The Landau Problem and non-Classicality
arXiv:2009.07171 · doi:10.1142/S0217751X20501481
Abstract
Exploring the concept of the extended Galilei group G. Representations for field theories in a symplectic manifold have been derived in association with the method of the Wigner function. The representation is written in the light-cone of a de Sitter space-time in five dimensions. A Hilbert space is constructed, endowed with a symplectic structure, which is used as a representation space for the Lie algebra of G. This representation gives rise to the spin-zero Schrödinger (Klein-Gordon-like) equation for the wave functions in phase space, such that the dependent variables have the content of position and linear momentum. This is a particular example of a conformal theory, such that the wave functions are associated with the Wigner function through the Moyal product. We construct the Pauli-Schrödinger (Dirac-like) equation in phase space in its explicitly covariant form. In addition, we analyze the gauge symmetry for spin 1/2 particles in phase space and show how to implement the minimal coupling in this case. We applied to the problem of an electron in an external field, and we recovered the non-relativistic Landau Levels. Finally, we study the parameter of negativity associated with the non-classicality of the system.
arXiv comment11 pages, 4 figures, 1 table
References in corpus (13)
- Negativity of the Wigner function as an indicator of nonclassicality
- An introduction to the tomographic picture of quantum mechanics
- Features of Time-independent Wigner Functions
- Photon distribution for one mode mixed light with generic gaussian Wigner function
- Multidimensional Hermite polynomials and photon distribution for polymode mixed light
- Equivalence between contextuality and negativity of the Wigner function for qudits
- On the Tomographic Picture of Quantum Mechanics
- Noncommutative Geometry and Symplectic Field Theory
- Galilei covariance and (4,1) de Sitter space
- Realization of the Noncommutative Seiberg-Witten Gauge Theory by Fields in Phase Space
- Galilean Covariance and the Gravitational Field
- The Noncommutative Harmonic Oscillator based in Simplectic Representation of Galilei Group
- Symplectic Field Theory of the Galilean Covariant Scalar and Spinor Representations