19.2 Bose–Einstein Condensation
131
not on the field algebra A, as displayed by the violation of the cluster property by
the correlation functions of A. The symmetry breaking emerges when one makes a
decomposition into the pure states of which is a mixture.
The model is an interesting example of the mechanism of spontaneous breaking
of a gauge symmetry, which by definition reduces to the identity on the observables.
19.3 Massless Field in d ≥ 3
For the free massless field in space-time dimensions d ≥ 3 (see Chap. 2; for d = 2
the model is infrared singular
109 ), the “gauge” transformations β
λ
(ϕ(x)) = ϕ(x) + λ
define internal algebraic symmetries, which are spontaneously broken in any
physically relevant representation, with symmetry breaking order parameter
< ϕ > 0 =< β
λ
(ϕ) > 0.
109 See, e.g. F. Strocchi, Selected Topics on the General Properties of Quantum Field Theory, World
Scientific 1993, Sect. 17.2.
131
not on the field algebra A, as displayed by the violation of the cluster property by
the correlation functions of A. The symmetry breaking emerges when one makes a
decomposition into the pure states of which is a mixture.
The model is an interesting example of the mechanism of spontaneous breaking
of a gauge symmetry, which by definition reduces to the identity on the observables.
19.3 Massless Field in d ≥ 3
For the free massless field in space-time dimensions d ≥ 3 (see Chap. 2; for d = 2
the model is infrared singular
109 ), the “gauge” transformations β
λ
(ϕ(x)) = ϕ(x) + λ
define internal algebraic symmetries, which are spontaneously broken in any
physically relevant representation, with symmetry breaking order parameter
< ϕ > 0 =< β
λ
(ϕ) > 0.
109 See, e.g. F. Strocchi, Selected Topics on the General Properties of Quantum Field Theory, World
Scientific 1993, Sect. 17.2.