Reading a spectrum from a Euclidean correlator

Euclidean time turns time evolution into a filter: high-energy components decay faster than low-energy components. This simple fact connects measured correlation functions to the spectrum.

Prerequisites: physical states and eigenstate expansions in quantum mechanics. The spectral theorem bridge supplies the general operator language.

By the end:

• Derive a spectral sum in a controlled example.
• Interpret decay rates and effective masses.

Begin with a quantum-mechanical system

Let $H$ be a nonnegative self-adjoint Hamiltonian and $\Omega$ a unique normalized vacuum (up to phase) with zero energy. Choose an operator $O$ whose vacuum-subtracted action is a well-defined vector,

$$\widetilde O=O-\langle\Omega,O\Omega\rangle, \qquad \psi=\widetilde O\Omega.$$

The vacuum subtraction makes $\psi$ orthogonal to $\Omega$. Uniqueness matters here: with degenerate zero-energy states, subtract the projection onto the whole kernel of $H$ before removing zero from the spectral integral. Define the Euclidean correlator

$$C_O(t)=\langle\psi,e^{-tH}\psi\rangle, \qquad t>0.$$

The unitary real-time operator $e^{-itH}$ has become the decaying semigroup $e^{-tH}$. This formula is a Hilbert-space statement; when used for a field theory, the required physical Hilbert space must first exist.

If the spectrum is discrete, insert a complete energy basis:

$$C_O(t)=\sum_{n>0} |\langle n|\widetilde O|\Omega\rangle|^2e^{-E_nt}.$$

Every term has nonnegative weight. The slowest surviving exponential comes from the lowest energy with nonzero overlap.

The representation does not require particles

An infinite-volume field theory typically has continuous spectral contributions. The spectral theorem replaces the sum by a positive measure:

$$C_O(t)=\int_{(0,\infty)}e^{-Et}\,d\mu_O(E).$$

Here $\mu_O(B)=\langle\psi,P_H(B)\psi\rangle$, where $P_H(B)$ projects onto energies in a set $B$. This is a compact way of describing how much of the vector $\psi$ lies in each energy range.

For quantum fields, use renormalized gauge-invariant observables with suitable smearing and domains. A pointwise product of singular fields is not automatically such an operator. Spatial momentum projection is useful for spectroscopy, but in infinite volume it also requires the usual distributional or limiting interpretation.

If a physical gap $\Delta$ exists, the measure of every vacuum-orthogonal vector has no support below it. Consequently,

$$0\leq C_O(t)\leq e^{-\Delta t}\|\psi\|^2.$$

This proves a direction of implication: a spectral gap gives exponential bounds on suitable correlators. Reversing the implication requires control over a sufficiently rich class of vectors and the relevant constants. A single correlator is insufficient.

Thresholds and isolated states

Suppose the measure contains an isolated contribution $Z\delta(E-m)$ and all other contributions begin above $m$. Then the leading large-time behavior is $Ze^{-mt}$, subject to its nonzero overlap.

A continuous threshold has a different prefactor. As an elementary example, suppose near $E_*$ the density behaves as

$$\frac{d\mu_O}{dE}\simeq A(E-E_*)^\alpha, \qquad \alpha>-1.$$

Changing variable to $u=t(E-E_*)$ gives

$$C_O(t)\sim A\Gamma(\alpha+1)\frac{e^{-E_*t}}{t^{\alpha+1}}.$$

There is still an exponential threshold, but no isolated one-particle pole has been assumed. This distinction matters when interpreting the word “mass” in a general spectral statement.

The polynomial factor is also a practical warning: fitting a threshold correlator to one exponential over a short time interval can return a time-dependent apparent energy. The asymptotic form and the accessible data range must be considered together.

Effective mass

For a differentiable positive correlator, define

$$m_{\mathrm{eff}}(t)=-\frac{d}{dt}\log C_O(t) =\frac{\int E e^{-Et}\,d\mu_O(E)} {\int e^{-Et}\,d\mu_O(E)}.$$

It is an energy average with time-dependent positive weights. Under suitable moment conditions,

$$\frac{d m_{\mathrm{eff}}}{dt} =-\left(\langle E^2\rangle_t-\langle E\rangle_t^2\right)\leq0.$$

The effective mass decreases toward the lowest energy visible to the operator. In lattice units a common discrete estimator is $\log[C(n)/C(n+1)]$; divide by $a$ to obtain physical energy.

Finite periodic time extent adds backward propagation. Noisy estimates can be negative or nonmonotone even when the exact correlator has positive spectral weights. These effects must be handled before interpreting a plateau. Explore the underlying operator-overlap issue in the correlator lab.

Exercises

1. The oscillator

For a harmonic oscillator with frequency $\omega$, shift the vacuum energy to zero. What is the decay rate of $\langle0|x(t)x(0)|0\rangle$?

Hint. Write $x=(a+a^\dagger)/\sqrt{2M\omega}$.

Solution. Only $|1\rangle$ is created from the vacuum, so the correlator is $(2M\omega)^{-1}e^{-\omega t}$. It measures the gap because this operator overlaps with the first excitation.

2. A continuum threshold

If $C(t)\sim A e^{-E_*t}/t^{3/2}$, find the leading effective mass.

Hint. Differentiate its logarithm.

Solution. $m_{\mathrm{eff}}(t)=E_*+3/(2t)+o(1/t)$ under the stated asymptotic control. A slow approach need not signal a second isolated particle.

Continue at your own depth

A gap and correlation decay develops this lesson without adding a prerequisite to the next lesson.

The reconstruction references must be read as a pair: Osterwalder–Schrader I (1973) and II (1975). The latter extends and corrects the earlier formulation. The spectral and Euclidean-data reasoning here assumes the required physical Hilbert space exists; it does not replace the corrected regularity, covariance and reconstruction hypotheses. See the reconstruction guide for the scope of that statement.

Read further

• Osterwalder and Schrader, Axioms for Euclidean Green’s functions, the reconstruction framework.
• Blossier and collaborators, the generalized eigenvalue method, spectral extraction from correlation matrices.

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