It is perhaps the most fundamental principle of Quantum Mechanics that the system of states forms a linear manifold,1 in which a unitary scalar product is defined.2 The states are generally represented by wave functions3 in such a way that φ and constant multiples of φ represent the same physical state. It is possible, therefore, to normalize the wave function, i.e., to multiply it by a constant factor such that its scalar product with itself becomes 1. Then, only a constant factor of modulus 1, the so-called phase, will be left undetermined in the wave function. The linear character of the wave function is called the superposition principle. The square of the modulus of the unitary scalar product (ψ,Φ) of two normalized wave functions ψ and Φ is called the transition probability from the state ψ into Φ, or conversely. This is supposed to give the probability that an experiment performed on a system in the state Φ, to see whether or not the state is ψ, gives the result that it is ψ. If there are two or more different experiments to decide this (e.g., essentially the same experiment, performed at different times) they are all supposed to give the same result, i.e., the transition probability has an invariant physical sense.

The Theory Of Relativity

On Unitary Representations of the Inhomogeneous Lorentz Group

Publisher Summary This chapter describes the photon wave function, and explains the basic properties of a well-defined mathematical object—a six-component function of space-time variables—that describes the quantum state of the photon. The most essential property that does not hold for the photon wave function is that the argument of the wave function cannot be directly associated with the position operator of the photon. The position operator for the photon simply does not exist. However, one should remember that for massive particles also, the true position operator exists only in the nonrelativistic approximation. The concept of localization associated with the Newton-Wigner position operator is not relativistically invariant. The photon wave function is not restricted to the wave mechanics of photons. The same wave functions also appear as mode functions in the expansion of the electromagnetic field operators.

V Photon Wave Function

Photon wave function is a controversial concept. Controversies stem from the fact that photon wave functions can not have all the properties of the Schroedinger wave functions of nonrelativistic wave mechanics. Insistence on those properties that, owing to peculiarities of photon dynamics, cannot be rendered, led some physicists to the extreme opinion that the photon wave function does not exist. I reject such a fundamentalist point of view in favor of a more pragmatic approach. In my view, the photon wave function exists as long as it can be precisely defined and made useful.

https://arxiv.org/pdf/quant-ph/0508202.pdf

Photon wave function

The Cosmological Woes of Newtonian Gravitation Theory

Gyrogroup theory and its applications is introduced and explored, exposing the fascinating interplay between Thomas precession of special relativity theory and hyperbolic geometry. The abstract Thomas precession, called Thomas gyration, gives rise to grouplike objects called gyrogroups [A, A. Ungar, Am. J. Phys.59, 824 (1991)] the underlying axions of which are presented. The prefix gyro extensively used in terms like gyrogroups, gyroassociative and gyrocommutative laws, gyroautomorphisms, and gyrosemidirect products, stems from their underlying abstract Thomas gyration. Thomas gyration is tailor made for hyperbolic geometry. In a similar way that commutative groups underlie vector spaces, gyrocommutative gyrogroups underlie gyrovector spaces. Gyrovector spaces, in turn, provide a most natural setting for hyperbolic geometry in full analogy with vector spaces that provide the setting for Euclidean geometry. As such, their applicability to relativistic physics and its spacetime geometry is obvious.

The Theory Of Relativity

Citations

On Unitary Representations of the Inhomogeneous Lorentz Group

V Photon Wave Function

Photon wave function

The Cosmological Woes of Newtonian Gravitation Theory

Thomas precession: Its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics

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