Finite-Volume Spectra and Amplitudes

Choose a finite-volume branch from the physics before choosing a formula. First ask whether the interaction is short ranged and the relevant state is isolated; then count open channels and particles, fix the total momentum and lattice irrep, decide whether a current is inserted, and name any analytic continuation. Massive stable-state wrapping, elastic two-body levels, coupled channels, one-to-two currents, three-particle levels, and finite-volume QED are distinct problems. This chapter develops the durable assumptions and checks that connect discrete spectra or matrix elements to infinite-volume quantities without letting one branch borrow another branch’s validity.

Enter this chapter

The common starting point is a Euclidean correlation function in a finite spatial box. Its real, discrete energies are observables of that regulated system. An infinite-volume amplitude is not read from an energy directly: a quantization or residue relation removes the universal on-shell volume dependence under stated range, channel, symmetry, and particle-number hypotheses. A resonance pole requires one more operation—analytic continuation of a fitted real-axis amplitude to a named sheet.

This chapter develops that finite-to-infinite-volume chain. General scattering normalization and analytic structure belong to Scattering, physical hadron and nuclear interpretation belongs to Gauge Theories and the Standard Model, thermal compactification belongs to Thermal and Nonequilibrium QFT, and theorem-level finite-volume analysis belongs to Mathematical QFT. Mutable coupled-channel, current, and three-body status belongs to the Lattice and Hamiltonian Field Theory Research area.

Check your preparation

No page is a hard prerequisite merely because it is an overview. Use these observable checks to choose an entry.

Can you do this?	If ready	If unsure	Repair
Separate lattice spacing, interaction range, correlation length, bound-state size, $L$ , and $T$	Enter the finite-volume hierarchy and exponential pages	Write the six dimensionless ratios and state which limit changes each	Lattice Geometry, Boundaries, and Anisotropy
Extract several correlated energies from one symmetry sector	Enter the correlation-matrix page	Diagonalize a two-state exact correlator and identify the omitted-state gap	Operator Bases, Effective Masses, and Excited-State Control
Translate between $E,\mathbf P$ and $E^,k^$ , and state elastic unitarity	Enter two-body quantization	Verify $E^{*2}=E^2-\mathbf P^2$ and $\operatorname{Im}t^{-1}=-\rho$	Relativistic Scattering Kinematics and Partial-Wave Unitarity
Distinguish a finite-volume irrep from continuum $J^P$	Enter moving frames	Decompose $\ell=2$ into $E^+\oplus T_2^+$ at rest	Representations, Intertwiners, Invariants, and Tensor Decomposition
Name a threshold cut, sheet, and continuation path	Enter pole or coupled-channel inference	Classify a bound-state pole and a second-sheet resonance pole	Resonance Poles, Riemann Sheets, and Unstable States
Keep state normalization and covariance through a current matrix element	Enter one-to-two transitions	Convert a unit finite-volume one-particle state to relativistic normalization	Three-Point Functions, Matrix Elements, and Disconnected Contributions

Choose a route

Goal	Suggested route	Capability at the end
First graduate encounter	Controlled deformation → exponential effects → correlation matrices → elastic quantization	Classify the volume effect and invert a controlled elastic level
Long-range or charged observable	Controlled deformation → massless fields and finite-volume QED	Specify zero modes and derive a prescription-specific power correction
Elastic resonance	Correlation matrices → elastic quantization → moving frames → amplitudes and poles	Fit correlated levels and continue a unitary amplitude to a named sheet
Coupled threshold problem	Elastic and moving-frame route → amplitudes and poles → coupled channels	Write the channel determinant and diagnose identifiability
Current-induced transition	Elastic quantization → finite-volume matrix elements	Apply the density or residue factor with complete normalization and covariance
Three particles	Coupled-channel discipline → three-body quantization	Separate $\mathcal K_2$ , $\mathcal K_{\mathrm{df},3}$ , and the physical amplitude
Re-enter an evolving method	Stable branch background → Lattice and Hamiltonian Field Theory Research area	Read dated scope, validation, and open-interface status

Arrows in this table are recommended reading order. The hard prerequisites are those stated on each leaf.

