Quantum field theory on curved manifolds

This paper discusses how particle production from the vacuum can be explained by local analysis when the field theory is defined by differential geometry on curved manifolds. We have performed the local analysis in a mathematically rigorous way, respecting the Markov property. The exact WKB is used as a tool for extracting non-perturbative effect from the local system. After a serious application of the differential geometry and the exact WKB to particle production, we show that entanglement does not appear in the Unruh effect as far as the standard formulation by the differential geometry is valid. This result should not be attributed to a consistency problem between the ``entanglement state'' and the ``standard field theory by differential geometry'', but to the fact that the conventional calculation of the Unruh effect is done by extrapolation which is not consistent with the differential geometry. The situation is similar to that of the Dirac monopole, but topology is not relevant and the basis for building field theories in differential geometry is strongly involved.