EARLY ANALYSIS OF THE FOUNDATIONS OF QUANTUM MECHANICS

WEYL, HERMANN.

Quantenmechanik und Gruppentheorie.

Berlin, Julius Springer, 1927. 8vo. In contemporary half cloth with gilt lettering to spine. In "Zeitschrift für Fhysik", vol. 46. Entire volume offered. Library stamp to title page. Pp. 1-46. [Entire volume: VII, (1), 902 pp.].

First printing of Weyl's exceedingly important paper which initially did not attract much attention but "its repercussion turned out to be remarkably strong in the long range". (Scholz, Weyl Entering the ’New’ Quantum Mechanics Discourse , p. 14). In it Weyl put forth an analysis of the foundations of quantum mechanics.

"Weyl's (1927) paper, referred to by Yang above, is entitled Quantenmechanik und Gruppentheorie (Quantum Mechanics and Group Theory). In it, Weyl provides an analysis of the foundations of quantum mechanics and he emphasizes the fundamental role Lie groups play in that theory. Weyl begins the paper by raising two questions: (1) how do I arrive at the self-adjoint operators, which represent a given quantity of a physical system whose constitution is known, and (2), what is the physical interpretation of these operators and which physical consequences can be derived from them? Weyl suggests that while the second question has been answered by von Neumann, the first question has not yet received a satisfactory answer, and Weyl proposes to provide one with the help of group theory." (SEP)

Weyl’s approach to quantization was so general that for decades to come it did not attract much attention of physicists. At the beginning it even attracted very few successor investigations inside mathematics and was not noticed in the foundation of QM discourse, which was exclusively shaped by the Hilbert and von Neumann view until the 1950s. Although the immediate reception of Weyl’s early contributions to QM until about 1927, in particular his (Weyl 1927), was very sparse, its repercussion turned out to be remarkably strong in the long range:

"1. A first and immediate next step was made by Marshall Stone and John von Neumann. They both took up Weyl’s statement of a uniquely determined structure of irreducible unitary ray representations. The result of this work is (for finite n) the now famous Stone/von Neumann representation theorem.
2. A second line of repercussions may be seen in that part of the work of E. Wigner and V. Bargmann, which dealt with unitary and semi-unitary ray representations. In particular Wigner’s now famous work (at the time among physicists completely neglected) on the irreducible unitary ray representations of the Poincar´e group (Wigner 1939) looks like a next step beyond Weyl’s non-relativistic quantum kinematics from 1927.
3. A third impact is clearly to be seen in George Mackeys’s work. Mackey expressedly took up Weyl’s perspective (Mackey 1949) and developed it into a broader program for the study of irreducible unitary representations of group extensions.
4.Finally, Weyl quantization was taken up by mathematical physicists from the later 1960s onwards with the rise of deformation quantization (Pool 1966). Here the starting point was the idea to translate the operator product introduced by Weyl’s own quantization.
The last two points lead straight into very recent developments of mathematical physics." (Scholz, Weyl Entering the ’New’ Quantum Mechanics Discourse).

Order-nr.: 48927