The finite-volume decision structure

The chapter can be reconstructed from five typed relations:

Requires: a correlation matrix and its covariance produce controlled finite-volume levels before any amplitude inference.
Classifies: range, mass gap, zero modes, channel count, current insertion, particle number, frame, and irrep choose the applicable branch.
Maps: a branch-specific quantization condition or residue maps discrete data to a real-axis infinite-volume quantity.
Validates: forward spectrum closure, held-out volumes or frames, partial-wave and parametrization alternatives, and analytic benchmarks test that map.
Continues: only a fitted analytic amplitude, with a named path and sheet, may be continued to a pole.

The central category error is to collapse these relations. A good correlator fit does not validate an elastic hypothesis; an elastic spectrum fit does not validate a three-body kernel; and a precise real-axis amplitude does not by itself make a distant pole precise. Lüscher’s foundational analyses separate massive stable-particle exponentials from two-particle on-shell power laws (Lüscher 1986, Part I, pp. 177–206; Part II, pp. 153–188), while the torus quantization condition makes the elastic map explicit (Lüscher 1991, pp. 531–578). A modern lattice-QCD synthesis of spectra, amplitudes, and resonance inference is Briceño, Dudek, and Young 2018, §§ II–IV.

Guide to the ten pages

Finite Volume as a Controlled Deformation asks when a spatial box is a controlled infrared change. It produces the hierarchy and the chapter’s hypothesis–branch–uncertainty table; continue according to the classified branch.
Exponential Finite-Volume Effects derives massive image sums and shows why stable particles and compact bound states often have exponentially small shifts. It stops at massless, threshold, and on-shell failures.
Massless Fields, Long-Range Forces, and Finite-Volume QED explains Gauss’s law, zero-mode prescriptions, power corrections, and matched comparisons. It is the correct branch when exponential locality fails.
Spectra from Euclidean Correlation Matrices constructs a GEVP in a declared finite-volume sector and challenges missing states, basis conditioning, and covariance. Its output is a correlated spectrum, not an amplitude.
Elastic Two-Body Quantization Conditions matches a short-range one-channel spectrum to a phase shift, fixes the branch and scattering-length sign, and requires forward spectrum closure.
Moving Frames, Cubic Irreducible Representations, and Partial-Wave Mixing subduces continuum angular momentum into finite-volume irreps and tests partial-wave truncation. It prevents a lattice level from receiving an unsupported unique $J^P$ label.
Scattering Amplitudes and Resonance Poles from Finite-Volume Spectra fits correlated elastic levels with alternative unitary forms and continues them to named sheets. It separates real-axis evidence from pole-model dependence.
Coupled-Channel Quantization and Inference adds channel matrices, threshold sheets, and sensitivity analysis. It shows which amplitude combinations sparse levels cannot identify.
Finite-Volume Matrix Elements and 1→2 Transitions converts a current matrix element with state norms, density or residue derivatives, renormalization, and joint covariance explicit.
Three-Body Quantization and Decay Amplitudes declares one spectator formalism, distinguishes $\mathcal K_{\mathrm{df},3}$ from the physical amplitude, and requires scheme cancellation and independent benchmarks.

The first six pages form the durable finite-volume core. The final four use stable conceptual contracts but send comparative validity, implementation coverage, and present consensus to Research rather than freezing those claims here.

Conventions that must survive every handoff

The site-wide metric, Fourier transform, and analytic conventions remain in force. This chapter adds recurring local data:

Quantity	Chapter convention and invariant check
Spatial box	Periodic cube unless stated; $\mathbf P=2\pi\mathbf d/L$ ; reproduce the noninteracting momentum spectrum
Finite states	Unit normalized; restore $\sqrt{2EL^3}$ for each relativistically normalized stable one-particle state
Two-body amplitude	State $\rho$ , partial-wave normalization, and scattering-length sign; check $S^\dagger S=1$ on the elastic real axis
Lattice symmetry	State momentum star, little group, $\Lambda,\mu$ , multiplicities, and partial waves; check projector idempotence and free-level multiplicities
Quantization branch	Track the integer root continuously; reproduce the observed spectrum by solving forward
Coupled sheets	Record one momentum-branch sign per channel and the crossed-cut path; pole positions must survive channel rephasing
Current map	State current projection and renormalization plus the derivative residue; recover the noninteracting density-of-states limit
Three-body scheme	State spectator cutoff and subtraction; scheme changes may move $\mathcal K_{\mathrm{df},3}$ but not matched spectra or $\mathcal M_3$
Limit order	Separate $T\to\infty$ , $a\to0$ along constant physics, $L\to\infty$ , massless, and threshold limits

One notation deserves special care: a mass gap controls the decay of bulk correlations, an interaction range controls when the exterior wave equation is free, and a binding momentum controls the size of a shallow bound state. They can differ by orders of magnitude.

