Euclidean Correlators and Schwinger Functions

Euclidean Schwinger functions are Euclidean $n$-point distributions: at a finite regulator they are moments generated by a normalized Euclidean source functional, and in a constructed continuum theory they are moments of a random distribution. For the free massive scalar they form a Gaussian hierarchy with covariance $(-\Delta_E+m^2)^{-1}$ and agree with the controlled continuation of the Lorentzian vacuum Feynman correlators. That agreement does not turn arbitrary Euclidean data into Lorentzian correlators: the ordering chamber, analytic domain, positivity conditions, regularity, and reconstruction or continuation theorem remain part of the claim.

Required background. Wick Rotation and Analytic Continuation supplies the legal contour deformation and the raw-correlator phase used in the free two-point check. The Generating Functional supplies normalization, source differentiation, and the distinction between full and connected correlators.

Helpful background. Probability Spaces, Random Variables, and Conditional Expectation supplies expectation and positivity language. Gaussian Vectors, Processes, Random Distributions, and Wick Structure supplies the covariance and pairing rules used below.

Schwinger functions are Euclidean moments

Work first in $d$-dimensional Euclidean space with positive-definite metric $\delta_{\mu\nu}$, coordinates $x_E=(\tau,\mathbf x)$, a real bosonic scalar, and a finite ultraviolet and infrared regulator $\Lambda$. The regulator leaves finitely many real variables or an otherwise controlled integral. Let $J$ be a real test source in the class for which the integral exists, and use the source sign inherited from the preceding page:

$$\begin{aligned} \mathcal Z_{E,\Lambda}[J] &= \int_{\mathcal C_\Lambda}\!\mathrm d\mu_\Lambda(\phi)\, \exp\!\left[-S_{E,\Lambda}[\phi]+(J,\phi)_\Lambda\right],\\ Z_{E,\Lambda}[J] &= \frac{\mathcal Z_{E,\Lambda}[J]} {\mathcal Z_{E,\Lambda}[0]}, \qquad W_{E,\Lambda}[J]=\log Z_{E,\Lambda}[J]. \end{aligned}$$

Here $\mathcal C_\Lambda$ is the declared real or complex integration cycle and $(J,\phi)_\Lambda$ is the regulated source pairing. For a positive real scalar weight at finite $\Lambda$, normalized expectation is

$$\langle F\rangle_{E,\Lambda} = \frac{1}{\mathcal Z_{E,\Lambda}[0]} \int_{\mathcal C_\Lambda}\!\mathrm d\mu_\Lambda(\phi)\, F[\phi]e^{-S_{E,\Lambda}[\phi]}.$$

The full and connected Schwinger functions are

$$\begin{aligned} S_{n,\Lambda}(x_1,\ldots,x_n) &= \left. \frac{\delta^n Z_{E,\Lambda}[J]} {\delta J(x_1)\cdots\delta J(x_n)} \right|_{J=0},\\ S^{\mathrm c}_{n,\Lambda}(x_1,\ldots,x_n) &= \left. \frac{\delta^n W_{E,\Lambda}[J]} {\delta J(x_1)\cdots\delta J(x_n)} \right|_{J=0}. \end{aligned}$$

There are no Lorentzian factors of $1/i$: the Euclidean weight contains $+(J,\phi)_\Lambda$. In continuum notation these objects are distributions. Their meaningful values are smearings such as

$$S_n(f_1,\ldots,f_n) = \int \prod_{r=1}^n\mathrm d^d x_r\, S_n(x_1,\ldots,x_n) \prod_{r=1}^n f_r(x_r),$$

not unrestricted products of point values. Zinn-Justin develops Euclidean source derivatives and the Gaussian hierarchy in Zinn-Justin 2021, §§ 2.5–2.6, pp. 27–32. His source notation is translated here to the declared $+J\phi$ convention.

