Generating all Wigner functions

Abstract

In the context of phase-space quantization, matrix elements and observables result from integration of c-number functions over phase space, with Wigner functions serving as the quasiprobability measure. The complete sets of Wigner functions necessary to expand all phase-space functions include off-diagonal Wigner functions, which may appear technically involved. Nevertheless, it is shown here that suitable generating functions of these complete sets can often be constructed, which are relatively simple, and lead to compact evaluations of matrix elements. New features of such generating functions are detailed and explored for integer-indexed sets, such as for the harmonic oscillator, as well as continuously indexed ones, such as for the linear potential and the Liouville potential. The utility of such generating functions is illustrated in the computation of star functions, spectra, and perturbation theory in phase space.

Generalized coherent state approach to star products and applications to the fuzzy sphere

“…(An alternative connection between coherent states and the Moyal star product is deduced in [18], [19]. )…”

Aharony–Bergman–Jafferis–Maldacena Wilson Loops in the Fermi Gas Approach

“…Since this commutator commutes in turn with B, one can use a simpler version of BCH which says that (see for example [46])…”

“…The coefficients of these terms can be determined in closed form by exploiting some particular cases of the BCH formula (see [46] for examples of such calculations). We will now determine these coefficients.…”

Efficient method to generate time evolution of the Wigner function for open quantum systems

“…The Wigner function is a useful tool in understanding the relationship between quantum systems and their classical counterparts [1][2][3][4][5], especially for chaotic systems in which visualization in phase-space has been crucial in enabling breakthroughs [6]. The Wigner function is also very useful for studying the quantum-to-classical transition, the process in which classical dynamics emerges as an effective theory from the underlying quantum mechanics [7][8][9][10][11][12], and for which open systems play an important role [13][14][15][16][17].…”