Hilbrand J. Groenewold
Encyclopedia
Hilbrand Johannes 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.
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 Holland 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.
His seminal 1946 thesis paper firmly 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 Weyl correspondence 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.
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
. 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.
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
in phase space
Phase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
known as phase-space quantization.
He graduated from the University of Groningen
University of Groningen
The University of Groningen , located in the city of Groningen, was founded in 1614. It is one of the oldest universities in the Netherlands as well as one of its largest. Since its inception more than 100,000 students have graduated...
, with a major in
physics and minors in mathematics and mechanics in 1934. After a visit to Cambridge to interact with John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
(1934-5) on the links between classical and quantum mechanics, and a checkered career working with Frits Zernike
Frits Zernike
Frits Zernike was a Dutch physicist and winner of the Nobel prize for physics in 1953 for his invention of the phase contrast microscope, an instrument that permits the study of internal cell structure without the need to stain and thus kill the cells....
in Groningen, then Leiden, the Hague, De Bilt, and several addresses in the North of Holland during World War II, he earned his Ph.D. degree in 1946, under the tutelage of Léon Rosenfeld
Léon Rosenfeld
Léon Rosenfeld was a Belgian physicist. He obtained a PhD at the University of Liège in 1926, and he was a collaborator of the physicist Niels Bohr. He did early work in quantum electrodynamics that predates by two decades the work by Dirac and Bergmann. He coined the name lepton...
at Utrecht University
Utrecht University
Utrecht University is a university in Utrecht, Netherlands. It is one of the oldest universities in the Netherlands and one of the largest in Europe. Established March 26, 1636, it had an enrollment of 29,082 students in 2008, and employed 8,614 faculty and staff, 570 of which are full professors....
. 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.
His seminal 1946 thesis paper firmly laid the foundations of quantum mechanics in phase space
Weyl quantization
In mathematics and physics, in the area of quantum mechanics, Weyl quantization is a method for systematically associating a "quantum mechanical" Hermitian operator with a "classical" kernel function in phase space invertibly...
, in (unwitting) parallel with J. E. Moyal
José Enrique Moyal
José Enrique Moyal was a mathematical physicist who contributed to aeronautical engineering, electrical engineering and statistics, among other fields...
. This treatise was the first to achieve full understanding of
the Weyl correspondence as an invertible transform, rather than as an unsatisfactory quantization rule. Significantly, this work further formulated and first appreciated the all-important star-product
Moyal product
In mathematics, the Moyal product, named after José Enrique Moyal, is perhaps the best-known example of a phase-space star product: an associative, non-commutative product, ∗, on the functions on ℝ2n, equipped with its Poisson bracket .This particular star product is also sometimes called...
, 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.
Moreover, Groenewold first understood and demonstrated that the Moyal bracket
Moyal bracket
In physics, the Moyal bracket is the suitably normalized antisymmetrization of the phase-space star product.The Moyal Bracket was developed in about 1940 by José Enrique Moyal, but Moyal only succeeded in publishing his work in 1949 after a lengthy dispute with Dirac...
is isomorphic to the quantum
Quantum
In physics, a quantum is the minimum amount of any physical entity involved in an interaction. Behind this, one finds the fundamental notion that a physical property may be "quantized," referred to as "the hypothesis of quantization". This means that the magnitude can take on only certain discrete...
commutator
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
, and thus that the latter cannot be made to faithfully correspond to the Poisson bracket
Poisson bracket
In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time-evolution of a Hamiltonian dynamical system...
, as had been envisioned by Paul Dirac
Paul Dirac
Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
. 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
Léon Van Hove
Léon Van Hove was a Belgian physicist and a former Director General of CERN. He developed a scientific career spanning mathematics, solid state physics, elementary particle and nuclear physics to cosmology.-Biography:...
theorem.