The free scalar is one Gaussian object in two descriptions

At finite regulator, write the centered free action as

$$S_{E,\Lambda}(\varphi) =\frac12\varphi^{\mathsf T}A_\Lambda\varphi, \qquad A_\Lambda=A_\Lambda^{\mathsf T}>0, \qquad C_\Lambda=A_\Lambda^{-1}.$$

The normalized Gaussian measure is

$$\mathrm d\mu_{C_\Lambda}(\varphi) = \frac{(\det A_\Lambda)^{1/2}}{(2\pi)^{N_\Lambda/2}} \exp\!\left(-\frac12\varphi^{\mathsf T}A_\Lambda\varphi\right) \mathrm d^{N_\Lambda}\varphi,$$

and completing the square gives

$$Z_{E,\Lambda}[J] = \exp\!\left(\frac12J^{\mathsf T}C_\Lambda J\right).$$

Consequently

$$\begin{aligned} S_{1,\Lambda}&=0, &S_{2,\Lambda}(a,b)&=(C_\Lambda)_{ab},\\ S_{4,\Lambda}(1,2,3,4) &=C_{12}C_{34}+C_{13}C_{24}+C_{14}C_{23}, &S^{\mathrm c}_{n,\Lambda}&=0\quad(n\ge3). \end{aligned}$$

This is an exact finite-dimensional statement. It is not yet a claim that a symbolic determinant defines a countably additive measure on continuum field configurations.

For the translation-invariant massive free scalar, take $m>0$ so that the zero-momentum infrared mode is controlled. The continuum covariance suggested by the regulated limit is

$$\begin{aligned} S_E[\phi] &= \frac12\int\mathrm d\tau\,\mathrm d^{d-1}\mathbf x\, \left[(\partial_\tau\phi)^2+(\boldsymbol\nabla\phi)^2+m^2\phi^2\right],\\ K_E&=-\Delta_E+m^2, \qquad Z_E[J]=\exp\!\left[\frac12(J,K_E^{-1}J)\right]. \end{aligned}$$

This Euclidean scalar functional, its vacuum interpretation, and the need for ultraviolet control are treated in Zinn-Justin 2021, § 6.5, pp. 118–120. Its two-point covariance is

$$\begin{aligned} C_m(\tau,\mathbf x) &= \int\frac{\mathrm dp_4\,\mathrm d^{d-1}\mathbf p}{(2\pi)^d} \frac{e^{-ip_4\tau+i\mathbf p\cdot\mathbf x}} {p_4^2+\mathbf p^2+m^2},\\ \widetilde C_m(p_E) &= \frac1{p_E^2+m^2}, \qquad p_E^2=p_4^2+\mathbf p^2. \end{aligned}$$

It obeys the Euclidean distributional equation

$$(-\Delta_E+m^2)C_m(x_E)=\delta_E^{(d)}(x_E), \qquad \Delta_E=\partial_\tau^2+\boldsymbol\nabla^2.$$

The normalization passes two independent checks. In mass units,

$$[\phi]=\frac{d-2}{2}, \qquad [J]=\frac{d+2}{2}, \qquad [C_m(x_E)]=d-2, \qquad [\widetilde C_m(p_E)]=-2,$$

so both the action and source pairing are dimensionless. Moreover, for every real Schwartz test function $f$,

$$C_m(f,f) = \int\frac{\mathrm d^d p_E}{(2\pi)^d} \frac{|\widetilde f(p_E)|^2}{p_E^2+m^2} \ge0.$$

This is covariance positivity for the free Gaussian field. It is not yet the reflected positivity condition used in reconstruction.

Compare this with the site’s Lorentzian raw correlator equation $(\Box+m^2)D_F=-i\delta$. After the legal deformation on the preceding page, the finite-$\epsilon$ holomorphic representative gives

$$\widetilde C_m(p_4,\mathbf p) = \widetilde S_2(p_E) = i\,\widetilde D_F(p^0=i p_4,\mathbf p) = \frac1{p_4^2+\mathbf p^2+m^2}.$$

The middle expression is conventional shorthand for the controlled deformation, not pointwise evaluation of the boundary distribution $\widetilde D_F$. Thus the free Euclidean Gaussian field and the continued Lorentzian vacuum theory have the same two-point data only after the contour, state, source, and limiting prescriptions have been matched. Srednicki gives the regulated scalar action rotation and its positive Euclidean quadratic form in Srednicki 2007, § 29, pp. 185–186.

Gaussian integration by parts also checks the full free hierarchy. With all variables treated independently and the identity read after smearing,

$$(-\Delta_x+m^2) S_{n+1}(x,x_1,\ldots,x_n) = \sum_{r=1}^{n} \delta_E^{(d)}(x-x_r) S_{n-1}(x_1,\ldots,\widehat{x_r},\ldots,x_n).$$

for $n\ge1$, with $S_0=1$. Away from every partial diagonal the right-hand side vanishes. On a regulator, $-\Delta+m^2$ and $\delta_E$ are replaced by the regulated kernel and identity matrix; the continuum formula is the controlled distributional limit.

