The task of quantizing general relativity raises serious questions about the meaning of the present formulation and interpretation of quantum mechanics when applied to so fundamental a structure as the space-time geometry itself. This paper seeks to clarify the foundations of quantum mechanics. It presents a reformulation of quantum theory in a form believed suitable for application to general relativity. The aim is not to deny or contradict the conventional formulation of quantum theory, which has demonstrated its usefulness in an overwhelming variety of problems, but rather to supply a new, more general and complete formulation, from which the conventional interpretation can be deduced. The relationship of this new formulation to the older formulation is therefore that of a metatheory to a theory, that is, it is an underlying theory in which the nature and consistency, as well as the realm of applicability, of the older theory can be investigated and clarified.

"relative State" Formulation of Quantum Mechanics

(b) Write out the equations for the time dependence of ρ11, ρ22, ρ12 and ρ21 assuming that a light field interaction V = -μ12Ee iωt + c.c. couples only levels |1> and |2>, and that the excited levels exhibit spontaneous decay. (8 marks) (c) Under steady-state conditions, find the ratio of populations in states |2> and |3>. (3 marks) (d) Find the slowly varying amplitude ̃ ρ 12 of the polarization ρ12 = ̃ ρ 12e iωt . (6 marks) (e) In the limiting case that no decay is possible from intermediate level |3>, what is the ground state population ρ11(∞)? (2 marks) 2. (15 marks total) In a 2-level atom system subjected to a strong field, dressed states are created in the form |D1(n)> = sin θ |1,n> + cos θ |2,n-1> |D2(n)> = cos θ |1,n> sin θ |2,n-1>

The Quantum Theory of Light

The Physics of Quantum Information

This Review article provides an overview of the fundamental origins and important applications of the main spin–orbit interaction phenomena in modern optics that play a crucial role at subwavelength scales. Light carries both spin and orbital angular momentum. These dynamical properties are determined by the polarization and spatial degrees of freedom of light. Nano-optics, photonics and plasmonics tend to explore subwavelength scales and additional degrees of freedom of structured — that is, spatially inhomogeneous — optical fields. In such fields, spin and orbital properties become strongly coupled with each other. In this Review we cover the fundamental origins and important applications of the main spin–orbit interaction phenomena in optics. These include: spin-Hall effects in inhomogeneous media and at optical interfaces, spin-dependent effects in nonparaxial (focused or scattered) fields, spin-controlled shaping of light using anisotropic structured interfaces (metasurfaces) and robust spin-directional coupling via evanescent near fields. We show that spin–orbit interactions are inherent in all basic optical processes, and that they play a crucial role in modern optics.

Spin-orbit interactions of light

Tables of integrals

New uncertainty relations in quantum mechanics are derived. They express restrictions imposed by quantum theory on probability distributions of canonically conjugate variables in terms of corresponding information entropies. The Heisenberg uncertainty relation follows from those inequalities and so does the Gross-Nelson inequality.

/pdf/uncertainty-relations-for-information-entropy-in-wave-8q3h06k5ms.pdf

Uncertainty relations for information entropy in wave mechanics

Nonlinear Wave Mechanics

The effective nonlinear Lagrangian derived by Heisenberg and Euler is used to describe the propagation of photons in slowly varying but otherwise arbitrary electromagnetic fields. The group and the phase velocities for both propagation modes are calculated, and it is shown that the propagation is always causal. The photon splitting processes are also studied, and it is shown that they do not play any significant role even in very strong magnetic fields surrounding neutron stars.

Nonlinear effects in quantum electrodynamics. photon propagation and photon splitting in an external field.

It is believed that certain matrix elements of the electromagnetic field operators in quantum electrodynamics, in close analogy with nonrelativistic quantum theory of massive particles, may be treated as photon wave functions. In this paper, I would like to push this interpretation even further by arguing that one can set up a consistent wave mechanics of photons that could be often used as a convenient tool in the description of electromagnetic fields, independently of the formalism of second quantization. In other words, in constructing quantum theories of photons we may proceed, as in quantum theory of massive particles, through two stages. At the first stage we introduce wave functions and a wave equation obeyed by these wave functions. At the second stage we upgrade the wave functions to the level of field operators in order to deal more effectively with the states involving many particles and to allow for processes in which the number of particles is not conserved. An important additional consequence of having a genuine wave function for the photon is a possibility to define an analog of the Wigner function for the photon with its semiclassical interpretation as a (quasi) distribution function in the phase space. The very concept of the photon wave function is not new, but strangely enough it has never been systematically explored. Some textbooks on quantum mechanics start the introduction to quantum theory with a discussion of photon polarization measurements (cf., for example [1–4]). Dirac, in particular, writes a lot about the role of the wave function in the description of quantum interference phenomena for photons (in this context he uses the now famous phrase: “Each photon interferes only with itself”), but in his exposition the photon wave function never takes on a specific mathematical form. It is true that in the description of polarization simple prototype two-component wave functions are often used to describe various polarization states of the photon and with their help the preparation and the measurement of polarization is thoroughly explained. However, after such a heuristic introduction to quantum theory, the authors go on to the study of massive particles and if they ever return to a quantum theory of photons it is always within the formalism of second quantization with creation and annihilation operators. In some textbooks (cf., for example [5–7]) one may even find statements that completely negate the possibility of introducing a wave function for the photon. I shall introduce a wave function for the photon by reviving and extending the mode of description of the electromagnetic field based on the complex form of the Maxwell equations. This form was known already at the beginning of the century [8,9] and was later rediscovered by Majorana [10] who explored the analogy between the Dirac equation and the Maxwell equations. The complex vector that appears in this description will be shown to have the properties that one would associate with a one-photon wave function, including also an acceptable probabilistic interpretation. Photons are much different from massive particles. They are also different from neutrinos since the photon number does not obey a conservation law. These differences are a source of complications, especially when photons propagate in a medium. However, even these complications have a certain value: they teach us something new about the nature of photons. The approach adopted in this paper is distinctly different from the line of investigation started by Landau and Peierls [11] and continued more recently by Cook [13]. The Landau-Peierls and Cook wave functions are nonlocal objects. The nonlocality is introduced by operating on the electromagnetic fields with the integral operator (−∆)−1/4,

/pdf/on-the-wave-function-of-the-photon-46jadgbzro.pdf

On the Wave Function of the Photon

Quantum-mechanical uncertainty relations for position and momentum are expressed in the form of inequalities involving the R\'enyi entropies. The proof of these inequalities requires the use of the exact expression for the $(p,q)$-norm of the Fourier transformation derived by Babenko and Beckner. Analogous uncertainty relations are derived for angle and angular momentum and also for a pair of complementary observables in $N$-level systems. All these uncertainty relations become more attractive when expressed in terms of the symmetrized R\'enyi entropies.

https://arxiv.org/pdf/quant-ph/0608116.pdf

Iwo Bialynicki-Birula

Papers

Uncertainty relations for information entropy in wave mechanics

Nonlinear Wave Mechanics

Nonlinear effects in quantum electrodynamics. photon propagation and photon splitting in an external field.

On the Wave Function of the Photon

Formulation of the uncertainty relations in terms of the Rényi entropies