Generating functional (or partition function)

[crackjack]

I am reading a book (Di Francesco's "CFT", pg 337) in which it is given that if we take the operator that translates the system along some direction (which is a combination of time and space) as 'A', then the partition function is just trace(A).
How do we get this?

 

[crackjack]

Any one?

 

[Avodyne]

I don't have that book, but the spacetime translation operator is exp(iPa), where P is the four-momentum vector operator, and a is a constant four-vector (that you are translating by). The partition function is usually Tr[exp(-beta H)]. So except for a factor of i, this is the translation operator with a translation in the time direction by beta.