Light does not travel in a perfectly straight line when it passes near massive objects. In this work, we calculate the gravitational deflection of light using the gauge theory of unified gravity [Rep. Prog. Phys. 88, 057802 (2025)], formulated as an extension of the Standard Model. The nonlinear graviton-graviton interaction is accounted for in the lowest order. The dynamical equations of light in external gravitational field in unified gravity are essentially different from the dynamical equations obtained using the curved metric of general relativity. The goal of the present work is to compare the predictions of unified gravity and general relativity for the gravitational deflection of light. Since the calculation of the gravitational deflection of light in unified gravity is fundamentally different from the pertinent calculation in general relativity, we devote ample space for approximations used. The deflection angles obtained from unified gravity and general relativity are equal in the first post-Newtonian (PN) order, and agree with previous experiments available for the Sun. However, the second PN order terms of unified gravity reveal measurable relative differences of 7/30≈23.3% and 4/15≈26.7% for the out-of-plane and in-plane polarizations, respectively, in comparison with the polarization-independent value of general relativity. Therefore, we expect that experimentally differentiating between the two theories will become possible in the near future. For future experiments, we also make calculations for other astrophysical objects, such as for a neutron star, which is in the limit of the PN approximation. The application of unified gravity to black holes, for which the PN approximation is not valid, is left as a topic of further work.

We derive the perihelion precession of planetary orbits using quantum field theory extending the Standard Model to include gravity. Modeling the gravitational bound state of an electron via the Dirac equation of unified gravity [Rep. Prog. Phys. 88, 057802 (2025)], and taking the classical planetary state limit, we obtain orbital dynamics exhibiting a precession in agreement with general relativity. This demonstrates that key general relativistic effects in planetary motion can emerge directly from quantum field theory without invoking the geometric framework of general relativity.

Gravitational interaction unavoidably influences atoms and their electromagnetic radiation field in strong gravitational fields. Theoretical description of such effects using the curved metric of general relativity is limited due to the classical nature of the metric and the assumption of the local inertial frame, where gravitational interaction is absent. Here we apply unified gravity extension of the Standard Model [Rep. Prog. Phys. 88, 057802 (2025)] to solve the Dirac equation for hydrogen-like atoms in the 4xU(1) gravity gauge field, which appears alongside all other quantum fields. We show that the gravity gauge field shifts the atomic Dirac energy levels by an amount that agrees with the experimentally observable gravitational redshift. Our result for the redshift follows directly from quantum field theory and is strictly independent of the metric-based explanation of general relativity. Furthermore, we present how gravitational potential gradient breaks the symmetry of the electric potential of the atomic nucleus, thus leading to splitting of otherwise degenerate spectral lines in strong gravitational fields. Enabling detailed spectral line analysis, our work opens novel possibilities for future investigations of quantum photonics phenomena in strong gravitational fields.

The Standard Model of particle physics describes electromagnetic, weak, and strong interactions, which are three of the four known fundamental forces of nature. The unification of the fourth interaction, gravity, with the Standard Model has been challenging due to incompatibilities of the underlying theories—general relativity and quantum field theory. While quantum field theory utilizes compact, finite-dimensional symmetries associated with the internal degrees of freedom of quantum fields, general relativity is based on noncompact, infinite-dimensional external space-time symmetries. The present work aims at deriving the gauge theory of gravity using compact, finite-dimensional symmetries in a way that resembles the formulation of the fundamental interactions of the Standard Model. For our eight-spinor representation of the Lagrangian, we define a quantity, called the space-time dimension field, which enables extracting four-dimensional space-time quantities from the eight-dimensional spinors. Four U(1) symmetries of the components of the space-time dimension field are used to derive a gauge theory, called unified gravity. The stress-energy-momentum tensor source term of gravity follows directly from these symmetries. The metric tensor enters in unified gravity through geometric conditions. We show how the teleparallel equivalent of general relativity in the Weitzenböck gauge is obtained from unified gravity by a gravity-gauge-field-dependent geometric condition. Unified gravity also enables a gravity-gauge-field-independent geometric condition that leads to an exact description of gravity in the Minkowski metric. This differs from the use of metric in general relativity, where the metric depends on the gravitational field by definition. Based on the Minkowski metric, unified gravity allows us to describe gravity within a single coherent mathematical framework together with the quantum fields of all fundamental interactions of the Standard Model. We present the Feynman rules for unified gravity and study the renormalizability and radiative corrections of the theory at one-loop order. The equivalence principle is formulated by requiring that the renormalized values of the inertial and gravitational masses are equal. In contrast to previous gauge theories of gravity, all infinities that are encountered in the calculations of loop diagrams can be absorbed by the redefinition of the small number of parameters of the theory in the same way as in the gauge theories of the Standard Model. This result and our observation that unified gravity fulfills the Becchi–Rouet–Stora–Tyutin (BRST) symmetry and its coupling constant is dimensionless suggest that unified gravity can provide the basis for a complete, renormalizable theory of quantum gravity.