Euclidean time order is encoded by chambers

For a bosonic scalar represented by commuting integration variables, source differentiation makes $S_n$ symmetric under permutations of its arguments. Some texts write $T_\tau$, but the Euclidean functional integral does not impose Lorentzian causal ordering.

If a Lorentzian vacuum Hamiltonian $H\ge0$ is already known, its vacuum energy has been shifted so that $H|\Omega\rangle=0$, and the continuation exists, then in the chamber

$$\tau_1>\tau_2>\cdots>\tau_n$$

the same data can be written schematically as

$$\begin{aligned} S_n(x_1,\ldots,x_n) =\langle\Omega|\, &\phi(\mathbf x_1)e^{-(\tau_1-\tau_2)H} \phi(\mathbf x_2)\cdots\\ &\cdots e^{-(\tau_{n-1}-\tau_n)H} \phi(\mathbf x_n)|\Omega\rangle. \end{aligned}$$

This formula explains the ordering information; it does not construct $H$ from arbitrary Euclidean functions. Another strict ordering of the $\tau_r$ selects another analytic chamber. The permutation-symmetric Euclidean distribution glues these chambers, while their Lorentzian boundary values recover different Wightman orderings and the time-ordered combination.

For the free two-point function,

$$S_2(\tau,\mathbf p)=\frac{e^{-E_{\mathbf p}|\tau|}}{2E_{\mathbf p}}, \qquad E_{\mathbf p}=\sqrt{\mathbf p^2+m^2}.$$

The $\tau>0$ half continues through its own holomorphy domain to the positive-time Feynman branch; the $\tau<0$ half continues through the opposite domain. The absolute value is therefore an assembly of two boundary domains, not a single entire function of complex time.

The reverse free check uses two analytic functions, not a complex absolute value:

$$\begin{aligned} \operatorname{Re}\tau>0:\quad F_+(\tau,\mathbf p) &=\frac{e^{-E_{\mathbf p}\tau}}{2E_{\mathbf p}},\\ \lim_{\epsilon\downarrow0} F_+(\epsilon+it,\mathbf p) &=\frac{e^{-iE_{\mathbf p}t}}{2E_{\mathbf p}}, \qquad t>0,\\[1mm] \operatorname{Re}\tau<0:\quad F_-(\tau,\mathbf p) &=\frac{e^{+E_{\mathbf p}\tau}}{2E_{\mathbf p}},\\ \lim_{\epsilon\downarrow0} F_-(-\epsilon+it,\mathbf p) &=\frac{e^{+iE_{\mathbf p}t}}{2E_{\mathbf p}}, \qquad t<0. \end{aligned}$$

On the real Euclidean axis, $S_2(\tau,\mathbf p)$ is the restriction of $F_+$ for $\tau>0$ and of $F_-$ for $\tau<0$. Their displayed boundary values are precisely the two time branches of the vacuum Feynman correlator. A different Lorentzian boundary—retarded, advanced, Wightman, or thermal—requires its own continuation and state data.

Coincident Euclidean times lie on the walls between chambers. There the Schwinger functions remain distributions, and composite insertions or renormalized time-ordered products may require local contact or subtraction terms. Analytic continuation of separated points does not determine those local extensions automatically.

A spectral representation survives as smoothed Euclidean data

Suppose a centered Hermitian scalar operator is evaluated in a Poincaré-invariant vacuum of a positive physical Lorentzian Hilbert space with forward-spectrum support. Assume first that its scalar spectral measure $\rho_{\mathcal O}$ makes the displayed integral convergent. A separately renormalized time ordering may also contain a local polynomial $P_E$. The continued Euclidean two-point function then has the polynomial-plus-Stieltjes form

$$\widetilde S_{2,\mathcal O}(p_E) = P_E(p_E^2) +\int_{[0,\infty)} \frac{\rho_{\mathcal O}(\mathrm d\mu^2)} {p_E^2+\mu^2}.$$

