Translating Observables Across Analytic and Regulated Methods

Analytic, Euclidean, finite-volume, Hamiltonian, and real-time methods compute the same observable only when their operator definitions, states, normalizations, renormalization schemes, boundary conditions, analytic continuations, and limit orders are connected by controlled maps. A common label such as “mass” or “string tension” is not enough. The translation succeeds when a diagram from each method’s native output to one renormalized target commutes within declared uncertainties; it fails at the first missing matching coefficient, continuation, or limit.

Required background. Claim status across methods separates a definition, controlled calculation, regulated result, and open implication. Spectral decomposition of two-point functions supplies the pole, continuum, and operator-overlap language used below.

Helpful background. Functional-method validation shows why a Euclidean solution’s branch, tensor basis, and reconstruction uncertainty remain part of a real-time claim.

Shared comparisons. The evidence-triangulation graph shows where translations enter a multi-method conclusion; the exact and rigorous status comparison prevents an exact native object from lending that label to an uncontrolled translation. The functional-method comparison records the corresponding closure and validation data.

Observable equality as a commuting diagram

Let method $i$ produce native data $D_i$—for example a Euclidean correlator at finite lattice spacing and volume, a truncated continuum propagator, or a Hamiltonian spectrum in a box. A comparison to a physical observable $O$ requires a map

$$D_i \xrightarrow{\;R_i\;} O_i^{\rm ren} \xrightarrow{\;C_i\;} O,$$

where $R_i$ includes renormalization and regulator removal and $C_i$ includes analytic continuation, state and normalization matching, and any volume or kinematic limit. Two methods agree on $O$ only if

$$C_1R_1(D_1)=C_2R_2(D_2)$$

within uncertainties propagated through both maps. This equation is meaningful only after its domain is specified. A continuation known for a reflection-positive two-point function below threshold need not exist for a gauge-fixed propagator, a noisy finite data set, or a multiparticle amplitude at arbitrary kinematics.

A reproducible translation record contains:

1. the theory, global data, state or ensemble, and boundary conditions;
2. the renormalized operator, including mixing and normalization;
3. the native estimator or analytic object and its regulator;
4. matching factors and renormalization scheme and scale;
5. the continuation or spectral representation used;
6. the continuum, volume, temperature, and kinematic limits in order;
7. statistical, discretization, truncation, fitting, and continuation uncertainties; and
8. a round-trip or solvable-limit check.

This is stronger than comparing central values. It asks whether applying a known inverse map, where one exists, recovers the method’s native quantity and whether a second observable responds correctly when a shared parameter changes.

Five quantities called a mass

For an operator $\mathcal O$ with the desired quantum numbers, the following quantities answer related but nonidentical questions.

Quantity	Native definition	Equality to a stable zero-temperature particle mass requires	Typical failure
Real-time pole mass	Isolated physical-sheet pole of a causal two-point function at $p^2=m^2$	A stable state, correct sheet, residue, and renormalized operator channel	A resonance pole lies on another sheet; a gauge-fixed pole need not be physical
Euclidean temporal decay rate	Lowest energy coupled to $\mathcal O$ in $C_E(\tau,\mathbf 0)$ at large $\tau$	Reflection-positive reconstruction, infinite temporal extent, nonzero overlap, and isolation from lower states	Excited-state contamination or loss of positivity
Spatial screening mass	Exponential decay rate of a static spatial correlator	Zero-temperature Euclidean symmetry or a separate argument equating spatial and temporal channels	At nonzero temperature it is generally not a real-time pole mass
Transfer-matrix gap	Difference $E_n(L)-E_0(L)$ in a regulated finite spatial volume	A physical transfer matrix, identified channel, continuum limit, and controlled $L\to\infty$ behavior	Level mixing and scattering states can dominate the operator basis
Finite-volume energy level	Discrete eigenvalue obtained from a box correlator or Hamiltonian	A finite-volume quantization or stable-particle relation connecting it to infinite-volume data	A level is not itself a resonance pole or infinite-volume mass

For a stable particle in a massive theory, the finite-volume mass shift can be exponentially small at large $L$, with a coefficient related to infinite-volume scattering data. That conclusion has hypotheses; it is not a license to ignore volume dependence for thresholds or massless modes Lüscher 1986, §§ 2–4, pp. 177–206.

A scalar round trip

The free massive scalar gives a minimal translation in which all decisive maps are visible. In infinite-volume Euclidean space,

$$G_E(p_4,\mathbf p) =\frac{1}{p_4^2+\mathbf p^2+m^2}.$$

At zero spatial momentum,

$$C_E(\tau) =\int_{-\infty}^{\infty}\frac{\mathrm dp_4}{2\pi} \frac{e^{ip_4\tau}}{p_4^2+m^2} =\frac{e^{-m|\tau|}}{2m}.$$

Analytic continuation of the boundary value $p_4=-i(p^0+i0)$ gives

$$G_F(p^0,\mathbf p) =\frac{i}{(p^0)^2-\mathbf p^2-m^2+i0},$$

whose positive-energy pole is $p^0=\sqrt{\mathbf p^2+m^2}$. Thus the temporal Euclidean decay rate at $\mathbf p=0$, the Hamiltonian one-particle gap, and the real-time pole mass all equal $m$. The equality used translation invariance, the vacuum state, a stable isolated pole, the infinite-volume limit, and the standard continuation. It did not follow from the shared letter $m$.

