Quantum Theory of Light Rays

• Quantum theory of light rays is a framework where classical rays emerge as coherent or quasi-classical limits of quantized electromagnetic fields and photon wave packets.
• The subject elucidates how phenomena like diffraction and interference are reinterpreted as detector-coupled mode selections rather than fixed trajectories.
• Experimental setups, from slit diffraction to structured beams and curved spacetime analyses, illustrate practical quantum limits on traditional ray propagation.

Quantum theory of light rays denotes the set of quantum-optical and quantum-field-theoretic descriptions in which a “ray” is not a fundamental line-like entity, but an emergent construct associated with a quantized electromagnetic field, a photon wave packet, a directionally concentrated family of modes, or a null-geodesic observable. In this perspective, light is fundamentally a quantized electromagnetic field whose excitations are photons; classical rays arise only in appropriate limits, such as coherent or quasi-classical states, narrow momentum distributions, and geometric-optics propagation (Yajnik, 2019, Karnieli et al., 2020). The same topic also includes quantum descriptions of diffraction, interference, emission, structured beams, and propagation in curved, discrete, or quantum spacetime, where the ray concept is replaced by probability amplitudes, detector-coupled bright modes, or fluctuating null trajectories (Cheng et al., 18 Oct 2025, Jia, 2022).

1. From corpuscles and waves to quantized light

The modern subject sits at the intersection of the Newtonian corpuscular picture, the Huygens–Young wave picture, Maxwell electrodynamics, and photon quantization. Maxwell’s theory established light as an electromagnetic wave phenomenon, but black-body radiation, Einstein’s photon hypothesis, and Bose’s derivation of the Planck spectrum reintroduced the corpuscular aspect at the level of quantized field excitations (Yajnik, 2019). In the formulation centered on coherent states and Sudarshan’s diagonal representation, classical optics is not external to quantum optics: classical states of light are subsumed within the full quantum description without taking the limit $h\to 0$, and a density operator can be written as

$$\hat{\rho}=\int d^2\alpha\,P(\alpha)\,|\alpha\rangle\langle\alpha|,$$

with classical behavior recovered when $P(\alpha)$ is a bona fide probability distribution (Yajnik, 2019).

At the single-particle level, the photon is treated as a massless spin-1 boson with helicity $\lambda=\pm1$, and its momentum-space wave function can be written as

$$\mathfrak f(\mathbf{k})= \begin{pmatrix} f_+(\mathbf{k})\ f_-(\mathbf{k}) \end{pmatrix}, \qquad \int \frac{d^3k}{k}\,|\mathfrak f(\mathbf{k})|^2=1.$$

In this representation, $|f_\pm(\mathbf{k})|^2$ is the probability density to find a photon with momentum $\hbar\mathbf{k}$ and helicity $\pm$, and the same functions $f_\pm(\mathbf{k})$ appear as Fourier coefficients of the classical Riemann–Silberstein field, making wave-particle duality a precise correspondence between photon amplitudes and Maxwell fields rather than a primitive dual ontology (Bialynicki-Birula et al., 2019).

This foundation already constrains the ray concept. In geometric optics, a ray is a line normal to a wavefront and aligned with mean energy flow. In the quantum description, that role is assumed by the expectation value of momentum or by a narrow momentum-space packet, while exact localization is limited. The photon theory discussed in this literature introduces a position operator identified with the “center of photon energy,” $\mathbf{R}=i\mathbf{D}_{\mathbf{k}}$, and derives a photon uncertainty relation

$$\hat{\rho}=\int d^2\alpha\,P(\alpha)\,|\alpha\rangle\langle\alpha|,$$0

which already indicates that perfectly sharp rays are not fundamental objects (Bialynicki-Birula et al., 2019).

