Beyond Infinity: Reconciling Quantum Fields and Gravity

Author: Denis Avetisyan

New research reveals a surprising link between the symmetries governing quantum electrodynamics and gravity, offering a path towards resolving long-standing infrared divergence problems.

This work establishes a framework based on asymptotic symmetries, supermomentum, and dressed states to provide a consistent description of scattering amplitudes in QED and gravity.

Resolving infrared divergences in quantum field theories remains a foundational challenge, prompting a re-examination of underlying symmetries and scattering methodologies. This is the focus of ‘Infrared physics of QED and gravity from representation theory’, which demonstrates how the unitary irreducible representations of asymptotic symmetry groups-extending the Poincaré group to include supertranslations-encode universal infrared features. Specifically, the work establishes a connection between conserved supermomentum and the construction of infrared-finite scattering amplitudes via dressed states, offering a consistent framework for describing interactions in both QED and gravity. Could this representation-theoretic approach ultimately provide a deeper understanding of the interplay between quantum mechanics, gravity, and the very fabric of spacetime?

Unveiling the Hidden Symmetries at Spacetime’s Edge

Poincaré symmetry has long served as a cornerstone of modern physics, describing the fundamental symmetries of spacetime – translations, rotations, and boosts – and underpinning calculations in quantum field theory. However, when examining the behavior of fields and particles at spatial and temporal infinity, this symmetry proves inadequate. Specifically, calculations involving scattering processes – where particles collide and interact – reveal inconsistencies when relying solely on Poincaré transformations. These issues arise because the standard symmetry group fails to fully account for the infinite degrees of freedom present at infinity, leading to divergences and unphysical results. This breakdown highlights the necessity of extending the framework to encompass a more comprehensive symmetry structure capable of accurately describing the long-range behavior of physical systems, paving the way for investigations into infinite-dimensional asymptotic symmetries like the $QED_{AsymptoticSymmetryGroup}$ and $BMSSymmetryGroup$ .

Beyond the well-established Poincaré symmetry, modern theoretical physics reveals a far more extensive framework for understanding spacetime symmetries at infinity. Investigations have identified infinite-dimensional asymptotic symmetry groups, prominently the QEDAsymptoticSymmetryGroup and the BMSSymmetryGroup, which dramatically expand upon traditional notions of symmetry. These groups aren’t simply larger versions of Poincaré symmetry; their structure is fundamentally linked to the CelestialSphere, a concept treating distant observations as projections onto this sphere. This connection implies that symmetries at infinity aren’t localized to specific directions, but are instead parameterized by functions on the celestial sphere, effectively creating an infinite set of conserved quantities that govern the long-range behavior of gravitational and electromagnetic fields. The implications are profound, suggesting a holographic correspondence where information about spacetime can be encoded on its distant boundary, and offering a novel perspective on the fundamental laws governing the universe.

The symmetries extending beyond traditional Poincaré invariance are not merely abstract mathematical constructs; they fundamentally govern how fields and particles interact at vast distances. These infinite-dimensional symmetries are intricately linked to functions defined on the celestial sphere – an infinite surface representing the sky as seen from the location of the observer. The behavior of these functions, captured within a mathematical quantity called ConformalDensity, dictates the long-range correlations and asymptotic forms of fields. Consequently, understanding these symmetries allows physicists to predict how particles scatter and interact when their separation approaches infinity, resolving inconsistencies that arise in conventional calculations. This framework suggests that the universe, at its largest scales, exhibits a structure profoundly shaped by these celestial symmetries, offering a pathway to a deeper understanding of spacetime and its fundamental constituents.

The discovery of infinite-dimensional asymptotic symmetries necessitates a profound reassessment of established conservation laws in physics. Traditionally, Noether’s theorem links symmetries to conserved quantities – Poincaré symmetry, for example, guarantees conservation of momentum and charge. However, these newly identified symmetries, extending beyond Poincaré, imply the existence of an infinite set of additional, previously unrecognized, conserved quantities. These are not simply higher-order corrections, but represent fundamentally new, global constraints on physical processes, particularly at the boundaries of spacetime. Consequently, a complete description of scattering amplitudes and the long-range behavior of fields requires tracking not only energy-momentum and charge, but also these novel, celestial-sphere-dependent conserved quantities, prompting a shift in how fundamental interactions and the very structure of spacetime are understood.

Supermomentum: A New Signature of Symmetry

Supermomentum is a conserved quantity stemming from the extended asymptotic symmetries of both Quantum Electrodynamics (QED) and General Relativity. These symmetries represent infinite-dimensional generalizations of the Poincaré group, which governs spacetime translations and Lorentz transformations. Specifically, supermomentum can be understood as a ‘translation’ acting on the celestial sphere, a construct representing the observable universe at spatial infinity. Mathematically, it emerges from conserved charges associated with these asymptotic symmetries and dictates a conservation law analogous to the conservation of linear momentum in conventional physics, but defined with respect to this celestial sphere. Its existence is not a postulated principle, but a consequence of the underlying symmetries of the theories when considered at null infinity.

