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On the interplay between boundary conditions and the Lorentzian Wetterich equation

Claudio Dappiaggi, Filippo Nava, Luca Sinibaldi

Abstract

In the framework of the functional renormalization group and of the perturbative, algebraic approach to quantum field theory (pAQFT), in [DDPR23] it has been derived a Lorentian version of a flow equation à la Wetterich, which can be used to study non linear, quantum scalar field theories on a globally hyperbolic spacetime. In this work we show that the realm of validity of this result can be extended to study interacting scalar field theories on globally hyperbolic manifolds with a timelike boundary. By considering the specific examples of half Minkowski spacetime and of the Poincaré patch of Anti-de Sitter, we show that the form of the Lorentzian Wetterich equation is strongly dependent on the boundary conditions assigned to the underlying field theory. In addition, using a numerical approach, we are able to provide strong evidences that there is a qualitative and not only a quantitative difference in the associated flow and we highlight this feature by considering Dirichlet and Neumann boundary conditions on half Minkowski spacetime.

On the interplay between boundary conditions and the Lorentzian Wetterich equation

Abstract

In the framework of the functional renormalization group and of the perturbative, algebraic approach to quantum field theory (pAQFT), in [DDPR23] it has been derived a Lorentian version of a flow equation à la Wetterich, which can be used to study non linear, quantum scalar field theories on a globally hyperbolic spacetime. In this work we show that the realm of validity of this result can be extended to study interacting scalar field theories on globally hyperbolic manifolds with a timelike boundary. By considering the specific examples of half Minkowski spacetime and of the Poincaré patch of Anti-de Sitter, we show that the form of the Lorentzian Wetterich equation is strongly dependent on the boundary conditions assigned to the underlying field theory. In addition, using a numerical approach, we are able to provide strong evidences that there is a qualitative and not only a quantitative difference in the associated flow and we highlight this feature by considering Dirichlet and Neumann boundary conditions on half Minkowski spacetime.
Paper Structure (19 sections, 3 theorems, 152 equations, 23 figures)

This paper contains 19 sections, 3 theorems, 152 equations, 23 figures.

Table of Contents

  1. Introduction
  2. Setting
  3. Geometry
  4. Functional approach to AQFT
  5. Interacting Theory
  6. S-Matrix
  7. Quantum field theory in presence of time-like boundaries
  8. Scalar quantum field theory on $\textnormal{P}\mathbb{A}\textnormal{d}\mathbb{S}_{d+1}$ --
  9. Scalar quantum field theory on $({\mathbb{H}}^{d+1},\eta)$ --
  10. Lorentzian Wetterich Equation in the pAQFT formalism
  11. Generating Functionals
  12. Functional Renormalization
  13. Lorentzian Wetterich Equation
  14. Applications
  15. Flow on half Minkowski with Dirichlet boundary conditions
  16. ...and 4 more sections

Key Result

Theorem 1

Let $\Delta_{+,D/N}^{\textnormal{P}\mathbb{A}\textnormal{d}\mathbb{S}_{d+1}}\in\mathcal{D}'(\textnormal{P}\mathbb{A}\textnormal{d}\mathbb{S}_{d+1}\times\textnormal{P}\mathbb{A}\textnormal{d}\mathbb{S}_{d+1})$ be as per Equation eq:twopointfunct_pads_2sol. Then its wave-front set reads where $k_x\,\vartriangleright 0$ if $k_x$ is future directed, while $(x,k_x)\sim_\pm (y,-k_y)$ if there exist $\g

Figures (23)

  • Figure 1: Scaling of the coupling constants on the whole Minkowski spacetime $({{\mathbb{M}}}^4,\eta)$. The arrows point in the infrared limit.
  • Figure 2: Comparison between the results obtained with a non-local regulator (orange) and the local regulator (blue).
  • Figure 3: Scaling of the coupling constants on Minkowski upper half-space. Dimensionless distance from the boundary: $\widetilde{z}=1$
  • Figure 4: Scaling of the coupling constants on Minkowski upper half-space. Dimensionless distance from the boundary: $\widetilde{z}=0.01$
  • Figure 5: Plot of $\mathfrak{B}(\widetilde{z},\widetilde{M})$ at fixed $M$, with respect to $\widetilde{z}$.
  • ...and 18 more figures

Theorems & Definitions (34)

  • Remark 1
  • Example 1
  • Definition 1: Off-shell field configurations
  • Remark 2
  • Definition 2: Regular and local functionals
  • Definition 3: Functional derivative
  • Definition 4: Micro-causal functional
  • Definition 5: Quantum *-algebra structure on ${\mathcal{F}_{reg}(\mathcal{M})}$
  • Definition 6
  • Definition 7: Quantum *-algebra structure on ${\mathcal{F}_{\mu c}(\mathcal{M})}$
  • ...and 24 more