Quantum mechanics of a photon

Abstract

A first-quantized free photon is a complex massless vector field A = (A µ) 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.

Maxwell quantum mechanics

“…This requires a number density that is positive definite for both positive and negative frequency fields. Such an inner product does indeed exist, and its properties were investigated by Mostafazadeh and co-workers [7,8]. Here we extend their photon density to a four-current that transforms as a Lorentz vector.…”

“…The plane wave basis satisfies an analogous completeness relation that gives a k-space density ρ (k). Babaei and Mostafazadeh defined a photon inner product [8] that is closely related to (33). The parameter g, called l in [8], is unspecified and they use a real symmetric/antisymmetric basis written in terms of A and ∂ t A ∝ E. Their position eigenvectors are of the Newton-Wigner form, the Coulomb gauge is used, and their photon number density is not the time-like component of a conserved four-current.…”

“…Recently, derivation of a photon position operator with commuting components [20,21] and the Mostafazadeh particle density that is positive definite for both positive and negative frequency waves [7] have led to photon quantum mechanics [8,22] with a Hilbert space consisting of solutions to Maxwell's equations and an invariant inner product. However, this is not an entirely satisfactory first quantized theory since in [8] the position operator and its eigenvectors are of the noncovariant Newton-Wigner form, while in [22] the position operator is not Hermitian and biorthogonality is derived from QFT. Also, in [8,22] and in [1], no photon density that transforms like a Lorentz vector was found.…”

“…However, this is not an entirely satisfactory first quantized theory since in [8] the position operator and its eigenvectors are of the noncovariant Newton-Wigner form, while in [22] the position operator is not Hermitian and biorthogonality is derived from QFT. Also, in [8,22] and in [1], no photon density that transforms like a Lorentz vector was found. Here we derive a photon number density that transforms as the zeroth component of a Lorentz vector.…”

Maxwell meets Reeh–Schlieder: The quantum mechanics of neutral bosons

“…Two sources of nonlocality have contributed to the perception that there is no relativistic QM or number density: Wave functions are assumed to be of positive frequency while Hegerfeldt's theorem [19] tells us that this restriction leads to instantaneous spreading, and the Newton-Wigner (NW) position eigenvectors [20] are localized in the sense that they are orthogonal but their relationship to the physical fields and to current sources is nonlocal in configuration space. We will argue here that both of these sources of nonlocality are nonphysical: In biorthogonal QM [21] the nonlocal transformation to the NW basis is not required [22] and a scalar product exists [23] that does not require separation of the fields into their nonlocal [5] positive and negative frequency parts [24]. As a consequence real fields are allowed and the paradoxical observer dependence of par-ticle density on acceleration [7,25] can be avoided.…”