Recently, a gauge theory of unified gravity [Rep. Prog. Phys. 88, 057802 (2025)] has been developed to extend the Standard Model to include gravity. Here we present unified gravity using the ordinary four-vector and tensor field notation of the Standard Model. The main goal of the present work is to extend the original Minkowski spacetime formulation of the theory to account for graviton-graviton interaction. This is a necessary extension for problems involving interactions between gravitational fields, for example, in the propagation of gravitational waves in external gravitational potentials. The 4xU(1) gauge invariance of unified gravity is preserved in this extension.

Quantum electrodynamics (QED) is the most accurate of all experimentally verified physical theories. How QED and other theories of fundamental interactions couple to gravity through special unitary symmetries, on which the standard model of particle physics is based, is, however, still unknown. Here we develop a coupling between the electromagnetic field, Dirac electron-positron field, and the gravitational field based on an eight-component spinorial representation of the electromagnetic field. Our spinorial representation is analogous to the well-known representation of particles in the Dirac theory but it is given in terms of 8x8 bosonic gamma matrices. In distinction from earlier works on the spinorial representations of the electromagnetic field, we reformulate QED using eight-component spinors. This enables us to introduce the generating Lagrangian density of gravity based on the special unitary symmetry of the eight-dimensional spinor space. The generating Lagrangian density of gravity plays, in the definition of the gauge theory of gravity and its symmetric stress-energy-momentum tensor source term, a similar role as the conventional Lagrangian density of the free Dirac field plays in the definition of the gauge theory of QED and its electric four-current density source term. The fundamental consequence, the Yang-Mills gauge theory of unified gravity, is studied in a separate work [arXiv:2310.01460], where the theory is also extended to cover the other fundamental interactions of the standard model. We devote ample space for details of the eight-spinor QED to provide solid mathematical basis for the present work and the related work on the Yang-Mills gauge theory of unified gravity.

The measurement of the spin angular momentum of circularly polarized light by Beth [Phys. Rev. 50, 115 (1936)] can be explained by using a microscopic torque density. However, the experiment does not resolve the space- and time-dependent evolution of the spin density of light and the wave plate and the covariant form of the microscopic torque density. Here we focus on the covariant description of the helicity, spin, and torque densities of light in materials using the chiral-stress-energy-momentum tensor. We also perform simulations of Gaussian light pulses in quarter-wave-plate geometries made of birefringent and dielectric materials.

Electrostriction, the deformation of dielectric materials under the influence of an electric field, is of continuous interest in optics. The classic experiment by Hakim and Higham [Proc. Phys. Soc. 80, 190 (1962)] for a stationary field supports a different formula of the electrostrictive force density than the recent experiment by Astrath et al. [Light Sci. Appl. 11, 103 (2022)] for an optical field. In this work, we study the origin of this difference by developing a time-dependent covariant theory of optical force densities in photonic materials. When a light pulse propagates in a bulk dielectric, the field-induced force density consists of two parts: (i) The optical wave momentum force density carries the wave momentum of light and drives forward a mass density wave of the covariant coupled field-material state of light. (ii) The optostrictive force density arises from the atomic density dependence of the electric and magnetic field energy densities. It represents an optical Lorentz-force-law-based generalization of the electro- and magnetostrictive force densities well known for static electromagnetic fields and derived from the principle of virtual work. Since the work done by the optostrictive force density is not equal to the change of the field energy density during the contraction of the material, we have to describe this difference by optostriction-related dissipation terms to fulfill the energy conservation. The detailed physical model of the dissipation is left for further works. The optostrictive force density can be understood in terms of field-induced pair interactions inside the material. Because of the related action and reaction effects, this force density cannot contribute to the net momentum transfer of the optical field. We also use the theory to simulate the propagation of a Gaussian light pulse through a dielectric material.

