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Canadian Journal of Mathematics

Elementary Algebraic Treatment of the Quantum Mechanical Symmetry Problem

Hermann Weyl

Published Feb 1, 1949• 6 citations
Publisher

Abstract

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.

DOI
10.4153/cjm-1949-005-7

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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.
The modern state of the Pauli Exclusion Principle (PEP) is discussed. PEP can be considered from two viewpoints. On the one hand, it asserts that particles with half-integer spin (fermions) are described by antisymmetric wave functions, and particles with integer spin (bosons) are described by symmetric wave functions. This is the so-called spin-statistics connection (SSC). As we will discuss, the physical reasons why SSC exists are still unknown. On the other hand, according to PEP, the permutation symmetry of the total wave functions can be only of two types: symmetric or antisymmetric, both belong to one-dimensional representations of the permutation group, all other types of permutation symmetry are forbidden; whereas the solution of the Schrödinger equation may have any permutation symmetry. It is demonstrated that the proof in some widespread textbooks on quantum mechanics that only symmetric and antisymmetric states (one-dimensional representations of the permutation group) can exist is wrong. However, the scenarios, in which an arbitrary permutation symmetry (degenerate permutation states) is permitted lead to contradictions with the concepts of particle identity and their independence. Thus, the existence in our nature particles only in nondegenerate permutation states (symmetric and antisymmetric) is not accidental and so-called symmetrization postulate may not be considered as a postulate, since all other symmetry options for the total wave function may not be realized. From this an important conclusion follows: we may not expect that in future some unknown elementary particles can be discovered that are not fermions or bosons.
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