The spectral form makes the generalization precise:

$$C_E(\tau,\mathbf 0) =\int_0^\infty \mathrm d\mu\, \rho(\mu)\,\frac{e^{-\sqrt\mu\,|\tau|}}{2\sqrt\mu}.$$

An isolated contribution $Z\,\delta(\mu-m^2)$ controls the asymptotic decay if it is the lowest spectral support with nonzero operator overlap. A continuum beginning at $\mu_0$ instead brings threshold powers multiplying $e^{-\sqrt{\mu_0}|\tau|}$; a resonance need not appear as a real-axis delta function. The inverse problem is also unstable: many spectral densities can approximate a finite, noisy set of Euclidean data. Positivity, asymptotics, sum rules, resolution tests, and held-out correlator data constrain the reconstruction but do not turn it into a unique analytic continuation by declaration.

Osterwalder–Schrader conditions identify when a complete family of Euclidean correlation functions reconstructs a relativistic theory Osterwalder and Schrader 1973, pp. 83–112. A finite set of samples with errors is not that complete input. Likewise, infinite-volume Euclidean correlators do not yield arbitrary scattering amplitudes through a naive large-time limit; the Maiani–Testa obstruction explains why finite-volume spectral methods or other additional structure are needed away from threshold Maiani and Testa 1990, pp. 585–590.

Translation workflow for a regulated spectrum

Suppose a lattice or Hamiltonian calculation reports several energies in a channel. A defensible route to an infinite-volume observable is:

1. Define the target. Decide whether the result sought is a stable mass, a resonance pole, a screening length, a string-breaking energy, or a finite-volume level.
2. Fix the operator map. State the continuum operator, lattice or basis realization, mixing pattern, and normalization. Verify the symmetry representation and quantum numbers.
3. Resolve the native spectrum. Vary the operator basis and temporal fit range; test excited-state and threshold alternatives rather than assigning a plateau automatically.
4. Renormalize and set scales. Give the scheme, matching order, scale-setting observable, and correlations introduced by common calibration.
5. Remove the regulator. Vary discretization or basis cutoff over a range that resolves the predicted leading corrections. A good fit with no resolved scaling direction is not a continuum test.
6. Control volume and kinematics. Use stable-particle asymptotics or a finite-volume quantization condition appropriate to the spectrum. Take the infinite-volume limit in the stated order relative to the continuum and infrared limits.
7. Translate uncertainty. Propagate covariance from the raw estimator through fits, matching, scale setting, and extrapolation. Keep continuation or model-choice uncertainty separate from sampling error.
8. Close a round trip. Reconstruct a native correlator or a second level not used in the fit and compare it with the measured quantity.

A functional or analytic method follows the same logic with different native data. Its tensor basis and closure replace the lattice operator and discretization; its branch choice and complex-momentum continuation replace the spectral fit and volume quantization. The target remains fixed.

When comparisons must be refused

Do not combine results when any of the following remains unresolved:

• one method reports a gauge-fixed propagator scale and another a gauge-invariant spectral gap;
• one mass is a thermal spatial screening rate and the other a zero-temperature pole;
• the renormalization schemes are unmatched and the quantity is scheme dependent;
• one observable is at finite volume while the other assumes an infinite-volume branch or threshold;
• one continuation selects a pole sheet or boundary value that the other method does not define;
• a dimensionful comparison inherits the same scale-setting input on both sides but treats its uncertainty as independent; or
• error bars omit the dominant truncation, fit-model, continuation, or limit uncertainty.

Refusal is constructive: it identifies the missing matching coefficient, additional volume, alternate operator, complex-momentum solution, or theorem needed to make a future comparison meaningful. Once the translation is explicit, correlated evidence and triangulation determines how much independent information the comparison adds.

Common pitfalls

Equating exponential decay with a particle pole without hypotheses. A Euclidean decay rate identifies the lowest spectral support visible to an operator. Stability, positivity, infinite volume, and analytic structure decide whether that support is an isolated physical pole.

Treating analytic continuation as a substitution. The $+i0$ boundary value, singularity structure, and growth conditions are part of the continuation. Noisy discrete data require a reconstruction problem with resolution and stability tests.

Hiding scale-setting correlations. Two dimensionful results expressed using the same calibration observable share its uncertainty. That common input must remain a shared ancestor in the comparison.

Exercises

In the free scalar calculation, integrate $C_E(\tau)$ back over $\tau$ and recover $G_E(p_4,\mathbf0)$.

Solution

Using evenness,

$$\begin{aligned} \int_{-\infty}^{\infty}\mathrm d\tau\, e^{-ip_4\tau}\frac{e^{-m|\tau|}}{2m} &=\frac1m\int_0^\infty\mathrm d\tau\, e^{-m\tau}\cos(p_4\tau)\\ &=\frac1m\frac{m}{p_4^2+m^2} =\frac1{p_4^2+m^2}. \end{aligned}$$

This round trip checks the normalization as well as the decay exponent.

A thermal calculation finds a spatial screening rate $m_{\rm scr}$ while a real-time calculation finds a pole at $\omega=m_{\rm pole}$ for zero spatial momentum. Explain why disagreement does not by itself falsify either method.

Solution

The heat bath breaks Lorentz symmetry, so spatial and temporal correlation functions probe different limits of the self-energy. The screening rate solves a static spatial condition, whereas the pole solves a causal frequency condition on a specified sheet. A valid comparison first derives the thermal relation, if any, between those limits and matches gauge, channel, and renormalization conventions.

References

• Lüscher, Martin. “Volume Dependence of the Energy Spectrum in Massive Quantum Field Theories. I. Stable Particle States.” Communications in Mathematical Physics 104, no. 2 (1986): 177–206. DOI.
• Maiani, Luciano, and Massimo Testa. “Final State Interactions from Euclidean Correlation Functions.” Physics Letters B 245, no. 3–4 (1990): 585–590. DOI.
• Osterwalder, Konrad, and Robert Schrader. “Axioms for Euclidean Green’s Functions.” Communications in Mathematical Physics 31, no. 2 (1973): 83–112. DOI. Open PDF.