Spontaneously broken symmetries

Abstract

Symmetries are automorphisms of the observable algebra A. There are two notions of spontaneous symmetry breaking. The stronger refers to a state ω over A such that the symmetry does not extend to the strong closure π_ω(A)″ where π_ω is the GNS-representation associated to ω. The standard example for that are the Schwinger terms [j(x), j(x′)]=iδ′(x−x′), x ∈ R which are not invariant under the parity j(x) → j(−x) though the j’s are constructed from fermion fields ψ(x) as strong limits and ψ(x) → ψ(−x) is a symmetry of the fermion algebra. The weaker notion which can also be realized in elementary quantum mechanics refers to a time evolution τ _t ∈ Aust A. A symmetry σ is said to be spontaneously broken if there is a state ω with ω o τ _t=ω, ω o σ≠ω though [σ, τ _t]=0.

We are interested in the latter case, i.e. in states that are invariant under time evolution but not under a gauge symmetry that commutes with time evolution. The possibility, if such states exist, can be decided on the level of the gauge invariant subalgebra by using the crossed product construction and there reduces to an eigenvalue problem. The method is applied to the Fermi algebra and spin systems. In particular, it is found that a translation invariant state cannot break the symmetry ψ → −ψ where ψ is a fermionic field operator.

To appear in the “Annales de l’Institut Henri Poincaré, Physique Théorique”.