Supermomentum’s existence is fundamentally connected to the $\text{QEDAsymptoticSymmetryGroup}$ and $\text{BMSSymmetryGroup}$ , which represent an extension of the Poincaré symmetry group. Poincaré symmetry, a cornerstone of modern physics, describes the fundamental symmetries of spacetime – translations, rotations, and Lorentz transformations. The QEDAsymptoticSymmetryGroup and BMSSymmetryGroup broaden this framework by incorporating infinite-dimensional asymptotic symmetries arising from the behavior of fields at null infinity. These extended symmetries introduce new generators beyond the Poincaré group, and supermomentum corresponds to the charges associated with specific translation generators within these larger symmetry groups. Therefore, the conservation of supermomentum is not an independent postulate, but rather a direct consequence of demanding consistency with these generalized symmetries at the asymptotic boundaries of spacetime.

The theoretical framework predicting supermomentum conservation encounters a challenge with specific symmetry representations known as HardRepresentationQED and HardRepresentationGravity. These representations, derived from the QEDAsymptoticSymmetryGroup and BMSSymmetryGroup respectively, demonstrably fail to conserve supermomentum under their associated transformations. This non-conservation is not an artifact of calculation, but a fundamental property of these particular representations, creating a tension between the predicted existence of a conserved quantity – supermomentum – and its absence within a mathematically consistent, albeit seemingly contradictory, segment of the symmetry group.

The observed non-conservation of supermomentum within specific “hard” representations of the QEDAsymptoticSymmetryGroup and BMSSymmetryGroup presents a challenge to established physical principles. These representations, while mathematically valid within the extended asymptotic symmetry framework, do not adhere to the expected conservation laws, specifically regarding supermomentum. Current research focuses on determining if these non-conserving representations represent a limitation of the mathematical framework, a signal of previously unknown physical mechanisms, or an artifact of the chosen representation itself. Potential resolutions involve modifications to the symmetry group, alternative definitions of supermomentum within these representations, or the identification of physical conditions under which these symmetries are broken, thereby allowing for non-conservation. Investigating this discrepancy is crucial for a consistent theoretical description of both quantum electrodynamics and gravity.

Taming the Infinite: Dressing States for Physical Consistency

Infrared divergences arise in perturbative calculations within both Quantum Electrodynamics (QED) and general relativity due to the contribution of massless gauge bosons – photons in QED and gravitons in gravity. Specifically, these divergences are generated by the emission of arbitrarily low-energy (soft) particles $\omega \rightarrow 0$ . The mathematical origin lies in the integrals over the phase space of these soft particles, which become ill-defined. These divergences are not physical predictions of the theory, but rather artifacts of the perturbative approach and signal a breakdown in the approximation scheme when considering infinitely many soft emissions. While seemingly problematic, these divergences are amenable to systematic treatment through techniques like renormalization and, as explored further, through the construction of DressedStates which effectively account for the soft emissions.

FKDressing is a perturbative method designed to systematically construct infrared-finite scattering amplitudes in quantum field theory. It achieves this by defining ‘DressedStates’ which incorporate the effects of emitted soft photons or gravitons directly into the asymptotic states used in scattering calculations. This construction involves a series of dressings, adding successive emissions to the initial and final state particles, and re-expressing the scattering amplitude in terms of these dressed states. The procedure is formally defined through a set of integral equations, allowing for the systematic calculation of corrections at each order in perturbation theory and ensuring that all contributions to the amplitude remain finite, even in the limit of zero momentum transfer for the emitted particles. The method’s efficacy is demonstrated through calculations in both Quantum Electrodynamics (QED) and gravity, yielding finite results for observable quantities.

DressedStates are constructed by incorporating the emission of soft photons or gravitons directly into the definition of the particle’s state. This process involves summing over all possible soft emissions, effectively modifying the original particle’s properties to include the effects of these emissions. The resulting DressedState possesses a finite scattering amplitude, resolving the infrared divergences present in standard perturbative calculations. Specifically, the emitted particles are not treated as external radiation but are considered integral components of the particle’s asymptotic state, altering its momentum and charge distribution in a manner that cancels the divergent terms arising from unobserved soft emissions. This screening effect is crucial for obtaining physically meaningful and finite results in quantum electrodynamics and gravity calculations.

Dressed states represent a generalization of Supermomentum Eigenstates, providing a framework to consistently incorporate soft emissions into calculations while maintaining conservation laws. Traditional Supermomentum Eigenstates define states with fixed total momentum, but do not account for the continuous emission of soft gravitons or photons; dressed states explicitly include these emissions as integral components of the particle’s state. This extension allows for the rigorous treatment of infrared divergences, as the soft emissions are no longer viewed as external disturbances but as intrinsic to the defined state. Consequently, calculations utilizing dressed states demonstrate equivalence to established soft graviton and photon theorems, which dictate the form of emitted radiation in the zero-energy limit $\omega \rightarrow 0$ , thereby validating the approach and providing a means to derive these theorems from first principles.

