Abstract
We study the relationship between the quantization of a massless scalar field on the two-dimensional Einstein cylinder and in a spacetime with a time machine. We find that the latter picks out a unique prescription for the state of the zero mode in the Einstein cylinder. We show how this choice arises from the computation of the vacuum Wightman function and the vacuum renormalized stress-energy tensor in the time-machine geometry. Finally, we relate the previously proposed regularization of the zero mode state as a squeezed state with the time-machine warp parameter, thus demonstrating that the quantization in the latter regularizes the quantization in an Einstein cylinder.
Highlights
Due to the presence of closed timelike curves, these spacetimes generally exhibit Cauchy horizons
A well-known example is furnished by a scalar field on the Einstein cylinder, with topology R×S1, which in this construction is equivalent to the same scalar field in Minkowski space withperiodic boundary conditions applied along the spatial direction
Inspired by the study of time machines, we propose a way to solve the inherent ambiguity of the zero-mode quantization in quantum field theory (QFT) living in spacetimes with spatial periodicity
Summary
We review the construction of a (1 + 1) wormhole and its conversion into a time machine. On we will concentrate on the canonical time machine with parameters A and L for which the metric in coordinates (t, y) in the universal covering space M is given by ds2 = −e−2Wydt2 + dy, log A This geometry has constant negative curvature with Ricci scalar R = −2W2, it is locally isometric to a two-dimensional Anti-de Sitter spacetime AdS2. In order to develop a quantum field theory in section, it is essential that M can be viewed as the universal covering space for M This universal covering technique is a useful tool to work with multiply-connected spacetimes since in the universal covering space the functions are simpler to evaluate and it is possible to define global Killing fields (if the time machine is locally static). For a slightly more formal review of the general machinery underlying this construction see appendix B
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