Euclidean Continuation Caveats

For a semibounded self-adjoint Hamiltonian, Euclidean evolution and real-time evolution are connected by the spectral calculus. The real-time group is a strong boundary value of a holomorphic semigroup. This precise statement does not imply that arbitrary Euclidean data have a quantum reconstruction, or that reconstruction from noisy data is stable. Euclidean and Imaginary-Time Path Integrals explains physical Wick rotation and the kernels involved.

Required background. Functional Calculus defines functions of self-adjoint operators. Feynman–Kac Formula supplies a probabilistic realization when its potential hypotheses hold.

A lower spectral bound gives a holomorphic half-plane

Use $\hbar=1$. Suppose $H$ is self-adjoint and $H\geq E_0I$. Set $K=H-E_0I\geq0$. For $\operatorname{Re}z>0$, define

$$T(z)=e^{-zK}=\int_{[0,\infty)}e^{-z\lambda}\,dE_K(\lambda).$$

Since $|e^{-z\lambda}|\leq1$, this is a contraction. On compact subsets of the right half-plane, the functions $\lambda^ne^{-z\lambda}$ are uniformly bounded for each fixed $n$. The spectral calculus therefore makes $T$ operator-norm holomorphic there, with derivatives $(-K)^ne^{-zK}$.

For real $t$, the boundary limit satisfies

$$\lim_{\varepsilon\downarrow0}T(\varepsilon+it)\psi =e^{-itK}\psi$$

in Hilbert norm for every $\psi$. Indeed, its squared error is

$$\int_{[0,\infty)}|e^{-\varepsilon\lambda}-1|^2 \,d\langle\psi,E_K(\lambda)\psi\rangle,$$

which tends to zero by dominated convergence. The phase $e^{-itE_0}$ restores evolution under $H$.

If $K$ is unbounded above, this boundary convergence is not in operator norm: $\|e^{-\varepsilon K}-I\|=1$ for every $\varepsilon>0$. High-energy vectors prevent a uniform approximation even while each fixed state converges.

Exact Euclidean matrix elements determine their continuation

For fixed vectors, the scalar function

$$F_{\phi,\psi}(z)=\langle\phi,e^{-zK}\psi\rangle$$

is holomorphic for $\operatorname{Re}z>0$. Its exact values on any real positive interval determine that holomorphic function uniquely by the identity theorem, and the strong boundary theorem specifies its real-time limit.

This is a uniqueness statement for data known to arise from the stated spectral construction. It is not an algorithmic stability estimate. Nor does it justify rotating contours through singularities in a separate integral representation without checking that representation’s analytic domain and large-contour behavior.

If the Hamiltonian is unbounded below, $e^{-\tau H}$ is generally an unbounded operator for $\tau>0$. One must then specify domains of matrix elements and a different continuation theorem; the bounded holomorphic-semigroup argument above no longer applies.

Euclidean damping can hide large real-time differences

Consider the positive spectral measures $\mu_n=\delta_n$ and $\nu_n=\delta_{2n}$ for odd positive integers $n$. They can be realized by energy eigenvectors of a fixed diagonal nonnegative Hamiltonian. Their Euclidean transforms are $e^{-n\tau}$ and $e^{-2n\tau}$.

For every fixed $\tau_0>0$, their difference is at most $e^{-n\tau_0}$ uniformly on $\tau\geq\tau_0$. Yet at real time $t=\pi$ their transforms are $-1$ and $+1$, differing by two. Both measures have total mass one.

Thus exponentially small differences in Euclidean data away from the boundary can conceal order-one real-time differences. Spectral cutoffs, prior bounds, and regularization can make an inverse problem useful, but exact uniqueness alone supplies no stable error bound.

Reflection positivity is stronger than a positive covariance

For a normalized vector $\psi$, let $F(\tau)=\langle\psi,e^{-\tau K}\psi\rangle$. For any nonnegative times $t_j$ and complex coefficients $c_j$,

$$\sum_{i,j}\overline{c_i}c_jF(t_i+t_j) =\left\|\sum_jc_je^{-t_jK}\psi\right\|^2\geq0.$$

This is a simple quantum-mechanical reflection-positivity condition: reflecting one time variable changes a difference into a sum. It is a necessary property of these Euclidean correlations. In addition, their spectral representation makes them completely monotone on positive times.

A positive probability measure on Euclidean functions does not by itself impose this reflected positivity. Full field-theoretic reconstruction involves further hypotheses; the finite matrix test above already detects some incompatible candidate data without invoking that broader theorem.

Exercise: a positive Gaussian covariance can fail the reflected test

The stationary Gaussian covariance $C(s-t)=e^{-(s-t)^2}$ is positive definite. Test whether $F(\tau)=e^{-\tau^2}$ can equal $\langle\psi,e^{-\tau K}\psi\rangle$ for a normalized vector and $K\geq0$.

Solution

At times $0$ and $a>0$, the reflected matrix has entries $1,e^{-a^2},e^{-4a^2}$. Its determinant is $e^{-4a^2}-e^{-2a^2}<0$, so it is not positive semidefinite. It cannot have the stated Hilbert-space semigroup representation. Ordinary covariance positivity tests $C(t_i-t_j)$, while the required reflected condition tests $F(t_i+t_j)$; these are different conditions.

References

• Osterwalder, Konrad, and Robert Schrader, “Axioms for Euclidean Green’s functions,” Communications in Mathematical Physics 31, 83–112 (1973), doi:10.1007/BF01645738 — reflection positivity in Euclidean reconstruction.
• Reed, Michael, and Barry Simon, Methods of Modern Mathematical Physics I: Functional Analysis, revised and enlarged ed., Academic Press, 1980 — spectral calculus and semigroup boundary limits.
• Simon, Barry, Functional Integration and Quantum Physics, 2nd ed., AMS Chelsea Publishing, 2005, doi:10.1090/chel/351 — Euclidean quantum mechanics and reconstruction.