[Paper Review] Infrared Finite Scattering Theory in Quantum Field Theory and Quantum Gravity

Kartik Prabhu, Gautam Satishchandran+1 more|arXiv (Cornell University)|Mar 27, 2022

Black Holes and Theoretical PhysicsPhysics and Astronomy96 references54 citations

TL;DR

This paper demonstrates that the Faddeev-Kulish dressing procedure—previously used to resolve infrared divergences in massive QED—fails catastrophically in massless QED, Yang-Mills theory, and quantum gravity due to infinite energy flux and non-conservation of large gauge charges. The authors argue that a fundamental reformulation of scattering theory is required, proposing an algebraic framework that avoids pre-defined Hilbert spaces and ensures infrared finiteness by directly incorporating memory effects and asymptotic symmetries.

ABSTRACT

Infrared (IR) divergences arise in scattering theory with massless fields and are manifestations of the memory effect. There is nothing singular about states with memory, but they do not lie in the standard Fock space. IR divergences are artifacts of trying to represent states with memory in the standard Fock space. For collider physics, one can impose an IR cutoff and calculate inclusive quantities. But, this approach cannot treat memory as a quantum observable and is highly unsatisfactory if one views the S-matrix as fundamental in QFT and quantum gravity, since the S-matrix is undefined. For a well-defined S-matrix, it is necessary to define in/out Hilbert spaces with memory. Such a construction was given by Faddeev and Kulish (FK) for QED. Their construction "dresses" momentum states of the charged particles by pairing them with memory states of the electromagnetic field to produce states of vanishing large gauge charges at spatial infinity. However, in massless QED, due to collinear divergences, the "dressing" has an infinite energy flux so these states are unphysical. In Yang-Mills theory the "soft particles" used for dressing also contribute to the current flux, invalidating the FK procedure. In quantum gravity, the analogous FK construction would attempt to produce a Hilbert space of eigenstates of supertranslation charges at spatial infinity. However, we prove that there are no eigenstates of supertranslation charges except the vacuum. Thus, the FK construction fails in quantum gravity. We investigate some alternatives to FK constructions but find that these also do not work. We believe that to treat scattering at a fundamental level in quantum gravity - as well as in massless QED and YM theory - it is necessary to take an algebraic viewpoint rather than shoehorn the in/out states into some fixed Hilbert space. We outline the framework of such an IR finite scattering theory.

Motivation & Objective

• To resolve the long-standing problem of infrared divergences in scattering amplitudes involving massless fields in quantum field theory and quantum gravity.
• To investigate the viability of the Faddeev-Kulish construction—used to define S-matrices in massive QED—for theories with massless charged particles and gauge fields.
• To analyze whether eigenstates of large gauge charges (e.g., supertranslation charges in gravity) can be constructed in quantum gravity and massless gauge theories.
• To demonstrate that standard Fock space representations are insufficient for describing physical scattering states with memory, and to propose an alternative algebraic framework for scattering theory.

Proposed method

• Analyzes classical and quantum phase space structures of massless fields, focusing on memory effects at null infinity.
• Applies asymptotic quantization to free fields and extends the algebra to include large gauge charges and Poincaré generators.
• Constructs Faddeev-Kulish-type states by dressing charged particles with soft fields to cancel large gauge charges.
• Demonstrates that in massless QED and Yang-Mills, the required dressing states have infinite energy flux, rendering them unphysical.
• Proves that in quantum gravity, no non-vacuum eigenstates of supertranslation charges exist at spatial infinity, invalidating the Faddeev-Kulish approach.
• Proposes an algebraic scattering framework where 'in' and 'out' states are not embedded in pre-defined Hilbert spaces, but defined via observable algebras and asymptotic symmetries.

Experimental results

Research questions

• RQ1 Can the Faddeev-Kulish dressing procedure be consistently applied to QED with massless charged particles?
• RQ2 Do eigenstates of large gauge charges (e.g., supertranslation charges) exist in quantum gravity beyond the vacuum?
• RQ3 Why does the standard Fock space fail to describe physical scattering states with memory in massless field theories?
• RQ4 What is the role of soft theorems and asymptotic symmetries in constructing a well-defined S-matrix in quantum gravity?
• RQ5 Is it possible to formulate an infrared-finite scattering theory without relying on Fock space representations?

Key findings

• The Faddeev-Kulish dressing procedure fails in massless QED because the required soft dressing states carry infinite energy flux, making them unphysical.
• In Yang-Mills theory, the soft gluons used for dressing contribute to the Yang-Mills charge-current flux, invalidating the construction of large gauge charge eigenstates.
• In quantum gravity, there are no non-vacuum eigenstates of supertranslation charges at spatial infinity, proving the Faddeev-Kulish approach is fundamentally inapplicable.
• The standard S-matrix is ill-defined in massless QFT and quantum gravity due to the inability to represent memory states in Fock space.
• The authors construct a new algebraic framework for scattering theory that avoids Hilbert space embeddings and ensures infrared finiteness by directly incorporating asymptotic symmetries and observable algebras.
• The proposed framework allows for the treatment of memory as a quantum observable and provides a manifestly infrared-finite formulation of scattering in quantum gravity and massless gauge theories.