A Deeper Symmetry: Implications for Our Understanding of the Universe

The persistence of supermomentum conservation within the framework of dressed states offers compelling support for the physical significance of asymptotic symmetries. Dressed states, which incorporate the effects of interactions with the quantum fields, aren’t simply mathematical constructs; their adherence to this fundamental conservation law suggests these symmetries aren’t merely abstract mathematical properties, but rather define the genuine, long-range behavior of physical systems. This consistency indicates that the symmetries, observable only at spatial or temporal infinity, are intrinsically linked to the dynamics of the system and play a crucial role in determining its physical properties. Essentially, the conservation of supermomentum within dressed states provides a rigorous test-and a positive confirmation-that these asymptotic symmetries are not just theoretical curiosities, but are integral to a complete description of spacetime and particle interactions, potentially reshaping how fundamental laws are understood and applied.

The discovery of representations like HardRepresentationQED and HardRepresentationGravity presents a nuanced challenge to the conventional link between symmetry and conserved quantities. Traditionally, symmetries dictate conserved quantities via Noether’s theorem; however, these specific representations demonstrate that a symmetry can exist without a corresponding conserved supermomentum. This isn’t a contradiction, but rather an indication that the relationship is more subtle than previously understood. It suggests that conserved quantities aren’t simply defined by symmetry, but emerge from a specific interplay between the symmetry and the physical system’s dynamics. These representations necessitate a careful examination of how conservation laws are derived and applied, potentially requiring a refinement of the criteria used to identify truly physical states and observables within a given theoretical framework.

Current theoretical frameworks often rely on the Fock space to describe physical states, but recent investigations demonstrate its inadequacy in fully capturing the symmetries present at spacetime infinity. The necessity of extending beyond this conventional Hilbert space implies that defining physical observables – quantities meant to represent measurable outcomes – requires careful consideration of these asymptotic symmetries. Traditional definitions, implicitly assuming a limited state space, may not accurately reflect the true physical content of a system, potentially leading to inconsistencies or incomplete descriptions of particle interactions. Consequently, a re-evaluation of how observables are formulated is crucial; they must be constructed to respect the extended symmetry structure, ensuring that measurements are consistent with the fundamental laws governing spacetime and avoiding ambiguities arising from the infinite-dimensional nature of the full symmetry group.

This innovative framework achieves a reconciliation between the seemingly disparate concepts of symmetry and conservation laws, offering a novel perspective on the fundamental structure of the universe. By successfully integrating representations of the Bondi-Metzner-Sachs (BMS) group – which describes asymptotic symmetries at null infinity – into the theoretical landscape, it provides a more complete description of spacetime and its interactions with matter. The approach suggests that symmetries, previously considered abstract mathematical constructs, are deeply intertwined with conserved quantities, influencing how particles behave and interact over vast distances. Consequently, this integration has the potential to refine current models of gravity and quantum field theory, offering insights into phenomena ranging from black hole physics to the early universe and potentially revealing previously hidden connections between spacetime geometry and particle dynamics.

The pursuit of consistent frameworks in physics, as demonstrated by this work on QED and gravity, echoes a fundamental principle of understanding any complex system. Rigorous analysis of asymptotic symmetries and infrared divergences requires careful attention to the boundaries of applicability, much like checking data boundaries to avoid spurious patterns. As Georg Wilhelm Friedrich Hegel observed, “The truth is the whole.” This research, by consistently addressing the challenges posed by infrared divergences through the use of dressed states and supermomentum, strives for that ‘whole’-a complete and internally consistent description of physical reality, where seemingly paradoxical behaviors resolve within a broader, more comprehensive understanding of the underlying principles.

Looking Ahead

The connection established between quantum electrodynamics and gravity, viewed through the lens of asymptotic symmetries, naturally directs attention toward the persistent question of information. The resolution of infrared divergences via dressed states, while elegant, does not entirely dissolve the conceptual tension between unitarity and the seemingly boundless degrees of freedom at null infinity. Future work must address whether this framework can meaningfully constrain the ‘soft’ gravitons often invoked in information paradox discussions, or if they remain perpetually outside a fully predictive theory.

A fruitful avenue lies in exploring the limitations of the BMS group itself. The current formalism largely treats spacetime as a classical background, against which symmetries are defined. However, a fully quantum treatment of gravity likely demands a dynamical spacetime, potentially necessitating modifications or extensions to the BMS group to account for quantum fluctuations of the metric. The search for non-trivial, physically realizable representations beyond those currently employed also represents a significant challenge.

Ultimately, the true test of this approach will be its ability to interface with observable phenomena. While the infrared structure of scattering amplitudes is undeniably important, bridging the gap between theoretical consistency and experimental verification remains elusive. Perhaps the most intriguing possibility is that the subtle effects predicted by this framework – the precise form of soft theorems, for example – could manifest in high-precision measurements of gravitational waves, offering a unique window into the quantum nature of spacetime.

Original article: https://arxiv.org/pdf/2603.06297.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

Unveiling the Hidden Symmetries at Spacetime’s Edge

Supermomentum: A New Signature of Symmetry

Taming the Infinite: Dressing States for Physical Consistency

A Deeper Symmetry: Implications for Our Understanding of the Universe

Looking Ahead

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