Photon Position Operator Research Articles

The Explicit Form of the Unitary Representation of the Poincaré Group for Vector-Valued Wave Functions (Massive and Massless), with Applications to Photon Localization and Position Operators

We geometrically derive the explicit form of the unitary representation of the Poincaré group for vector-valued wave functions and use it to apply speed-of-light boosts to a simple polarization basis to end up with a Hawton–Baylis photon position operator with commuting components. We give explicit formulas for other photon boost eigenmodes. We investigate the underlying affine connections on the light cone in momentum space and find that while the Pryce connection is metric semi-symmetric, the flat Hawton–Baylis connection is not semi-symmetric. Finally, we discuss the localizability of photon states on closed loops and show that photon states on the circle, both unnormalized improper states and finite-norm wave packet smeared-over washer-like regions are strictly localized not only with respect to Hawton–Baylis operators with commuting components but also with respect to the noncommutative Jauch–Piron–Amrein POV measure.

Construction of a photon position operator with commuting components from natural axioms

A general form of the photon position operator with commuting components fulfilling some natural axioms is obtained. This operator commutes with the photon helicity operator, is Hermitian with respect to the Bia\l{}ynicki--Birula scalar product, and is defined up to a unitary transformation preserving the transversality condition. It is shown that, using the procedure analogous to the one introduced by T. T. Wu and C. N. Yang for the case of the Dirac magnetic monopole, the photon position operator can be defined by a flat connection in some trivial vector bundle over ${\mathbb{R}}^{3}\ensuremath{\setminus}{(0,0,0)}$. This observation enables us to reformulate the quantum mechanics of a single photon on $({\mathbb{R}}^{3}\ensuremath{\setminus}{(0,0,0)})\ifmmode\times\else\texttimes\fi{}{\mathbb{C}}^{2}$.

Geometrical interpretation of the photon position operator with commuting components

It is shown that the photon position operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\stackrel{P\vec}{X}}$ with commuting components can be written in the momentum representation as $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\stackrel{P\vec}{X}}=i\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\stackrel{P\vec}{D}}$, where $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\stackrel{P\vec}{D}}$ is a flat connection in the tangent bundle $T({\mathbb{R}}^{3}\ensuremath{\setminus}{(0,0,{k}_{3})\ensuremath{\in}{\mathbb{R}}^{3}:{k}_{3}\ensuremath{\ge}0})$ over ${\mathbb{R}}^{3}\ensuremath{\setminus}{(0,0,{k}_{3})\ensuremath{\in}{\mathbb{R}}^{3}:{k}_{3}\ensuremath{\ge}0}$ equipped with the Cartesian structure. Moreover, $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\stackrel{P\vec}{D}}$ is such that the tangent 2-planes orthogonal to the momentum are propagated parallel with respect to $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\stackrel{P\vec}{D}}$ and also $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\stackrel{P\vec}{D}}$ is an anti-Hermitian (i.e., $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\stackrel{P\vec}{X}}$ is Hermitian) operator with respect to the scalar product $\ensuremath{\langle}\mathrm{\ensuremath{\Psi}}|{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}^{\ensuremath{-}2s}|\mathrm{\ensuremath{\Phi}}\ensuremath{\rangle}$. The eigenfunctions ${\mathrm{\ensuremath{\Psi}}}_{\stackrel{P\vec}{X}}(\stackrel{P\vec}{x})$ of the position operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\stackrel{P\vec}{X}}$ are found.

Maxwell quantum mechanics

We extend classical Maxwell field theory to a first quantized theory of the photon by deriving a conserved Lorentz four-current whose zero component is a positive definite number density. Fields are real and their positive (negative) frequency parts are interpreted as absorption (emission) of a positive energy photon. With invariant plane wave normalization, the photon position operator is Hermitian with instantaneously localized eigenvectors that transform as Lorentz four-vectors. Reality of the fields and wave function ensure causal propagation and zero net absorption of energy in the absence of charged matter. The photon probability amplitude is the real part of the projection of the photon's state vector onto a basis of position eigenvectors and its square implements the Born rule. Manifest covariance and consistency with quantum field theory is maintained through use of the electromagnetic four-potential and the Lorenz gauge.

Photon position eigenvectors, Wigner’s little group, and Berry’s phase

We show that the cylindrical symmetry of the eigenvectors of the photon position operator with commuting components, x^, reflects the E(2) symmetry of the photon little group. The eigenvectors of x^ form a basis of localized states that have definite angular momentum, J^, parallel to their common axis of symmetry. This basis is well suited to the description of “twisted light” that has been the subject of many recent experiments and calculations. Rotation of the axis of symmetry of this basis results in the observed Berry phase displacement. We prove that x^1,x^2,J^3 is a realization of the two dimensional Euclidean e2 algebra that effects genuine infinitesimal displacements in configuration space.

