# Deformation Quantization: Quantum Mechanics Lives and Works in Phase-Space

**Cosmas Zachos**

ANL

Wigners 1932 quasi-probability Distribution Function in phase-space is a special (Weyl) representation of the density matrix. It has been useful in describing quantum flows in: semiclassical limits; quantum optics; nuclear physics; decoherence (eg, quantum computing); quantum chaos; "Welcher Weg" puzzles. It is also of importance in signal processing (time-frequency analysis). Nevertheless, a remarkable aspect of its internal logic, pioneered by the late J Moyal, has only emerged in the last quarter century: It furnishes a third, alternate, formulation of Quantum Mechanics, independent of the conventional Hilbert Space, or Path Integral formulations, and perhaps more intuitive, since it shares language with classical mechanics. It is logically complete and self-standing, and accommodates the uncertainty principle in an unexpected manner: it thrives on it. Simple illustrations of this fact will be detailed.