Researchers develop semi-local electromagnetism framework for finite Cauchy lenses with boundaries and corners

The fundamental nature of boundaries and corners within spacetime presents a long-standing challenge for physicists, impacting areas from entropy calculations to the complexities of quantum field theory and gravity. Christopher J. Fewster, Daan W. Janssen, and Kasia Rejzner, all from the University of York, address this problem by developing a new algebraic framework for understanding electromagnetism in spaces with defined boundaries. Their research introduces the concept of ‘semi-local’ observables, which effectively account for boundary effects, and demonstrates how these observables behave under large gauge transformations, transformations that are particularly sensitive to the presence of boundaries. Crucially, the team recovers gauge invariance by employing the notion of quantum reference frames, treating boundary degrees of freedom as external perspectives, and establishes a state-independent method for relating different theoretical descriptions of these systems, offering new tools for tackling complex problems in gauge theory and potentially paving the way for a more complete understanding of spacetime itself.

Including entanglement entropy, the infrared problem in quantum field theory and quantum gravity present significant challenges to standard local quantum field theory, which struggles to accommodate such boundary-sensitive observables. This paper develops an algebraic framework for semi-local quantum electromagnetism on finite Cauchy lenses, a class of compact spacetimes possessing boundaries and corners. At the classical level, the research establishes a decomposition of the reduced covariant phase space into bulk closed-loop and surface sectors, demonstrating how the covariant phase space approach relates to the Peierls bracket construction commonly used in perturbative algebraic quantum field theory. Upon quantisation, the work obtains a Weyl C∗-algebra of semi-local electromagnetism, providing a foundation for further investigation into boundary effects and non-local phenomena.

Operator Algebras in Quantum Field Theory

This research builds upon a foundation of established work in quantum field theory and algebraic quantum field theory (AQFT), drawing upon key references such as Rejzner’s introduction to AQFT, Slawny’s foundational work on canonical commutation relations, and Takesaki’s comprehensive theory of operator algebras. Advanced mathematical tools, including those presented by Van Daele, are also crucial for understanding the structure of these algebras, alongside research exploring the connection between symmetries and quantum field theory, such as Strominger’s investigations into asymptotic symmetries. Additional references cover the study of infrared divergences and asymptotic symmetries, as seen in the work of Strominger and Riello’s investigations into edge modes and symplectic reduction of Yang-Mills theory with boundaries. Research by Rejzner and Schiavina applies the Batalin-Vilkovisky formalism to study asymptotic symmetries, while their work on Hamiltonian gauge theory with corners further explores the complexities of boundaries.

Wen’s foundational paper on edge states in the fractional quantum Hall effect provides crucial context, alongside Strohmaier’s studies of the photon field in non-compact manifolds with boundaries. The research also draws upon foundational work in general relativity, such as Regge and Teitelboim’s introduction of surface integrals into the Hamiltonian formulation of general relativity, and Penrose’s work on the existence of gravitational waves. Taylor’s reference on partial differential equations provides essential mathematical tools for solving Einstein’s equations, while Satishchandran and Wald’s work explores the asymptotic behavior of massless fields and the memory effect. Further contributions come from the study of gauge theory and topology, including Sniatycki’s investigations into boundary conditions for Yang-Mills fields, and Zuckerman’s exploration of action principles and global geometry, alongside Wilson’s introduction of the Wilson loop.

Mathematical tools such as Hodge decomposition, as presented by Schwarz, and topological vector spaces, as described by Schaefer and Wolff, are also essential. Research into boundary conditions, superselection, and edge modes, including Staruszkiewicz’s investigations into surface contributions to the photon number integral, and Dappiaggi, Hack, and Sanders’ studies of the Aharonov-Bohm effect, further informs the work. The research also considers quantum reference frames and quantum information, as explored by Vanrietvelde et al. Finally, references to Souriau’s work on dynamical systems, Robinson’s study of symplectic pathology, and Penrose’s work on gravitational waves provide a broader context, with key themes including asymptotic symmetries, boundary conditions, and edge modes, all crucial for understanding the behaviour of fields in complex spacetimes.

Boundary Contributions to Spacetime Electromagnetism Revealed

Researchers have developed a novel algebraic framework for understanding electromagnetism in complex spacetimes featuring boundaries and corners, termed semi-local electromagnetism. This work addresses limitations in standard field theory when dealing with boundary-sensitive phenomena, such as entropy and infrared problems in quantum field theory and gravity. The team established a decomposition of the electromagnetic phase space into sectors representing bulk and surface contributions, demonstrating a connection to established methods in perturbative field theory. The research yields a Weyl algebra of observables that transform in a specific way under large gauge transformations, symmetries that are non-trivial at the boundary of the spacetime.

To restore gauge invariance, scientists invoked the concept of quantum reference frames (QRFs), treating auxiliary surface degrees of freedom as these frames for the large gauge transformations. Crucially, this construction of QRFs occurs directly at the level of algebras, making it independent of specific quantum states. This approach provides new tools for understanding gauge theories on manifolds with boundaries and offers a solution to the problem of combining theories defined on different regions of spacetime. The findings demonstrate that edge modes can be understood as QRFs, providing a fundamental link between these modes and large gauge transformations.

By introducing an auxiliary “surface field” that transforms covariantly, researchers constructed a new algebra of invariants related to the original semi-local algebra via a “relativisation map. ” This map effectively dresses gauge-covariant quantities, converting them into gauge-invariant ones. Furthermore, the team showed that projections onto this relativised algebra define superselection sectors, providing a criterion for identifying distinct physical states. This analysis differs from previous work by focusing on external fluxes, allowing for interpolation between states with different internal fluxes. Finally, the research establishes a connection between the algebraic relativisation map and the operational QRF formalism, demonstrating that the map can be constructed from large gauge covariant projection measures. This QRF perspective not only offers conceptual clarity but also provides practical advantages, particularly in the treatment of gluing algebras of observables across common boundaries, demonstrating how algebras defined on separate regions of spacetime can be combined, offering a pathway to constructing states on the combined system.

Semi-Local Observables and Boundary Quantisation

This work presents a new algebraic framework for understanding electromagnetism in spacetimes with boundaries and corners, addressing limitations in standard approaches to quantum field theory. The researchers develop a method for constructing and analysing ‘semi-local’ observables, quantities that capture effects related to boundaries, such as electric fluxes, which are not easily accommodated by conventional local quantum field theory. The core of this approach involves decomposing the relevant physical space into sectors and demonstrating how these relate to established mathematical tools used in perturbative field theory. The team successfully quantises the system, obtaining a mathematical structure that describes how these semi-local observables transform under large gauge transformations, symmetries that are non-trivial at the boundary of the spacetime.

To ensure gauge invariance, they introduce the concept of ‘reference frames’ and a ‘relativisation map’, effectively treating boundary degrees of freedom as reference points for these transformations. This construction is notably state-independent, offering a robust and general approach to the problem. The framework also provides new tools for ‘gluing’ together theories defined on different regions of spacetime with shared boundaries, which is crucial for constructing more complex physical models. The authors acknowledge that their current work focuses on pure electromagnetism and does not.