Abstract
We unify and generalize the notions of vacuum and amplitude in linear quantum field theory in curved spacetime. Crucially, the generalized notion admits a localization in spacetime regions and on hypersurfaces. The underlying concept is that of a Lagrangian subspace of the space of complexified germs of solutions of the equations of motion on hypersurfaces. Traditional vacua and traditional amplitudes correspond to the special cases of definite and real Lagrangian subspaces respectively. Further, we introduce both infinitesimal and asymptotic methods for vacuum selection that involve a localized version of Wick rotation. We provide examples from Klein-Gordon theory in settings involving different types of regions and hypersurfaces to showcase generalized vacua and the application of the proposed vacuum selection methods. A recurrent theme is the occurrence of mixed vacua, where propagating solutions yield definite Lagrangian subspaces and evanescent solutions yield real Lagrangian subspaces. The examples cover Minkowski space, Rindler space, Euclidean space and de Sitter space. A simple formula allows for the calculation of expectation values for observables in the generalized vacua.
Highlights
In nonrelativistic quantum theory a vacuum state can be identified with a lowest-energy state
The key insight is that the solutions of a sufficiently simple field theory in a spacetime region form a Lagrangian subspace of the space of germs of solutions on the boundary
We show on the classical level that the definite Lagrangian subspaces are naturally associated to “sufficiently” noncompact regions of spacetime, complementing the real Lagrangian subspaces for compact and “mildly” noncompact regions
Summary
In nonrelativistic quantum theory a vacuum state can be identified with a lowest-energy state. The key insight is that the solutions of a sufficiently simple field theory in a spacetime region form a Lagrangian subspace of the space of germs of solutions on the boundary.. In order to motivate our proposal we showcase the natural occurrence of generalized vacua in simple examples and demonstrate the application of our vacuum selection methods This is partly in the spirit of the reverse engineering approach to quantum field theory, where we use known tools and methods to extract underlying structure [11]. While we focus the discussion in this work on standard quantum field theories and the methods of vacuum selection proposed in Sec. VI rely to some extent on a metric, the framework of Sec. V is in principle applicable in the absence of a metric. A more precise treatment of some of the mathematical structures used is provided in Appendix B
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