Operator Algebras and Quantum Statistical Mechanics: by Professor Ola Bratteli, Professor Derek W. Robinson (auth.)

By Professor Ola Bratteli, Professor Derek W. Robinson (auth.)

For nearly twenty years this has been the classical textbook on functions of operator algebra idea to quantum statistical physics. It describes the final constitution of equilibrium states, the KMS-condition and balance, quantum spin structures and non-stop systems.
Major adjustments within the re-creation relate to Bose--Einstein condensation, the dynamics of the X-Y version and questions about part transitions. Notes and feedback were significantly augmented.

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Operator Algebras and Quantum Statistical Mechanics: Equilibrium States. Models in Quantum Statistical Mechanics

For nearly 20 years this has been the classical textbook on purposes of operator algebra idea to quantum statistical physics. It describes the overall constitution of equilibrium states, the KMS-condition and balance, quantum spin platforms and non-stop platforms. significant adjustments within the re-creation relate to Bose--Einstein condensation, the dynamics of the X-Y version and questions about section transitions.

Extra info for Operator Algebras and Quantum Statistical Mechanics: Equilibrium States. Models in Quantum Statistical Mechanics

Example text

Im Iw(Arx(B)) - w(A)w(rx(B))1 = 0 Ixl~oo for all A, B E 21, is equivalent to the three-body cluster property lim Iw(Arx(B)C) - w(AC)w(rx(B))1 = 0 Ixl~oo for all A, B, CE 21. If, however, w is IR" -invariant and analytic the following are equivalent: (I) w is strongly mixing for space translations, (2) limlxl~oo wr(<1>w(fJ) , ... , <1>W(fIl) , rx(<1>W(gJ)) , ... , rx(<1>w(gm))) = 0 for all n, m 2 land all PROOF. (I) =} 1" ... 1'1> g" . , gm E L2(1R 1'), where rx(<1>w(f)) = <1>co(Ut (2). For 9 E L 2 (1R") and all xE IR I ' the vector-valued functions I).

L for all A. ~ O. In particular this means that D(T:;I/2) ~ D(T~I/2) by the definition of the form ordering. 13 implies that (T2 for all A. ~)-1/2 > o. These points will be used in proving the following result. 13. D(t) = {I/I; 1/1 E (lD(1<1), s~p t

PROOF. It follows from property (2) of the W(f) that W(f)W(O) = W(f) = W(O)W(f) and W( - f) W(f) = W(O) = W(f) W( - f) As the W(j) are nonzero one eonc1udes that W(O) is the identity 1 and property (I) implies that W(f) is unitary. Next let us regard the linear spaee H as a diserete additive abelian group. The W(f) give a unitary representation of H up to a phase, or multiplier b(j, g) = exp{ -ja(j, g)/2}. , + 12, g) = b(fl, g)b(h, g) . Our aim is to prove that the C*-algebras 21;, i = 1,2, are '-isomorphie.

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Operator Algebras and Quantum Statistical Mechanics: by Professor Ola Bratteli, Professor Derek W. Robinson (auth.)
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