Abstract
Chapter 2 explains the algebraic approach to quantum statistical mechanics in the (very) simple case of finite quantum systems. Here, we extend this approach to a large class of infinitely extended (infinite-dimensional) quantum systems that are homogeneous in space. The physical interactions considered in this chapter are short-range, in a sense, that is, they do not contain additional mean-field interaction terms, this kind of long-range interactions being the subject of Chap. 6. Crucial phenomenological aspects of thermodynamic equilibrium of infinite systems, like existence of first-order phase transitions and spontaneous symmetry breaking, can be derived in the short-range setting, as simple corollaries of classical (deep) results of convex analysis. In this chapter, we additionally contribute a brief introduction to the mathematical foundations of the Hartree-Fock theory, which is very popular in molecular and condensed matter physics.
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Notes
- 1.
We adopt this terminology, because of the formal analogy to the classical case: Recall that the invariant probability measures of a given classical dynamical systems are ergodic, in the usual sense, iff they are extreme in the convex set of all invariant measures of the system.
- 2.
In the present construction, one may even take the group U(N) of all unitary N × N matrices. However, we restrict ourselves to the so-called special unitary matrices, because this case is the most relevant one in physics.
- 3.
Recall that a homeomorphism is a bijection (one-to-one and onto) such that both the function and its inverse are continuous.
- 4.
By definition, Ξ is a mapping \(\mathcal {A}\rightarrow C(E(\mathcal {A}); \mathbb {C})\). It is naturally seen here as a mapping \(\mathcal {A}\rightarrow C(E;\mathbb {C})\), by restriction of continuous functions on \(E(\mathcal {A})\) to E.
- 5.
That is, \(\mathrm {e}_{\mathrm {s},x}(\tilde {\mathrm {s}},\tilde {x})=1\) if \(( \mathrm {s},x)=(\tilde {\mathrm {s}},\tilde {x})\) and \(\mathrm {e}_{\mathrm {s},x}( \tilde {\mathrm {s}},\tilde {x})=0\), else.
- 6.
In some papers, only (approximating) Slater determinants are considered.
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Bru, JB., Alberto Siqueira Pedra, W.d. (2023). Thermodynamic Equilibrium in Infinite Volume. In: C*-Algebras and Mathematical Foundations of Quantum Statistical Mechanics. Latin American Mathematics Series(). Springer, Cham. https://doi.org/10.1007/978-3-031-28949-1_5
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