Abstract Quantum Mechanics and classical optics feature similar phenomena such as superposition, interference and even entanglement. Hence, techniques from optics can be used in quantum mechanics and vice versa. In this article I address the question: What can we learn from formulating optics in the language of quantum mechanics? It is argued that the solutions of the wave equations for the electromagnetic field form a tensor product of Hilbert spaces corresponding to the degrees of freedom of classical light. Therefore, it comprises non-separable solutions reminiscent of entanglement. Moreover, the two spatial degrees of freedom each carry non-commuting position and momentum variables forming a Heisenberg algebra like quantum particles moving in a two dimensional space. An analogy between the dynamics of a quantum harmonic oscillator and paraxial light propagating through a converging lens is drawn. This article presents a modern formulation of optics in the language of state vectors and operators (based on Dirac’s notation) along the lines of an earlier contribution [1], specifying and explaining its results.
Abstract Quantum Mechanics and classical optics feature similar phenomena such as superposition, interference and even entanglement. Hence, techniques from optics can be used in quantum mechanics and vice versa. In this article I address the question: What can we learn from formulating optics in the language of quantum mechanics? It is argued that the solutions of the wave equations for the electromagnetic field form a tensor product of Hilbert spaces corresponding to the degrees of freedom of classical light. Therefore, it comprises non-separable solutions reminiscent of entanglement. Moreover, the two spatial degrees of freedom each carry non-commuting position and momentum variables forming a Heisenberg algebra like quantum particles moving in a two dimensional space. An analogy between the dynamics of a quantum harmonic oscillator and paraxial light propagating through a converging lens is drawn. This article presents a modern formulation of optics in the language of state vectors and operators (based on Dirac’s notation) along the lines of an earlier contribution [1], specifying and explaining its results.
The phase space representation is used in the classical mechanics of particles but is not appropriate for describing interference phenomena that occur in classical field theory such as classical optics. The methods of classical optics and quantum mechanics are amazingly similar. That is principally because both use Hilbert spaces to represent states. The only difference is in the use of non-commuting operators to represent observables in quantum mechanics. The quantum theory of light predicts perfect anti-coincidence for single detections on the two sides of a beam splitter, whereas classical light, however weak, would be divided on a beam splitter and produce a minimum degree of correlated or coincidence counts. In mathematics the Hermite polynomials are classical orthogonal polynomial sequences that arise in probability and numerical analysis as Gaussian quadrature. They are also used in systems theory in connection with non-linear operations on Gaussian noise and other sciences.
The similarities between quantum mechanics and paraxial optics were already well-known to the founding fathers of quantum mechanics; indeed knowledge of paraxial optics partly informed quantum mechanics as a wave theory. Likewise quantum mechanical methods have been employed to better understand optics, for example, to determine which optical transformations are in principle realisable and which not. Recently the notion of classical entanglement has been mooted, ushering in a new avenue to explore, perhaps bridging the classical-quantum divide. These developments have raised questions as to which quantum tasks could be implemented with classical light, taking advantage of the wealth of four centuries of experience in optics. In this article we review the similarities as well as differences between optics and quantum mechanics, providing a quantum notation for classical light. We review the evidence for cross-fertilisation between quantum mechanics and classical optics, in particular considering the issue of classical entanglement and its exploitation for quantum tasks. Our work provides a concise theoretical framework punctuated with relevant examples, and critically assesses the current state of the field and its limits.
Physical processes are usually described using four-dimensional vector quantities - coordinate vector, momentum vector, current vector. But at the fundamental level they are characterized by coordinate and momentum spinors. The propagation of fields and their interaction takes place at the spinor level, and since each spinor uniquely corresponds to a certain vector, the results of physical processes appear before us in vector form. For example, the relativistic Schrödinger equation and the Dirac equation are formulated by means of coordinate vectors, momentum vectors and quantum operators corresponding to them. In the Dirac equation the wave function is a spinor with complex components, but still coordinates and momentum are vectors. For a closed description of nature using only spinor quantities, it is necessary to have an equation similar to the Dirac equation in which momentum, coordinates and operators are spinors. It is such an equation that is presented in this paper. Quantum mechanics describes interactions using complex wave functions that cannot be observed directly; only upon measurement do they translate into probabilities expressed as the squares of the magnitudes of the wave functions. In the same way spinor quantities are not observable, but they completely determine observable vectors. In Section 2 of the paper, we analyze the quadratic form for an arbitrary four-component complex vector based on Pauli matrices. The form is invariant with respect to Lorentz transformations including any rotations and boosts. The invariance of the form allows us to construct on its basis an equation for a free particle combining the properties of the relativistic wave equation and the Dirac equation. For an electron in the presence of an electromagnetic potential it is shown that taking into account the commutation relations between the momentum and coordinate components allows us to obtain from this equation the known results describing the interactions of the electron spin with the electric and magnetic field. In the presence of a potential the momentum components cease to commute with each other. To neutralize this effect, the Schrödinger equation is supplemented by several equations with mixed partial derivatives on coordinates. Section 3 presents a version of quantum theory in spinor coordinate space. Section 4 presents a modification of the theory of the path integral, which consists in considering the path integral in the spinor coordinate space. An equation of motion for the electron is proposed that is relativistically invariant. This novel equation permitted the construction of an actually invariant procedure for the second quantization of the fermion field in spinor coordinate space. Furthermore, it is demonstrated that the field operators are a combination of plane waves in spinor or vector space, with the coefficients of which being pseudospinors or pseudovectors. Furthermore, each plane wave possesses an additional coefficient in the form of a creation or annihilation operator. In vector space, these operators commute, whereas in spinor space they anticommutate. The paper presents the spinor and vector representations of the field operators in explicit form, comprising sets of 16 pseudospinors or 4 pseudovectors. An explicit form of the symmetric traceless tensor with spin two, zero mass and two polarizations are presented, which can serve as a model of the graviton. Section 5 presents Einstein's inhomogeneous equation, which features the Riemann tensor rather than the Ricci tensor, with the energy-momentum tensor of a charged particle having four indices rather than two. In the case where a particle of matter has a charge, another Riemann tensor is added to Einstein’s equation, in addition to the usual Riemann curvature tensor, which describes the additional curvature created for a charged particle by an electromagnetic field.
