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Register a new userNo Rescue for the No Boundary Proposal
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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.
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In previous works, we have demonstrated that the path integral for {it real, Lorentzian} four-geometries in Einstein gravity yields sensible results in well-understood physical situations, but leads to uncontrolled fluctuations when the no boundary condition proposed by Hartle and Hawking is imposed. In order to circumvent our result, new definitions for the gravitational path integral have been sought, involving specific choices for a class of {it complex} four-geometries to be included. In their latest proposal, Diaz Dorronsoro {it et al.}~cite{DiazDorronsoro:2018wro} advocate integrating the lapse over a complex circular contour enclosing the origin. In this note we show that, like their earlier proposal, this leads to mathematical and physical inconsistencies and thus cannot be regarded as a basis for quantum cosmology. We also comment on Vilenkin and Yamadas recent modification of the tunneling proposal, made in order to avoid the same problems. We show that it leads to the breakdown of perturbation theory in a strong coupling regime.
We specify the semiclassical no-boundary wave function of the universe without relying on a functional integral of any kind. The wave function is given as a sum of specific saddle points of the dynamical theory that satisfy conditions of regularity on geometry and field and which together yield a time neutral state that is normalizable in an appropriate inner product. This specifies a predictive framework of semiclassical quantum cosmology that is adequate to make probabilistic predictions, which are in agreement with observations in simple models. The use of holography to go beyond the semiclassical approximation is briefly discussed.
We identify a fundamental obstruction to any theory of the beginning of the universe, formulated as a semiclassical path integral. Hartle and Hawkings no boundary proposal and Vilenkins tunneling proposal are examples of such theories. Each may be formulated as the quantum amplitude for obtaining a final 3-geometry by integrating over 4-geometries. We introduce a new mathematical tool - Picard-Lefschetz theory - for defining the semiclassical path integral for gravity. The Lorentzian path integral for quantum cosmology with a positive cosmological constant is meaningful in this approach, but the Euclidean version is not. Framed in this way, the resulting framework and predictions are unique. Unfortunately, the outcome is that primordial tensor (gravitational wave) fluctuations are unsuppressed. We prove a general theorem to this effect, in a wide class of theories.
We implement the no-boundary proposal for the wave function of the universe in an exactly solvable Bianchi IX minisuperspace model with two scale factors. We extend our earlier work (Phys. Rev. Lett. 121, 081302, 2018 / arXiv:1804.01102) to include the contribution from the $mathbb{C}text{P}^2 setminus B^4$ topology. The resulting wave function yields normalizable probabilities and thus fits into a predictive framework for semiclassical quantum cosmology. We find that the amplitude is low for large anisotropies. In the isotropic limit the usual Hartle-Hawking wave function for the de Sitter minisuperspace model is recovered. Inhomogeneous perturbations in an extended minisuperspace are shown to be initially in their ground state. We also demonstrate that the precise mathematical implementation of the no-boundary proposal as a functional integral in minisuperspace depends on detailed aspects of the model, including the choice of gauge-fixing. This shows in particular that the choice of contour cannot be fundamental, adding weight to the recent proposal that the semiclassical no-boundary wave function should be defined solely in terms of a collection of saddle points. We adopt this approach in most of this paper. Finally we show that the semiclassical tunneling wave function of the universe is essentially equal to the no-boundary state in this particular minisuperspace model, at least in the subset of the classical domain where the former is known.
There are various no-go results forbidding self-interactions for a single partially massless spin-2 field. Given the photon-like structure of the linear partially massless field, it is natural to ask whether a multiplet of such fields can interact under an internal Yang-Mills like extension of the partially massless symmetry. We give two arguments that such a partially massless Yang-Mills theory does not exist. The first is that there is no Yang-Mills like non-abelian deformation of the partially massless symmetry, and the second is that cubic vertices with the appropriate structure constants do not exist.
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