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I am looking at the Wikipedia article on Partition function -- As a measure. Unfortunately the article has no relevant references or reading suggestions.

I am looking for books or other resources that cover the topic of this subsection in detail and in particular explain and derive the conditions that the ground state not be degenerate. Quoting from the subsection:

... given a specific configuration $(x_1,x_2,\ldots)$, $$P(x_1,x_2,\dots) = \frac{1}{Z(\beta)} \exp \left(-\beta H(x_1,x_2,\dots) \right)$$ is the probability of the configuration $(x_1,x_2,\ldots)$ occurring in the system.

...

There exists at least one configuration $(x_1,x_2,\ldots)$ for which the probability is maximized; this configuration is conventionally called the ground state. If the configuration is unique, the ground state is said to be non-degenerate, and the system is said to be ergodic; otherwise the ground state is degenerate. The ground state may or may not commute with the generators of the symmetry; if commutes, it is said to be an invariant measure. When it does not commute, the symmetry is said to be spontaneously broken.

Conditions under which a ground state exists and is unique are given by the Karush–Kuhn–Tucker conditions; these conditions are commonly used to justify the use of the Gibbs measure in maximum-entropy problems.

As usual for Wikipedia articles, the description is not too coherent (e.g. what does it mean that "the ground state commutes with ..."? Previously the ground state was defined as a set of values.) I'd appreciate suggestions to introductory books on this topic, especially ones written for people with physics training (familiarity with the canonical ensemble in stat. phys., etc.)

Szabolcs
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  • I think the "commute with" means the ground state simply obeys the same symmetry laws as the probability function of which it is a ground state. It's probably an abuse of language as the probability function will commute with the symmetry generators in the sense that the Poisson bracket of $P$ and a generator will be zero. Or if it's in a quantum context, it's literally the commutator of the operators. But you can always make a variable out of a state by considering the projector on that state. So a Poisson bracket or commutator can make sense. – Raskolnikov Jun 18 '13 at 10:22
  • But, in the case of spontaneous symmetry breaking, the ground state will not commute with all the symmetry generators. Think of the sombrero potential which has circle symmetry, but any ground state lacks the symmetry. Applying a rotation just turns one ground state into another. – Raskolnikov Jun 18 '13 at 10:24
  • That statement about the Karush-Kuhn-Tucker conditions is misleading, I think. The KKT conditions are necessary conditions for a local optimum in a constrained continuous system. They say little about a global minimum, and nothing at all about a discrete system. – Robert Israel Jun 18 '13 at 10:39

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