Acta Phys. Hung. B 20/1–2 (2004) 87–90 QUANTUM ELECTRONICS Wigner, Moyal, and Precursors to Canonical Coherent States John R. Klauder Departments of Physics and Mathematics, University of Florida Gainesville, FL 32611, USA Received 30 July 2003 Abstract. The phase space formulation of quantum mechanics pioneered by Wigner and developed by Moyal is shown to contain relevant concepts useful in the study of canonical coherent states. Keywords: canonical coherent states, group orthogonality relations PACS: 03.65.-w The 1932 paper of E. Wigner [1] in which the so-called Wigner quantum mechani- cal phase space distribution function was introduced started a recognizable cottage industry. It was Moyal [2], however, who expanded the ideas of Wigner and demon- strated how they could be used to provide a reformulation of quantum mechanics that exclusively uses phase space functions. The concept of the Moyal bracket and the implicit star product that are the quantum replacements for the Poisson bracket and the usual product of classical mechanics have been precursors to the re- cent program of deformation quantization [3]. Even more contemporary is an effort to revive and extend the Moyal program as a tool to analyze situations involving noncommuting geometry [4]. The goal of this short note is far more modest. In particular, our goal is to show how a calculation of Moyal already implies the validity of the resolution of unity formula for canonical coherent states. In Moyal’s paper he was naturally led to consider functions of the form ( = 1) Φ kl (x, k) ≡  φ k (x - q/2) ∗ e -ikq φ l (x + q/2) dq (1) 1589-9535/04/ $ 20.00 c  2004 Akad´ emiai Kiad´o, Budapest