By Ola Bratteli, Derek W. Robinson

For nearly 20 years, this has been the classical textbook on purposes of operator algebra idea to quantum statistical physics. significant alterations within the new version relate to Bose-Einstein condensation, the dynamics of the X-Y version and questions about part transitions.

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**Extra resources for Operator Algebras and Quantum Statistical Mechanics 2 : Equilibrium States. Models in Quantum Statistical Mechanics (Texts and Monographs in Physics)**

**Sample text**

15 identifies the regulär states on the CCR algebra ^(1^) over a general pre-Hilbert space l) with the states which are locally normal with respect to the quasi-local structure defined by the finite-dimensional complex subspaces of l). Thus, äs far äs regulär states are concerned, one could define the CCR algebra äs a closure of a union of algebras of the form ^(5+(^^)), where g_^ (M) is the Symmetrie Fock space over the finite-dimensional subspace M and ^(g_^(M)) is the weak closure of ^(M) in the Fock representation.

Implication (3) => (2) is a consequence of Observation 2 il/ e D(n(,j} is cyclic, then )(wco) contains the dense Thus if linear combinations of T^ü)(W(f]]\l/,f G and the set linearity of by finite formed I). OBSERVATION 3 a^(f]D(Nll'~]CD(N,,,] and for ijj e D(Nl^~) N,a,,(f]ilj Observation 2 implies = a,,(f](N,-^)ilj . ^(r(//)>,iA)ll/llV2 for q) G D(N(o) and with / replaced by ij/ G D(Nj~]. D(^,,(f)}. ^rD(afo(f]]. o(/)(A^a. - for all cp G D(N(o} where we have used tinuity of the right-hand side in (p.

O-, ^co) is regulär. Note that on \\(n,,(W(tf)]-^]7L,,(W(cj))^,,\\^ 2 e-^^^'^^^^^^^(D(W(tf]] e^^^^^^^^^^a}(W(-tf)) = and ^ G [R hence ^ it - - follows easily G)(W(tf)] G to denote the infinitesimal with that is regulär if, and only if, / G i). We use the notation ^,o(f] unitary group Ti^,:,(W(tf]) associated co C is continuous for all generator of the regulär state. _^(f)). One by the easily calculates that cof,(W(f)) Even if co is a = regulär state, (Q, W(f)ü) there are = ^-ll/ll'/4 . certain technical domain problems which complicate the introduction of annihilation and creation operators.