Computational field theory onboarding

Use this pathway when the research decision is not merely how to run a code, but which finite calculation can support a stated QFT conclusion. Leave with one result that another person can rerun, an uncertainty analysis that follows the actual dependencies, and a scientific cross-check whose main assumptions are not identical to those of the primary calculation.

A successful execution establishes that a specified algorithm produced an output. It does not by itself establish that the finite problem represents the intended theory, that a continuum or truncation limit is controlled, that a quoted uncertainty has the claimed meaning, or that the result describes the physical regime of interest.

Define the observable before the algorithm

Take these three actions before selecting a package or computing allocation.

1. Write the claim as a quantity and a domain. Name the action or Hamiltonian, state and boundary conditions, renormalization conditions, observable, units, kinematics, and target precision. Prefer a dimensionless ratio or state the reference scale that supplies dimensions.
2. Choose the finite representation and its limits. List every lattice spacing, volume, basis cutoff, bond dimension, closure, perturbative order, grid, precision, and sampling limit that separates the calculation from the target QFT. State which limits commute only if you have evidence that they do.
3. Fix tests and provenance before the production result. Run an analytic or exactly solvable fixture, predict the leading refinement behavior, choose a genuinely independent check, and record inputs, environment, random streams, tolerances, and stopping rules.

Keep four conclusions distinct:

Conclusion	Evidence it needs
The algorithm executed	Inputs, environment, command, termination state, and raw output
The finite calculation is numerically verified	Analytic fixtures, residuals, precision and refinement scans, invariants, and defect-sensitive tests
A limiting prediction is controlled	A justified continuum, volume, basis, closure, or perturbative sequence with correlated inference and omitted terms
The scientific claim is supported	All preceding evidence plus the correct observable map, physical inputs, domain, and a cross-check with materially different failure modes

Agreement with one published number can be useful, but it may reproduce the same convention, code lineage, tuning input, or modeling error. Independence is a property to document, not a label to assume.

Check preparation by capability

Enter at the first operation below that you cannot perform for your chosen quantity.

• Formulate the target. Use functional integrals and correlators if you cannot derive the measure, state prescription, source normalization, and correlator that the computation estimates.
• Control regulators and scales. Use loops and regularization, renormalization and the renormalization group, and effective field theory and matching when bare parameters, scheme dependence, scale setting, or omitted operators are not yet separated.
• Work with finite linear objects. Use the mathematics diagnostic when spectra, generalized eigenproblems, tensor contractions, conditioning, or Fourier modes block the calculation.
• Analyze dependent data. Use the statistics diagnostic when an estimator, covariance, autocorrelation time, likelihood, or uncertainty interval is unclear.
• Test and preserve a computation. Use the computational-evidence diagnostic or the focused numerical and reproducibility review when convergence, roundoff, defect-sensitive tests, clean reruns, or provenance are the missing capability.

Do not repeat all of Core QFT because one numerical step is unfamiliar. Repair the operation that blocks the stated observable, then return to the same finite problem.

Running example: the free scalar as a computational fixture

Consider the Euclidean free scalar in $d$ dimensions,

$$S_E = \frac12 \int\mathrm d^dx\, \left[ (\partial_\mu\phi)^2+m_0^2\phi^2 \right].$$

In units with $\hbar=c=1$,

$$[x]=-1, \qquad [m_0]=1, \qquad [\phi]=\frac{d-2}{2},$$

so the action is dimensionless. On an isotropic periodic lattice with spacing $a$ and extents $L_\mu=N_\mu a$, take

$$S_a = \frac{a^d}{2} \sum_x \left[ \sum_\mu \left( \frac{\phi_{x+a\hat\mu}-\phi_x}{a} \right)^2 +m_0^2\phi_x^2 \right].$$

The factors of $a$ are not optional bookkeeping: they make every term dimensionless before anyone sets $a=1$. The exact free lattice propagator is

$$G_a(p) = \frac{1}{\widehat{p}^{2}+m_0^2}, \qquad \widehat{p}^{2} = \frac{4}{a^2} \sum_\mu \sin^2\!\left(\frac{ap_\mu}{2}\right).$$

At fixed physical momentum,

$$\widehat{p}^{2} = p^2 -\frac{a^2}{12}\sum_\mu p_\mu^4 +O(a^4p^6).$$

Thus this action has a leading free cutoff effect of order $(aQ)^2$ for a characteristic momentum $Q$. Interacting cutoff effects require the complete operator content allowed by the lattice symmetries, not just this tree-level Taylor term; that is the role of Symanzik’s effective-action expansion Symanzik 1983, Part I, §§2–3, pp. 187–204.