The photon detection operator and complementarity between an electric detector and a magnetic detector

It has been a long-standing problem that there is no consistent definition of the photon position operator or photon number density in the context of quantum theory. In this paper, we derive the photon detection operator, which defines the location of photon absorption, by applying the theory of indirect measurement to quantum electrodynamics. It is shown that the photon detection probability depends on the properties of a photon-absorbing atom, in particular, on both electric and magnetic dipole moments of the atom. An experiment is proposed in which the complementarity of wave–particle nature of light will be tested. It is also discussed that the complementarity is related to the non-commutativity of the electric and the magnetic fields. (When this work was presented at the conference CEWQO2013, it had the title ‘Photon position operators and complementarity between electric detector and magnetic detector’.)

Uncertainty Relation for Photons

The uncertainty relation for the photons in three dimensions that overcomes the difficulties caused by the nonexistence of the photon position operator is derived in quantum electrodynamics. The photon energy density plays the role of the probability density in configuration space. It is shown that the measure of the spatial extension based on the energy distribution in space leads to an inequality that is a natural counterpart of the standard Heisenberg relation. The equation satisfied by the photon wave function in momentum space which saturates the uncertainty relations has the form of the Schrödinger equation in coordinate space in the presence of electric and magnetic charges.

Why photons cannot be sharply localized

Physicists have pondered over the problem of photon localization and the related problem of the photon wave function for almost 80 years now, beginning with the work of Landau and Peierls 1. An extensive review of the photon localization problem was recently presented by Keller 2. The problem of photon localization is closely related to the widely studied problem of the photon position operator. In this paper we introduce an operational definition of partial localization based on the measurements of correlation functions for electric or magnetic fields. For a want of better names, we shall use the terms electric localization and magnetic localization even though this might erroneously suggest the presence of some electric or magnetic devices that confine the photons. Since a sharp localization of photons according to our operational definition of localization is not possible, a photon position operator compatible with this definition does not exist. Earlier studies of the photon localization emphasized the mathematical aspects see, for example, 3‐5. In this paper we emphasize the physical properties of the electromagnetic field. In particular, we exhibit the role of the photon helicity and the symmetry between the electric and magnetic fields. We proceed in the footsteps of Glauber 6‐9 who was the first to recognize the significance of the space-dependent creation and annihilation operators. This approach was recently summarized and expanded in an extensive paper by Smith and Raymer 10. In our work we concentrate on the analysis of the photon localization in terms of the electric and magnetic field operators. We treat these fields after smearing over space-time regions as bona fide observables. We put emphasis on the field aspect that is complementary to the particle aspect. It might have a weaker connection with experiments usually based on photon counting as highlighted by Glauber, but it is more precise as was explained in detail by Bohr and Rosenfeld 11,12 The standard method of quantization of the free electromagnetic field based on the decomposition into monochromatic modes is not well suited for the discussion of localizability because the monochromatic mode functions are not localized. To overcome this problem we further developed an alternative method of quantization that does not require a mode decomposition. The essential mathematical tools in our analysis are the Riemann-Silberstein RS vector and the helicity operator. This formulation has some merits of its own, and it can also be used to study other general properties of

Photon wave mechanics and position eigenvectors

One and two photon wave functions are derived by projecting the quantum state vector onto simultaneous eigenvectors of the number operator and a recently constructed photon position operator [Phys. Rev A 59, 954 (1999)] that couples spin and orbital angular momentum. While only the Landau-Peierls wave function defines a positive definite photon density, a similarity transformation to a biorthogonal field-potential pair of positive frequency solutions of Maxwell's equations preserves eigenvalues and expectation values. We show that this real space description of photons is compatible with all of the usual rules of quantum mechanics and provides a framework for understanding the relationships amongst different forms of the photon wave function in the literature. It also gives a quantum picture of the optical angular momentum of beams that applies to both one photon and coherent states. According to the rules of qunatum mechanics, this wave function gives the probability to count a photon at any position in space.

Photon position operators and localized bases

We extend a procedure for construction of the photon position operators with transverse eigenvectors and commuting components [Phys. Rev. A 59, 954 (1999)] to body rotations described by three Euler angles. The axial angle can be made a function of the two polar angles, and different choices of the functional dependence are analogous to different gauges of a magnetic field. Symmetries broken by a choice of gauge are re-established by transformations within the gauge group. The approach allows several previous proposals to be related. Because of the coupling of the photon momentum and spin, our position operator, like that proposed by Pryce, is a matrix that does not commute with the spin operator. Unlike the Pryce operator, however, our operator has commuting components, but the commutators of these components with the total angular momentum require an extra term to rotate the matrices for each vector component around the momentum direction. Several proofs of the nonexistence of a photon position operator with commuting components are based on overly restrictive premises that do not apply here.

Photon position operator with commuting components

A photon position operator with commuting components is constructed, and it is proved that it equals the Pryce operator plus a term that compensates for the adiabatic phase. Its eigenkets are transverse and longitudinal vectors, and thus states can be selected that have definite polarization or helicity. For angular momentum and boost operators defined in the usual way, all of the commutation relations of the Poincar\'e group are satisfied. This new position operator is unitarily equivalent to the Newton-Wigner-Pryce position operator for massive particles.

Optimal photon localization

We show that there is only one operator having some minimal properties enabling it to be a one-photon position operator. These properties are stated, and the solution is shown to be the photon position operator proposed by Pryce. This operator has non-commuting components. Nevertheless, it is shown that one can find states localized with an arbitrary precision.