2. Rays as approximations: diffraction, uncertainty, and three-dimensional visualization

The transition from rays to distributed amplitudes is especially explicit in slit diffraction. For a slit of width $$\hat{\rho}=\int d^2\alpha\,P(\alpha)\,|\alpha\rangle\langle\alpha|,$$1, classical diffraction gives a central angular width

$$\hat{\rho}=\int d^2\alpha\,P(\alpha)\,|\alpha\rangle\langle\alpha|,$$2

so a narrower slit produces a broader outgoing beam. In quantum language this is the visually embodied statement that narrowing the transverse localization $$\hat{\rho}=\int d^2\alpha\,P(\alpha)\,|\alpha\rangle\langle\alpha|,$$3 increases the transverse momentum spread $$\hat{\rho}=\int d^2\alpha\,P(\alpha)\,|\alpha\rangle\langle\alpha|,$$4, in accord with

$$\hat{\rho}=\int d^2\alpha\,P(\alpha)\,|\alpha\rangle\langle\alpha|,$$5

while a photon’s momentum magnitude remains $$\hat{\rho}=\int d^2\alpha\,P(\alpha)\,|\alpha\rangle\langle\alpha|,$$6 (Logiurato et al., 2018). A ray then becomes only the central direction of a broader momentum distribution.

A direct experimental realization of this replacement of rays by distributed amplitudes was given with a He–Ne laser of $$\hat{\rho}=\int d^2\alpha\,P(\alpha)\,|\alpha\rangle\langle\alpha|,$$7 and wavelength

$$\hat{\rho}=\int d^2\alpha\,P(\alpha)\,|\alpha\rangle\langle\alpha|,$$8

combined with single slits of $$\hat{\rho}=\int d^2\alpha\,P(\alpha)\,|\alpha\rangle\langle\alpha|,$$9, $P(\alpha)$0, and $P(\alpha)$1, a double slit of width $P(\alpha)$2 and center-to-center separation $P(\alpha)$3, and a fog chamber filled with very fine, stable droplets produced by an ultrasonic mist-maker (Logiurato et al., 2018). Because droplets scatter light wherever the local optical field $P(\alpha)$4 has appreciable intensity, the luminous fog directly maps

$P(\alpha)$5

throughout the volume. Instead of a two-dimensional screen pattern, one sees the entire beam path in three dimensions: single-slit diffraction appears as a luminous fan, and double-slit propagation shows both the single-slit envelope and the fine interference fringes developing in space (Logiurato et al., 2018).

This same apparatus illustrates the quantum reinterpretation of the ray picture. The bright regions can be read as the probability landscape for photon detections,

$P(\alpha)$6

with the mathematical identity between $P(\alpha)$7 and the single-photon rule $P(\alpha)$8 providing the bridge from classical optics to quantum theory (Logiurato et al., 2018). This suggests that “rays” are large-scale summaries of beam propagation only when diffraction is negligible and the momentum distribution is narrowly peaked.

3. Diffraction and interference as mode selection

A central contemporary reformulation treats diffraction not as wave cancellation but as detector-dependent mode selection. In the continuous-mode single-slit construction, a localized slit mode $P(\alpha)$9 is assigned to each point $\lambda=\pm1$0 across the slit, with

$\lambda=\pm1$1

A far-field detector at angle $\lambda=\pm1$2 couples only to the collective operator

$\lambda=\pm1$3

and this defines a detector-oriented basis consisting of one bright state and infinitely many dark states (Cheng et al., 18 Oct 2025).

For each $\lambda=\pm1$4, the bright state is the unique collective mode that couples to the detector, while dark states are orthogonal modes annihilated by $\lambda=\pm1$5. If the incident single-photon state is expanded as

$\lambda=\pm1$6

then the detection probability is simply $\lambda=\pm1$7. For a uniformly illuminated slit, the coefficient satisfies

$\lambda=\pm1$8

which reproduces the classical Fraunhofer envelope exactly (Cheng et al., 18 Oct 2025). On this reading, photons at a dark fringe do not disappear; they populate dark modes that are undetectable for that detector geometry.

