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Journal of Physics A Mathematical and Theoretical

Gauge transformation of quantum states in probability representation

Ya. A. Korennoy, Vladimir I. Man’ko

Published Feb 9, 2017• 7 citations
Publisher

Abstract

Abstract The gauge invariance of the evolution equations of tomographic probability distribution functions of quantum particles in an electromagnetic field is illustrated. Explicit expressions for the transformations of ordinary tomograms of states under a gauge transformation of electromagnetic field potentials are obtained. Gauge-independent optical and symplectic tomographic quasi-distributions and tomographic probability distributions of states of quantum system are introduced, and their evolution equations have the Liouville equation in corresponding representations as the classical limits are found.

DOI
10.1088/1751-8121/aa5f64

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Abstract The gauge invariance of the evolution equations of tomographic probability distribution functions of quantum particles in an electromagnetic field is illustrated. Explicit expressions for the transformations of ordinary tomograms of states under a gauge transformation of electromagnetic field potentials are obtained. Gauge-independent optical and symplectic tomographic quasi-distributions and tomographic probability distributions of states of quantum system are introduced, and their evolution equations have the Liouville equation in corresponding representations as the classical limits are found.
The gauge invariance of the evolution equations of tomographic probability distribution functions of quantum particles in an electromagnetic field is illustrated. Explicit expressions for the transformations of ordinary tomograms of states under a gauge transformation of electromagnetic field potentials are obtained. Gauge-independent optical and symplectic tomographic quasi-distributions and tomographic probability distributions of states of quantum system are introduced, and their evolution equations having the Liouville equation in corresponding representations as the classical limit are found.
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The problem of moving of a charged particle in electromagnetic field is considered in terms of tomographic probability representation. The coherent and Fock states of a charge moving in varying homogeneous magnetic field are studied in the tomographic probability representation of quantum mechanics. The states are expressed in terms of quantum tomograms. The Fock state tomograms are given in the form of probability distributions described by multivariable Hermite polynomials with time-dependent arguments. The obtained results are generalized in the present work and are applied to determining the transition probabilities between Landau levels. Transition probabilities are calculated using the symplectic tomograms instead of the wave functions. The same method is used in obtaining the transition probabilities between the Landau levels possessed by a charge moving in varying electromagnetic field.
The von Neumann evolution equation for density matrix and the Moyal equation for the Wigner function are mapped onto evolution equation for optical tomogram of quantum state. The connection with known evolution equation for symplectic tomogram of the quantum state is clarified. The stationary states corresponding to quantum energy levels are associated with the probability representation of the von Neumann and Moyal equations written for the optical tomograms. Classical Liouville equation for optical tomogram is obtained. Example of parametric oscillator is considered in detail.
We present a new formulation of conventional quantum mechanics, in which the notion of a quantum state is identified via a fair probability distribution of the position measured in a reference frame of the phase space with rotated axes. In this formulation, the quantum evolution equation as well as the equation for finding energy levels are expressed as linear equations for the probability distributions that determine the quantum states. We also give the integral transforms relating the probability distribution (called the tomographic-probability distribution or the state tomogram) to the density matrix and the Wigner function and discuss their connection with the Radon transform. Qudit states are considered and the invertible map of the state density operators onto the probability vectors is discussed. The tomographic entropies and entropic uncertainty relations are reviewed. We demonstrate the uncertainty relations for the position and momentum and the entropic uncertainty relations in the tomographic-probability representation, which is suitable for an experimental check of the uncertainty relations.
Partial symplectic conditional and joint probability representations of quantum mechanics are considered. The correspondence rules for most interesting physical operators are found and the expressions of the dual symbols of operators are derived. Calculations were made by use of general formalism of quantizers and dequantizers determining the star product quantization scheme in these representations. Taking the Gaussian functions as the distributions of the tomographic parameters the examples of joint probability representations were considered. Evolution equations and stationary states equations for partial symplectic conditional and joint probability distributions are obtained.