A new look at the position operator in quantum theory

Abstract

Abstract: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.

Symmetries in Foundation of Quantum Theory and Mathematics

“…This conclusion is based on the assumption that coordinate and momentum representations in quantum theory are related to each other by the Fourier transform. However, as shown in Reference [3], this assumption is based neither on strong theoretical arguments nor on experimental data.…”

“…In the latter case it is known, for example, as the Newton-Wigner position operator [2] or its modification (see e.g. Reference [3]). As pointed out even in textbooks on quantum theory, the coordinate description of elementary particles can work only in some approximations.…”

“…For example, a section in the known textbook [5] is titled "Impossibility of introducing the photon wave function in coordinate representation". On the other hand, a detailed discussion of the photon position operator in papers by Margaret Hawton [6,7,8] and references therein indicates that it is possible to define the photon coordinate wave function ψ(r, t) but the description with such a wave function can have a good accuracy only in semiclassical approximation (see also Reference [3]), and coordinates cannot be directly measured with the accuracy better than the size of the hydrogen atom.…”

Fundamental quantal paradox and its resolution

“…A more detailed discussion can be found in subsection 1.1 of Ref. [3] titled "Historical reasons for choosing standard form of position operator".…”

“…Position operators discussed by other authors differ from the Newton-Wigner one only by spin terms and by the behavior at distances less than the wave length (see Refs. [12,3] for a more detailed discussion).…”