Gauge transformation of quantum states in probability representation

The gauge invariance of evolution equations of tomographic probability distribution functions of quantum particles in an electro-magnetic field will be illustrated [1]. Optical and symplectic tomograms, which are determined by conventional gauge-independent dequantizers (see [2]), are gauge-dependent and are converted by means of an integral transformation simultaneously with a gauge transformation of the 4-potential of the electro-magnetic field, and the evolution equations of such tomograms, being gauge-invariant, are dependent on the gauge. Contrary to the quantum case, optical and symplectic tomograms of a classical distribution function in the phase space possess of the property of gauge-independence, and their evolution equations (Liouville equation in corresponding representations) are also gauge-independent. To decide this problem we introduced gauge-independent optical and symplectic tomographic quasi-distributions and tomographic probability distributions, and obtained their gauge-independent evolution equations, which are converted in the classical limit to the Liouville equation in corresponding tomographic representations.

References

1. Ya.A.Korennoy, V.I.Man'ko, Gauge transformation of quantum states in probability representation // arXiv:1511.00364.
2. G.G.Amosov, Ya.A.Korennoy, V.I.Man'ko, Description and measurement of observables in the optical tomographic probability representation of quantum mechanics // Phys.Rev.A 85,052119 (2012).