Abstract
In this chapter we discuss some contemporary extensions of quantum field theory to mixed states and to curved manifolds. There can be a connection between these two topics as quantum fields in some moving frames (accelerated frames) can be viewed as thermal fields (the Unruh effect) and quantum fields on a black hole background have the thermal spectrum. We derive the formula for the correlation functions of the scalar quantum field in the quantum Gibbs (thermal) state. Then, we approach the problem of a quantization of the scalar field defined on a manifold. We formulate canonical commutation rules for quantization. We solve the heuristic eigenvalue equation to find a Gaussian ground state in a static metric. The ground state wave function is used to construct a Gaussian measure which defines the Hilbert space of square integrable functions (the Schrödinger representation of QFT).
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