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arXiv:2610.14088

How long is a photon?

Sky Nelson-Isaacs

quant-phSubmitted Oct 8, 2026• Announced Oct 9, 2026• 0 citations
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Abstract

A photon has the strange property that it travels along a null interval with zero proper time, hence its emission and absorption points are, in a sense, a singular event. For their part, the emission and absorption processes have a finite duration which determines the spectral content of the photon. The question in the title points to the conceptual difficulty in `stitching' together these three intervals, which all correspond to the same physical entity but have fundamentally incompatible temporal descriptions. A previously developed model for quantum wavefunction propagation solves this apparent dilemma by formulating the life cycle of a photon in terms of recursive Fourier transforms on dual spaces. Because it doesn't treat space as dependent on time, and it is built on the Fourier transform whose phase is an invariant in special relativity as well as the foundation of the uncertainty principle, the model is a promising candidate framework for making quantum mechanics compatible with relativity. The model proposes to serve as an improved Klein-Gordon equation, or relativistically compatible replacement for Schrödinger's equation.

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A photon has the strange property that it travels along a null interval with zero proper time, hence its emission and absorption points are, in a sense, a singular event. For their part, the emission and absorption processes have a finite duration which determines the spectral content of the photon. The question in the title points to the conceptual difficulty in `stitching' together these three intervals, which all correspond to the same physical entity but have fundamentally incompatible temporal descriptions. A previously developed model for quantum wavefunction propagation solves this apparent dilemma by formulating the life cycle of a photon in terms of recursive Fourier transforms on dual spaces. Because it doesn't treat space as dependent on time, and it is built on the Fourier transform whose phase is an invariant in special relativity as well as the foundation of the uncertainty principle, the model is a promising candidate framework for making quantum mechanics compatible with relativity. The model proposes to serve as an improved Klein-Gordon equation, or relativistically compatible replacement for Schrödinger's equation.
A physical theory is proposed that obeys both the principles of special relativity and of quantum mechanics. As a key feature, the laws are formulated in terms of quantum events rather than of particle states. Temporal and spatial coordinates of a quantum event are treated on equal footing, namely as self-adjoint operators on a Hilbert space. The theory is not based upon Lagrangian or Hamiltonian mechanics, and breaks with the concept of a continuously flowing time. The physical object under consideration is a spinless particle exposed to an external potential. The theory also accounts for particle-antiparticle pair creation and annihilation, and is therefore not a single-particle theory in the usual sense. The Maxwell equations are derived as a straightforward consequence of certain fundamental commutation relations. In the non-relativistic limit and in the limit of vanishing time uncertainty, the Schrödinger equation of a spinless particle exposed to an external electromagnetic field is obtained.
A physical theory is proposed that obeys both the principles of special relativity and of quantum mechanics. Time and space are treated on exactly the same footing, namely as quantum mechanical observables on a Hilbert space. The theory is not based upon Lagrangian or Hamiltonian mechanics and cannot be formulated within the framework of unitary dynamics. As a most important aspect, the theory breaks with the concept of a continuously flowing time in favour of a discrete jump process in spacetime. All physical statements are formulated in terms of detector events rather than particle states. The physical object under consideration is a spinless particle in empty space. Yet the theory also accounts for particle-antiparticle pair creation and annihilation, and is therefore not a single-particle theory in the strict sense. The Maxwell equations are derived as a straightforward consequence of some fundamental commutation relations. In the non-relativistic limit, and in the limit of infinitely small time uncertainties, one obtains the familiar Schr\\"odinger equation of a spinless particle exposed to an external electromagnetic field.
The present study explores a theoretical extension of relativistic quantum wave equations through the introduction of complex space-time coordinates. The motivation for this approach arises from persistent conceptual challenges in relativistic quantum mechanics, particularly those related to probability interpretation and particle localization. By reformulating the Klein?Gordon and Dirac equations within a complex coordinate framework, the paper examines how the mathematical structure of these equations can be broadened without compromising fundamental physical principles. The analysis shows that treating the wave function as an analytic entity in complex space-time provides additional flexibility in handling boundary conditions and singularities. It also offers a refined perspective on the interpretation of solutions, especially in cases where traditional real-valued formulations lead to ambiguities. The study further demonstrates that Lorentz invariance can be preserved through analytic continuation, ensuring consistency with relativistic requirements. Although the framework remains theoretical, it suggests that complexification of space-time may serve as a useful tool in addressing certain limitations of relativistic quantum theory. The findings indicate that this approach has the potential to contribute to ongoing developments in quantum field theory and may provide a conceptual bridge toward more generalized formulations of quantum physics.
In quantum mechanics, light cannot be described in terms of waves and particles at the same time. However, the question about light "as it really is" implies a description that is independent from the constraints of measurement. It is rather preferable to incorporate both aspects in a causal ontological theory, especially if it simplifies explanation and improves predictions. Hence, the photon can be described as an extended longitudinal association of discrete indivisible constituents, which are held together by the effect of their own waves on each other. The same waves can then be used to interpret the force of interactions between different pulses of light, or between light and matter. The concept of wavemediated interaction between sub-photonic entities is entirely consistent with the observations of quantum mechanics. The key is to assume that elementary sources of waves never collide, and that they have constant parameters. Novel experimental implications follow from this approach. For example, the speed of propagation of changes within magnetic or electrostatic fields may not be equal to the speed of light. Also, interference fringes should be detectable beyond the volume of overlap of two non-entangled beams, but only in special experimental settings.
Solitary-particle quantum mechanics' inherent compatibility with special relativity is implicit in Schroedinger's postulated wave-function rule for the operator quantization of the particle's canonical three-momentum, taken together with his famed time-dependent wave-function equation that analogously treats the operator quantization of its Hamiltonian. The resulting formally four-vector equation system assures proper relativistic covariance for any solitary-particle Hamiltonian operator which, together with its canonical three-momentum operator, is a Lorentz-covariant four-vector operator. This, of course, is always the case for the quantization of the Hamiltonian of a properly relativistic classical theory, so the strong correspondence principle definitely remains valid in the relativistic domain. Klein-Gordon theory impairs this four-vector equation by iterating and contracting it, thereby injecting extraneous negative-energy solutions that are not orthogonal to their positive-energy counterparts of the same momentum, thus destroying the basis of the quantum probability interpretation. Klein-Gordon theory, which thus depends on the square of the Hamiltonian operator, is as well thereby cut adrift from Heisenberg's equations of motion. Dirac theory confuses the space-time symmetry of the four-vector equation system with such symmetry for its time component alone, which it fatuously imposes, thereby breaching the strong correspondence principle for the free particle and imposing the starkly unphysical momentum-independence of velocity. Physically sensible alternatives, with external electromagnetic fields, to the Klein-Gordon and Dirac equations are derived, and the simple, elegant symmetry-based approach to antiparticles is pointed out.