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José Enrique Moyal

José Enrique Moyal (Hebrew: יוסף הנרי מויאל‎; born 1 October 1910 Jerusalem, Ottoman Empire – died 22 May 1998 Canberra, Australia) was a mathematical physicist who contributed to aeronautical engineering, electrical engineering and statistics, among other fields.[1][2]

He helped establish the phase space formulation of quantum mechanics in 1949 by bringing together the ideas of Hermann Weyl, John von Neumann, Eugene Wigner, and Hip Groenewold.[3] This formulation is statistical in nature[4] and makes logical connections between quantum mechanics and classical statistical mechanics, enabling a natural comparison between the two formulations. "Phase Space Quantization," is a synonym for "Moyal Quantization" and largely avoids use of operators for quantum mechanical observables prevalent in the canonical formulation.[5] Quantum-mechanical evolution in phase space is specified by a Moyal bracket.

Moyal grew up in Tel Aviv, and attended the Herzliya Hebrew Gymnasium. He studied in Paris in the 1930s, at the École Supérieure d'Electricité, Institut de Statistique, and, finally, at the Institut Henri Poincaré. His work was carried out in wartime England in the 1940s, while employed at the de Havilland Aircraft company. He was a professor of mathematics at Macquarie University, a colleague of John Clive Ward at the former School of Mathematics and Physics, having previously worked at Argonne National Laboratory in Illinois.[2]


  • Works 1
  • References 2
  • External links 3
  • See also 4


  • Moyal, J. E.; Bartlett, M. S. (1949). "Quantum mechanics as a statistical theory". Mathematical Proceedings of the Cambridge Philosophical Society 45: 99.  
  • J.E. Moyal, "Stochastic Processes and Statistical Physics" Journal of the Royal Statistical Society B, 11, (1949), 150-210.


  1. ^ * J. Gani, "Obituary: José Enrique Moyal," J. Appl. Probab. 35 (1998), no. 4, 1012–1017.
  2. ^ a b Ann Moyal, Maverick Mathematician: The Life and Science of J.E. Moyal, ANU E-press, 2006; ISBN 978-1920942588, accessible online
  3. ^ Curtright, T. L.; Zachos, C. K. (2012). "Quantum Mechanics in Phase Space". Asia Pacific Physics Newsletter 01: 37.  
  4. ^ M.S. Bartlett, "José Enrique Moyal," The Statistician 48 (1998), 273–274. (Bartlett acknowledges how much he gained from Moyal's wide knowledge of the European literature on probability.)
  5. ^ C. K. Zachos, D. B. Fairlie, and T. L. Curtright, "Quantum Mechanics in Phase Space", World Scientific, Singapore, 2005; ISBN 981-238-384-0

External links

  • Maverick Mathematician: The Life and Science of J.E. Moyal
  • Obituary by Alan McIntosh and photographs
  • Moyal Medal awarded annually by Macquarie University for research contributions to mathematics, physics or statistics
  • José Enrique Moyal from the Mathematics Genealogy Project

See also

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