Chapter 22
∗ Thermal States
The physically relevant representations discussed in Chap. 15 are characterized by
the existence of a lowest energy or ground state and are supposed to describe states of
an infinitely extended isolated system. The situation changes if one wants to describe
states of a system at non-zero temperature (thermal states), i.e. states of a system in
thermal equilibrium with a reservoir. The stability of the system is now guaranteed
by the reservoir and there is no need for the energy spectral condition. The role of the
ground state is now taken by the equilibrium state and one is therefore led to discuss
representations of the canonical or observable algebra defined by equilibrium states.
As for the zero temperature case, one expects substantial differences with respect
to the finite dimensional case; for infinitely extended systems, the Gibbs factor
becomes meaningless in general, because the formal (Fock) Hamiltonian becomes
ill defined in the infinite volume limit. The strategy is to extract from the finite
dimensional case those properties which survive the thermodynamical limit.
121
As the first step we shall discuss the characterization of the equilibrium states,
Sects. 22.1–22.3; then we shall identify those states which describe pure phases,
Sect. 22.4.
121 Here we give a sketchy account, in view of the discussion of symmetry breaking at non-zero
temperature. For a beautiful and more detailed presentation, see R. Haag, Local Quantum Physics,
2nd ed., Springer 1996, Chap.V and H.M. Hugenholtz, in Mathematics of Contemporary Physics,
R.F. Streater ed., Academic Press 1972.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0_22
147
∗ Thermal States
The physically relevant representations discussed in Chap. 15 are characterized by
the existence of a lowest energy or ground state and are supposed to describe states of
an infinitely extended isolated system. The situation changes if one wants to describe
states of a system at non-zero temperature (thermal states), i.e. states of a system in
thermal equilibrium with a reservoir. The stability of the system is now guaranteed
by the reservoir and there is no need for the energy spectral condition. The role of the
ground state is now taken by the equilibrium state and one is therefore led to discuss
representations of the canonical or observable algebra defined by equilibrium states.
As for the zero temperature case, one expects substantial differences with respect
to the finite dimensional case; for infinitely extended systems, the Gibbs factor
becomes meaningless in general, because the formal (Fock) Hamiltonian becomes
ill defined in the infinite volume limit. The strategy is to extract from the finite
dimensional case those properties which survive the thermodynamical limit.
121
As the first step we shall discuss the characterization of the equilibrium states,
Sects. 22.1–22.3; then we shall identify those states which describe pure phases,
Sect. 22.4.
121 Here we give a sketchy account, in view of the discussion of symmetry breaking at non-zero
temperature. For a beautiful and more detailed presentation, see R. Haag, Local Quantum Physics,
2nd ed., Springer 1996, Chap.V and H.M. Hugenholtz, in Mathematics of Contemporary Physics,
R.F. Streater ed., Academic Press 1972.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0_22
147