Relative entropy of an interval for a massless boson at finite temperature

Abstract: We compute Araki's relative entropy associated to a bounded interval $I=(a,b)$ between a thermal state and a coherent excitation of itself in the bosonic U(1)-current model, namely the (derivative of the) chiral boson. For this purpose we briefly review some recent results on the entropy of standard subspaces and on the relative entropy of non-pure states such as thermal states. In particular, recently Bostelmann, Cadamuro and Del Vecchio have obtained the relative entropy at finite temperature for the unbounded interval $(-\infty,t)$, using previous results of Borchers and Yngvason, mainly a unitary dilation that provides the modular evolution in the negative half-line. Here we find a unitary rotation in order to make use of the full PSL$(2,\mathbb{R})$ symmetries and obtain the modular group, modular Hamiltonian and the relative entropy $S$ of a bounded interval at finite temperature. Such relative entropy entails both a Bekenstein-like bound and a QNEC-like bound, but violates $S''\geq 0$. Finally, we extend the results to the free massless boson in $1+1$ dimensions with analogous bounds.

• The paper presents a detailed derivation of Araki's relative entropy for coherent excitations within a bounded interval in the bosonic U(1)-current model.
• It employs modular theory and PSL(2,R) symmetries to obtain explicit modular Hamiltonian and entropy bounds resembling Bekenstein and QNEC limits.
• The findings provide critical insights into entropy dynamics at finite temperature, paving the way for further studies in conformal field theories.

Relative Entropy of an Interval for a Massless Boson at Finite Temperature

Introduction

The paper "Relative entropy of an interval for a massless boson at finite temperature" (2209.00035) provides a comprehensive analysis of Araki's relative entropy applied to a bounded interval between a thermal state and a coherent excitation in the bosonic U(1)-current model. This investigation extends previous work on the entropy of standard subspaces and the relative entropy of non-pure states, particularly focusing on thermal states at finite temperatures. The study leverages conformal symmetries to derive expressions for the modular group, Hamiltonian, and relative entropy within bounded intervals, and it also explores entropy bounds akin to Bekenstein and QNEC, noting their potential violations.

Modular Framework and Relative Entropy

The paper begins by detailing the adoption of Araki's framework for defining relative entropy in quantum mechanics and quantum field theory (QFT), emphasizing its utility for states associated with von Neumann algebras. The researchers focus on extending the definition to thermal states, necessitating the use of unitary dilations and modular groups to handle non-trivial spacetime regions like a bounded interval $(a,b)$. The intricate geometric symmetries of the model, particularly PSL$(2,\mathbb{R})$, are pivotal in deriving the modular Hamiltonian and thus the relative entropy for bounded intervals.

Main Results and Bounds

The main analytical undertaking is to derive the relative entropy for coherent excitations on a bounded interval, leveraging symmetries to tap into the full PSL$(2,\mathbb{R})$ group. The authors identify two critical bounds for the relative entropy:

• Bekenstein-like Bound: The paper establishes a bound related to the energy stored in the interval, denoting a proportional relationship between the relative entropy and energy, modulated by the interval's length.
• QNEC-like Bound: The Quantum Null Energy Condition (QNEC) bound emerges in the exploration of curvature in entropy changes, indicating conditions under which classical energy distribution could lead to violations.

The results reveal significant theoretical implications, particularly in understanding entropy in free massless boson systems in 1+1 dimensions, where analogous bounds and violations manifest.

Implications and Future Directions

The findings have notable implications for theoretical explorations in conformal field theories and their associated entropy characteristics. By formulating precise mathematical definitions within these systems, the study elucidates entropy dynamics at finite temperatures, paving the way for new insights into entropy in more complex quantum systems. The proposed methodology and resulting bounds establish a foundation for further exploration into interacting theories and potential applications in higher dimensions, presenting intriguing possibilities for advancing our understanding of thermal properties in QFT.

Conclusion

The paper "Relative entropy of an interval for a massless boson at finite temperature" provides a crucial extension to current knowledge about entropy in quantum thermal states, with significant theoretical advances in modular theory applications. The derived entropy bounds and the discussion on their violations contribute to a deeper understanding of relative entropy in conformal field theory, offering foundational insights for further research into complex quantum systems and their entropy characteristics.