Symmetries in quantum field theory and quantum gravity (1810.05338v2)

Abstract: In this paper we use the AdS/CFT correspondence to refine and then establish a set of old conjectures about symmetries in quantum gravity. We first show that any global symmetry, discrete or continuous, in a bulk quantum gravity theory with a CFT dual would lead to an inconsistency in that CFT, and thus that there are no bulk global symmetries in AdS/CFT. We then argue that any "long-range" bulk gauge symmetry leads to a global symmetry in the boundary CFT, whose consistency requires the existence of bulk dynamical objects which transform in all finite-dimensional irreducible representations of the bulk gauge group. We mostly assume that all internal symmetry groups are compact, but we also give a general condition on CFTs, which we expect to be true quite broadly, which implies this. We extend all of these results to the case of higher-form symmetries. Finally we extend a recently proposed new motivation for the weak gravity conjecture to more general gauge groups, reproducing the "convex hull condition" of Cheung and Remmen. An essential point, which we dwell on at length, is precisely defining what we mean by gauge and global symmetries in the bulk and boundary. Quantum field theory results we meet while assembling the necessary tools include continuous global symmetries without Noether currents, new perspectives on spontaneous symmetry-breaking and 't Hooft anomalies, a new order parameter for confinement which works in the presence of fundamental quarks, a Hamiltonian lattice formulation of gauge theories with arbitrary discrete gauge groups, an extension of the Coleman-Mandula theorem to discrete symmetries, and an improved explanation of the decay $\pi^0\to\gamma \gamma$ in the standard model of particle physics. We also describe new black hole solutions of the Einstein equation in $d+1$ dimensions with horizon topology $\mathbb{T}^p\times \mathbb{S}^{d-p-1}$.

• The paper establishes that global symmetries cannot exist in quantum gravity by leveraging evidence from the AdS/CFT correspondence.
• It demonstrates that a complete charge spectrum is essential, ensuring all finite-dimensional representations appear within any gauge group.
• It argues that gauge groups in quantum gravity must be compact, supported by analyses using entanglement wedge reconstruction and lattice gauge theory.

Overview of Symmetries in Quantum Field Theory and Quantum Gravity

The paper "Symmetries in Quantum Field Theory and Quantum Gravity" undertakes a comprehensive investigation into the role and nature of symmetries within the frameworks of quantum field theory (QFT) and quantum gravity, leveraging the AdS/CFT correspondence as a pivotal tool. The authors, Daniel Harlow and Hirosi Ooguri, aim to refine longstanding conjectures about symmetries in quantum gravity and subsequently provide evidence from the AdS/CFT correspondence model that not only refines but also supports these conjectures.

Key Conjectures and Investigations

Three primary conjectures drive the discourse within the field of quantum gravity. The authors aim to address the following:

1. Nonexistence of Global Symmetries: It has been conjectured that no global symmetries can exist in a consistent theory of quantum gravity. The authors argue that the presence of any global symmetry would lead to inconsistency within the conformal field theory (CFT) due to stringent constraints arising from the holographic dual description in AdS/CFT.
2. Completeness of Charge Spectrum: This conjecture posits that if a quantum gravity theory supports a gauge group, physical states must exist for all finite-dimensional irreducible representations of that gauge group. This notion is supported through arguments that AdS/CFT implies the presence of such states in the boundary CFT as a result of the duality.
3. Compactness of Gauge Groups: The authors explore the idea that gauge groups within quantum gravity must be compact. They discuss conditions under which CFTs admit only compact internal global symmetry groups, drawing connections with broader mathematical frameworks.

Analytical Approach and Techniques

The authors delve into various mathematical and physical tools to analyze symmetries, including:

• Entanglement Wedge Reconstruction: This framework, part of holography, is crucial in analyzing how different parts of a CFT can give insight into the bulk structure, thereby affecting the possible symmetry structures therein.
• Lattice Gauge Theory: The paper utilizes lattice gauge theory to unpack the structure of gauge symmetries and facilitate a detailed understanding of the spatial topology and associated algebra within quantum systems.
• Noether’s Theorem and Anomalies: Noether's theorem and anomalies are considered from traditional continuous symmetry contexts and are extended to address discrete groups and higher-form symmetries in this quantum framework.

Results and Implications

The findings of Harlow and Ooguri have profound theoretical implications:

• The absence of global symmetries in quantum gravity adds credibility to the broader conjecture against such symmetries, backed by difficulties reconciling them within the CFT context.
• The completeness of charge spectrum influences the understanding of compact versus noncompact groups, highlighting a need for a complete representation of possible charges within quantum gravitational systems.
• Speculative discussions around possible future directions in AI or machine learning could revolve around understanding symmetry constraints within complex systems, adapting methods developed in holography to alternative domains of research.

Concluding Remarks

This paper stands as a pivotal analysis within the landscape of theoretical physics, providing detailed mathematical frameworks and physical discussions that enhance our comprehension of how symmetries operate in quantum field theories and the elusive domain of quantum gravity. Additionally, it proposes that future advancements in AdS/CFT and quantum field theories could significantly dictate our understanding of fundamental forces and structures in physics, guiding the development of coherent, symmetry-based models in quantum gravity.