A closely related two-mode formulation identifies bright and dark collective operators

$\lambda=\pm1$9

so that a two-level atom coupled by

$$\mathfrak f(\mathbf{k})= \begin{pmatrix} f_+(\mathbf{k})\ f_-(\mathbf{k}) \end{pmatrix}, \qquad \int \frac{d^3k}{k}\,|\mathfrak f(\mathbf{k})|^2=1.$$0

interacts only with the bright mode $$\mathfrak f(\mathbf{k})= \begin{pmatrix} f_+(\mathbf{k})\ f_-(\mathbf{k}) \end{pmatrix}, \qquad \int \frac{d^3k}{k}\,|\mathfrak f(\mathbf{k})|^2=1.$$1, while the dark mode $$\mathfrak f(\mathbf{k})= \begin{pmatrix} f_+(\mathbf{k})\ f_-(\mathbf{k}) \end{pmatrix}, \qquad \int \frac{d^3k}{k}\,|\mathfrak f(\mathbf{k})|^2=1.$$2 is completely uncoupled (Villas-Boas et al., 2021). Perfectly dark states have no photons in $$\mathfrak f(\mathbf{k})= \begin{pmatrix} f_+(\mathbf{k})\ f_-(\mathbf{k}) \end{pmatrix}, \qquad \int \frac{d^3k}{k}\,|\mathfrak f(\mathbf{k})|^2=1.$$3, maximally superradiant states have all photons in $$\mathfrak f(\mathbf{k})= \begin{pmatrix} f_+(\mathbf{k})\ f_-(\mathbf{k}) \end{pmatrix}, \qquad \int \frac{d^3k}{k}\,|\mathfrak f(\mathbf{k})|^2=1.$$4, and intermediate entangled multimode number states generate interference behavior that cannot be captured by classical theory alone (Villas-Boas et al., 2021).

The same distinction appears in a broader QED treatment of diffraction. At first order, many quantum states produce only two basic classes of diffraction images, so classical coherent and incoherent wave superposition remain effectively degenerate with the photon description. At second order, the degeneracy is lifted: coherent, entangled, number, phase-diffused, and chaotic states produce distinct two-photon diffraction patterns, and the “wave-particle equivalence breaks down” because the images directly reveal the quantum substructure of light (Stöhr, 2020). A plausible implication is that the ray concept remains reliable only for low-order, coarse-grained observables; at higher correlation order, detector-coupled mode structure becomes the more fundamental object.

4. Emission, coherence, and quantum limits on ray-like radiation

Quantum theory of light rays is not limited to free propagation; it also concerns how apparently classical beams and shockwaves are produced. In a quantum treatment of cathodoluminescence, the classical point current

$$\mathfrak f(\mathbf{k})= \begin{pmatrix} f_+(\mathbf{k})\ f_-(\mathbf{k}) \end{pmatrix}, \qquad \int \frac{d^3k}{k}\,|\mathfrak f(\mathbf{k})|^2=1.$$5

is replaced by the quantum current operator

$$\mathfrak f(\mathbf{k})= \begin{pmatrix} f_+(\mathbf{k})\ f_-(\mathbf{k}) \end{pmatrix}, \qquad \int \frac{d^3k}{k}\,|\mathfrak f(\mathbf{k})|^2=1.$$6

and the emitted light becomes entangled with the emitter’s final state (Karnieli et al., 2020). Tracing over the unobserved emitter yields a mixed photonic state, so the first-order field correlation

$$\mathfrak f(\mathbf{k})= \begin{pmatrix} f_+(\mathbf{k})\ f_-(\mathbf{k}) \end{pmatrix}, \qquad \int \frac{d^3k}{k}\,|\mathfrak f(\mathbf{k})|^2=1.$$7

depends on the emitter’s probability density and coherence structure. In Cherenkov radiation this leads to a generalized uncertainty relation for the shockwave,

$$\mathfrak f(\mathbf{k})= \begin{pmatrix} f_+(\mathbf{k})\ f_-(\mathbf{k}) \end{pmatrix}, \qquad \int \frac{d^3k}{k}\,|\mathfrak f(\mathbf{k})|^2=1.$$8

showing that a short, coherent, shockwave-like “ray” requires sufficient coherent momentum uncertainty of the emitting particle (Karnieli et al., 2020). Spectral autocorrelations of the emitted photons then encode the emitter’s wavepacket size and coherence, even when the power spectrum remains classically unchanged (Karnieli et al., 2020).

