Note on Wigner's Theorem on Symmetry Operations

V.  Bargmann

doi:10.1063/1.1704188

Abstract

Wigner's theorem states that a symmetry operation of a quantum system is induced by a unitary or an anti‐unitary transformation. This note presents a detailed proof which closely follows Wigner's original exposition.

REFERENCES

1. Here f corresponds of course to a wavefunction ψ. In this note vectors will be denoted by italics and scalars by lower‐case Greek letters. The product of a vector f by the scalar λ will be written $fλ.$ Google Scholar
2. If superselection rules hold for S, K will be considered a coherent subspace of the Hilbert space of all states. See the discussion in Wightman [A. S. Wightman, Nuovo Cimento Suppl. 14 (1959), p. 81]. Google Scholar
3. E. P. Wigner, Gruppentheorie (Frederick Vieweg und Sohn, Braunschweig, Germany, 1931), pp. 251–254; Google Scholar
Group Theory (Academic Press Inc., New York, 1959), pp. 233–236. Google Scholar
4. Although Wigner did not explicitly formulate his theorem in terms of rays, it is essentially equivalent to the one stated here and certainly follows from the theorem proved below. Google Scholar
5. For a bibliography and a critical analysis of the proofs, see Uhlhorn, Ref. 6. Google Scholar
The recent paper by Lomont and Mendelson [J. S. Lomont and P. Mendelson, Ann. Math. 78, 548 (1963).] should be added to his list. Google ScholarCrossref
6. U. Uhlhorn, Arkiv Fysik 23, 307 (1963). Google Scholar
7. This criticism does not apply at all to the very interesting papers by Emch and Piron [G. Emch and C. Piron, J. Math. Phys. 4, 469 (1963)] and by Uhlhorn, who start from more general premises and, consequently, obtain more comprehensive results. Google ScholarScitation
8. Many authors define a ray differently, by including all multiples $f_{0} λ$ a fixed vector $f_{0}$ (irrespective of $|λ|$ ) in one ray—provided $f_{0} ≠0$ and $λ≠0$ —as is suggested by projective geometry. It seems to the writer that in the present context the definition of the text is more convenient. Google Scholar
9. By definition, a unitary (anti‐unitary) mapping is a linear (antilinear) isometry which has an inverse. Google Scholar
10. dim K denotes the dimension of K. Google Scholar
11. The selection of $e′$ constitutes the only arbitrary choice in the construction of U. All other definitions are uniquely determined by $e′.$ Google Scholar
12. e and f are orthonormal, so are $e′$ and $f′.$ Google Scholar
13. The definitions A and B are the crucial steps in Wigner’s construction. Google Scholar
14. If $m = 1,$ Sec. 4.5 may be omitted since Eq. (16) reduces then to Eq. (14). Google Scholar
15. In terms of the construction in Sec. 3.2 this merely means that A is replaced by $U_{2} e = e′θ$ (see footnote 11). It follows from Sec. 1.4 that dim $K ⩾2$ is a necessary assumption. Google Scholar
16. D. Finkelstein, J. M. Jauch, S. Schiminovich, and D. Speiser, J. Math. Phys. 3, 207 (1961). Google ScholarScitation
17. The reader is assumed to be familiar with quaternions. These introductory remarks fix the notation and review Beveral relations to be used later on. Google Scholar
18. It should be added that the remarks in Sec. 1.5 do not apply to the quaternion case. The proof that Δ does not depend on the choice of the representatives $a_{i}$ uses commutativity of multiplication. Google Scholar
19. Uhlhorn’s contrary assertion (Ref. 6, pp. 335, 336) is incorrect. He overlooked the second solution in (A5). Google Scholar

Note on Wigner's Theorem on Symmetry Operations

REFERENCES

Article Metrics

Altmetric

Resources

General Information

236

REFERENCES

Article Metrics

Altmetric