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arXiv:quant-ph/9811071

An Alternative Commutation Relation Between Position and Momentum Operators of Massless Particles

H. Razmi

Published Nov 25, 1998• 0 citations
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Abstract

It seems that the problem of finding a suitable position operator for photon has been solved in a recently published work which is based on a new commutation relation between position and momentum operators of massless particles[1]. Although the authors of [1] have presented a classical reasoning in favour of the new commutator,here we are going to find this new commutator based on a quantum mechanical argument.

DOI
10.48550/arxiv.quant-ph/9811071

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It seems that the problem of finding a suitable position operator for photon has been solved in a recently published work which is based on a new commutation relation between position and momentum operators of massless particles[1]. Although the authors of [1] have presented a classical reasoning in favour of the new commutator,here we are going to find this new commutator based on a quantum mechanical argument.
A general proof of several formulas used in three-dimensional quantum problems is presented. By considering the two vector operators A and B having Cartesian components that satisfy the canonical commutation relations—as the position and momentum operators do—we obtain commutation rules and relations for the projection onto each another. All of these rules and relations have symmetry under A↔B exchange. This symmetry suggests, in the particular case of position and momentum operators, a new operator, the projection of r onto p, which seems to be useful in dissipative quantum problems.
The postulate that coordinate and momentum representations are related to each other by the Fourier transform has been accepted from the beginning of quantum theory by analogy with classical electrodynamics. As a consequence, an inevitable effect in standard theory is the wave packet spreading (WPS) of the photon coordinate wave function in directions perpendicular to the photon momentum. This leads to several paradoxes. The most striking of them is that coordinate wave functions of photons emitted by stars have cosmic sizes and strong arguments indicate that this contradicts observational data. We argue that the above postulate is based neither on strong theoretical arguments nor on experimental data and propose a new consistent definition of the position operator. Then WPS in directions perpendicular to the particle momentum is absent and the paradoxes are resolved. Different components of the new position operator do not commute with each other and, as a consequence, there is no wave function in coordinate representation. Implications of the results for entanglement, quantum locality and the problem of time in quantum theory are discussed.
There has been an extended debate regarding the existence of a spin-orbital decomposition of the angular momentum of photons and other massless particles. It was recently shown that there are both geometric and topological obstructions preventing any such decomposition. Here we show that any geometric connection on a particle's state space induces a splitting of the angular momentum into two operators. These operators are well-defined angular momentum operators if and only if the connection has zero curvature. Massive particles have two canonical curved connections corresponding to boosts and rotations, respectively. These can be uniquely combined to produce a flat connection, and this gives a novel derivation of the Newton-Wigner position operator and the corresponding spin and orbital angular momenta for relativistic massive particles. When the mass is taken to zero, transverse boosts and rotations degenerate, leaving only a single connection for massless particles. This connection produces a commonly proposed splitting of the massless angular momentum into two operators. However, the connection is not flat, explaining why these operators do not satisfy the angular momentum commutation relations and are thus not true spin and orbital angular momentum operators.