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キーワード:
High Energy Physics - Theory, hep-th,General Relativity and Quantum Cosmology, gr-qc
要旨:
In recent work, we introduced Picard-Lefschetz theory as a tool for defining
the Lorentzian path integral for quantum gravity in a systematic semiclassical
expansion. This formulation avoids several pitfalls occurring in the Euclidean
approach. Our method provides, in particular, a more precise formulation of the
Hartle-Hawking no boundary proposal, as a sum over real Lorentzian
four-geometries interpolating between an initial three-geometry of zero size,
{\it i.e}, a point, and a final three-geometry. With this definition, we
calculated the no boundary amplitude for a closed universe with a cosmological
constant, assuming cosmological symmetry for the background and including
linear perturbations. We found the opposite semiclassical exponent to that
obtained by Hartle and Hawking for the creation of a de Sitter spacetime "from
nothing". Furthermore, we found the linearized perturbations to be governed by
an {\it inverse} Gaussian distribution, meaning they are unsuppressed and out
of control. Recently, Diaz Dorronsoro {\it et al.} followed our methods but
attempted to rescue the no boundary proposal by integrating the lapse over a
different, intrinsically complex contour. Here, we show that, in addition to
the desired Hartle-Hawking saddle point contribution, their contour yields
extra, non-perturbative corrections which again render the perturbations
unsuppressed. We prove there is {\it no} choice of complex contour for the
lapse which avoids this problem. We extend our discussion to include
backreaction in the leading semiclassical approximation, fully nonlinearly for
the lowest tensor harmonic and to second order for all higher modes.
Implications for quantum de Sitter spacetime and for cosmic inflation are
briefly discussed.