Rayleigh scattering provides a complementary example. The incident ray is modeled as a selected free-space mode coupled to a two-level atom by the Jaynes–Cummings Hamiltonian

$$\mathfrak f(\mathbf{k})= \begin{pmatrix} f_+(\mathbf{k})\ f_-(\mathbf{k}) \end{pmatrix}, \qquad \int \frac{d^3k}{k}\,|\mathfrak f(\mathbf{k})|^2=1.$$9

while all other free-space modes form a reservoir

$|f_\pm(\mathbf{k})|^2$0

with interaction

$|f_\pm(\mathbf{k})|^2$1

In this formulation, scattered light is the population transferred from the selected mode to the reservoir, and the atom plus selected mode form entangled Jaynes–Cummings eigenstates rather than passing through a phenomenological “virtual level” (Vinogradov et al., 2020). In the large-detuning Rayleigh regime, the scattered spectrum remains centered at the selected-mode frequency and has linewidth

$|f_\pm(\mathbf{k})|^2$2

which is why the process is effectively elastic (Vinogradov et al., 2020).

Strong-field high harmonic generation adds a third layer. For coherent and Fock driving fields, the established cutoff law is retained,

$|f_\pm(\mathbf{k})|^2$3

For thermal and bright squeezed vacuum driving, the heavy tails of the Husimi distribution shift the dominant effective field amplitudes and substantially extend the cutoff. The derived laws are

$|f_\pm(\mathbf{k})|^2$4

and

$|f_\pm(\mathbf{k})|^2$5

so the plateau and cutoff of the emitted beam are directly sensitive to the photon statistics of the driving light (Gorlach et al., 2022). This suggests that in extreme nonlinear optics a “ray” is better understood as an ensemble of strong-field trajectories weighted by a quantum phase-space distribution, rather than by a single classical field amplitude.

5. Structured beams, beam photons, and correlation imaging

A different strand of the subject concerns beam-like and structured monochromatic light. Starting from the vector Helmholtz equation in Coulomb gauge, beam-photon operators are constructed by transforming plane-wave operators into angular-spectrum modes,

$|f_\pm(\mathbf{k})|^2$6

where $|f_\pm(\mathbf{k})|^2$7 labels the structured mode, including Laguerre–Gaussian beams carrying orbital angular momentum $|f_\pm(\mathbf{k})|^2$8 (Punnoose et al., 2017). The corresponding beam states are genuine eigenstates of the number operator and Hamiltonian, but the beam-photon operators do not in general satisfy canonical commutation relations because the free-field theory excludes evanescent components, yielding

$|f_\pm(\mathbf{k})|^2$9

with a nontrivial overlap matrix $\hbar\mathbf{k}$0 (Punnoose et al., 2017). For Laguerre–Gaussian beams, different $\hbar\mathbf{k}$1 values remain orthogonal, while radial indices $\hbar\mathbf{k}$2 are not generally orthogonal at fixed $\hbar\mathbf{k}$3 (Punnoose et al., 2017). In the language of rays, this replaces idealized line-like propagation by directionally concentrated, mode-resolved single-photon states whose propagation is still wave-equation driven but whose orthogonality structure is genuinely quantum.

Quantum imaging with X-rays provides a direct experimental realization of ray-like behavior emerging from biphoton correlations. In that work, spontaneous parametric down-conversion in a diamond crystal pumped at $\hbar\mathbf{k}$4 generates correlated X-ray photon pairs satisfying

$\hbar\mathbf{k}$5

together with phase-matching relations that connect energy fraction $\hbar\mathbf{k}$6 to emission angle $\hbar\mathbf{k}$7 (Goodrich et al., 2024). At a detector distance $\hbar\mathbf{k}$8, the conical SPDC distribution forms a ring with

$\hbar\mathbf{k}$9

and the energy–radius relation

$\pm$0

induces a mapping $\pm$1 between signal and idler radii (Goodrich et al., 2024). The experiment reports an unprecedented detection rate of about $\pm$2 pairs per hour and the observation of energy anti-correlation for the X-ray photon pairs (Goodrich et al., 2024). In correlation imaging, only the signal photon interacts with the object, while the ghost image is reconstructed from the joint distribution $\pm$3, so the effective ray geometry is encoded in biphoton energy-angle correlations rather than in independent classical trajectories (Goodrich et al., 2024).

