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Elementary Algebraic Treatment of the Quantum Mechanical Symmetry Problem

Published online by Cambridge University Press:  20 November 2018

Hermann Weyl
Hermann Weyl*
Affiliation:
Institute for Advanced Study Princeton, New Jersey
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A function η(i1, …, if) of ƒ quantities i, varying over the finite range i = 1,2, …, n, is usually called an n-dimensional tensor of rank ƒ. Any permutation p: 1 → 1, … , ƒ → ƒ… changes this tensor into a tensor pη according to the equation pη(i1, ...,if) = η(i1', ...,if')Thus the permutation p appears as a linear operator p in the n-dimensional space Σ = Σn,ƒ of all n-dimensional tensors of rank ƒ, η is symmetric if pη = η for all permutations p, it is antisymmetric if pη δp.η where δp = + 1 for the even and — 1 for the odd permutations.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 1 , Issue 1 , 01 February 1949 , pp. 57 - 68
Copyright
Copyright © Canadian Mathematical Society 1949

References

1 We shall adhere to this terminology and not use the word symmetric in the sense presently to be mentioned -under the name Hermitean.

2 Weyl, Cf. H., Gruppentheorie una Quantenmechanik (2nd ed. Leipzig, 1931)Google Scholar [quoted as GQ], chap. V, §§ 1-7 and 13-14.

3 In passing we notice that the order of the algebra may now be evaluated as Σ(v + l)2, the sum extending over the non-negative v of the sequence ƒ‘, ƒ ‘ — 2, … where ƒ ‘ = min (n - d, n + d) = min (ƒ, 2n - ƒ ) , and hence equals (ƒ’ + l)(ƒ + 2)(ƒ’ + 3)/l-2-3. It should be easily possible to confirm this directly.

4 The dot under a letter merely serves to indicate that it stands for a symmetry quantity.

5 By using deeper algebraic resources than we care to employ in this elementary approach, Theorem I could be obtained as an immediate consequence of the following two facts: (a) Every representation a -> a of the group ring of the symmetric group breaks up into irreducible parts (is ‘fully reducible“) ; (β) A fully reducible ma trie algebra coincides with the commutator algebra of its commutator algebra (R. Brauer).—Another variant: Explicit construction by means of Young's symmetry operators shows that the inequivalent irreducible parts of the representation a -> a are absolutelyirreducible and inequivalent, and consequently (a) yields a complete decomposition. With this additional knowledge (β) can be replaced by the trivial fact that complete decomposition of a matric algebra implies its identity with the commutator algebra of its commutator algebra.