The zero-spatial-momentum pole gives an exact second check. Continue the Euclidean temporal momentum to $p_0=iE_a$. The pole condition is

$$\frac{4}{a^2} \sinh^2\!\left(\frac{aE_a}{2}\right) =m_0^2,$$

so

$$E_a = \frac{2}{a} \operatorname{arsinh}\!\left(\frac{am_0}{2}\right), \qquad \frac{E_a}{m_0} = 1-\frac{(am_0)^2}{24}+O\!\left((am_0)^4\right).$$

The ratio $E_a/m_0$ is dimensionless, has continuum target $1$, and predicts both the sign and order of the leading artifact. It is therefore more useful as a fixture than an unexplained agreement at one lattice spacing.

For a periodic temporal extent $T$, a zero-momentum correlator has the single-state form

$$C_{\mathrm{model}}(t) = A \left[ e^{-E_at}+e^{-E_a(T-t)} \right]$$

when one state is sufficient. Monte Carlo estimates of $C(t)$ at different times are correlated, and successive configurations can be autocorrelated. After forming $B$ approximately independent block estimates $C_b(t)$, an estimate of the covariance of the ensemble mean is

$$\Sigma_{tt'} = \frac{1}{B(B-1)} \sum_{b=1}^{B} \left[C_b(t)-\overline C(t)\right] \left[C_b(t')-\overline C(t')\right].$$

Fit with

$$\begin{aligned} \Delta C(t) &= C(t)-C_{\mathrm{model}}(t), \\ \chi^2 &= \sum_{t,t'} \Delta C(t) \left(\Sigma^{-1}\right)_{tt'} \Delta C(t'). \end{aligned}$$

when the estimated covariance is sufficiently stable to invert. If it is not, reduce the fit dimension or use a declared regularization whose effect is propagated and tested; silently discarding the off-diagonal entries changes the inference. Repeat after changing the block length, fit window, covariance treatment, volume, and lattice spacing. A stable central value with an unstable covariance estimate is not a stable result. Autocorrelation-aware estimation and window choices are developed in Sokal 1997, pp. 131–192.

For this fixture, fitting $E_a/m_0$ against $(am_0)^2$ and evaluating the fit at $a=0$ is a continuum extrapolation. It tests the implementation and the inference procedure against a known answer. It does not validate an interacting action, a difficult sampler, or a real-time observable. Those need their own fixtures and limits.

Choose the method by the finite object

The four tracks below can support the same physical question, but they do not share one error model. Combining methods strengthens a conclusion only when their shared actions, parameters, software, and calibration inputs are made explicit.

Lattice field theory

Choose this track when the finite object is a Euclidean lattice measure, transfer matrix, or gauge ensemble and the target is a continuum observable. Begin with Lattice Regulators and Continuum Targets, then use Sampling Algorithms for Lattice Fields and Statistical Inference and Error Budgets.

Record the lattice action and exact symmetries, boundary conditions, bare parameters and tuning observables, scale setting, operator normalization, update algorithm, equilibration and autocorrelation evidence, and the joint sequence in $a$, $L$, statistics, and solver tolerances. The continuum fit belongs only to points plausibly inside its predicted asymptotic regime; precise coarse-lattice data do not become continuum data by receiving large weights. Continue to Lattice Observables and Continuum Inference for lines of constant physics, renormalized operators, spectra, and continuum fits.

For the running example, the analytic $G_a(p)$, exact pole mass, positivity of $\widehat{p}^{2}+m_0^2$, and $O(a^2)$ trend are separate tests. A sampler can reproduce all four and still fail on an interacting or topological target, so add a benchmark that exercises the difficult part of the production method.

Tensor networks and Hamiltonian methods

Choose this track when the finite object is a regulated Hamiltonian, projected Hilbert space, matrix-product or projected-entangled-pair state, or contracted Euclidean tensor network. Start with Hamiltonian Lattice Field Theory, Hamiltonian Truncation and Variational Methods, or Tensor Networks for QFT and Many-Body Systems according to the finite object.

