User:Jheald/Wigner's theorem

More basic theorem
eg Swarthmore lecture (Olivier Pfister. sim  cf, p.65)

If G is the symmetry group of a Hamiltonian H, then every degenerate eigensubspace of H is globally invariant under G i.e. constitutes a representation of the group G. ... All IR’s of the symmetry group G of a Hamiltonian H correspond to degenerate eigensubspaces of H. The reverse is not true: in a given degenerate eigensubspace, the matrices of a nonAbelian G may or may not be completely diagonalizable. However, it turns out that the diagonalizable case is extremely rare in experimental observations, and that there is a de facto equivalence between energy degeneracies and IR’s of the symmetry group, which yield what are called the essential degeneracies. Direct use can therefore be made of the extensive knowledge of the IR’s of the group to label the energy levels of a system and predict its degeneracies.
 * 2.4 Wigner’s Theorem

Then, the extremely rare occurrence of a reducible degenerate subspace is usually the sign of an overlooked larger symmetry, and this case is called accidental degeneracy

A.S. Wightman, in an obituary of Wigner for the Notices of the AMS : He recognized that if the Hamiltonian commutes with the action on wave functions of permutations of coordinates or with the action on wave functions of rotations of coordinates, then the linear subspace spanned by the eigenfunctions of a fixed eigenvalue is left invariant by these actions and, in the subspace, yields a unitary representation of the permutation and the rotation group. Such a representation is a direct sum of irreducibles, so the dimension of the linear space spanned by the eigenfunctions must be a sum with possible multiplicities of the dimensions of the irreducible representations of the groups.

This is the elementary group theoretical explanation for the ubiquitous appearance in atomic physics of degenerate multiplets of multiplicity 2j + 1 where j is a positive integer or half-odd integer. Later on, in the context of nuclear physics, this argument led to a theory of super-multiplets in which the group SU(2) is replaced by the group SU(4).

Arianna Borrelli (2009): Since the Schrödinger equation was itself linear, under a linear transformation its solutions became linear combinations of each other. In group theory, Wigner explained, the coefficients of such linear combinations were called ‘‘representations’’ of the relevant group (Wigner, 1927c, p. 627). He then introduced the notion of ‘‘irreducible representations’’, which I shall not discuss here further, and showed how one could classify the solutions of a given Schrödinger equation in terms of the irreducible representations of the symmetry group of the equation (Scholz, 2006, pp. 448–451; Wigner, 1927c, p. 629). Each complete, linearly independent set of solutions corresponding to the same energy transformed according to one of the irreducible representations of the symmetry group, so that its members only combined with each other under any transformation leaving the Schrödinger equation invariant.


 * ref: Scholz, E. (2006). Introducing groups into quantum theory (1926–1930). Historia Mathematica, 33, 440–490.

Heine (1960):
 * If a Hamiltonian is invariant under a group G of symmetry transformations, then the eigenfunctions corresponding to one energy level form a basis for a representation of G.
 * Corollary: If a Hamiltonian is invariant under a group G of transformations, then eigenfunctions of the Hamiltonian transforming according to one irreducible representation of G belong to the same energy level.
 * If the group G includes all possible symmetry transformations of the Hamiltonian, then the eigenfunctions of each energy level transform irreducibly under G, apart from accidental degeneracy.

Existing article (Wigner's theorem)
The theorem specifies how physical symmetries such as rotations, translations, CPT, etc. act on the Hilbert space of states. According to the theorem any symmetry acts as a unitary or anti-unitary transformation in the Hilbert space.

More precisely, it states that a surjective map $$T:H\rightarrow H$$ on a complex Hilbert space $$H$$, which satisfies
 * $$|\langle Tx,Ty\rangle|=|\langle x,y\rangle|$$

for all $$x,y \in H$$, has the form $$Tx=\varphi(x)Ux$$ for all $$x\in H$$, where $$\varphi:H\rightarrow \mathbb{C}$$ is unimodular and $$U:H\rightarrow H$$ is either unitary or antiunitary.

additional
cf also

ref

 * Wigner (1959), pp. 233-236

General

 * Volker Heine (1960), Group theory in quantum mechanics: an introduction to its present usage‎, pp. 41 et seq
 * Michael Tinkham (1964), Group theory and quantum mechanics, p. 32
 * JF Cornwell (1997), Group theory in Physics: an introduction, p. 93-97


 * W Ludwig and C Falter, Symmetries in Physics: Group Theory Applied to Physical Problems, Springer Series in Solid-State Sciences.
 * JP Elliot and PG Dawber (1979), Symmetry in Physics (vol 1 and 2), Palgrave Macmillan. (out of print.) -- vol.1 p.90 ?
 * HF Jones (1998), Groups, Representations and Physics (2e), Taylor & Francis.


 * Atkins, Molecular Quantum Mechanics, (1e, 1970), p. 152; (2e, 1983 - more explicit), pp.162-163, (3e, 1997) pp 156-57, (4e, 2005) pp 159-160, (5e, 2011), p. 158

Lecture notes

 * D. Vvedensky, Group theory course - chap. 6, Imperial College, 2001