Using mathematical tools developed by Hermann Weyl, the Wigner classification of group‐representations and co‐representations is clarified and extended. The three types of representation, and the three types of co‐representation, are shown to be directly related to the three types of division algebra with real coefficients, namely, the real numbers, complex numbers, and quaternions. The author's theory of matrix ensembles, in which again three possible types were found, is shown to be in exact correspondence with the Wigner classification of co‐representations. In particular, it is proved that the most general kind of matrix ensemble, defined with a symmetry group which may be completely arbitrary, reduces to a direct product of independent irreducible ensembles each of which belongs to one of the three known types.
REFERENCES
1.
See also, E. P. Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra (Academic Press Inc., New York, 1959), English edition, Chaps 24 and 26.
2.
H. Weyl, The Classical Groups, Their Invariants and Representations (Princeton University Press, Princeton, New Jersey, 1939). Chapter 3 of this book contains the essential theorems on which all of our arguments hang.
For Weyl’s treatment of semilinear representations, see
Duke Math. J.
3
, 200
(1937
).3.
4.
F. J.
Dyson
, J. Math. Phys.
3
, 140
, 157
and
166
(1962
). This series of three papers includes references to earlier work by others in the same field. Paper IV in the series is being written in collaboration with Dr. M. L. Mehta and will be published later. The present paper should logically be considered to be number zero in the series, since it provides an improved mathematical and logical foundation for the rest of the series. Since Roman numerals contain no symbol for zero, we preferred to publish the present paper under a separate title.5.
A sketch of the historical development is to be found in the section headed “Remembrance of Things Past” in Weyl’s book (reference 2), p. 27.
6.
Chapter 24 of Wigner’s book (reference 1). This classification was discovered by
A.
Loewy
, Trans. Am. Math. Soc.
4
, 171
(1903
).See also
G.
Frobenius
and I.
Schur
, Sitzber. Preuss. Akad. Wiss., Physik.‐Math. Kl.
186
, (1906
).7.
Chapter 26 of Wigner’s book (reference 1).
8.
V. Bargmann (private communication).
9.
L. E. Dickson, Linear Algebras (Cambridge University Press, New York, 1914), p. 10.
10.
The restriction to associative algebras is forced by the fact that the rule of matrix multiplication is associative. In all applications of group theory to quantum mechanics we identify the operation of multiplication with ordinary matrix multiplication. It is well‐known that a fourth division algebra over the real number field exists, namely the algebra of octonions, if multiplication is allowed to be nonassociative. It is interesting to speculate upon possible physical interpretations of the octonion algebra [see
A.
Pais
, Phys. Rev. Letters
7
, 291
, 1961
]. We have tried, and failed, to find a natural way to fit octonions into the mathematical framework developed in this paper.11.
The general formalism of quantum mechanics over a real ground field has been worked out by
E. C. G.
Stueckelberg
, Helv. Phys. Acta
32
, 254
(1959
);Two further papers by Stueckelberg and collaborators have been circulated as preprints and will appear in Helv. Phys. Acta. These papers have many points of contact with the present work. For a brief summary of Stueckelberg’s conclusions, see also the paper of Finkelstein et al. (reference 12).
12.
D.
Finkelstein
, J. M.
Jauch
, S.
Schiminovich
, and D.
Speiser
, J. Math. Phys.
3
, 207
(1962
).13.
This is theorem (3.5B) on p. 95 of Weyl’s book (reference 2), combined with the theorem that every group ring is fully reducible (p. 101 of the same book).
14.
This lemma could probably be deduced as a special case from the general theorems of
A. H.
Clifford
, Ann. Math.
38
, 533
(1937
), concerning the connections between representations of groups and subgroups. However, it seemed simpler to give a direct and elementary proof of the lemma without appeal to Clifford’s work.15.
See Wigner (reference 1), p. 75, Theorem 2.
16.
See Wigner (reference 1), p. 78.
17.
G. Frobenius and I. Schur, reference 6.
18.
V. Bargmann (private communication).
19.
E. P. Wigner, Proceedings of the 4th Canadian Mathematics Congress (University of Toronto Press, Toronto, Canada, 1959), p. 174.
20.
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