Synthesis: what each finite-volume signal means

An isolated stable level in a massive theory changes through virtual images, so its leading correction is exponential with a derived exponent and algebraic prefactor. A two-particle scattering level changes by powers of $1/L$ because two internal lines can go on shell; the power dependence is useful signal, not contamination. A massless propagator also produces powers, but through long-range infrared modes and a prescription-dependent zero-mode structure; it must not be inserted into a short-range Lüscher formula.

Rotational symmetry reduction changes which amplitude entries one level sees. Channel coupling changes the dimension of the amplitude and the threshold sheets. A current changes the problem from a determinant zero to a determinant residue. Three particles add spectator exchange and scheme-dependent divergence-free intermediates. These are logical changes in the map, not merely larger matrices.

The chapter’s stopping boundary is equally important. It does not infer a physical resonance from a bump, equate a fitted kernel with an observable, claim a charged exclusive S matrix in a periodic box, or summarize current method consensus without a dated Research assessment.

Review the chapter

Each prompt gives a concise success criterion and a repair route.

Retrieval and explanation

Prompt. State the analytic reason massive stable-state effects are exponential while elastic two-particle effects are power law.

Answer criterion

Your answer must mention Poisson images, contour displacement to a massive singularity, and the on-shell pinch that prevents that displacement for a two-particle state. If either range or threshold hypotheses are missing, repair with Finite Volume as a Controlled Deformation.

Derivation and representation change

Prompt. Starting from $\delta(k)+\phi(kL/2\pi)=n\pi$ , derive the level density used in a one-to-two matrix-element factor and explain what changes in a coupled irrep.

Answer criterion

Differentiate to obtain $\mathrm dn/\mathrm dE=\pi^{-1}\partial_E(\delta+\phi)$ . In a coupled irrep, replace the scalar derivative by the residue of the full quantization matrix in channel–partial-wave space. Repair with Finite-Volume Matrix Elements and 1→2 Transitions.

Comparison and failure diagnosis

Prompt. A charged mass and an elastic two-particle level both vary as a power of $1/L$ . Give a test that distinguishes the branches.

Answer criterion

Identify the massless zero-mode prescription and its point-particle coefficient for the charged mass; identify an elastic on-shell channel, irrep, and quantization root for the two-particle level. Boundary-condition or charge dependence tests the first, while forward phase-shift closure across frames tests the second. A power law alone does not choose the branch.

Transfer and synthesis

Prompt. You receive three correlated levels near a second-channel threshold, one current matrix element, and a desired resonance pole. Write the minimum claim chain.

Answer criterion

The chain must include sector and basis stability; a coupled-channel determinant with partial waves and threshold branches; alternative unitary fits and held-out-level closure; a matrix residue plus current renormalization and joint covariance; and a separate named-sheet continuation. It must state which amplitude combinations are unidentified. Current status and comparison claims then exit to Research.

You have met the chapter’s exit test when you can build the full hypothesis-to-amplitude table for a new calculation and refuse an elastic, short-range, or two-body formula when its assumptions fail.

Continue from here

Continue within the volume to Complete Lattice Error Budgets when the finite-volume stage must be combined with continuum, scale, renormalization, and sampling uncertainties.
Use Scattering for amplitude normalization, unitarity, analyticity, and resonance-sheet grammar.
Use Gauge Theories and the Standard Model for physical hadron, nuclear, QED, and decay interpretation.
Consult the Lattice and Hamiltonian Field Theory Research area for dated coupled-channel, current, and three-body status.

References

Briceño, Raúl A., Jozef J. Dudek, and Ross D. Young. “Scattering Processes and Resonances from Lattice QCD.” Reviews of Modern Physics 90 (2018): 025001. DOI. Open PDF.
Lüscher, Martin. “Two-Particle States on a Torus and Their Relation to the Scattering Matrix.” Nuclear Physics B 354 (1991): 531–578. DOI.
Lüscher, Martin. “Volume Dependence of the Energy Spectrum in Massive Quantum Field Theories. I. Stable Particle States.” Communications in Mathematical Physics 104 (1986): 177–206. DOI.
Lüscher, Martin. “Volume Dependence of the Energy Spectrum in Massive Quantum Field Theories. II. Scattering States.” Communications in Mathematical Physics 105 (1986): 153–188. DOI.