States of Low Energy (SLEs) are exact Hadamard states defined on arbitrary Friedmann–Lemaître spacetimes. They are constructed from a fiducial state by minimizing the Hamiltonian’s expectation value after averaging with a temporal window function. We show the SLE to be expressible solely in terms of the (state independent) commutator function. They also admit a convergent series expansion in powers of the spatial momentum, both for massive and for massless theories. In the massless case, the leading infrared behavior is found to be Minkowski-like for all scale factors. This provides a new cure for the infrared divergences in Friedmann–Lemaître spacetimes with accelerated expansion. As a consequence, massless SLEs are viable candidates for pre-inflationary vacua, and in a soluble model, they are shown to entail a qualitatively correct primordial power spectrum.
Graph rules for the linked cluster expansion of the Legendre effective action Γ[φ ] are derived and proven in D ≥ 2 Euclidean dimensions. A key aspect is the weight assigned to articulation vertices which is itself shown to be computable from labeled tree graphs. The hopping interaction is allowed to be long ranged and scale dependent, thereby producing an in principle exact solution of Γ[φ ]'s functional renormalization group equation.
A lattice version of the widely used Functional Renormalization Group (FRG) for the Legendre effective action is solved -in principle exactly -in terms of graph rules for the linked cluster expansion. Conversely, the FRG induces nonlinear flow equations governing suitable resummations of the graph expansion. The (finite) radius of convergence determining criticality can then be efficiently computed as the unstable manifold of a Gaussian or non-Gaussian fixed point of the FRG flow. The correspondence is tested on the critical line of the Lüscher-Weisz solution of the φ 4 4 theory and its φ 4 3 counterpart.
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