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By Spivak, M

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This condition allows us to formulate mathematically, in quantum statistical mechanics, the problem of the coexistence of distinct phases at given temperature T , that is, the problem of the uniqueness of ϕ , given (αt ) and β . We shall give, in Chapter V Section 11, an explicit example of a phase transition with spontaneous symmetry breaking coming from the statistical theory of prime numbers. This same condition has played an essential role in the modular theory of operator algebras. It has thus become an indisputable point of interaction between theoretical physics and pure mathematics.

In fact, the finite Borel measures on X correspond exactly to the continuous linear forms on the Banach space C(X) of continuous functions on X, equipped with the norm ||f || = supx∈X |f (x)|. e. the linear forms ϕ such that ϕ f¯f ≥ 0 for all f ∈ C(X). To go into matters more deeply, it is necessary to understand how this theory arises naturally from the spectral analysis of selfadjoint operators in Hilbert space, and thus becomes a special commutative case of the theory of von Neumann algebras (Chapter 5), which is itself the natural extension of linear algebra to infinite dimensions.

THE METRIC ASPECT OF NONCOMMUTATIVE GEOMETRY 37 Yang-Mills action and the fermionic part is easy to get since we are given the Dirac operator to start with. Thus we can keep track of the above modifications of the Lagrangian as modifications of the geometry of space-time. e. a triple (A, H, D) as above, such that the associated Maxwell-Dirac action functional produces the standard model of electroweak and strong interactions with all its refinements dictated by experimental results. The result that we obtain is canonically derived from the standard model considered as a phenomenological model.

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Calculus on manifolds by Spivak, M

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