Keep the spacetime regulator separate from the additional Hilbert-space, local-state, bond, environment, or contraction truncation. A small eigenpair residual says that a finite matrix was solved accurately; it does not bound the omitted-state correction. A variational upper bound on an energy does not automatically bound a gap, matrix element, or real-time response. Vary volume, lattice spacing, energy cutoff or local dimension, bond dimension, contraction environment, time step, and optimization seed independently where possible. Exact symmetry-sector dimensions, sum rules, transfer-matrix spectra, cross-basis comparisons, and held-out observables provide stronger tests than one energy plateau. The structure and limitations of MPS and PEPS approximations are reviewed in Cirac et al. 2021, §§II–IV, while a field-theory truncation construction is given in Hogervorst, Rychkov, and van Rees 2015, §§2–4.

Continuum and functional numerics

Choose this track when the starting point is a Schwinger–Dyson hierarchy, $n$PI stationarity equation, Bethe–Salpeter problem, functional-RG flow, or another continuum integral or differential equation. Begin with Continuum Functional Equations and Controlled Truncations and finish the chosen calculation with Functional-Method Validation and Error Control.

Separate the exact identity from the finite closure used to solve it. Record the retained and omitted fields, vertices, tensor structures, momentum dependence, regulator and renormalization conditions, nonlinear branch, quadrature and grid, domain boundaries, and solver tolerances. Then vary the closure as well as the numerical grid. A residual of $10^{-12}$ inside one closure is compatible with a percent-level truncation error. Ward identities, known perturbative limits, direct complex-momentum solutions, regulator and projection variation, and observables from a method with different closure assumptions are appropriate checks. The Wetterich equation, for example, is exact before a finite theory-space projection is chosen Wetterich 1993, Eqs. (1)–(7), pp. 90–94.

Perturbative and amplitude automation

Choose this track when the target is a fixed-order amplitude, master integral, matching coefficient, phase-space integral, or infrared-safe observable. Loop Calculations organizes regularization, reduction, analytic structure, and renormalization. Numerical Evaluation and Validation of Loop Integrals and Phase-Space Integration and Monte Carlo Estimators provide the numerical checks.

Automation can generate graphs, perform algebra, reduce integral families, solve differential equations, evaluate special functions, and integrate phase space. It does not decide whether the observable is infrared safe, whether the renormalization and factorization conventions match, or whether an unstable numerical point represents a physical singularity. Check UV and IR poles, Ward identities, crossing and factorization limits, real–virtual or subtraction cancellation, precision stability, independent integral representations, and phase-space normalization. Vary perturbative order, renormalization and factorization scales, subtraction parameters, integration maps, and random streams without treating their spreads as automatically independent probabilities. Adaptive Monte Carlo controls integration variance through its sampling map; its original construction and error logic are given in Lepage 1978, pp. 192–203.

If the main output is a collider observable rather than a computational-method study, continue through Scattering calculations for phenomenology.

Distinguish interpolation, extrapolation, and continuation

These operations answer different questions and need different validation.

Operation	Example	Main evidence
Interpolation	Estimate $C(t)$ between sampled times inside the same Euclidean domain	Approximation order, held-out points, smoothness assumptions, and local refinement
Extrapolation	Infer $E_0$ from $E(a)=E_0+c_2a^2+\cdots$ as $a\to0$	A theory-based asymptotic form, several points in its regime, covariance, fit-window and model stability, and omitted terms
Analytic continuation or inverse reconstruction	Infer a pole or spectral density from Euclidean correlator data	Analytic hypotheses, a regularization or prior, synthetic recovery tests, resolution limits, alternative representations, and stability to uncertain data

For a common zero-temperature spectral representation,

$$G_E(\tau) = \int_0^\infty \mathrm d\omega\, \rho(\omega)e^{-\omega\tau}.$$

Evaluating a fitted $G_E$ at another Euclidean time is interpolation or extrapolation in $\tau$. Inferring $\rho(\omega)$ or a Lorentzian response is an inverse analytic problem. With finitely many uncertain Euclidean samples, many spectral functions can agree within the data errors, so a smooth curve through every point does not establish a unique spectrum. The ill conditioning and the role of prior information are analyzed in Jarrell and Gubernatis 1996, §§II–IV, pp. 133–195. Use Complex Momentum, Spectral, and Real-Time Information when the continuation is part of a functional calculation.

