Infrared physics of QED and gravity from representation theory

Imagine you are trying to send a message across a vast, stormy ocean. In the world of physics, this "message" is a particle collision (like two electrons smashing together), and the "ocean" is the fabric of space and time.

For decades, physicists have been struggling with a problem: when they try to calculate the outcome of these collisions using standard math, the numbers blow up to infinity. This is called an Infrared (IR) Divergence. It happens because the forces involved (like electromagnetism and gravity) have an infinite reach. Even if a particle is far away, it still feels a tiny tug, and mathematically, summing up all these tiny tugs from the infinite universe creates a mess.

This paper, by Laura Donnay and Yannick Herfray, proposes a radical new way to fix this mess. They suggest we stop looking at particles the way we always have and start seeing them through the lens of symmetry.

Here is the story, broken down with simple analogies.

1. The Problem: The "Perfectly Calm" Ocean Doesn't Exist

In standard physics, we assume particles travel through empty space like cars on a highway. We calculate their paths and collisions based on this "empty highway" assumption.

But in reality, the highway is never empty. It's filled with a fog of invisible, low-energy waves (photons for light, gravitons for gravity).

The Old View: We tried to ignore the fog or count it only if it was "loud" enough to be heard. But the fog is always there, whispering infinitely.
The Result: When we try to calculate the collision, the whispers add up to a scream (infinity). The math breaks.

2. The New View: The "Symmetry Orchestra"

The authors say: "Stop trying to ignore the fog. Instead, realize that the fog is part of the music."

They introduce the idea of Asymptotic Symmetries. Think of the universe not as a static stage, but as a giant, infinite drum.

The Drum: The "celestial sphere" (the sky as seen from a particle).
The Symmetry: You can tap this drum in infinite different ways. In the past, we only cared about the big, loud beats (standard movements like moving forward or rotating). But there are infinite subtle, soft taps (called Supertranslations and Large Gauge Transformations) that we ignored.

The paper argues that the universe has a hidden "Orchestra" of these infinite taps. Every particle is actually a musician in this orchestra, playing a specific note.

3. The "Hard" vs. "Soft" Mistake

The authors divide particles into two types:

Hard Particles: These are the loud, energetic notes. They are the particles we usually see (electrons, protons). In the old math, we thought these were the only things that mattered.
Soft Particles: These are the faint, background hums. They carry almost no energy but are everywhere.

The Big Realization:
The paper shows that if you only count the "Hard" particles, the music is out of tune. The "Hard" notes don't respect the rules of the infinite orchestra. They violate a conservation law called Supermomentum.

Imagine trying to balance a scale. If you only put the heavy weights (Hard particles) on one side, the scale tips. The universe demands that you also account for the invisible air pressure (Soft particles) to keep it balanced.

4. The Solution: "Dressed" Particles

To fix the math, the authors propose a new definition of what a particle is.

Old Definition: A particle is a single, isolated point of energy.
New Definition: A particle is a "Dressed" entity. It is the loud note plus its own personal cloud of soft whispers.

Think of it like a celebrity walking through a crowd.

The Old View: We just counted the celebrity (the Hard particle).
The New View: The celebrity is surrounded by a swarm of paparazzi and fans (the Soft cloud). You cannot separate the celebrity from the swarm. The "particle" is the celebrity and the swarm together.

When you calculate collisions using these "Dressed" particles, the math suddenly works. The infinities cancel out, and you get a finite, sensible answer.

5. The "Goldstone" Glitch

There is one tricky part. To make this math work, the authors have to introduce something called a Goldstone Operator.

The Analogy:
Imagine you are trying to describe the position of a boat on a lake. Usually, you use a map with fixed coordinates (North, South, East, West).
But in this new theory, the "map" itself is shifting. The Goldstone Operator is like a special tool that tells you how the map is moving relative to the boat.

The authors admit this tool feels a bit "hacky" or "ad hoc" (like using duct tape to fix a spaceship). It works, but it suggests that our current way of doing physics (Quantum Field Theory) is a bit like trying to drive a car with the steering wheel removed. We are forcing the car to go straight by holding the wheel with our hands (the Goldstone operator) instead of fixing the steering mechanism.

