Infrared Finite Scattering Theory in Quantum Field Theory and Quantum Gravity

The foundation tool for studying quantum field theory and quantum gravity is the “S-matrix”. The S-matrix maps incoming states to outgoing states and yields predictions that have been verified to incredibly precise predictions in collider experiments. Despite these successes, the S-matrix is plagued by “infrared divergences” and is actually not defined. These infrared divergences are manifestations of a classical observable known as the “memory effect”. States with memory are perfectly well-defined states (and are generically produced in any scattering process) but simply do not live in the standard Fock space used for scattering theory. 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 quantum field theory and quantum gravity, since the S-matrix is undefined. To have a well-defined scattering theory, one needs to include states with memory into a Hilbert space.

Such a construction was provided for QED (electromagnetism coupled to massive charge fields) by Faddeev and Kulish in the 1970’s [1]. In this article we recast their construction in terms of intrinsic observables (i.e. charges and memory) of the theory. This allows us to generalize their construction to all theories so we can directly see if their construction is valid in general. In particular, their construction corresponds to constructing a Hilbert space of eigenstates of the conserved charges at spatial infinity which yields electrons “dressed” with a “cloud of soft photons”. We show that the same construction can be done in QED with massless charged fields since the corresponding “dressing” is singular and has infinite energy! The same dressing procedure does not apply since the Yang-Mills field contributes to its own dressing which fails to yield an eigenstate. For Yang-Mills we also show that any other procedure to construct eigenstates of the charge at spatial infinity cannot result in a Hilbert space of states. Finally, in quantum gravity, the gravitational field always contributes to its own dressing and one cannot replicate the Faddeev-Kulish procedure. However, we further show that the only eigenstate of charge at spatial infinity is the vacuum state (i.e. the state with no gravitational radiation)! Therefore, the Faddeev-Kulish dressing fails catastrophically in quantum gravity.

We also consider alternatives to Faddeev and Kulish representations and find that these also do not work. Therefore, despite the fact that there exist many well-defined states with memory, they generically do not fit into a single Hilbert space! We believe that if one wishes to treat scattering at a fundamental level in quantum gravity — as well as in massless QED and Yang-Mills theory — it is necessary to approach it from an algebraic viewpoint on the “in” and “out” states, wherein one does not attempt to “shoehorn” these states into some pre-chosen “in” and “out” Hilbert spaces. We outline the framework of such a scattering theory, which would be manifestly infrared finite