Route bounds, inverse problems, and certification separately

Some computational questions are organized more naturally by the kind of inference than by the four implementation tracks:

• Numerical Bootstrap is the route for positivity constraints, finite derivative or spin truncations, optimization certificates, and boundary stability.
• Response, Transport, and Inference is the route for Kubo relations, analytic continuation, inverse problems, and transport extraction.
• Quantum-Matter Probes, Inference, and Evidence is the route for model-to-probe maps and comparisons among competing phase interpretations.
• Inference, Certification, and Evidence is the route for reconstructed states or channels, entropic quantities, and certification under incomplete data.

A solver certificate, a continuum extrapolation, a posterior sensitivity, and an analytic-continuation test address different possible failures. Report the one your claim actually uses.

A compact computational work product

Use this plain-text template so the result can be assessed without opening the code repository:

scientific question and strongest supported claim:
theory, state, boundary conditions, and renormalization inputs:
observable, normalization, units, and reference scale:
finite representation and all cutoff or truncation parameters:
algorithm, implementation revision, dependencies, and hardware-sensitive choices:
inputs, raw-output locations, checksums, random generator, streams, and seed policy:
estimators, covariance, autocorrelation, and fit or inversion assumptions:
refinement sequences, expected orders, stopping rules, and first omitted terms:
analytic fixture and defect-sensitive tests:
independent cross-check and shared assumptions:
clean-rerun command and observed result:
domain in which the conclusion is supported:

Store the data needed to recompute the reported estimator and covariance, not only a plot or final fit table. A checksum identifies bytes; the surrounding record explains their physical meaning.

Check your understanding

1. Derive the scalar cutoff effect

Starting from the one-dimensional lattice momentum $\widehat{p}^{2}=4\sin^2(ap/2)/a^2$, derive its small-$a$ expansion. Then derive $E_a/m_0$ through $O(a^2)$ and check that every correction is dimensionless.

Solution

Using $\sin z=z-z^3/6+O(z^5)$ gives

$$\sin^2 z = z^2-\frac{z^4}{3}+O(z^6).$$

With $z=ap/2$,

$$\widehat{p}^{2} = \frac{4}{a^2} \left[ \frac{a^2p^2}{4} -\frac{a^4p^4}{48} +O(a^6p^6) \right] = p^2-\frac{a^2p^4}{12}+O(a^4p^6).$$

For the pole mass, use $\operatorname{arsinh}z=z-z^3/6+O(z^5)$ with $z=am_0/2$:

$$\begin{aligned} E_a &= \frac{2}{a} \left[ \frac{am_0}{2} -\frac{(am_0)^3}{48} +O((am_0)^5) \right] \\ &= m_0-\frac{a^2m_0^3}{24}+O(a^4m_0^5). \end{aligned}$$

Therefore

$$\frac{E_a}{m_0} = 1-\frac{(am_0)^2}{24}+O((am_0)^4).$$

Because $[a]=-1$ and $[m_0]=1$, every power of $am_0$ is dimensionless. The sign and coefficient are exact for this free nearest-neighbor fixture; they are not a universal interacting continuum-fit formula.

2. Quantify a correlated uncertainty

Two estimates $Y_1,Y_2$ have the same expectation $A$, variance $\sigma^2$, and correlation $\rho$. Find the variance of their average. For $\rho=0.8$, compare the correct standard error with the value obtained by treating them as independent.

Solution

The covariance matrix is

$$\Sigma = \sigma^2 \begin{pmatrix} 1 & \rho\\ \rho & 1 \end{pmatrix}.$$

For $\overline Y=(Y_1+Y_2)/2$,

$$\begin{aligned} \operatorname{Var}(\overline Y) &= \frac14 \left[ \operatorname{Var}(Y_1) +\operatorname{Var}(Y_2) +2\operatorname{Cov}(Y_1,Y_2) \right] \\ &= \frac{\sigma^2}{2}(1+\rho). \end{aligned}$$

At $\rho=0.8$, the correct standard error is

$$\operatorname{SE}(\overline Y) = \sigma\sqrt{0.9} \approx0.949\sigma.$$

The independent formula gives $\sigma/\sqrt2\approx0.707\sigma$, smaller by a factor $\sqrt{1.8}\approx1.34$. This calculation addresses the sampling covariance only. It says nothing about lattice spacing, volume, fit-model, calibration, or physical-model error.

3. Classify three numerical inferences

Classify each operation and name one validation that belongs specifically to it:

1. evaluating a spline for $C(\tau)$ halfway between two measured Euclidean times;
2. fitting $R(a)=R_0+c_2a^2+c_4a^4$ and evaluating it at $a=0$; and
3. reconstructing $\rho(\omega)$ from finitely many noisy values of $G_E(\tau)$.

