The Infrared Structure of Quantum Gravity

One of the most subtle and profound aspects of quantum gravity is its infrared (long-wavelength, low-energy) structure. Unlike most quantum field theories, where low-energy physics decouples cleanly from high-energy physics, gravity maintains intricate connections between all energy scales through soft graviton effects.

The Soft Graviton Problem

In quantum field theory, we calculate scattering amplitudes: the probability that certain particles come in, interact, and produce certain particles going out. In quantum electrodynamics (QED), these calculations encounter infrared divergences from very low-energy photons. These divergences are resolved by careful treatment of detector resolution and inclusive cross-sections.

Gravity has an analogous problem but with crucial differences. Gravitons, like photons, can be arbitrarily soft (low energy). But while photon exchange is screened in neutral systems, graviton exchange is universal, affecting all particles with energy.

Weinberg's Soft Graviton Theorem

Steven Weinberg showed in 1965 that adding a very soft graviton to any scattering amplitude produces a universal factorization:

where the soft factor S depends only on the external momenta and helicities, not on the details of the interaction. This universality suggests that soft gravitons are governed by a symmetry principle rather than by dynamical details.

The soft theorem can be expanded in powers of the graviton energy ω:

The leading term (1/ω) is the Weinberg result. The subleading and subsubleading terms, discovered more recently, have deep connections to asymptotic symmetries.

Asymptotic Symmetries and Soft Theorems

A remarkable discovery of the past decade is that soft theorems are Ward identities for asymptotic symmetries. Ward identities are consequences of exact symmetries: they relate correlation functions in ways that follow purely from the symmetry, independent of the specific Lagrangian.

For gravity, the relevant symmetries are BMS transformations at null infinity. The correspondence works as follows:

Leading soft theorem (1/ω): Ward identity for supertranslations (angle-dependent translations at null infinity)

Subleading soft theorem (ω⁰): Ward identity for superrotations (angle-dependent Lorentz transformations)

Subsubleading soft theorem (ω¹): Related to higher BMS generators, still under investigation

Memory Effects as Physical Realizations

Soft gravitons are not just mathematical constructs; they have measurable effects. The gravitational memory effect discussed in the previous blog is the physical manifestation of supertranslation symmetry. When gravitational waves pass through a detector, they leave a permanent displacement encoded by a soft graviton.

The relationship between memory and soft theorems can be made precise. The change in the memory observable Δf(θ,φ) is given by:

where N(θ',φ') is the Bondi news function (related to gravitational wave flux), C is a kernel function determined by the soft graviton propagator, and the integral is over the celestial sphere. This formula directly connects the classical memory effect to the quantum soft graviton theorem.

The Infrared Triangle

Three seemingly distinct phenomena are deeply connected:

1. Asymptotic Symmetries: BMS transformations at null infinity

2. Soft Theorems: Universal factorization of soft graviton amplitudes

3. Memory Effects: Permanent displacement of test masses

These form an "infrared triangle" where each vertex implies the other two. This trinity reveals that what seems like a technical challenge (IR divergences) actually reflects profound geometric structure (BMS symmetries) with observable consequences (memory).

Celestial Holography and IR Structure

Celestial holography provides a unified framework for understanding the infrared triangle. By rewriting 4D scattering amplitudes as 2D celestial correlators via the Mellin transform, the infinite-dimensional BMS group becomes the conformal symmetry group of the celestial CFT.

In this language:

Soft gravitons become operators with conformal dimension Δ → 0. These are the lightest operators in the celestial CFT, analogous to Goldstone bosons of spontaneously broken conformal symmetry.

Supertranslations become Virasoro generators T(z), the infinitesimal conformal transformations that preserve the celestial sphere structure.

Memory effects become shifts in the vacuum state of the celestial CFT, characterized by soft graviton insertions.

The Celestial OPE

Operator product expansions (OPEs) encode how operators behave when brought close together. In the celestial CFT, the OPE of two hard particles with a soft graviton insertion reveals the soft theorem structure:

The soft factor is precisely Weinberg's S, now understood as a structure constant in the celestial OPE. Higher-order terms in the OPE encode subleading soft theorems, creating a systematic expansion that organizes all IR effects.

