Title: Generalized Partition Functions and Quantum Distillations

Abstract: I describe a generalization of the notion of partition function in general quantum field theories. The thermal partition function is a state sum over the Hilbert space, which typically has non-analyticities, and can
be used to study phase transitions, and thermodynamics. I will construct graded states sums which may potentially be analytic. I will show that this notion is useful to understand the general aspects of quantum
field theory. I will provide simple examples.