The Standard Model of particle physics describes electromagnetic, weak, and strong interactions, which are three of the four known fundamental forces of nature. The unification of the fourth interaction, gravity, with the Standard Model has been challenging due to incompatibilities of the underlying theories - general relativity and quantum field theory. While quantum field theory utilizes symmetries associated with internal vector spaces of quantum fields, general relativity is based on external space-time symmetries. Here we employ an internal special unitary symmetry of quantum fields to develop the Yang-Mills gauge theory, called unified gravity, which is an approach towards unification of gravity with the Standard Model. The theory is based on rewriting the Lagrangian density in terms of eight-spinors and using the Yang-Mills gauge theory to obtain a locally gauge-invariant Lagrangian density in which a tensor gauge field describes gravitational interaction. In this Lagrangian density, the gauge field of gravity appears together with the quantum fields of all other fundamental interactions of the Standard Model within a single coherent mathematical framework. The quantization of unified gravity is left as a topic of further work.

The momentum and radiation pressure of light in negative-index metamaterials (NIMs) are commonly expected to reverse their direction from what is observed for normal materials. The negative refraction and inverse Doppler effect of light in NIMs have been experimentally observed, but the equally surprising phenomenon, the negative radiation pressure of light, still lacks experimental verification. We show by simulating the exact position- and time-dependent field-material dynamics in NIMs that the momentum and radiation pressure of light in NIMs can be either positive or negative depending on their subwavelength structure. In NIMs exhibiting negative radiation pressure, the negative total momentum of light is caused by the sum of the positive momentum of the electromagnetic field and the negative momentum of the material. The negative momentum of the material results from the optical force density, which drives atoms backward and reduces the local density of atoms at the site of the light field. In contrast to earlier works, light in NIMs exhibiting negative radiation pressure has both negative total momentum and energy. For the experimental discovery of the negative radiation pressure, one must carefully design the NIM structure and record the joint total pressure of the field and material momentum components.

We present a position- and time-dependent optical force theory for optomechanics of dispersive 3D photonic materials and devices. The theory applies to media including material interfaces, waveguides, and general photonic crystal structures. The theory enables calculation of the dynamical state of the coupled field-material system and the interference of this state with other excitations of the material, such as surface acoustic waves or phonons. As an example, we present computer simulations of energy and momentum flows through a silicon crystal with anti-reflective structured interfaces. Using commercially available simulation tools, the theory can be applied to analyze optical forces in complex photonic materials and devices.