6. Curved spacetime, discrete models, and quantum-spacetime rays

In curved spacetime, the classical backbone of any quantum theory of light rays is the null geodesic. A generalized Fermat principle states that for light emitted from an event and detected on a timelike worldline, the arrival time is stationary for the null geodesic among nearby future-directed null curves (Frolov, 2013). Writing the spacetime metric in generalized ultrastationary form,

$\pm$4

the null condition gives

$\pm$5

and the resulting optimal-control Hamiltonian reproduces the null-geodesic equations exactly (Frolov, 2013). In a stationary spacetime this reduces to a travel-time functional

$\pm$6

so rays are still extremals of an action, now with direct relevance for Hamilton–Jacobi and eikonal constructions (Frolov, 2013).

The causal geometry of null rays also supports notions of refocusing and strong refocusing. In a strongly causal spacetime, refocusing at an event $\pm$7 means that for any sufficiently small neighborhood of $\pm$8, there exists a distant event $\pm$9 such that all null geodesics through $f_\pm(\mathbf{k})$0 enter that neighborhood; strong refocusing requires all null geodesics through $f_\pm(\mathbf{k})$1 to pass through $f_\pm(\mathbf{k})$2 itself (Kinlaw, 2010). The set of refocusing points is closed, Lorentz covering spaces preserve strong causality and refocusing, and in dimensions $f_\pm(\mathbf{k})$3 and $f_\pm(\mathbf{k})$4, Perelman’s geometrization theorem implies that every globally hyperbolic refocusing spacetime admits a globally hyperbolic strongly refocusing metric (Kinlaw, 2010). These results remain classical, but they determine the geodesic skeleton on which semiclassical or quantum propagation is built.

Two nonstandard microscopic extensions push the ray concept further. In a quantum cellular automaton model of light, free electrodynamics emerges from two Weyl QCAs on a BCC lattice, and the photon is introduced as a composite particle made of a pair of correlated massless Fermions (Bisio et al., 2014). The emergent field obeys Maxwell’s equations for small wave vector $f_\pm(\mathbf{k})$5, but the dispersion relation becomes

$f_\pm(\mathbf{k})$6

leading to dispersive propagation in vacuum, a small longitudinal polarization, and a saturation effect of fermionic origin (Bisio et al., 2014). In Lorentzian simplicial quantum gravity, by contrast, a test light ray is traced through each simplicial geometry in a gravitational path integral, and one computes amplitudes for different landing positions on the far boundary. The result is a probability distribution for the ray endpoint rather than a single null trajectory, with fluctuations that become large when the coupling constants are relatively small in absolute value and, in $f_\pm(\mathbf{k})$7 and $f_\pm(\mathbf{k})$8 dimensions, increase as the boundary size is decreased (Jia, 2022). A plausible implication is that in quantum spacetime a ray is not merely broadened by diffraction; its very causal path becomes geometry-dependent and probabilistic.

A different point of view proposes a free-space quantization based on gauge duality, generalized fluxes, and virtual charges confined to “virtual electrodes,” with four bosons introduced to encode a proper spin-1 pseudo-vector together with parity- and charge-related operators; in that proposal, real photons are the helicity bosons and virtual ones correspond to a parity charge (Collin, 10 Oct 2025). This is not the standard QED construction, but it exemplifies an ongoing controversy about whether the usual potential-based quantization is the most satisfactory starting point for a quantum theory of free-space light.

Taken together, these approaches converge on a common conclusion. Quantum theory does not abolish the ray concept, but it relocates it. A ray is a large-scale, state-dependent, detector-dependent, and sometimes geometry-dependent emergent object: a mean momentum direction, a bright collective mode, a sharply supported correlation channel, or a semiclassical null characteristic. Diffraction, interference, emission, and propagation then cease to be properties of ideal lines and become properties of quantum states, their correlations, and the structures that couple them to matter and spacetime.