Finite-Volume Spectra and Amplitudes
A finite box turns scattering states into discrete energy levels. Their volume dependence can constrain infinite-volume amplitudes, but the conversion depends on particle number, open channels, interaction range, momentum, and inserted currents. This chapter develops those distinctions before interpreting a resonance.
Enter this chapter
Section titled “Enter this chapter”Begin with Finite Volume as a Controlled Deformation to identify the infrared deformation and the physical limits it requires.
Topics
Section titled “Topics”- Finite Volume as a Controlled Deformation — Defines the geometry, state, and interaction assumptions.
- Exponential Finite-Volume Effects — Derives wrapping effects for massive isolated states.
- Massless Fields, Long-Range Forces, and Finite-Volume QED — Explains why massless forces produce different corrections.
- Spectra from Euclidean Correlation Matrices — Resolves finite-volume levels through operator overlaps.
- Elastic Two-Body Quantization Conditions — Relates elastic energies to phase shifts.
- Moving Frames, Cubic Irreducible Representations, and Partial-Wave Mixing — Tracks reduced rotational symmetry and wave mixing.
- Scattering Amplitudes and Resonance Poles from Finite-Volume Spectra — Infers an amplitude before continuing to resonance poles.
- Coupled-Channel Quantization and Inference — Generalizes quantization to communicating channels.
- Finite-Volume Matrix Elements and 1→2 Transitions — Converts finite-volume current matrix elements.
- Three-Body Quantization and Decay Amplitudes — Separates three-body kernels from physical amplitudes.
How the ideas fit together
Section titled “How the ideas fit together”A finite-volume level is real spectral data of the box; a resonance is a complex pole of an infinite-volume amplitude. The inference first uses a valid quantization condition to constrain the amplitude on real energies, then analytically continues a tested parameterization. An avoided crossing alone does not determine a pole. Exponentially small wrapping corrections require a massive, short-range setting; massless exchange and additional on-shell channels change the derivation. Briceño, Dudek, and Young 2018, §§ II–III.
Where to go next
Section titled “Where to go next”Continue to Finite Density and Sign Problems to derive how chemical potential modifies the Euclidean weight.