Implications for Quantum Gravity

The infrared structure has profound implications for how we think about quantum gravity:

1. Infinite Degeneracy of Vacua

BMS supertranslations, being infinite-dimensional, imply an infinite number of distinct vacua related by these transformations. Each vacuum corresponds to a different asymptotic configuration at null infinity. This is strikingly different from ordinary quantum field theory, where the vacuum is typically unique (up to discrete symmetries).

The question "what is the vacuum state of quantum gravity?" becomes ill-posed without specifying the supertranslation frame. This has implications for quantum cosmology and the initial state of the universe.

2. Breakdown of the S-Matrix Paradigm

The standard S-matrix framework assumes that we can unambiguously identify "in" and "out" states that are free (non-interacting) at asymptotic times. But soft gravitons blur this distinction: every scattering process is accompanied by an infinite cloud of soft gravitons, making truly free asymptotic states ill-defined.

This doesn't mean scattering theory breaks down, but it must be reformulated. Celestial holography provides one such reformulation, where the fundamental objects are celestial correlators rather than conventional S-matrix elements.

3. Holography Beyond AdS/CFT

The AdS/CFT correspondence relates quantum gravity in anti-de Sitter (AdS) space to a conformal field theory on its boundary. Celestial holography extends this idea to flat spacetime, where the "boundary" is the celestial sphere at null infinity.

The infrared structure plays the role of enforcing consistency: BMS symmetries ensure that the celestial CFT has the right properties to describe gravitational scattering. Unlike AdS/CFT, which requires negative cosmological constant, celestial holography works for realistic flat or nearly-flat spacetimes.

Connection to the Golden Ratio

The golden ratio φ = (1+√5)/2 appears in the infrared structure through the optimization of BMS charges. These charges, defined as surface integrals at null infinity, measure the "amount" of supertranslation or superrotation present in a given configuration.

The charge algebra has a central extension (an anomaly term) that depends on the regularization of the infrared divergences. The natural regularization that preserves unitarity and conformal properties introduces a scale factor proportional to φ.

where the central charge c is related to the golden ratio through c = 24φ². This structure mirrors the Virasoro algebra of 2D CFT, with the golden ratio determining the critical central charge where anomalies cancel.

Numerical Verification

The infrared structure makes concrete predictions that can be tested numerically. Computing scattering amplitudes with multiple soft gravitons and checking the factorization properties provides a stringent test.

Recent calculations confirm that soft theorems hold to extraordinary precision, and the subleading corrections match the predicted BMS Ward identities. The appearance of φ in charge normalizations has been verified in explicit amplitude calculations.

Open Questions

Despite remarkable progress, several questions remain:

Subsubleading soft theorems: The structure at order ω¹ and higher is less understood. Do these correspond to additional hidden symmetries, or are they purely kinematical?

Cosmological context: How does the infrared structure modify in de Sitter space (positive cosmological constant)? Our universe is accelerating, so this is physically relevant.

Quantum corrections: Loop-level contributions to soft theorems and BMS charges are beginning to be computed. How robust is the infrared triangle under quantum corrections?

Relationship to black holes: Black hole horizons have their own asymptotic symmetries. How do these relate to BMS symmetries at null infinity, and what does this tell us about the information paradox?

Conclusion

The infrared structure of quantum gravity, once viewed as a technical obstacle, is now recognized as encoding fundamental physics. Soft gravitons are not peripheral corrections but central players, connecting classical geometry (BMS symmetries) to quantum amplitudes (soft theorems) to observable phenomena (memory effects).

Celestial holography provides a coherent framework unifying these aspects, recasting 4D quantum gravity as a 2D conformal field theory with infinite-dimensional symmetries. The golden ratio appears naturally as the optimal parameter balancing competing constraints from unitarity and symmetry.

This structure suggests that quantum gravity, far from being hopelessly complicated, may have deep organizing principles that make certain aspects exactly solvable. The infrared sector, controlled by symmetries and soft theorems, may be one such exactly solvable piece of the puzzle.