By Ola Bratteli, Derek W. Robinson
For nearly 20 years, this has been the classical textbook on functions of operator algebra concept to quantum statistical physics. significant alterations within the re-creation relate to Bose-Einstein condensation, the dynamics of the X-Y version and questions about part transitions
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Additional info for Operator algebras and quantum statistical mechanics 2
Example text
26 (,,^, TT^J) is a countable representation since the Fock representation is Continuous irreducible. Let N^j N = operator and choose ij/ orthogonal, and ij/ be N {'A}>o D(Nia). Thus G = direct a of sum such that Sw ^ i/^ Quantum Systems. (//)^f . ^^ ^ {/,}CF {//}CF irreducibility of the Fock representation ij/^ ensure that ij/ is cyclic for 7rf,,(5I(^)). Fock space. But the orthogonality of the components Thus condition (3) is valid. and the (1) for the comment on the easier case of the CAR algebra.
I / sum {J\," ,Jn} (Ji 72, -,/) = K^ is over and , is does not affect the examples follows. recursively by F(/)=^8(^;)J]Fr(J) where the are äs function from the nonempty numbers. ^/ -l of / into ordered according odd. Note that äs the // odd character of the are even their permutation. 5) - + , Fr(a, ß)FT(j, S) Fr(a, y'JFriß, <5) + subsets, permutation even to whether the Fr(a, d]Fr(ß, y) interchange simplest The Continuous and by Inversion one obtains FT in terms of connecting the F and Fj, Again F, I 43 simple in- Quantum Systems.
Basically the criterion for normality is the existence of a selfadjoint number operator but äs this operator is almost always unbounded some care has to be (cü5 ^oj5 ^cü) being a direct sum of taken in its definition. On Fock space the number operator A^ posed in terms of operators Nf of the form Nf a^(f)a(f) = and it is consistent with the creation operators that Moreover, one easily Nf = -(g,f)a(f) = (f,g)a*(f) interpretation be decom- . It follows from the CARs and CCRs that these operators [Nf,a(g)] [Nf,a*(g)] can of (/) satisfy the relations , and a* (g) äs annihilation and the number of particles in the state measures / G l).