Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Optical science and engineering

A Photon Viewed from Wigner Phase Space

Holger Mack, Wolfgang P. Schleich

Published Jul 25, 2008• 3 citations
Publisher

Abstract

We present a brief history of the photon and summarize the canonical procedure to quantize the radiation field. Our answer to the question what is a photon?. springs from the Wigner representation of quantum mechanics as applied to a single photon number state.
DOI
10.1201/9781420044256.ch5

Related papers

Four-dimensional photon cavities

K. B. Wharton

The question "What is a photon?" motivates an analysis of the interface between quantum field theory and non-relativistic quantum mechanics (QM). Despite the predictive success of quantum field theory, it retains many non-relativistic concepts while throwing out the one aspect of QM that might give a spatially-well-defined answer to the original question. A novel approach to quantizing relativistic wavefunctions is then discussed, leading to an identification of the electromagnetic vector potential as a possible single-photon wavefunction. The key missing ingredient is a relativistically-correct measurement theory, in which a photon can be partially constrained/measured everywhere along a closed 3-D hypersurface in 4-D space-time.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Aug 26, 2007•0 citations

Photon Localizationat the Nanoscale

Klaus D. Sattler

You might think that the great success of quantum electrodynamics (QED) would settle the debate on the nature of light to provide a clear view of its behavior, where the photon is regarded as the unit of excitation associated with a quantized mode of the electromagnetic (radiation) œeld. However, Heisenberg’s uncertainty principle tells us that a state of deœnite momentum, energy, and polarization associated with a plane wave used as a basis function of quantization must be completely indeœnite in space and time. It suggests the diµculty of spatial localization of a photon as a particle. In fact, Newton and Wigner showed that a free photon, as a massless particle with spin 1, has no localized states on the basis of natural invariance requirements that localized states for which operators of the Lorentz group apply should be orthogonal to the undisplaced localized states, a§er a translation (Newton and Wigner 1949). According to them, one can obtain a general expression for a position operator for massive particles and for massless particles of spin 0 or 1/2, not for massless particles with œnite spin, which indicates that there is no probability density for the position of the photon, and thus a position-representation wave function cannot be consistently introduced. It has also been shown that photons are not localizable, on the basis of imprimitive representations of the Euclidean group (Wightman 1962). It is now believed that photons are only weakly localizable, although single-photon states with arbitrarily fast asymptotic fallo˜ of energy density exist, and that a lack of strict localizability is directly related to the absence of a position operator for a photon in free space and a positionrepresentation photon wave function (Hawton 1999).

CRC Press eBooks•Sep 17, 2010•0 citations

Quantum mechanics without quantization: Representing a universal kinematical group

Gerald A. Goldin

A certain infinite-dimensional Lie group, obtained originally by regularizing and then exponentiating a singular local current algebra constructed from canonical fields, has been shown over several decades to describe a wide variety of quantum systems, via its continuous unitary representations. The group is a semidirect product of Schwartz space functions under pointwise addition with a diffeomorphism group under composition. Previously unsuspected possibilities were predicted this way, including anyons and nonabelian anyons in two-dimensional space. Last year David Sharp and I offered a much simpler, direct construction. We proposed fundamental reasons why this group serves as a universal kinematical group for quantum systems with mass in an arbitrary physical space. This paper beyond summarizing that construction, discusses more extensively what it means to obtain quantum mechanics without actually quantizing classical observables in phase space, and answers some fundamental questions. Directions are suggested for obtaining still greater generality of kinematical and dynamical possibilities by modifying the initial assumptions.

International Journal of Geometric Methods in Modern Physics•Mar 27, 2025•0 citations

What is a photon?

Oleg N Krokhin

Certain properties of photons viewed as quanta (particles of an electromagnetic field) are discussed. Specifically, the nature of localization (size) of photons along and across the direction of wave propagation is examined.

Bulletin of the Russian Academy of Sciences Physics•May 1, 2016•0 citations

Wheeler-DeWitt quantization can solve the singularity problem

Nelson Pinto-Neto, F. T. Falciano, Roberto Pereira B +1 more

We study the Wheeler-DeWitt quantum cosmology of a spatially flat Friedmann cosmological model with a free massless scalar field. We compare the consistent histories approach with the de Broglie--Bohm theory when applied to this simple model under two different quantization schemes: the Schr\"odinger-like quantization, which essentially takes the square root of the resulting Klein-Gordon equation through the restriction to positive frequencies and their associated Newton-Wigner states, or the induced Klein-Gordon quantization, that allows both positive and negative frequencies together. We show that the consistent histories approach can give a precise answer to the question concerning the existence of a quantum bounce if and only if one takes the single frequency approach within a single family of histories; namely, a family containing histories concerning properties of the quantum system at only two specific moments of time: the infinity past and the infinity future. In that case, as shown by Craig and Singh [Phys. Rev. D 82, 123526 (2010)], there is no quantum bounce. In any other situation, the question concerning the existence of a quantum bounce has no meaning in the consistent histories approach. On the contrary, we show that if one considers the de Broglie--Bohm theory, there are always states where quantum bounces occur in both quantization schemes. Hence, the assertion that the Wheeler-DeWitt quantization does not solve the singularity problem in cosmology is not precise. To address this question, one must specify not only the quantum interpretation adopted but also the quantization scheme chosen.

arXiv:1206.4021•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Sep 5, 2012•43 citations

What Is a Photon

R. Loudon

The concept of the photon is introduced by discussion of the process of electromagnetic field quantization within a closed cavity or in an open optical system. The nature of a single-photon state is clarified by consideration of its behaviour at an optical beam splitter. The importance of linear superposition or entangled states in the distinctions between quantum-mechanical photon states and classical excitations of the electromagnetic field is emphasized. These concepts and the ideas of wave-particle duality are illustrated by discussions of the effects of single-photon inputs to Brown-Twiss and Mach-Zehnder interferometers. Both the theoretical predictions and the-confirming experimental observations are covered. The defining property of the single photon in terms of its ability to trigger one. and only one, photodetection event is discussed.

