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The localization of photons

Scott E. Hoffmann

Published Aug 31, 2019• 0 citations
OpenAlex

Abstract

We propose a position operator for the photon. The components commute and commute with helicity, and the two operators form a complete set of commuting observables. However, the simultaneous eigenvectors of this position operator and helicity do not rotate locally. For a normalized wavepacket state vector, the position-helicity amplitudes give a complete characterization of the physical properties of the state vector, yet their moduli-squared have rotation properties that do not have a simple interpretation. Thus we call these pseudo-amplitudes. We show that there is a regime of wavepacket state vectors, those with a small fractional spread around an average momentum, where the position-helicity probability pseudo-densities rotate very nearly locally as scalar functions. This is just the regime that is needed to describe a scattering experiment involving photons. We consider other measures of localization for the photon, including those proposed by other authors. It is important to note that exactly the same problems arise if we try to characterize the state vector of a massive particle with spin by position and helicity. The problems are not the result of the masslessness of the photon but arise because of its limited helicity spectrum.

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We propose a position operator for the photon. The components commute and commute with helicity, and the two operators form a complete set of commuting observables. However, the simultaneous eigenvectors of this position operator and helicity do not rotate locally. For a normalized wavepacket state vector, the position-helicity amplitudes give a complete characterization of the physical properties of the state vector, yet their moduli-squared have rotation properties that do not have a simple interpretation. Thus we call these pseudo-amplitudes. We show that there is a regime of wavepacket state vectors, those with a small fractional spread around an average momentum, where the position-helicity probability pseudo-densities rotate very nearly locally as scalar functions. This is just the regime that is needed to describe a scattering experiment involving photons. We consider other measures of localization for the photon, including those proposed by other authors. It is important to note that exactly the same problems arise if we try to characterize the state vector of a massive particle with spin by position and helicity. The problems are not the result of the masslessness of the photon but arise because of its limited helicity spectrum.
The paper provides three main definitions of the Cartesian photon position operator based on: boost generator, the transversality condition and the helicity operator. In each case, the correctness of the definition and Hermitianness of the operator in the domain of physical states are proven. All considered definitions lead to the same form of the Cartesian position operator in the domain of physical states. Radial photon position operators were also defined, but they turned out to be non-equivalent. Nevertheless, the most useful two radial operators turned out to be twin counterparts in the sense of the transformation to the helical representation, which is an alternative positional representation. The components of the photon position operator do not commute, but in analogy to the problem of eigenangular momentum, its eigenstates do exist. The eigenstates of the two components of the position operator (including the radial component) are called photon strings. Photon strings on a straight line and on a circle have been studied. The usual, previously known photon string states were named electric strings, but new magnetic strings were also introduced. Exact helical photon strings on a straight line, as well as hybrid helical photon strings on a circle, were also considered. On the other hand, infinitely short photon magnetic strings turned out to be flat photon vortices on planes with a normalization formula looks like as point-localized particles.
We extend a procedure for the construction of 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 reestablished 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.
A first-quantized free photon is a complex massless vector field A=(Aμ) whose field strength satisfies Maxwell’s equations in vacuum. We construct the Hilbert space H of the photon by endowing the vector space of the fields A in the temporal-Coulomb gauge with a positive-definite and relativistically invariant inner product. We give an explicit expression for this inner product, identify the Hamiltonian for the photon with the generator of time translations in H, determine the operators representing the momentum and the helicity of the photon, and introduce a chirality operator whose eigenfunctions correspond to fields having a definite sign of energy. We also construct a position operator for the photon whose components commute with each other and with the chirality and helicity operators. This allows for the construction of the localized states of the photon with a definite sign of energy and helicity. We derive an explicit formula for the latter and compute the corresponding electric and magnetic fields. These turn out to diverge not just at the point where the photon is localized but on a plane containing this point. We identify the axis normal to this plane with an associated symmetry axis and show that each choice of this axis specifies a particular position operator, a corresponding position basis, and a position representation of the quantum mechanics of a photon. In particular, we examine the position wave functions determined by such a position basis, elucidate their relationship with the Riemann-Silberstein and Landau-Peierls wave functions, and give an explicit formula for the probability density of the spatial localization of the photon.
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.
We show that simple geometric requirements lead to a unique position operator for the photon. The different components of the position do not commute. We investigate also the problem of maximal localizability and we draw the conclusion that for every closed curve there is a photon state localized on this curve. Note: This paper was published in Bulletin de l’Academie Polonaise de Sciences, Serie des sciences math. astr. et phys., Vol XXI, No 5, 1972, pp. 477-483. The original paper has ”for every closed space-like curve there are exactly two independent photon states corresponding to helicities ±1 localized on this curve. ” The conclusion in this form is incorrect.