Abstract
We study the nonperturbative quantum evolution of the interacting O(N ) vector model at large-N , formulated on a spatial two-sphere, with time dependent couplings which diverge at finite time. This model - the so-called “E-frame” theory, is related via a conformal transformation to the interacting O(N ) model in three dimensional global de Sitter spacetime with time independent couplings. We show that with a purely quartic, relevant deformation the quantum evolution of the E-frame model is regular even when the classical theory is rendered singular at the end of time by the diverging coupling. Time evolution drives the E-frame theory to the large-N Wilson-Fisher fixed point when the classical coupling diverges. We study the quantum evolution numerically for a variety of initial conditions and demonstrate the finiteness of the energy at the classical “end of time”. With an additional (time dependent) mass deformation, quantum backreaction lowers the mass, with a putative smooth time evolution only possible in the limit of infinite quartic coupling. We discuss the relevance of these results for the resolution of crunch singularities in AdS geometries dual to E-frame theories with a classical gravity dual.
Highlights
Asymptotically AdS geometries, wherein Hamiltonian evolution of the boundary QFT ends at a singularity in finite time
We study the nonperturbative quantum evolution of the interacting O(N ) vector model at large-N, formulated on a spatial two-sphere, with time dependent couplings which diverge at finite time
The appearance of a bulk crunch singularity may be viewed as a singularity in the time evolution of the boundary field theory driven by the divergent classical couplings
Summary
The large-N limit of the interacting O(N ) vector model on R3 describes a flow between the UV free fixed point and an IR Gaussian fixed point where the operator. Extremizing with respect to the fields h and σE, we arrive at the large-N saddle point conditions: λ(τ ) h = 2 σE bare. This is the so-called gap equation for the mean field σE , but within a time dependent setting. The formal saddle point conditions can be written in the form of a well-defined initial value problem. This is easy to make precise in the canonical quantization approach since the dynamics about the large-N saddle point is Gaussian. In an initially equilibrium thermal state the occupations numbers n are given by the Bose-Einstein distribution function: n eβω
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