Optics and Photonics News•Dec 19, 2017•0 citations
The question "What is a photon?" motivates an analysis of the interface between quantum field theory and non-relativistic quantum mechanics (QM). Despite the predictive success of quantum field theory, it retains many non-relativistic concepts while throwing out the one aspect of QM that might give a spatially-well-defined answer to the original question. A novel approach to quantizing relativistic wavefunctions is then discussed, leading to an identification of the electromagnetic vector potential as a possible single-photon wavefunction. The key missing ingredient is a relativistically-correct measurement theory, in which a photon can be partially constrained/measured everywhere along a closed 3-D hypersurface in 4-D space-time.
You might think that the great success of quantum electrodynamics (QED) would settle the debate on the nature of light to provide a clear view of its behavior, where the photon is regarded as the unit of excitation associated with a quantized mode of the electromagnetic (radiation) œeld. However, Heisenberg’s uncertainty principle tells us that a state of deœnite momentum, energy, and polarization associated with a plane wave used as a basis function of quantization must be completely indeœnite in space and time. It suggests the diµculty of spatial localization of a photon as a particle. In fact, Newton and Wigner showed that a free photon, as a massless particle with spin 1, has no localized states on the basis of natural invariance requirements that localized states for which operators of the Lorentz group apply should be orthogonal to the undisplaced localized states, a§er a translation (Newton and Wigner 1949). According to them, one can obtain a general expression for a position operator for massive particles and for massless particles of spin 0 or 1/2, not for massless particles with œnite spin, which indicates that there is no probability density for the position of the photon, and thus a position-representation wave function cannot be consistently introduced. It has also been shown that photons are not localizable, on the basis of imprimitive representations of the Euclidean group (Wightman 1962). It is now believed that photons are only weakly localizable, although single-photon states with arbitrarily fast asymptotic fallo˜ of energy density exist, and that a lack of strict localizability is directly related to the absence of a position operator for a photon in free space and a positionrepresentation photon wave function (Hawton 1999).
A certain infinite-dimensional Lie group, obtained originally by regularizing and then exponentiating a singular local current algebra constructed from canonical fields, has been shown over several decades to describe a wide variety of quantum systems, via its continuous unitary representations. The group is a semidirect product of Schwartz space functions under pointwise addition with a diffeomorphism group under composition. Previously unsuspected possibilities were predicted this way, including anyons and nonabelian anyons in two-dimensional space. Last year David Sharp and I offered a much simpler, direct construction. We proposed fundamental reasons why this group serves as a universal kinematical group for quantum systems with mass in an arbitrary physical space. This paper beyond summarizing that construction, discusses more extensively what it means to obtain quantum mechanics without actually quantizing classical observables in phase space, and answers some fundamental questions. Directions are suggested for obtaining still greater generality of kinematical and dynamical possibilities by modifying the initial assumptions.
We study the Wheeler-DeWitt quantum cosmology of a spatially flat Friedmann cosmological model with a free massless scalar field. We compare the consistent histories approach with the de Broglie--Bohm theory when applied to this simple model under two different quantization schemes: the Schr\"odinger-like quantization, which essentially takes the square root of the resulting Klein-Gordon equation through the restriction to positive frequencies and their associated Newton-Wigner states, or the induced Klein-Gordon quantization, that allows both positive and negative frequencies together. We show that the consistent histories approach can give a precise answer to the question concerning the existence of a quantum bounce if and only if one takes the single frequency approach within a single family of histories; namely, a family containing histories concerning properties of the quantum system at only two specific moments of time: the infinity past and the infinity future. In that case, as shown by Craig and Singh [Phys. Rev. D 82, 123526 (2010)], there is no quantum bounce. In any other situation, the question concerning the existence of a quantum bounce has no meaning in the consistent histories approach. On the contrary, we show that if one considers the de Broglie--Bohm theory, there are always states where quantum bounces occur in both quantization schemes. Hence, the assertion that the Wheeler-DeWitt quantization does not solve the singularity problem in cosmology is not precise. To address this question, one must specify not only the quantum interpretation adopted but also the quantization scheme chosen.
The concept of the photon is introduced by discussion of the process of electromagnetic field quantization within a closed cavity or in an open optical system. The nature of a single-photon state is clarified by consideration of its behaviour at an optical beam splitter. The importance of linear superposition or entangled states in the distinctions between quantum-mechanical photon states and classical excitations of the electromagnetic field is emphasized. These concepts and the ideas of wave-particle duality are illustrated by discussions of the effects of single-photon inputs to Brown-Twiss and Mach-Zehnder interferometers. Both the theoretical predictions and the-confirming experimental observations are covered. The defining property of the single photon in terms of its ability to trigger one. and only one, photodetection event is discussed.