The relativistic theory of the time- and position-dependent energy and momentum densities of light in glasses and other low-loss dispersive media, where different wavelengths of light propagate at different phase velocities, has remained a largely unsolved challenge until now. This is astonishing in view of the excellent overall theoretical understanding of Maxwell's equations and the abundant experimental measurements of optical phenomena in dispersive media. The challenge is related to the complexity of the interference patterns of partial waves and to the coupling of the field and medium dynamics by the optical force on the medium atoms. In this work, we use the mass-polariton theory of light [Phys. Rev. A 96, 063834 (2017)] to derive the stressenergy-momentum (SEM) tensors of the field and the dispersive medium. Our starting point, the fundamental local conservation laws of energy and momentum densities in classical field theory, is close to that of a recent theoretical work on light in dispersive media by Philbin [Phys. Rev. A 83, 013823 (2011)], which, however, excludes the power-conversion and force density source terms describing the coupling between the field and the medium. In the general inertial frame, we present the SEM tensors in terms of Lorentz scalars, four-vectors, and field tensors that reflect in a transparent way the Lorentz covariance of the theory. The SEM tensors of the field and the medium are symmetric, form-invariant for all inertial observers, and in full accordance with the covariance principle of the special theory of relativity. When the power-conversion and force density source terms are accounted for, there is no need to introduce asymmetric SEM tensors or heuristic symmetrization procedures even for the field and medium subsystems. Therefore, asymmetric SEM tensors based on strictly classical energy and momentum densities, which have been studied in previous literature, do not account for all aspects in the field-medium coupling of electromagnetic waves. However, being based on classical field theory, the present work prompts for further groundwork on the classical limit of the quantum mechanical spin of light in a medium. The SEM tensor of the coupled field-medium state of light also has zero four-divergence. Therefore, light in a dispersive medium has a well-defined four-momentum and rest frame. The volume integrals of the total energy and momentum densities of light agree with the model of mass-polariton quasiparticles having a nonzero rest mass. The coupled field-medium state of light drives forward an atomic mass density wave, which makes the constant center-of-energy velocity law of an isolated system-a fundamental conservation law of nature-satisfied. This provides strong evidence for the consistency of the theory. The predictions of our work, such as the atomic mass density wave associated with light, are accessible to experiments.

K. Sääskilahti, ∗ J. Oksanen, S. Volz, 3, † and J. Tulkki Department of Biomedical Engineering and Computational Science, Aalto University, FI-00076 Aalto, Finland Ecole Centrale Paris, Grande Voie des Vignes, 92295 Châtenay-Malabry, France CNRS, UPR 288 Laboratoire d’Energétique Moléculaire et Macroscopique, Combustion (EM2C), Grande Voie des Vignes, 92295 Châtenay-Malabry, France (Dated: June 4, 2021)

It is common both in the classical and quantum optics to describe the optical field as a superposition of plane waves. However, it is well known that optically active materials emit photons vastly dominantly in the electric dipole approximation. The photons emitted in the electric dipole transitions are not plane waves, but spherical photon states corresponding to eigenstates J = 1 of the total angular momentum and M = +/- 1 of the z component of the total angular momentum. In addition, electric dipole photons are separated from the magnetic photons by the state index eta = e for electric photons in distinction to eta = m for magnetic photons. In this work, we study the far-field that is generated when a two-dimensional matrix of atoms emits electric dipole photons and compare this far-field at large distance from the emitting atomic matrix with a plane wave. The goal of our work is to find out if the light emitted from the electric dipole transitions carry the memory of being electric dipole photons when they are far from the emitting atoms. In particular, it is well known that a plane wave includes all angular momentum components corresponding to quantum numbers J = 1; 2; 3; ... , while the far field of the electric dipole photons emitted by the atomic matrix can include only the angular momentum component J = 1. Although the two-dimensional atomic matrix used as a light source in our simulations is certainly nontrivial to fabricate, it is nevertheless fully physical and we expect that with some modifications, the conclusions from the present simulations can be generalized to atomic light sources that are more easy to realize experimentally.

In the mass-polariton (MP) theory of light formulated by us recently [Phys. Rev. A 95, 063850 (2017)], light in a medium is described as a coupled state of the field and matter. The key result of the MP theory is that the optical force of light propagating in a transparent material drives forward an atomic mass density wave (MDW). In previous theories, it has been well understood that the medium carries part of the momentum of light. The MP theory is fundamentally different since it shows that this momentum is associated with the MDW that carries a substantial atomic mass density and the related rest energy with light. In this work, we prove the Lorentz covariance of the MP theory and show how the stress-energy-momentum (SEM) tensor of the MP transforms between arbitrary inertial frames. We also compare the MP SEM tensor with the conventional Minkowski SEM tensor and show how the well-known fundamental problems of the Minkowski SEM tensor become solved by the SEM tensor based on the MP theory. We have particularly written our work for non-expert readers by pointing out how the Lorentz transformation and various conservation laws and symmetries of the special theory of relativity are fulfilled in the MP theory.

