Abstract
The physically relevant representations discussed in Chap. 5 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 of 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.
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References
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.
See e.g. P.A.M. Dirac, The Principles of Quantum Mechanics, 4th ed., Claredon Press Oxford 1958, Sect. 33; K. Huang, loc. cit. 1987.
For the properties of trace class operators, see e.g. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I, Academic Press 1972, Sect. VI.6.
R. Haag, N.M. Hugenholtz and M. Winnink, Comm. Math. Phys. 5, 215 (1967), hereafter referred as [HHW].
R. Kubo, J. Phys. Soc. Jap. 12, 570 (1957); P.C. Martin and J. Schwinger, Phys. Rev. 115, 1342 (1959).
See e.g. M. Reed and B. Simon, loc. cit., Theorem VI.22.
[HHW]; see also the book by Haag (1996) and the London lectures by Hugenholtz (1972), quoted in footnote 121.
J. Dixmier, Von Neumann algebras, North-Holland 1981, Chap. 1.
Here we give a brief sketch; for a detailed proof, see e.g. O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Vol. II, Springer 1981, Proposition 5.3.29.
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Strocchi, F. (2008). Thermal States. In: Symmetry Breaking. Lecture Notes in Physics, vol 732. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73593-9_22
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DOI: https://doi.org/10.1007/978-3-540-73593-9_22
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