By Ola Bratteli
For virtually twenty years, this has been the classical textbook on functions of operator algebra thought to quantum statistical physics. significant adjustments within the new version relate to Bose-Einstein condensation, the dynamics of the X-Y version and questions about section transitions.
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Additional info for Operator Algebras and Quantum Statistical Mechanics: Equilibrium States Models in Quantum Statistical Mechanics
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21. Let m: be the CAR algebra over L 2 (1Rv). Rv of space translations acts as a strongly continuous group of *-automorphism t of m: whose action is such that (U ,J)(y) = f(y - x). Rv-abelian and w is automatically even. 5. It proceeds as follows. Rv)invariant vectors in f>ro and let m:_ and m:+ denote the odd and even elements of m:. It follows from the CARs and a monomial approximation that lim ll[tx{A), B]ll = 0 lxl-oo if either A or B is even. Rv. If, however, A is odd, then a similar argument gives lim ll{tx(A), A*} II lxl-oo =0 46 States in Quantum Statistical Mechanics and hence Since both terms in the anti-commutator are positive, one concludes that Ew nw(A)Ew = 0.
Lw)) by the asymptotic commutation and anti-commutation, and Ewnw(B 1)Ew = 0. Similarly for the second term of the commutator. Finally, one concludes that the following conditions are equivalent. (1) (2) (3) limlxl~oo w(Arx(B)) = w(A)w(B) limlxl~oo w(Arx(B)C) limlxl~oo Wr(B(f1 ), for all A, BE 21. = w(AC)w(B) for all A, B, C E 21. ,, gl, ... (a(f) + a*(f)). Tx(B(gm))) = 0 for all f 1, ••• , Clearly (2) = (1), but the converse follows by noting that if B = B- + B+ and C = c- + c+ are the decompositions of Band C into odd and even parts, then lim {w(Arx(B)C)- w(ACrx(B+))- w(AC+rx(B-)) + w(ACrx(B-))} = 0 lxl~oo by the asymptotic commutation relations cited above.
28. ;;;;; ~ from a directed set when ordered by inclusion. fi} is an orthonormal basis for F, define and As any two orthonormal bases for Fare related by a unitary matrix, it follows immediately that this definition of nco, F only depends on w and F. Continuous Quantum Systems. 13. If this latter form is densely defined, then it uniquely determines a selfadjoint number operator Nm on f)m. The following result shows that the existence of such an operator characterizes normality of m. The main burden in the proof is the demonstration of algebraic properties of N m analogous to properties of the number operator on Fock space.