Nine formulations of quantum mechanics

Abstract

Nine formulations of nonrelativistic quantum mechanics are reviewed. These are the wavefunction, matrix, path integral, phase space, density matrix, second quantization, variational, pilot wave, and Hamilton–Jacobi formulations. Also mentioned are the many-worlds and transactional interpretations. The various formulations differ dramatically in mathematical and conceptual overview, yet each one makes identical predictions for all experimental results.

Hermitian–Non-Hermitian Interfaces in Quantum Theory

“…In the conventional applications to quantum theory, the description of the unitary evolution of a given system S need not necessarily be performed in Schrödinger picture (cf., e.g., the compact review of its eight eligible alternatives in [24]). Naturally, once people decide to prefer the work in Schrödinger picture, they usually recall Stone's theorem [29] and conclude that the Hamiltonian (i.e., in our present notation, operator h ∈ F (퐻) acting in the conventional Hilbert space H (textbook) ) must necessarily be Hermitian (for the sake of brevity we spoke here about the HSP realization of Schrödinger picture).…”

“…An optimal formulation of quantum theory is, obviously, application-dependent [24]. Still, the so-called Schrödinger picture seems exceptional.…”

“…During the birth of quantum theory, its oldest (namely, the Heisenberg's "matrix") picture was quickly followed by Schrödinger's "wave-function" formulation which proved less intuitive but more economical [24]. The conventional, "textbook" alias "Hermitian" Schrödinger picture (HSP, [25]) was later complemented by its "non-Hermitian" Schrödinger picture (NHSP) alternative (cf.…”

A tomographic description for classical and quantum cosmological perturbations

“…In this picture, the fair nonnegative probability distribution, containing a complete information on the states, is used instead of the wave function or the density matrix. Though the new probability representation is essentially equivalent to all the other available representations adopted in quantum mechanics [11], it has its own merits. An important property of this representation is that the quantum state is associated to the same tomographic probability density W(X, µ, ν), connected with the Wigner function by the standard integral Radon transform [12].…”

Neoclassical Theory of X-Ray Scattering by Electrons

“…It is considering, that all formulations (and them more than 10) despite "differ dramatically in mathematical and conceptual overview, yet each one makes identical predictions for all results" (Styer et al, 2002).…”