Quantum optics and classical optics have coexisted for nearly a century as two distinct, self-consistent descriptions of light. What influences there were between the two domains all tended to go in one direction, as concepts from classical optics were incorporated into quantum theory's early development. But it's becoming increasingly clear that a significant quantum presence exists in classical territory-and, in particular, that the quintessential quantum attribute, entanglement, can be seen, studied and exploited in classical optics. This blurring of the classical-quantum boundary has opened up a potential new direction for frontier work in optics.
Physical processes are usually described using four-dimensional vector quantities - coordinate vector, momentum vector, current vector. But at the fundamental level they are characterized by coordinate and momentum spinors. The propagation of fields and their interaction takes place at the spinor level, and since each spinor uniquely corresponds to a certain vector, the results of physical processes appear before us in vector form. For example, the relativistic Schrödinger equation and the Dirac equation are formulated by means of coordinate vectors, momentum vectors and quantum operators corresponding to them. In the Dirac equation the wave function is a spinor with complex components, but still coordinates and momentum are vectors. For a closed description of nature using only spinor quantities, it is necessary to have an equation similar to the Dirac equation in which momentum, coordinates and operators are spinors. It is such an equation that is presented in this paper. Using the example of the interaction between an electron and an electromagnetic field, we can see that the spinor equation contains more detailed information about the interaction than the vector equations. This is not new for quantum mechanics, since it describes interactions using complex wave functions, which cannot be observed directly, and only when measured goes to probabilities in the form of squares of the moduli of the wave functions. In the same way spinor quantities are not observable, but they completely determine observable vectors. In Section 2 of the paper, we analyze the quadratic form for an arbitrary four-component complex vector based on Pauli matrices. The form is invariant with respect to Lorentz transformations including any rotations and boosts. The invariance of the form allows us to construct on its basis an equation for a free particle combining the properties of the relativistic wave equation and the Dirac equation. For an electron in the presence of an electromagnetic potential it is shown that taking into account the commutation relations between the momentum and coordinate components allows us to obtain from this equation the known results describing the interactions of the electron spin with the electric and magnetic field. In the presence of a potential the momentum components cease to commute with each other. To neutralize this effect, the Schrödinger equation is supplemented by several equations with mixed partial derivatives on coordinates. In section 3 of the paper this quadratic form is expressed through momentum spinors, which makes it possible to obtain an equation for the spinor wave function in spinor coordinate space by replacing the momentum spinor components by partial derivative operators on the corresponding coordinate spinor component. Section 4 presents a modification of the theory of the path integral, which consists in considering the path integral in the spinor coordinate space. The Lagrangian densities for the scalar field and for the electron field, along with their corresponding propagators, are presented. An equation of motion for the electron is proposed that is relativistically invariant. This novel equation permitted the construction of an actually invariant procedure for the second quantization of the fermion field in spinor coordinate space. Furthermore, it is demonstrated that the field operators are a combination of plane waves in spinor or vector space, with the coefficients of which being pseudospinors or pseudovectors. Each of these pseudovectors or pseudospinors corresponds to one of the particles presented in the theory of electrodynamics. Furthermore, each plane wave possesses an additional coefficient in the form of a creation or annihilation operator. In vector space, these operators commute, whereas in spinor space they anticommutate. The paper presents the spinor and vector representations of the field operators in explicit form, comprising sets of 16 pseudospinors or 4 pseudovectors corresponding to particles represented in electrodynamics. An explicit form of the symmetric traceless tensor with spin two, zero mass and two polarizations is presented, which can serve as a model of the graviton. The results obtained may prompt changes in some aspects of the construction of Feynman diagrams. Among other things, it presents a purely mathematical derivation of Maxwell's inhomogeneous equations without reference to empirical data on the action of electric current, which is usually referred to when deriving equations. Section 4 presents Einstein's inhomogeneous equation, which features the Riemann tensor rather than the Ricci tensor, with the energy-momentum tensor of a charged particle having four indices rather than two. In the case where a particle of matter has a charge, another Riemann tensor is added to Einstein’s equation, in addition to the usual Riemann curvature tensor, which describes the additional curvature created for a charged particle by an electromagnetic field.
Quantum mechanics describes the physical state of a microscopic system using vectors in a complex Hilbert space rather than deterministic numerical variables. This mathematical formulation provides a rigorous framework for understanding quantum superposition, probability, and measurement. The Hilbert space structure allows quantum states to be represented as normalized vectors, while inner products define transition probabilities through the Born rule. Linear operators acting on these vectors correspond to measurable physical observables such as position, momentum, and energy. This paper presents a concise mathematical treatment of quantum states in Hilbert space, emphasizing vector representation, superposition, normalization, and projection-based measurement. In addition, the mathematical formulation underlying the concept of quantum immortality is briefly discussed from the perspective of the Many-Worlds Interpretation. The study demonstrates how linear algebra serves as the fundamental language of modern quantum mechanics and provides the theoretical basis for advanced areas including quantum computation, quantum information theory, and quantum communication.