6. The Grand Conclusion: A New Kind of Particle

The paper's ultimate message is philosophical as much as it is mathematical:

We need to redefine what a "particle" is.

For 100 years, we defined a particle as a tiny object obeying the rules of the Poincaré group (standard space-time rules).
The authors argue: No. A particle should be defined as a unit of the Asymptotic Symmetry Group.

Old Particle: A dot on a graph.
New Particle: A unique, indivisible note in the infinite symphony of the universe's symmetries.

If we accept this new definition, the "infrared divergences" (the math breaking) disappear naturally. The universe isn't broken; our definition of the building blocks was just too small.

Summary in One Sentence

This paper argues that to fix the broken math of particle collisions, we must stop treating particles as lonely travelers and start treating them as "dressed" entities that carry their own infinite cloud of soft gravitational and electromagnetic whispers, governed by a hidden, infinite symphony of universal symmetries.

Here is a detailed technical summary of the paper "Infrared physics of QED and gravity from representation theory" by Laura Donnay and Yannick Herfray.

1. Problem Statement

The paper addresses the long-standing issue of infrared (IR) divergences in the scattering amplitudes of Quantum Electrodynamics (QED) and perturbative Quantum Gravity.

The Core Issue: In standard Quantum Field Theory (QFT), the S-matrix between conventional Fock states (built from free particles) is mathematically ill-defined due to IR divergences arising from massless bosons (photons and gravitons).
Limitations of Current Approaches:
- Inclusive Cross-sections: The traditional method of tracing over soft photons below a detector resolution yields finite observables but abandons a mathematically rigorous definition of the S-matrix.
- Faddeev-Kulish (FK) Construction: This approach constructs "dressed states" (charged particles surrounded by clouds of soft photons/gravitons) to diagonalize the asymptotic Hamiltonian. While these states yield IR-finite amplitudes, they suffer from significant drawbacks:
  - They are not normalizable elements of a standard Fock space.
  - They rely on ambiguous dressing factors.
  - They are not eigenstates of the full asymptotic symmetry generators.
The Gap: Recent developments have linked IR physics to asymptotic symmetries (Large U(1) transformations in QED and BMS supertranslations in gravity). However, a systematic framework connecting these symmetries to the construction of a unitary, IR-finite S-matrix via Unitary Irreducible Representations (UIRs) has been lacking.

2. Methodology

The authors employ representation theory of infinite-dimensional asymptotic symmetry groups to reformulate the problem of IR divergences.

Symmetry Groups:
- QED: The group is $SO(3,1) \ltimes (\mathbb{R}^{3,1} \times E[0])$ , where $E[0]$ represents smooth conformal densities of weight 0 on the celestial sphere (large U(1) transformations).
- Gravity: The group is the BMS group, $SO(3,1) \ltimes E[1]$ , where $E[1]$ represents supertranslations (conformal densities of weight 1).
Induced Representations: The paper utilizes Mackey's theory of induced representations for semi-direct product groups ( $SO(3,1) \ltimes A$ $S O (3, 1) ⋉ A$ ).
- States are classified by supermomenta (elements of the dual space of the abelian subgroup).
- The "little group" (stabilizer of the supermomentum) determines the spin and degrees of freedom.
Hard vs. Soft Decomposition: A central technical tool is the unique, Lorentz-invariant, but non-linear decomposition of a generic supermomentum $P$ into a "hard" part (determined by standard momentum/charge) and a "soft" part (determined by a function on the celestial sphere).

3. Key Contributions and Results

A. Classification of Representations

The authors systematically classify the representations of the asymptotic symmetry groups:

Hard Representations: These correspond one-to-one with standard Poincaré UIRs.
- For a particle with momentum $p^\mu$ and charge $q_e$ , the supermomentum is fixed (e.g., $Q(z, \bar{z}) \propto q_e \delta^{(2)}(z-\zeta)$ for massless QED).
- Crucial Finding: Hard representations cannot conserve supermomentum in scattering processes. Even if standard momentum and charge are conserved, the sum of hard supermomenta is non-zero due to a "soft obstruction" term $S(z, \bar{z})$ .
Soft Representations: These have zero standard momentum and charge but non-zero supermomentum (e.g., $Q = \partial \bar{\partial} N$ ). They form a Hilbert space.
Generic Representations: A generic state is a superposition of hard and soft components. The supermomentum is $P = P_{\text{hard}} + \partial \bar{\partial} N$ $P = P_{hard} + \partial \overset{ˉ}{\partial} N$ .
- The decomposition is non-linear: The sum of two supermomenta involves a correction term $S$ that depends on the momenta of the constituents. This non-linearity is the root cause of the failure of supermomentum conservation for standard Fock states.

