Abstract
Having obtained frozen quantum equations, another strategy is to find an emergent time once already at the quantum level. This further subdivides into, firstly, approaches making one of various parallels with quantum field theory, considered in the current chapter. Secondly, ones in which the emergent time only arises within a physical regime of interest, in particular semiclassical quantum cosmology (see the next two chapters).
We suggest that, prior to embarking on this chapter, the reader first consult Chap. 12 for a conceptual synopsis of these strategies in the context of a comparison with the other families of strategies considered to date. The brief current chapter then involves attempting a Schrödinger-type inner product, a Klein–Gordon-type inner product, or a ‘third quantization’ analogue of ‘second quantization’, in each case now for general relativity (including in affine formulation). The second of these is based on general relativity’s configuration space Riem being indefinite like special relativity’s spacetime is. The third’s name is unsatisfactory enough that we propose it henceforth be called ’operator-valued wavefunction of the universe’ strategy instead.
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Notes
- 1.
Isham [483] further asserted that \(\mathrm{d}\boldsymbol{\Sigma} _{ab}\) needs to be spacelike with respect to the GR kinetic metric \(\mbox{M}^{abcd}\), and that making this inner product rigorous is difficult. It is certainly only intended as a formal expression which has yet to take into account. For instance, this could be formally attained by projecting the inner product down to \(\boldsymbol {\mathfrak{s}}\mbox{uperspace}(\boldsymbol{\Sigma} )\). This matter is absent in the Minisuperspace examples which are used widely in such an approach. The conceptual core, however, is clear: the expression is “invariant under deformations of the ‘spatial’ hypersurface in \(\boldsymbol{\mathfrak{R}}\mbox{iem}(\boldsymbol{\Sigma} )\) ” [483]. This is (paraphrasing) the Quantum GR analogue of the normal Klein–Gordon inner product’s time-independence property.
- 2.
However, relativistic QFT’s motivation in terms of finding a satisfactory inner product is absent here, since the first Quantization’s Schrödinger inner product works just fine for RPMs.
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Anderson, E. (2017). Tempus Post Quantum. i. Paralleling QFT. In: The Problem of Time. Fundamental Theories of Physics, vol 190. Springer, Cham. https://doi.org/10.1007/978-3-319-58848-3_45
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