Method of coadjoint orbits in thermodynamics of non-compact manifolds

Abstract:
In presented work we take into consideration the main problem of thermodynamics of homogeneous spaces (quite synonymous to that quantum statistic mechanics) - evaluation of heat kernel and corresponding statistic sum or partition function. To solve that problem for given space one must integrate heat kernel or Bloch differential equation with specific boundary condition and then perform integration over the volume of the space. There is no applicable algorithm of doing it for arbitrary non-compact space because of sufficient divergences caused by infinite volume.

We suggest the method for solution of the stated problem for non-compact Lie group (non-compact homogeneous space with group structure) which is based on the framework of non-commutative integration, originating in theory of coadjoint orbits. That algorithm allows to find heat kernel and partition function avoiding inconveniences connected with infinite volume of the space. The method is illustrated with non-trivial examples.