At fixed spatial momentum this becomes

$$S_{2,\mathcal O}(\tau,\mathbf p) = S_{2,\mathrm{local}}(\tau,\mathbf p) +\int_{[0,\infty)} \rho_{\mathcal O}(\mathrm d\mu^2) \frac{e^{-E_{\mu,\mathbf p}|\tau|}} {2E_{\mu,\mathbf p}}, \qquad E_{\mu,\mathbf p}=\sqrt{\mathbf p^2+\mu^2}.$$

Here $S_{2,\mathrm{local}}$ is supported at coincident Euclidean points. Under the site’s raw-correlator convention, continuation maps the Lorentzian contact polynomial by

$$P_E(s)=iP_L(-s), \qquad s=p_E^2.$$

If the raw spectral integral is ultraviolet divergent, it must instead be replaced by a declared $N$-subtracted kernel. Adding a polynomial to an undefined integral does not make that integral convergent.

The Euclidean denominator and exponential smooth the mass distribution. An isolated atom produces a single exponential, while continuum support produces a superposition. Recovering detailed real-time spectral structure from finite, noisy Euclidean data is an inverse problem; it is not accomplished by replacing $p_4$ with a real frequency.

The positivity of $\rho_{\mathcal O}$ here came from the declared physical Hilbert-space assumptions. A generic Euclidean two-point function, a gauge-fixed auxiliary field, or an off-diagonal operator pair does not acquire a positive scalar spectral measure merely because it is written with a Euclidean momentum.

The positive-measure Euclidean spectral form and its relation to the physical scalar spectrum are developed in Zinn-Justin 2021, § 6.6, pp. 122–124. The page uses that result only under the stated diagonal, positive-metric assumptions.

Four positivity statements must remain distinct

The word “positive” refers to different quadratic forms:

Statement	Test	What it supports	What it does not prove
Positive Euclidean measure	$\int	F(\phi)	^2,\mathrm d\mu_E(\phi)\ge0$
Positive covariance	$\int\overline{f(x)}S_2(x,y)f(y)\,\mathrm d^dx\,\mathrm d^dy\ge0$	a centered Gaussian random field with that covariance, when the required continuity conditions hold	the reflection form or relativistic reconstruction
Reflection positivity	$\langle\overline{F[\theta\phi]}\,F[\phi]\rangle_E\ge0$ for positive-time $F$	a candidate inner product after time reflection and quotienting null vectors	the remaining Osterwalder–Schrader hypotheses or the full theorem
Lorentzian spectral positivity	$\rho_{\mathcal O}\ge0$ in a physical Hilbert space	nonnegative diagonal spectral weights	positivity of gauge-variant auxiliary correlators or arbitrary Euclidean data

Here $(\theta\phi)(\tau,\mathbf x)=\phi(-\tau,\mathbf x)$, and “positive-time $F$” means that $F$ depends only on fields smeared in $\tau>0$.

Ordinary pointwise positivity of $S_n(x_1,\ldots,x_n)$ is not one of these requirements. Nor is a Euclidean random-field configuration a measured Lorentzian history. The next page develops the reflection form rather than treating these four rows as synonyms.

Interacting Euclidean functionals require a claim boundary

For a finite collection of real variables and a stable real action,

$$S_{E,\Lambda}[\phi] = \frac12(\phi,A_\Lambda\phi) +V_\Lambda[\phi],$$

a coercive $S_{E,\Lambda}$ can make $e^{-S_{E,\Lambda}}\mathrm d^{N_\Lambda}\phi$ a genuine probability measure. Source derivatives then define honest regulated Schwinger moments. This useful fact still leaves separate questions:

• Does a family of regulated measures have a cutoff-removal limit?
• Are the limiting moments distributions with the required regularity and covariance?
• Does the hierarchy determine an underlying measure?
• Does it satisfy reflection positivity and the other reconstruction hypotheses?
• Which renormalized local terms are needed at coincident points?

An unbounded action, a complex integration cycle, a fermion determinant, a chemical potential, a topological phase, or gauge fixing can remove ordinary measure positivity while leaving a formal or algebraic Euclidean functional. Finite-temperature imaginary time also introduces a compact Euclidean-time circle and periodic or antiperiodic boundary conditions; it is not the vacuum setting used here.

Lattice Monte Carlo estimators, continuum extrapolation, and inverse-problem uncertainty belong to Euclidean Correlators and Spectral Information. Euclidean conformal correlators and radial quantization begin at Radial Time and Quantization on Spheres.

