Analytic Continuation between Euclidean and Lorentzian Domains

Euclidean and Lorentzian correlators are related as boundary values of analytic functions on specified complexified-spacetime domains. The relation is not the pointwise substitution $t=-i\tau$: one must state the ordering of imaginary times, the connected tube reached by continuation, the singular sets avoided, the approach direction, and the topology in which the boundary value exists.

Required background. Tube domains, complex Lorentz covariance, and analyticity supplies forward tubes and distributional boundary values. Osterwalder–Schrader reconstruction supplies the positive-energy Wightman hierarchy. Euclidean growth, regularity, and temperedness supplies the estimates that control the boundary.

Helpful background. Wick rotation and analytic continuation gives the physical overview. Contour deformation, pinches, and causal prescriptions supplies the contour conditions used in momentum-space examples.

Ordered Euclidean regions and forward tubes

Use the Lorentzian metric $(+---)$ and write a complexified point as

$$z=x-iy, \qquad y\in V_+,$$

where $V_+$ is the open future timelike cone. If a translation-invariant two-point distribution has positive-energy spectral measure $d\rho(p)$, then

$$F(z)=\int_{\overline V_+}e^{-ip\cdot z}\,d\rho(p) =\int_{\overline V_+}e^{-ip\cdot x}e^{-p\cdot y}\,d\rho(p)$$

is analytic in the lower forward tube. The damping factor, not a mnemonic rotation arrow, fixes the sign.

For an ordered $n$-point Wightman function $W_n(x_1,\ldots,x_n)$, translation invariance reduces the variables to successive differences. The primitive tube is characterized by

$$\operatorname{Im}(z_j-z_{j+1})=-(y_j-y_{j+1}), \qquad y_j-y_{j+1}\in V_+.$$

Purely imaginary time points in this tube have $\tau_1>\tau_2>\cdots>\tau_n$ for the displayed field order. A different permutation of field labels begins in a different ordered Euclidean region and a correspondingly permuted tube. Euclidean symmetry says the real Euclidean distributions agree under permutation where defined; Lorentzian locality and analytic continuation determine when the permuted analytic functions glue.

The extended tube obtained using complex Lorentz transformations is larger than the primitive tube, but it is not all of complex spacetime. Coincidence loci, light-cone singularities, and multiparticle singularities remain obstructions. The OS continuation proceeds through controlled domains and estimates; its real-analytic and “towards the real world” steps are set out in Osterwalder and Schrader 1975, §§V.1–V.2, pp. 291–297.

Boundary values are distributions

Let $F$ be analytic in a tube with the polynomial and cone-boundary bounds supplied by the reconstruction theorem. Its Wightman boundary value is

$$W=\lim_{y\to0,\ y\in C}F(\,\cdot-iy)$$

in $\mathcal S'$, for the declared cone $C$. This statement allows $W$ to be singular on the light cone and at coincident points. It does not imply pointwise convergence there.

Three distinctions follow.

• A Wightman function preserves a particular operator order and is the boundary value of its associated tube.
• A time-ordered function combines different Wightman boundary values with time-ordering distributions; in momentum space this produces the Feynman $i0$ prescription.
• A Euclidean Schwinger function is the common Euclidean restriction after the ordered analytic pieces have been related by symmetry. It does not carry a visible real-time ordering until an approach domain is chosen.

Analytic continuation is unique inside a connected analytic domain once exact data are known on an appropriate open set. Reconstructing it from noisy, finite Euclidean data is nevertheless ill-conditioned. The OS theorem is an existence-and-uniqueness theorem for exact distributions, not a stability theorem for numerical inversion.

First QFT application: three free scalar boundary values

For $\omega_{\mathbf p}=\sqrt{\mathbf p^2+m^2}$, define

$$F(z^0,\mathbf x)= \int\frac{d^{d-1}\mathbf p}{(2\pi)^{d-1}2\omega_{\mathbf p}} e^{-i\omega_{\mathbf p}z^0+i\mathbf p\cdot\mathbf x}, \qquad \operatorname{Im}z^0<0.$$

At $z^0=-i\tau$ with $\tau>0$,

$$F(-i\tau,\mathbf x)= \int\frac{d^{d-1}\mathbf p}{(2\pi)^{d-1}2\omega_{\mathbf p}} e^{-\omega_{\mathbf p}\tau+i\mathbf p\cdot\mathbf x},$$

which is the positive-time Euclidean covariance. Approaching $z^0=t-i0$ gives the positive-frequency Wightman distribution $W_+(t,\mathbf x)$. The reversed field order is the boundary value $W_-(t,\mathbf x)=W_+(-t,-\mathbf x)$ from the oppositely ordered tube. Combining them gives

$$\Delta_F(t,\mathbf x) =\theta(t)W_+(t,\mathbf x)+\theta(-t)W_-(t,\mathbf x),$$

whose momentum-space denominator is $p^2-m^2+i0$ up to the conventional overall factor. These three Lorentzian distributions share one Euclidean kernel but are not interchangeable.

Checks are direct. The sign $\operatorname{Im}z^0<0$ makes $e^{-\omega\tau}$ decay; the $W_+$ Fourier transform lies on the positive mass shell; and the discontinuity $W_+-W_-$ is the commutator distribution. The full relation is the free-field instance of Wick rotation and analytic continuation, with the domain and boundary prescriptions now explicit.

Adversarial test: a pinched contour

In a loop integral, rotating the energy contour is legitimate only while the contour can be continuously deformed without crossing singularities and while the arcs at infinity vanish. As external energy approaches a physical threshold, poles can approach the integration contour from opposite sides. Once they pinch it, no homotopy to the Euclidean contour exists within the analytic domain.

Continuing straight through that pinch either changes the boundary value by residues or produces an undefined expression. It is not OS analytic continuation. The correct answer is a boundary value on a specified side of the cut, and distinct sides encode the physical discontinuity. This obstruction is local in parameter space: a rotation valid below threshold need not remain valid after the external invariants cross a Landau singularity.

Independent continuation checklist

• Starting region: give the exact imaginary-time ordering and field permutation.
• Analytic domain: state the primitive or extended tube and any excluded singular variety.
• Path: exhibit a deformation that stays inside the domain; do not cross a pinch.
• Growth: verify the bound that gives a tempered boundary distribution.
• Approach: state $+i0$, $-i0$, or the relevant cone direction and identify the resulting ordering.
• Convention: check the Fourier phase and the $(+---)$ mass shell so the damping and energy support have the correct signs.

Exercise

Why does $F(t-i\epsilon,\mathbf x)$ converge for $\epsilon>0$, while the same integral with $t+i\epsilon$ does not define the same analytic function?

Solution

Substitution gives

$$e^{-i\omega(t-i\epsilon)}=e^{-i\omega t}e^{-\omega\epsilon},$$

so positive energy produces exponential damping. With $t+i\epsilon$, the factor is $e^{+\omega\epsilon}$ and grows on the positive mass shell. The latter boundary must instead be obtained from the oppositely ordered spectral representation, whose energy and field order are reversed. It is not another approach inside the same lower tube.

References

• Osterwalder, Konrad, and Robert Schrader. “Axioms for Euclidean Green’s Functions.” Communications in Mathematical Physics 31 (1973): 83–112. doi:10.1007/BF01645738. Open PDF.
• Osterwalder, Konrad, and Robert Schrader. “Axioms for Euclidean Green’s Functions II.” Communications in Mathematical Physics 42 (1975): 281–305. doi:10.1007/BF01608978. Open PDF.