Solution

The first is interpolation inside the sampled Euclidean domain. Held-out times or grid refinement can test the interpolation error under the stated smoothness assumptions.

The second is extrapolation to a boundary outside the finite-$a$ data. It needs a theoretically justified cutoff expansion, a demonstrated asymptotic window, the full covariance, and stability when fit ranges and omitted terms change. A good polynomial fit alone does not prove that the simulated spacings lie in that regime.

The third is an inverse analytic-continuation problem. It needs analytic and positivity assumptions where applicable, regularization or a prior, synthetic-spectrum recovery, resolution tests, and stability under data and method changes. Exact interpolation of the input points is not evidence that the reconstructed spectrum is unique.

4. State what a precise solver result proves

A functional equation is solved with residual $10^{-12}$. Doubling the grid changes the observable by $10^{-5}$, but adding the next allowed vertex tensor changes it by $3\%$. An independent perturbative limit agrees at weak coupling. What precision and conclusion are justified?

Solution

The residual and grid scan show that the chosen finite closure is solved much more accurately than one part in $10^5$ over the tested domain. They do not control the closure. The observed $3\%$ shift is currently the dominant scientific variation, so twelve digits—or even five—are not justified for the full result.

The weak-coupling comparison validates the equations, conventions, and branch in that limit. Away from it, the supported statement is that the observable is known within the declared closure with a closure sensitivity of order a few percent, subject to the other listed errors. A nested closure sequence or a cross-method benchmark is needed before interpreting the shift as a reliable error estimate or claiming higher precision.

Finish with one reproducible result

You are ready to leave this pathway when one result includes:

• a dimensionally consistent observable and a precise scientific domain;
• the finite theory, all independent limit parameters, and the order in which limits are taken;
• numerical refinement and precision checks that match a derived error model;
• correlation-aware statistical inference and separately named systematic or truncation effects;
• an analytic fixture plus at least one test shown to detect a plausible defect;
• a clean rerun with source, dependency, input, random-stream, raw-output, and analysis provenance; and
• an independent scientific cross-check together with its shared assumptions and the strongest conclusion both calculations support.

If the target is a Euclidean continuum observable, open Lattice Regulators and Continuum Targets. For spectra or real-time evolution, choose Hamiltonian Truncation and Variational Methods or Tensor Networks for QFT and Many-Body Systems. For a closed hierarchy or flow, open Continuum Functional Equations and Controlled Truncations. For automated amplitudes and phase space, open Loop Calculations.

References

• J. Ignacio Cirac, David Pérez-García, Norbert Schuch, and Frank Verstraete, “Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, Theorems,” Reviews of Modern Physics 93 (2021), 045003, doi:10.1103/RevModPhys.93.045003.
• Matthijs Hogervorst, Slava Rychkov, and Balt C. van Rees, “Truncated Conformal Space Approach in $d$ Dimensions: A Cheap Alternative to Lattice Field Theory?” Physical Review D 91 (2015), 025005, doi:10.1103/PhysRevD.91.025005.
• Mark Jarrell and J. E. Gubernatis, “Bayesian Inference and the Analytic Continuation of Imaginary-Time Quantum Monte Carlo Data,” Physics Reports 269 (1996), 133–195, doi:10.1016/0370-1573(95)00074-7.
• G. Peter Lepage, “A New Algorithm for Adaptive Multidimensional Integration,” Journal of Computational Physics 27 (1978), 192–203, doi:10.1016/0021-9991(78)90004-9.
• Alan D. Sokal, “Monte Carlo Methods in Statistical Mechanics: Foundations and New Algorithms,” in C. DeWitt-Morette, P. Cartier, and A. Folacci, eds., Functional Integration: Basics and Applications, pp. 131–192, Springer, 1997, doi:10.1007/978-1-4899-0319-8_6.
• Kurt Symanzik, “Continuum Limit and Improved Action in Lattice Theories. I. Principles and $\phi^4$ Theory,” Nuclear Physics B 226 (1983), 187–204, doi:10.1016/0550-3213(83)90468-6.
• Christof Wetterich, “Exact Evolution Equation for the Effective Potential,” Physics Letters B 301 (1993), 90–94, doi:10.1016/0370-2693(93)90726-X.