Reconstruction is a separate theorem

A symmetric, Euclidean-covariant hierarchy is not automatically a relativistic QFT. The Osterwalder–Schrader program additionally controls regularity or growth, reflection positivity, permutation symmetry, and clustering or vacuum structure, with the exact package depending on the theorem version. The original axiom list and reconstruction statement appear in Osterwalder and Schrader 1973, § 3, pp. 87–90 (Open PDF); the authors later extended and corrected the sufficient conditions in Osterwalder and Schrader 1975, Introduction, pp. 281–283, and § IV.1, pp. 287–288 (Open PDF).

This page supplies the physical hierarchy and its free scalar check. It does not construct the reflected Hilbert space, prove the reconstruction theorem, or establish a continuum measure.

Common mistakes

“Euclidean” is not a change of variable label. A Schwinger function includes its Euclidean domain, source convention, distributional meaning, and—when it comes from continuation—the original state, ordering chamber, and contour.

Permutation symmetry is not causal ordering. Symmetry of bosonic Euclidean moments does not imply retarded support, commutator vanishing, or a real-time response law.

A positive Boltzmann weight is not the whole reconstruction problem. It gives ordinary measure positivity at the declared regulator. Reflection positivity and the other reconstruction hypotheses remain independent checks.

Euclidean decay is not a fitted particle without assumptions. A clean exponential can indicate a spectral atom under the spectral and reconstruction hypotheses. Finite extent, excited-state contamination, continuum weight, and inverse-problem instability can mimic or obscure that behavior.

Check your understanding

1. Check the source phases

Differentiate $Z_E[J]=\exp(\tfrac12 JCJ)$ twice. Why is there no factor of $i$?

Answer

The first derivative is $(CJ)_aZ_E$ and the second at $J=0$ is $C_{ab}$. The Euclidean source appears as $+J\phi$ in $e^{-S_E+J\phi}$, so differentiation inserts $\phi$ directly. Lorentzian factors of $i$ arise from the different weight $e^{iS+iJ\phi}$.

2. Round-trip the free contact equation

Apply $-\Delta_E+m^2$ to the Fourier representation of $C_m$. What must be recovered?

Answer

The operator multiplies the integrand by $p_E^2+m^2$, leaving the Fourier representation of $\delta_E^{(d)}$. This checks both the positive Euclidean denominator and the absence of the Lorentzian factor $-i$.

3. Diagnose the positivity claim

A finite-dimensional scalar weight is everywhere nonnegative. May one conclude that its continuum Schwinger hierarchy reconstructs a Lorentzian QFT?

Answer

No. One still needs a controlled continuum hierarchy, reflection positivity, Euclidean covariance, symmetry, suitable regularity or growth, and the remaining hypotheses of the reconstruction theorem. Regulated ordinary measure positivity alone is insufficient.

4. Choose the continuation object

Why does substituting $p_4=-i\omega$ into tabulated Euclidean data not directly produce a retarded correlator?

Answer

The Euclidean data must first define an analytic function in a domain reaching the required boundary. Feynman and retarded functions approach different Lorentzian boundaries and encode different support and ordering data. With finite or noisy samples, analytic continuation is also an ill-conditioned inverse problem.

Where to continue

• Reflection Positivity within Osterwalder–Schrader Reconstruction develops the time-reflection quadratic form and a finite free-scalar test.
• Euclidean Random Fields and Schwinger Hierarchies treats measures on distribution spaces, moment hierarchies, and their converse problems at theorem-oriented depth.
• Analytic Continuation between Euclidean and Lorentzian Domains supplies the complex spacetime domains and boundary-value theorems.
• Wick Rotation and Analytic Continuation remains the place to diagnose a crossed singularity, nonvanishing arc, or pinch before using a Euclidean formula.

References

• Osterwalder, Konrad, and Robert Schrader. “Axioms for Euclidean Green’s Functions.” Communications in Mathematical Physics 31 (1973): 83–112. DOI. Open PDF.

• Osterwalder, Konrad, and Robert Schrader. “Axioms for Euclidean Green’s Functions II (with an Appendix by Stephen Summers).” Communications in Mathematical Physics 42 (1975): 281–305. DOI. Open PDF.

• Srednicki, Mark. Quantum Field Theory. Cambridge: Cambridge University Press, 2007. DOI.

• Zinn-Justin, Jean. Quantum Field Theory and Critical Phenomena. Fifth ed. Oxford: Oxford University Press, 2021. DOI.