Entanglement entropy and nonabelian gauge symmetry (1406.7304v1)

Abstract: Entanglement entropy has proven to be an extremely useful concept in quantum field theory. Gauge theories are of particular interest, but for these systems the entanglement entropy is not clearly defined because the physical Hilbert space does not factor as a tensor product according to regions of space. Here we review a definition of entanglement entropy that applies to abelian and nonabelian lattice gauge theories. This entanglement entropy is obtained by embedding the physical Hilbert space into a product of Hilbert spaces associated to regions with boundary. The latter Hilbert spaces include degrees of freedom on the entangling surface that transform like surface charges under the gauge symmetry. These degrees of freedom are shown to contribute to the entanglement entropy, and the form of this contribution is determined by the gauge symmetry. We test our definition using the example of two-dimensional Yang-Mills theory, and find that it agrees with the thermal entropy in de Sitter space, and with the results of the Euclidean replica trick. We discuss the possible implications of this result for more complicated gauge theories, including quantum gravity.

• The paper defines entanglement entropy for gauge theories by embedding the physical Hilbert space into a product space incorporating boundary degrees of freedom.
• Testing the definition on two-dimensional Yang-Mills theory yields results consistent with thermal entropy in de Sitter space and the replica trick.
• Extending the approach to lattice gauge theory decomposes entanglement entropy into contributions from local and boundary degrees of freedom.

Entanglement Entropy and Nonabelian Gauge Symmetry

The paper of entanglement entropy in quantum field theory, particularly in gauge theories, poses significant challenges due to the difficulty in defining entanglement entropy in systems where the physical Hilbert space does not factorize into tensor products according to spatial regions. This paper by William Donnelly addresses these challenges by reviewing a definition of entanglement entropy that applies to both abelian and nonabelian lattice gauge theories.

The definition presented in the paper involves embedding the physical Hilbert space into a product of Hilbert spaces associated with regions having boundaries. These Hilbert spaces incorporate degrees of freedom on entangling surfaces, which behave like surface charges under the gauge symmetry. The contribution of these degrees of freedom to the entanglement entropy is shown to depend on the form dictated by the gauge symmetry.

The definition is tested on two-dimensional Yang-Mills theory, where it is found that the entanglement entropy aligns with both the thermal entropy in de Sitter space and the results obtained from the Euclidean replica trick. This concordance indicates a robust foundation for extending the definition of entanglement entropy to more complex gauge theories, including quantum gravity.

Key Findings and Implications

1. Definition of Entanglement Entropy in Gauge Theories: The paper introduces a methodology for defining entanglement entropy in gauge theories where the Hilbert space is embedded into a product space associated with spatial regions while accommodating non-gauge-invariant surface charges. This approach provides clarity in situations where the conventional tensor-product partitioning of the Hilbert space fails.
2. Two-Dimensional Yang-Mills Theory: The paper provides an insightful analysis of two-dimensional Yang-Mills theory as a case paper. In this context, the proposed definition of entanglement entropy yields results that are consistent with existing methods, such as the thermal distribution in de Sitter space and the replica trick, offering a unified framework for understanding entanglement in gauge theories.
3. Extension to Lattice Gauge Theory: Extending the methodology to lattice gauge theories, the paper decomposes entanglement entropy into contributions from local and boundary degrees of freedom. This step is crucial for theories with local dynamics and enhances the possibility of applying these concepts to real-world systems.
4. Potential Application to Quantum Gravity: Although speculative, the paper touches on the implications for defining entanglement entropy in quantum theories of gravity. It postulates that gravitational theories may possess analogous boundary degrees of freedom, potentially leading to new insights into the thermodynamics of spacetime.

Future Directions

The implications of this paper are substantial, particularly for the field of quantum gravity, where understanding the entanglement entropy could elucidate the underlying microstate structure of horizons and contribute to resolving foundational issues such as the black hole information paradox. Furthermore, the results may foster better comprehension of holographic dualities, where the surface degrees of freedom play a pivotal role.

The approach needs to be rigorously examined in higher-dimensional and more complex gauge theories, potentially involving dynamical spacetime backgrounds. Also, exploring the connections between entanglement, mutual information, and topological entropy across diverse quantum field theories could yield profound insights.

Conclusion

William Donnelly's investigation into entanglement entropy within gauge theories provides a crucial step forward in understanding how non-factorizable systems can be systematically approached through the lens of quantum information theory. These findings may potentially shape future research directions in quantum gravity and quantum field theory, providing the tools necessary to explore the entanglement characteristics of more intricate and realistic systems.