In the recently introduced mass-polariton (MP) theory of light (Partanen et al 2017 Phys. Rev. A 95 063850), the optical force of light drives in a medium forward an atomic mass density wave. In this work, we present the Lagrangian formulation of the MP theory starting directly from the principle of least action and the well-known Lagrangian densities of the electromagnetic field and the medium within the special theory of relativity. The Lagrangian densities and the resulting Euler–Lagrange equations lead directly and without any further postulates to the unique expression of the optical Abraham force that dynamically couples the electromagnetic field and the medium in the MP theory of light. The field-medium coupling is symmetric and bi-directional and it fulfills the law of action and counteraction. The coupled dynamical equations also enable the exact description of the very small kinetic energy of the medium as a part of the total energy of the coupled state of light. Thus, the Lagrangian formulation of the present work is a complementary approach to Lorentz covariance properties of the MP theory discussed in our recent work (Partanen and Tulkki 2019 Phys. Rev. A 99 033852). We show how the coupled dynamical equations of the field and the medium can be solved analytically for a Gaussian light pulse. It is astonishing how the simple analytic results for the dynamical equations, the optical force, and the stress-energy-momentum tensor of the MP theory follow ab initio from the Lagrangian densities that have been well known for almost a century.

The recently introduced mass-polariton (MP) theory of light describes light in a medium as a coupled state of the field and matter [Phys. Rev. A 95, 063850 (2017)]. In the MP theory, the optical force density drives forward an atomic mass density wave (MDW) that accompanies electromagnetic waves in a medium. The MDW is necessary for the fulfilment of the conservation laws and the Lorentz covariance of light. In silicon at wavelength λ0 = 1550 nm, the atomic MDW carries 92% of the total momentum and angular momentum of light. The MDW of a light pulse having field energy E propagating in a dielectric also transfers a net mass equal to δM = (npng − 1)E/c2 , where np and ng are the phase and group refractive indices. In this work, we present a schematic experimental setup for the measurement of the MDW in a silicon crystal. This setup overcomes many challenges that have been present in previously introduced setups and that have made the experimental observation of the MDW effect difficult due to its smallness in comparison with other effects, such as the momentum transfer by absorption and reflections. The present setup also overcomes challenges with elastic relaxation effects while extending possible measurement time scales beyond the time scale of sound waves in the setup geometry. For the proposed setup, we also compare the predictions of the MP theory of light to the predictions of the conventional Minkowski theory, where the total momentum of light is carried by the electromagnetic field. We also aim at optimizing experimental studies of the MDW effect using the proposed setup.

We have recently developed the mass-polariton (MP) theory of light to describe propagation of light in dielectric materials [Phys. Rev. A 95, 063850 (2017)]. The MP theory considers a light wave simultaneously with the dynamics of the medium atoms driven by optoelastic forces between the field-induced dipoles and the electromagnetic field. The MP theory combines the well-known optical forces with the Newtonian dynamics of the medium. Therefore, it can be applied to any inhomogeneous, dispersive, and lossy materials. One of the key observations of the MP theory of light is that a light pulse propagating in a nondispersive dielectric transfers an increased atomic density such that the total transferred mass is equal to δM = (n2 − 1)E/c2 , where n is the refractive index and E is the electromagnetic energy of the pulse. This mass is transferred by an atomic mass density wave (MDW) where the atoms are spaced more densely inside the light pulse as a result of the optical force. Another key observation is that, in common semiconductors, most of the linear and angular momenta of light is transferred by the semiconductor atoms in the MDW moving under the influence of the optical force. In this work, we use the electric and magnetic fields of selected Laguerre-Gaussian mode beams to calculate the optical force density, which is used in the optoelastic continuum dynamics to simulate the dynamics of medium atoms in edge-supported free-standing thin film structures. The goal of our work is to find out how the different force components related to the reflection, transmission, absorption, and the atomic MDW bend and twist the film. The simulations also aim at optimizing experimental studies of the atomic dynamics in the thin film and to relating the measurements to the properties of incoming light.