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Asymptotic Weyl Symmetry and Its Anomaly in a Curved Spacetime

Jeongwon Ho, O-Kab Kwon, Sang-A Park, Sang-Heon Yi

TL;DR

This work identifies an asymptotic Weyl-like symmetry (AWS) in a (1+1)D scalar field theory with nonminimal coupling on curved spacetime and explains its quantum breaking as an asymptotic trace anomaly, captured by a universal curvature term with coefficient 1/(24π). By mapping the FTCS to an inhomogeneous field theory (IFT) on flat space, the authors interpret AWS as a hidden internal symmetry acting on a position-dependent mass m_eff^2(x), and they construct the renormalized Hadamard two-point function to compute ⟨T_{μν}⟩. They demonstrate that, in the right asymptotic region, ⟨T^μ{}_{μ}⟩_ren → (1/(24π)) R and provide explicit expressions for ⟨T_{μν}⟩_ren, including an Unruh-like negative energy density near the bubble wall and subleading pressure contributions in the IFT description. Overall, AWS provides a coherent framework linking boundary-term effects, trace anomalies, and quantum stress-energy profiles, with implications for bubble-wall dynamics and potential extensions to other inhomogeneous setups.

Abstract

We explore an unusual symmetry in a field theory on a specific (1+1)-dimensional curved spacetime, which has an interesting interpretation as an approximate asymptotic Weyl symmetry. Unlike the conventional Weyl symmetry, the boundary term under the variation plays a crucial role in understanding for its anomaly. After converting a two-dimensional field theory on curved spacetime to an inhomogeneous field theory, we obtain the vacuum expectation value of the energy-momentum tensor. Then, we show the existence of an Unruh-like effect in the bubble wall expansion at the zero temperature.

Asymptotic Weyl Symmetry and Its Anomaly in a Curved Spacetime

TL;DR

This work identifies an asymptotic Weyl-like symmetry (AWS) in a (1+1)D scalar field theory with nonminimal coupling on curved spacetime and explains its quantum breaking as an asymptotic trace anomaly, captured by a universal curvature term with coefficient 1/(24π). By mapping the FTCS to an inhomogeneous field theory (IFT) on flat space, the authors interpret AWS as a hidden internal symmetry acting on a position-dependent mass m_eff^2(x), and they construct the renormalized Hadamard two-point function to compute ⟨T_{μν}⟩. They demonstrate that, in the right asymptotic region, ⟨T^μ{}_{μ}⟩_ren → (1/(24π)) R and provide explicit expressions for ⟨T_{μν}⟩_ren, including an Unruh-like negative energy density near the bubble wall and subleading pressure contributions in the IFT description. Overall, AWS provides a coherent framework linking boundary-term effects, trace anomalies, and quantum stress-energy profiles, with implications for bubble-wall dynamics and potential extensions to other inhomogeneous setups.

Abstract

We explore an unusual symmetry in a field theory on a specific (1+1)-dimensional curved spacetime, which has an interesting interpretation as an approximate asymptotic Weyl symmetry. Unlike the conventional Weyl symmetry, the boundary term under the variation plays a crucial role in understanding for its anomaly. After converting a two-dimensional field theory on curved spacetime to an inhomogeneous field theory, we obtain the vacuum expectation value of the energy-momentum tensor. Then, we show the existence of an Unruh-like effect in the bubble wall expansion at the zero temperature.

Paper Structure

This paper contains 20 sections, 82 equations, 4 figures.

Table of Contents

  1. Introduction
  2. FTCS–IFT Conversion Beyond SUSY Backgrounds
  3. Review on conversion between FTCS and IFT
  4. A static background
  5. Non-supersymmetric spacetime background
  6. Global behavior:
  7. Qualitative behavior:
  8. Asymptotic Weyl Symmetry
  9. Classical AWS
  10. AWS and anomaly
  11. Interpretation in the IFT description
  12. Quantum Energy-momentum Tensor
  13. Renormalized Hadamard two-point function
  14. Quantum energy-momentum tensor and trace anomaly
  15. VEV of the energy-momentum tensor
  16. ...and 5 more sections

Figures (4)

  • Figure 1: Plot of $e^{2\omega(x)}$ for $b=1$ and $\alpha=0,0.5,1$.
  • Figure 2: Plot of $m_{\textrm{eff}}^{2}/m_{0}^{2}$ for $\alpha=0.5$ and $\xi b^{2}/m_{0}^{2}=0.1,1,1.9$.
  • Figure 3: Plot of $m_{\textrm{eff}}^{2}/m_{0}^{2}$ for $\alpha=0.5$ and $\xi b^{2}/m_{0}^{2}=0.1,1,1.9$.
  • Figure 4: Plot of the effective mass profile $m_{\mathrm{eff}}^{2}(x)$ for $\alpha = 0.5$ and $\xi b^{2}/m_{0}^{2} = 1$, satisfying the condition \ref{['paracon']}. From the expression in \ref{['effMassSq1']}, one finds that $\epsilon \sim \mathcal{O}(e^{-2 b x_\epsilon})$. Hence, the contribution of $c_{\omega}$ is subleading and can be safely neglected in our analysis.