Graph rules for the linked cluster expansion of the Legendre effective action

Banerjee, Rudrajit; Niedermaier, M.

doi:10.1063/1.5031429

Cited by 5 publications

(7 citation statements)

References 21 publications

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“…The direct solution of the recursion quickly becomes impractical, both manually and by computer. Instead, a graph theoretical approach was developed in [6] and leads to a closed formula for Γ l [φ] (without having to work out the lower orders first). A graph L = (V, E) comprises a set of vertices V, a set of edges (or lines) E, and the information that pairs of vertices are connected by (possibly several) edges.…”

Section: Linked Cluster Expansions and Frgmentioning

confidence: 99%

“…(d) Use the separate "dashed graph rule" invoking only labeled tree graphs to obtain µ v . In particular, The "dashed graph rule" producing µ v in (d) is described in Section 3 of [6] for Euclidean signature QFT on Z 1+d . It carries over with minor modifications to the spatial LCE at hand.…”

Section: Theorem 1 ([6]) (γ Graph Rule)mentioning

confidence: 99%

“…In combination, the "main" and the "dashed" graph rule give rise to a closed graph formula for Γ l [φ] analogous to the one for Euclidean signature [6]. Since it requires more graph theoretical notions and notations, we omit it here and instead highlight a number of structural properties:…”

Section: Theorem 1 ([6]) (γ Graph Rule)mentioning

confidence: 99%

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Anti-Newtonian Expansions and the Functional Renormalization Group

Niedermaier

2019

Universe

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Anti-Newtonian expansions are introduced for scalar quantum field theories and classical gravity. They expand around a limiting theory that evolves only in time while the spatial points are dynamically decoupled. Higher orders of the expansion re-introduce spatial interactions and produce overlapping lightcones from the limiting isolated world line evolution. In scalar quantum field theories, the limiting system consists of copies of a self-interacting quantum mechanical system. In a spatially discretized setting, a nonlinear “graph transform” arises that produces an in principle exact solution of the Functional Renormalization Group for the Legendre effective action. The quantum mechanical input data can be prepared from its 1 + 0 dimensional counterpart. In Einstein gravity, the anti-Newtonian limit has no dynamical spatial gradients, yet remains fully diffeomorphism invariant and propagates the original number of degrees of freedom. A canonical transformation (trivialization map) is constructed, in powers of a fractional inverse of Newton’s constant, that maps the ADM action into its anti-Newtonian limit. We outline the prospects of an associated trivializing flow in the quantum theory.

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Section: Linked Cluster Expansions and Frgmentioning

confidence: 99%

Section: Theorem 1 ([6]) (γ Graph Rule)mentioning

confidence: 99%

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Anti-Newtonian Expansions and the Functional Renormalization Group

Niedermaier

2019

Universe

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“…We expect it to be true generally whenever the full spatial FRG can be rendered well-defined. A systematic proposal to do so based on a spatial hopping expansion [10,11] will be presented elsewhere. The state dependence of the spatial EPA flow will be studied in Section 4.…”

Section: Introductionmentioning

confidence: 99%

The spatial Functional Renormalization Group and Hadamard states on cosmological spacetimes

Banerjee,

Niedermaier

2022

Preprint

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A spatial variant of the Functional Renormalization Group (FRG) is introduced on (Lorentzian signature) globally hyperbolic spacetimes. Through its perturbative expansion it is argued that such a FRG must inevitably be state dependent and that it should be based on a Hadamard state. A concrete implementation is presented for scalar quantum fields on flat Friedmann-Lemaître spacetimes. The universal ultraviolet behavior of Hadamard states allows the flow to be matched to the one-loop renormalized flow (where strict removal of the ultraviolet cutoff requires a tower of potentials, one for each power of the Ricci scalar). The state-dependent infrared behavior of the flow is investigated for States of Low Energy, which are Hadamard states deemed to be viable vacua for a pre-inflationary period. A simple time-dependent infrared fixed point equation (resembling that in Minkowski space) arises for any scale factor, with analytically computable corrections coding the non-perturbative ramifications of the Hadamard property in the infrared.

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“…The direct recursion turns out to become intractable beyond O(κ 6 ), say. However, a closed graph theoretical solution of the recursion can be obtained that yields Γ l for any l ≥ 1 [3]. Importantly, the series in (1.3) can be expected to have finite radius of convergence κ < κ c , at least as far as the associated vertex functions are concerned [9].…”

Section: Introductionmentioning

confidence: 99%

Critical behavior of the hopping expansion from the Functional Renormalization Group

Banerjee¹

2019

Proceedings of the 36th Annual International Symposium on Lattice Field Theory — PoS(LATTICE2018)

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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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Graph rules for the linked cluster expansion of the Legendre effective action

Cited by 5 publications

References 21 publications

Anti-Newtonian Expansions and the Functional Renormalization Group

Anti-Newtonian Expansions and the Functional Renormalization Group

The spatial Functional Renormalization Group and Hadamard states on cosmological spacetimes

Critical behavior of the hopping expansion from the Functional Renormalization Group

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