Evolution between slices in D>2 curved spacetimes shifts the state between unitarily inequivalent Fock representations; the physical representation can be fixed locally by a positive-frequency condition equivalent, in leading order, to the Hadamard condition.
Complex Structures for Klein-Gordon Theory on Globally Hyperbolic Spacetimes
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abstract
We develop a rigorous method to parametrize complex structures for Klein-Gordon theory in globally hyperbolic spacetimes that satisfy a completeness condition. The complex structures are conserved under time-evolution and implement unitary quantizations. They can be interpreted as corresponding to global choices of vacuum. The main ingredient in our construction is a system of operator differential equations. We provide a number of theorems ensuring that all ingredients and steps in the construction are well-defined. We apply the method to exhibit natural quantizations for certain classes of globally hyperbolic spacetimes. In particular, we consider static, expanding and Friedmann-Robertson-Walker spacetimes. Moreover, for a huge class of spacetimes we prove that the differential equation for the complex structure is given by the Gelfand-Dikki equation.
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Challenges for describing unitary evolution in nontrivial geometries: pictures and representations
Evolution between slices in D>2 curved spacetimes shifts the state between unitarily inequivalent Fock representations; the physical representation can be fixed locally by a positive-frequency condition equivalent, in leading order, to the Hadamard condition.