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Soft cutoffs in the covariant phase space of dynamical reference frames

Kang Liu, Wei Guo, Xiao-Mei Kuang

Abstract

We construct covariant theories incorporating fluctuating boundaries and soft cutoffs by introducing dynamical reference frames (DRFs). This framework generalizes the covariant action from a hard-cutoff to a soft-cutoff formulation, utilizing smearing functions and their corresponding operator expansions. This generalization initially leads to a loss of diffeomorphism covariance, which is recovered solely by restricting the DRFs, along with both their associated and linear MCFs, to specific forms, and by imposing suitable boundary conditions on the smearing functions. Satisfying these conditions restores covariance in relational spacetime, thereby enabling the consistent definition of subsystems. Within the covariant phase space formalism, we derive the charges of the soft-cutoff theory while explicitly addressing the inherent ambiguities arising from the boundary Lagrangian. We demonstrate that introducing an additional pointwise dependence is essential to resolve these ambiguities and ensure the integrability of the charges, even under fluctuating boundary conditions. Finally, in the context of General Relativity (GR), we establish the conditions under which holographic renormalization results at the asymptotic boundary coincide with the Noether charges derived from our soft-cutoff procedure.

Soft cutoffs in the covariant phase space of dynamical reference frames

Abstract

We construct covariant theories incorporating fluctuating boundaries and soft cutoffs by introducing dynamical reference frames (DRFs). This framework generalizes the covariant action from a hard-cutoff to a soft-cutoff formulation, utilizing smearing functions and their corresponding operator expansions. This generalization initially leads to a loss of diffeomorphism covariance, which is recovered solely by restricting the DRFs, along with both their associated and linear MCFs, to specific forms, and by imposing suitable boundary conditions on the smearing functions. Satisfying these conditions restores covariance in relational spacetime, thereby enabling the consistent definition of subsystems. Within the covariant phase space formalism, we derive the charges of the soft-cutoff theory while explicitly addressing the inherent ambiguities arising from the boundary Lagrangian. We demonstrate that introducing an additional pointwise dependence is essential to resolve these ambiguities and ensure the integrability of the charges, even under fluctuating boundary conditions. Finally, in the context of General Relativity (GR), we establish the conditions under which holographic renormalization results at the asymptotic boundary coincide with the Noether charges derived from our soft-cutoff procedure.
Paper Structure (11 sections, 129 equations, 1 figure)

This paper contains 11 sections, 129 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Dynamical reference frame and Maurer-Cartan form in relational spacetime
  3. Covariant theories with fluctuating and smearing boundaries
  4. Covariant phase space formalism
  5. Symplectic potentials
  6. Symplectic forms and integrable charges
  7. Noether charge at the asymptotic boundary of general relativity in the FG gauge
  8. Conclusions and discussions
  9. Proofs for equations regarding DRFs
  10. Calculation of expansion coefficients
  11. Solutions for $m_n$ in holographic renormalization

Figures (1)

  • Figure 1: Left panel: The brown region represents the range of $b$ in the original spacetime, while the green region denotes the relational spacetime extended via the boundary mapping induced by the DRFs. Within the interval $[b_L, b_R]$, $\mathcal{H}_b \approx 1$, whereas outside this region, it approaches the small value set by the thickness field. The fluctuating boundaries in spacetime are characterized by the deviation between $b_{L,R}$ and $\alpha_{L,R}$. Right panel: Plots of $\mathcal{H}_b$ evaluated from equation \ref{['example_G']} are displayed for various values of $\epsilon$, obtained by explicitly setting $F=0$.