Hilbrand J. Groenewold
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Hilbrand J. Groenewold

Hilbrand Johannes "Hip" Groenewold (1910-1996) was a Dutch theoretical physicist who pioneered the largely operator-free formulation of quantum mechanics in phase space known as phase-space quantization.


Groenewold was born on 29 June 1910 in Muntendam in the province of Groningen. He graduated from the University of Groningen, with a major in physics and minors in mathematics and mechanics in 1934. After a visit to Cambridge to interact with John von Neumann (1934-5) on the links between classical and quantum mechanics, and a checkered career working with Frits Zernike in Groningen, then Leiden, the Hague, De Bilt, and several addresses in the North of the Netherlands during World War II, he earned his Ph.D. degree in 1946, under the tutelage of Léon Rosenfeld at Utrecht University. In 1951, he obtained a position in Groningen in theoretical physics, first as a lecturer, then as a senior lecturer, and finally as a professor in 1955. He was the initiator and organizer of the Vosbergen Conference in the Netherlands for over two decades.

His 1946 thesis paper [1] laid the foundations of quantum mechanics in phase space, in unwitting parallel with J. E. Moyal. This treatise was the first to achieve full understanding of the Wigner-Weyl transform as an invertible transform, rather than as an unsatisfactory quantization rule. Significantly, this work further formulated and first appreciated the all-important star-product, the cornerstone of this formulation of the theory, ironically often also associated with Moyal's name, even though it is not featured in Moyal's papers and was not fully understood by Moyal.[2]

Moreover, Groenewold first understood and demonstrated that the Moyal bracket is isomorphic to the quantum commutator, and thus that the latter cannot be made to faithfully correspond to the Poisson bracket, as had been envisioned by Paul Dirac.[3] This observation and his counterexamples contrasting Poisson brackets to commutators have been generalized and codified to what is now known as the Groenewold - Van Hove theorem.[4] See Groenewold's theorem for one version.


  1. ^ H.J. Groenewold (1946), "On the Principles of elementary quantum mechanics", Physica 12, pp. 405-460. doi:10.1016/S0031-8914(46)80059-4
  2. ^ Curtright, T. L.; Zachos, C. K. (2012). "Quantum Mechanics in Phase Space". Asia Pacific Physics Newsletter. 01: 37. arXiv:1104.5269. doi:10.1142/S2251158X12000069.
  3. ^ Dirac, P. A. M. (1925). "The Fundamental Equations of Quantum Mechanics". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 109 (752): 642. Bibcode:1925RSPSA.109..642D. doi:10.1098/rspa.1925.0150.
  4. ^ See Hall 2013 Theorem 13.13 for one version of this result
  • Hall, Brian C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, 267, Springer.
  • Biographical note from U Groningen memorial conference, 2016.
  • Alternate open source of the 1946 article.

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