Gauge transformation of quantum states in probability representation

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 having the Liouville equation in corresponding representations as the classical limit are found.

Probability Representation of Quantum Mechanics Where System States Are Identified with Probability Distributions

“…Using the dequantizer-quantizer formalism, gauge invariance in the probability representation of quantum mechanics was considered in [21].…”

“…Here, the integral kernel is determined by the quantizer-dequantizer operators [21] K( γ, γ , t) = iTr Û ( γ) D ( γ )Ĥ −ĤD( γ ) .…”

“…We point out that different representations-like the Wigner function representation for the system states with discrete variables based on using the formalism of the quantizer operators-were studied in [17][18][19][20]. Gauge invariance of quantum mechanics in the probability representation was studied in [21].…”

Quantum process in probability representation of quantum mechanics

“…At present, the probability representation of quantum mechanics has been already deeply elaborated and better suited to study quantum systems in a continuous-variable domain, but yet extended to deal with spin systems [7,9] and photon number states [10,11]. The scope of applications addressed using the symplectic tomography includes not only such particular problems as considering free quantum particles [8,12], particles in an electromagnetic field [13,14] or quantum oscillators [15,16], but also fundamental themes: open quantum systems [17,18], measurements [19], quantum information theory [20] and general aspects of quantum theory [21][22][23][24]. Therefore, tomographic methods propose a rich toolkit for quantum research and constitute a self-consisted branch of quantum mechanics.…”

Continuous measurements in probability representation of quantum mechanics

“…The probability representation of quantum mechanics, introduced not so far and being actively developed in recent years, is attractive due to being one of the most promising to formulate quantum mechanics in the closest manner to statistical physics [1,2,3,4]. Essentially, this approach suggests that a family of probability distributions of a coordinate in linearly and homogeneously transformed phase space is employed to describe a quantum state instead of a density matrix.…”