B. Reinterpretation of IR Divergences

The paper re-derives standard IR results entirely within the representation theory framework:

Soft Theorems as Conservation Laws: Weinberg's soft photon/graviton theorems are shown to be equivalent to the conservation of supermomentum. The "soft factor" in the theorem is precisely the obstruction $S(z, \bar{z})$ required to balance the supermomentum equation.
Virtual Divergences: The exponentiation of virtual IR divergences (the factor $e^W$ $e^{W}$ ) is identified with the Lorentz-invariant norm squared of the obstruction function $S$ $S$ .
- For QED: $\Re(W) \propto -\|S\|^2$ .
- For Gravity: $\Re(W) \propto -\|S\|^2$ .
- This unifies the understanding of real (soft emission) and virtual divergences as consequences of failing to conserve the infinite-dimensional charges.

C. Dressed States vs. Supermomentum Eigenstates

The authors compare the Faddeev-Kulish (FK) dressed states with the natural supermomentum eigenstates arising from representation theory:

FK States: These are eigenstates of the soft charge but not the full supermomentum (they are not momentum eigenstates in the strict sense due to the infinite cloud).
Supermomentum Eigenstates: These are the natural UIRs of the asymptotic group. They are labeled by $(p^\mu, \partial \bar{\partial} N)$ .
The Connection: The authors demonstrate that FK dressed states can be turned into supermomentum eigenstates via a specific limiting procedure (taking the dressing cutoff to zero).
- The dressing effectively linearizes the supermomentum.
- If the dressing parameters are chosen correctly (satisfying specific conditions like $\Delta c = 0$ ), the conservation of standard momentum and charge becomes equivalent to the conservation of supermomentum.
- This explains why FK states are IR finite: they are constructed to satisfy the conservation laws of the asymptotic symmetry group.

D. The "Goldstone" Operator Issue

The paper highlights a tension in the FK construction: it relies on a "Goldstone operator" (conjugate to the soft charge) to generate the dressing.

This operator acts like a position operator in the space of vacua.
The authors argue that introducing such an operator is unnatural from a pure representation theory perspective (analogous to trying to define a position operator to generate momentum eigenstates in standard QFT).
Conclusion: The FK construction is an attempt to "shoehorn" the new notion of asymptotic particles (UIRs of the asymptotic group) into the old framework of standard QFT.

4. Significance and Outlook

New Definition of Particles: The paper advocates for a paradigm shift: Asymptotic particles should be defined as UIRs of the asymptotic symmetry group (BMS or QED asymptotic group), not the Poincaré group. This naturally incorporates the soft degrees of freedom required for IR finiteness.
Mathematical Rigor: It provides a mathematically well-defined framework for the S-matrix where states belong to a separable Hilbert space (unlike the non-normalizable FK states in some formulations), provided one accepts the extended symmetry group.
Unification: It unifies the treatment of massless and massive particles, and QED and Gravity, under a single representation-theoretic umbrella.
Future Directions:
- Constructing multi-particle states (tensor products) of these new asymptotic particles.
- Extending the analysis to higher dimensions ( $d > 4$ ) and non-abelian gauge theories.
- Investigating the inclusion of superrotations and subleading soft theorems.

In summary, Donnay and Herfray demonstrate that IR divergences are not merely technical nuisances to be canceled by inclusive cross-sections or ad-hoc dressings, but are fundamental signals that the standard particle definition is incomplete. The correct physical states are those that transform unitarily under the infinite-dimensional asymptotic symmetry groups, ensuring the conservation of supermomentum and rendering the S-matrix finite.