Local and global definitions of time: Cosmology and quantum theory

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conference paper description
I will give a broad overview of what has become the standard paradigm in cosmology. I will describe the relational (\`{a} la Leibnitz) notion of time that is often used in cosmological calculations and discuss how the local nature of Einstein's equations allows us to translate this notion into statements about `initial' data. Classically this relates our local definition of time to a quasi-local region of a particular spatial slice, however incorporating quantum theory comes at the expense of losing this (quasi-) locality entirely. This occurs due to the presence of two, apparently distinct, issues: (1) Seemingly classical issues to do with the infinite spatial volume of the universe and (2) Quantum field theory issues, which revolve around trying to apply renormalization in cosmology. Following the ‘cosmological principle’ - an extension of the ‘Copernicus principle’ - that physics at every point in our universe should look the same, we are lead to the modern view of cosmology. This procedure is reasonably well understood for an exactly homogeneous universe, however the inclusions of small perturbations over this homogeneity leads to many interpretational/ conceptual difficulties. For example, in an (spatially) infinite universe perturbations can be arbitrarily close to homogeneous. To any observer, such a perturbation would appear to be a simple rescaling of the homogenous background and hence, physically, would not be considered an inhomogeneous perturbation at all. However, any attempt to choose the physically relevant scale at which perturbations should be considered homogeneous will break the cosmological principle i.e. it will make the resulting physics observer dependent. It amounts to `putting the perturbations in a box' and a delicate practical issue is that the universe is not static, hence the scale of the box will be time dependent. Thus what appears ‘physically homogeneous’ to an observer at one time will not appear so at another. This issue is brought to the forefront by considering the canonical (space and time rather space-time) version of the theory. The phase space formulation of General Relativity, just as for any other theory, contains the shadow of the underlying quantum theory. This means that, although the formulation is still classical, many of the subtleties that are present in the quantum theory are already apparent. In the cosmological context the infinite spatial volume renders almost all expressions formal or ill-defined. In order to proceed, we must restrict our attention to a cosmology that has some finite spatial extent, on which our relational notion of time is everywhere definable. In particular, this would constrain the permissible data outside our `observable universe'. This difficulty is an IR or large (spatial) scale issues in cosmology, however in addition there are UV or short (spatial) scale problems that need to be tackled. There are the usual problems of renormalization, which are further complicated by the fact that the universe is not static. In the cosmological setting this leads to new IR problems which again prevent one from taking the spatial extent of the universe to infinity. The physical relevance of this problem, the consequence for defining a time variable, and the distinction of homogeneous and inhomogeneous IR issues will be discussed.