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Thermofield dynamics and Gravity

Thermofield dynamics is presented in terms of a path-integral using coherent states, equivalently, using a coadjoint orbit action. A field theoretic formulation in terms of fields on a manifold ${\mathcal M} \times {\tilde{\mathcal M}}$ where the two components have opposite orientation is also presented. We propose formulating gravitational dynamics for noncommutative geometry using thermofield dynamics, doubling the Hilbert space modeling the noncommutative space. We consider 2+1 dimensions in some detail and since ${\mathcal M}$ and ${\tilde{\mathcal M}}$ have opposite orientation, the commutative limit leads to the Einstein-Hilbert action as the difference of two Chern-Simons actions.