized to that of fuzzy-phase-space representation spaces L2(Γs), 0<s<∞, which are interpolated in between these two standard representations. It is shown that the wavepacket in L2(Γs) displays the familiar evanescence property from any region Ks × Ms in the fuzzy phase space Γs if that region is bounded in its configuration part Ks; also, that the probability of detecting the system in Ks × Ms has a finite asymptotic time limit if Ks is a (fuzzy) cone. For scattering states the existence of free states that are asymptotic in Γs is established, and a formula for differential cross section in Γs is derived.
Abstract It is a widespread belief, at least within the physics community, that there is no relativistic quantum theory of (localizable) particles; and, thus, that the only relativistic quantum theory is a theory of fields. This belief has received much support in recent years in the form of rigorous no­go theorems by Malament (1996) and Hegerfeldt (1998a,b). In particular, Hegerfeldt shows that in a generic quantum theory (relativistic or non­relativistic), if there are states with localized particles, and if there is a lower bound on the system’s energy, then superluminal spreading of the wavefunction must occur. Similarly, Malament shows the inconsistency of a few intuitive desiderata for a relativistic quantum mechanics of (localizable) par­
Non-classical light sources are now being developed in more and more laboratories. Their noise reduction properties are potentially important for technology. Examples are squeezed and antibunched light sources. It is tempting to picture in some of these processes single photons flying through space. In non-relativistic quantum mechanics it is easy to discuss the properties of particle localisation in a general context without recourse to a particular experimental configuration. A position operator exists and the localised states are just the eigenstates of this operator. The situation in relativistic quantum mechanics (or field theory) is not as straightforward. The photon is relativistic and it has the addi-tional property of being massless. It has no position operator associated with it. Yet, nevertheless, it is possible to pose the question of finding a generic construction for a wave packet for the photon which is localised as far as possible in some (imaginary) box in space. The difficulty of the construction follows from the fact that in an electromagnetic wave the polarisation is perpendicular to the direction of propagation. Thus we need to construct a vector field which is localised and has this transverse property related to the masslessness of the photon.