This paper gives a constructive answer to the question whether photon states can contain or not, and to what extent, the readings of rulers and clocks. The paper first shows explicitly that, along with the momentum representation, there is room in the one photon Hilbert space for an alternative position representation. This is made possible by the existence of a self-adjoint, involutive, position operator conjugate to the momentum operator (due to M. Hawton). Position and momenta are shown to satisfy the Heisenberg-Weyl quantization rules in the helicity basis, which is analyzed anew from this point of view. The paper then turns to the photon's time of arrival. By picking an appropriate photon Hamiltonian - using Maxwell equations as the photon Schroedinger equation - a conjugate time of arrival operator is built. Its interpretation, including the probability densities for the instant of arrival (at arbitrary points of 3-D space) of photon states with different helicities coming from arbitrary places, is discussed.
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.
The notation
ψ(x,t) is among the first pieces of quantum mechanics that students learn. It is also among the easiest to over-interpret. Because
x and
t occur as arguments of the same function, students may ask whether they have the same mathematical status. They may also ask whether
ψ(t) should require a generalized bra
⟨t∣ in the same way that
ψ(x)=⟨x⟩ψ is often written. A related question is whether the absence of a universal time operator follows simply from Pauli's argument. These questions mix several structures that are usually introduced in different parts of the curriculum. We present a unified pedagogical treatment built around two maps hidden in
ψ(x,t). Time evolution selects a state along a trajectory in Hilbert space. A spectral representation then maps that state to amplitudes labelled by outcomes of a chosen observable. We formulate the position representation without generalized eigenkets. We recover Dirac's
∣x⟩ notation as a controlled continuum shorthand and use a finite-grid limit to show where delta normalization enters. We distinguish background coordinates, translation parameters, observables, spectral labels, and physical records. We also clarify the Stone-theorem analogy, compare prescribed-time position measurements with arrival-time measurements, state what the strong form of Pauli's argument excludes, and exhibit an exactly solvable boundary case in which a canonical self-adjoint time observable exists. Spin, circuit-QED, and optical-clock examples provide experimentally grounded checks. The aim is not a new interpretation of time. It is a reusable teaching framework for separating mathematical role from notation.
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) eld. However, Heisenberg’s uncertainty principle tells us that a state of denite momentum, energy, and polarization associated with a plane wave used as a basis function of quantization must be completely indenite 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 nite 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).
From the point of view of purely mathematics, coordinates and momenta can be imaginary, and then it is not necessary for a Hamiltonian operator to have a semi-bounded spectrum, only if one imposes some physical conditions, a Hilbert space is defined and the spectrum of the Hamiltonian operator is determined. Pauli's Theorem just shows us a mathematical truth: via a time operator, one can find all the possible eigenstates of the Hamiltonian operator, but only a part of them satisfy the given physical conditions and represent the physical solutions, while the remainders represent the redundant solutions, they do not satisfy the given physical conditions and should be discarded. The transition probabilities between the physical solutions and the redundant solutions vanish. Therefore, the existence of a self-adjoint time operator actually presents no problem.
An alternative quantization rule, in which time becomes a self-adjoint operator and position is a parameter, was proposed by Dias and Parisio [Phys. Rev. A 95, 032133 (2017)]. In this approach, the authors derived a space-time-symmetric (STS) extension of quantum mechanics (QM) where a new quantum state (intrinsic to the particle) $|\ensuremath{\phi}(x)\ensuremath{\rangle}$ is defined at each point in space. The quantum state $|\ensuremath{\phi}(x)\ensuremath{\rangle}$ obeys a space-conditional (SC) Schr\"odinger equation and its projection on $|t\ensuremath{\rangle}, \ensuremath{\langle}t|\ensuremath{\phi}(x)\ensuremath{\rangle}$, represents the probability amplitude of the particle's arrival time at
x. In this work we provide an interpretation of the SC Schr\"odinger equation and the eigenstates of observables in the STS extension. Analogous to the usual QM, we propose that by knowing the initial state $|\ensuremath{\phi}({x}_{0})\ensuremath{\rangle}$, which predicts any measurement on the particle performed by a detector localized at
x0, the SC Schr\"odinger equation provides $|\ensuremath{\phi}(x)\ensuremath{\rangle}=\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbb{U}}(x,{x}_{0})|\ensuremath{\phi}({x}_{0})\ensuremath{\rangle}$, enabling us to predict measurements when the detector is at $x\ensuremath{\lessgtr}{x}_{0}$. We also verify that for space-dependent potentials, momentum eigenstates in the STS extension $|{P}_{b}(x)\ensuremath{\rangle}$ depend on position just as energy eigenstates in the usual QM depend on time for time-dependent potentials. In this context, whereas a particle in the momentum eigenstate in the standard QM, $|\ensuremath{\psi}(t)\ensuremath{\rangle}={|P\ensuremath{\rangle}|}_{t}$, at time
t, has momentum
P (and indefinite position), the same particle in the state $|\ensuremath{\phi}(x)\ensuremath{\rangle}=|{P}_{b}(x)\ensuremath{\rangle}$ arrives at position
x with momentum
Pb(x) (and indefinite arrival time). By investigating the fact that $|\ensuremath{\psi}(t)\ensuremath{\rangle}$ and $|\ensuremath{\phi}(x)\ensuremath{\rangle}$ describe experimental data of the same observables collected at
t and
x, respectively, we conclude that they provide complementary information about the same particle. Finally, we solve the SC Schr\"odinger equation for an arbitrary space-dependent potential. We apply this solution to a potential barrier and compare it with a generalized Kijowski distribution, showing that they can predict distinct traversal times.
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 quantum mechanics, this wave function gives the probability to count a photon at any position in space.