Entanglement entropy in gauge theories and the holographic principle for electric strings (0806.3376v1)

Abstract: We consider quantum entanglement between gauge fields in some region of space A and its complement B. It is argued that the Hilbert space of physical states of gauge theories cannot be decomposed into a direct product of Hilbert spaces of states localized in A and B. The reason is that elementary excitations in gauge theories - electric strings - are associated with closed loops rather than points in space, and there are closed loops which belong both to A and B. Direct product structure and hence the reduction procedure with respect to the fields in B can only be defined if the Hilbert space of physical states is extended by including the states of electric strings which can open on the boundary of A. The positions of string endpoints on this boundary are the additional degrees of freedom which also contribute to the entanglement entropy. We explicitly demonstrate this for the three-dimensional Z2 lattice gauge theory both numerically and using a simple trial ground state wave function. The entanglement entropy appears to be saturated almost completely by the entropy of string endpoints, thus reminding of a ``holographic principle'' in quantum gravity and AdS/CFT correspondence.

• The paper demonstrates that extending the Hilbert space to include cut electric strings resolves the factorization issue when computing entanglement entropy in gauge theories.
• It employs numerical analysis using a three-dimensional Z2 lattice gauge theory, confirming that entanglement entropy is mainly governed by string endpoints in accordance with the area law.
• The study connects gauge theory entanglement with holographic principles, suggesting broader implications for quantum gravity and black hole entropy.

Entanglement Entropy in Gauge Theories and the Holographic Principle for Electric Strings

The paper by Buividovich and Polikarpov offers a rigorous investigation into the intricacies of entanglement entropy in gauge theories, with a particular focus on the role of electric strings. By employing the framework of the holographic principle, this paper explores the nontrivial challenges posed by the absence of a straightforward decomposition of the Hilbert space in gauge theories.

The central thesis of the paper revolves around the entanglement between gauge fields within a region $A$ and its complement $B$. The authors contend that, unlike scalar field theories where the Hilbert space allows a direct factorization into $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$, such a decomposition is infeasible in gauge theories due to the nature of elementary excitations—electric strings that manifest as closed loops. These loops may intersect both regions $A$ and $B$, inherently complicating any direct product structures and the consequent procedure of reduction in terms of entanglement entropy.

The paper articulates a significant notion: addressing this issue necessitates an extension of the Hilbert space to incorporate states of electric strings that can effectively be 'cut' along region boundaries. This results in new degrees of freedom associated with the endpoints of these strings, contributing to the entanglement entropy. These findings resonate with concepts from the holographic principle and AdS/CFT correspondence, suggesting that in quantum gravity, the entanglement properties might correlate with boundary phenomena.

Numerical analysis is used to substantiate these theoretical claims, specifically through simulations of the three-dimensional $Z_{2}$ lattice gauge theory. The results demonstrate that entanglement entropy is, to an overwhelming extent, governed by the entropy of string endpoints. This observation aligns with the "area law," which asserts that entanglement entropy is proportional to the boundary area between regions $A$ and $B$, further suggesting a holographic perspective on gauge theories.

The implications of this research are profound. Practically, it offers a clarified path for computing entanglement entropy in gauge theories, an area clouded by the complexities introduced by electric strings. Theoretically, it proposes a structure where gauge theory entanglement can be understood in connection with boundary phenomena, inviting speculation on the influence of such frameworks on quantifying entropy in black hole physics and other areas where the holographic principle plays a pivotal role.

Future developments in AI related to this paper could include exploring algorithmic models that take these complex patterns of entanglement into account, providing new tools for simulating physical systems with intricate entanglement structures, such as those found in quantum computing and advanced material science contexts. There is also the enigmatic potential of generalizing these findings beyond Abelian gauge theories, which promises to extend these theoretical frameworks to more comprehensive models in quantum field theory and gravity.