A gap and correlation decay

Prerequisites: Euclidean correlators and operator overlap.

You will learn to derive a decay estimate from spectral support, diagnose a hidden light state, and state the assumptions behind a converse.

Which energies can a correlator see?

Euclidean evolution suppresses high-energy components of a state. It does not make an operator couple to a state that it could not create in the first place. This elementary fact separates a measured channel mass from the gap of a whole theory.

Assume a physical Hilbert space, a self-adjoint Hamiltonian $H\geq0$, and a normalized vacuum $\Omega$ with $H\Omega=0$. For this discussion the vacuum is unique. Choose a smeared gauge-invariant observable $O$ for which $O\Omega$ is a vector of finite norm. Its vacuum-subtracted state is

$$\psi=(O-\langle\Omega,O\Omega\rangle)\Omega.$$

Smearing and the domain requirement matter: a formal field at one point need not define such a vector. See operator domains when this issue is new.

The spectral theorem gives a positive finite measure $\mu_\psi$ with

$$C_\psi(t)=\langle\psi,e^{-tH}\psi\rangle =\int_{[0,\infty)}e^{-tE}\,d\mu_\psi(E),\qquad t\geq0.$$

Vacuum subtraction removes the atom at zero under our unique-vacuum assumption. It does not remove non-vacuum states of arbitrarily low positive energy.

The forward implication

Suppose the spectrum on $\Omega^\perp$ lies in $[\Delta,\infty)$, where $\Delta>0$. Since $e^{-tE}\leq e^{-t\Delta}$ on this interval,

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

This is a direct spectral estimate for time-separated states. Extending it to spatial clustering of local fields also uses the relativistic structure; it is not an instruction to replace $t$ by any separation in any model. The official Jaffe–Witten description, §§4–5 distinguishes the gap definition from its clustering consequences.

A continuum of energies can have a positive threshold. For example, the normalized model measure

$$d\mu(E)=\mathbf1_{E\geq m}e^{-(E-m)}\,dE$$

in units with its exponential scale equal to one gives

$$C(t)=\frac{e^{-mt}}{1+t}.$$

There is no isolated pole in this example. Nevertheless its support starts at $m>0$, and $-\partial_t\log C=m+1/(1+t)$ tends to $m$.

A worked hidden-state example

Use a three-dimensional Hilbert space with energies $0,1,3$ in chosen energy units. Let $\Omega$ be the zero-energy vector and let $O\Omega$ be only the energy-three vector. Then $C_O(t)=e^{-3t}$ even though the gap of this Hamiltonian is one. Another operator $P$ with $P\Omega$ equal to the energy-one vector gives $C_P(t)=e^{-t}$.

The first plateau was exact. The inference about the full spectrum was wrong. In a field theory there are also symmetry sectors, multiparticle states and continua; testing more operators improves evidence but does not automatically establish completeness. Explore the endpoint with zero low-state weight in the correlator lab.

When a converse is justified

There is a precise conditional converse. Suppose a dense set $D\subset\Omega^\perp$ satisfies, for a common $\Delta>0$ and every $\psi\in D$,

$$C_\psi(t)\leq K_\psi e^{-\Delta t}\quad(t\geq0),$$

where $K_\psi<\infty$ is independent of $t$. If $\mu_\psi((0,b])>0$ for some $b<\Delta$, positivity would give

$$C_\psi(t)\geq e^{-bt}\mu_\psi((0,b]),$$

contradicting the upper bound as $t\to\infty$. Thus each such low-energy spectral projection annihilates $D$. Since a spectral projection is bounded and $D$ is dense, it annihilates $\Omega^\perp$. Taking increasing $b<\Delta$ excludes spectrum in $(0,\Delta)$.

The hard field-theory work is in constructing $H$, identifying the physical states, establishing density, and proving the shared positive rate after all limits. A finite list of fitted correlators supplies none of those facts by itself.

Exercise: distinguish the threshold from the prefactor

Let $C(t)=0.001e^{-t}+0.999e^{-4t}$. Estimate when the two terms become equal. Does the small coefficient change the true lowest visible energy?

Solution. Equality gives $e^{3t}=999$, so $t=\log(999)/3\approx2.30$ in inverse-energy units. Before that time the heavier contribution can dominate. At sufficiently large time the lower energy wins because its coefficient is nonzero. The coefficient changes the crossover, not the threshold. If it were exactly zero, that operator would not reveal the energy-one state.

Scope and continuation

These arguments are exact for the stated spectral models and hypotheses, not a construction of four-dimensional Yang–Mills. Continue to reflection positivity for how Euclidean data can begin to supply a state space, then limits and uniformity for what a regulator-removal argument must preserve.