What one correlator can and cannot reveal

A correlation function is a measurement through a particular operator. Even perfect data can miss states that the operator does not create. Imperfect data can also hide states with a small overlap.

Prerequisite: the spectral representation.

By the end:

• Construct an example with a hidden lighter state.
• Explain the purpose and limits of an operator basis.

A completely soluble blind spot

Consider the harmonic oscillator, with its vacuum energy subtracted. Its first excitation has energy $\omega$. The position operator $x$ connects the vacuum to $|1\rangle$, so its correlator decays as $e^{-\omega t}$.

Now use the vacuum-subtracted even operator

$$Q=x^2-\langle0|x^2|0\rangle.$$

Writing $x=(a+a^\dagger)/\sqrt{2M\omega}$ gives

$$Q|0\rangle=\frac{\sqrt2}{2M\omega}|2\rangle, \qquad \langle0|Qe^{-tH}Q|0\rangle =\frac{1}{2M^2\omega^2}e^{-2\omega t}.$$

An exact, clean, single exponential with decay rate $2\omega$ has missed the actual gap $\omega$. Parity forbids the overlap with the first excitation.

The same logic applies to field-theory quantum numbers. A scalar operator does not probe every spin channel. A parity-even operator does not create parity-odd states from a parity-invariant vacuum. Measuring one channel, however accurately, does not establish a lower bound on all physical excitations.

A small overlap can imitate absence

Consider a synthetic correlator with two positive spectral weights,

$$C(t)=A e^{-E_1t}+B e^{-E_2t}, \qquad 0<E_1<E_2,\quad A,B>0.$$

Its continuous effective mass is

$$m_{\mathrm{eff}}(t)=E_1+ \frac{(E_2-E_1)r e^{-(E_2-E_1)t}} {1+r e^{-(E_2-E_1)t}}, \qquad r=B/A.$$

The limit is $E_1$, but if $A$ is very small the second contribution dominates for a long time. Their sizes cross near

$$t_\times=\frac{\log(B/A)}{E_2-E_1}$$

when $B>A$. For $A=0.01$, $B=0.99$, $E_1=1$, and $E_2=3$, the crossover is about $2.30$ in the chosen time units. Data ending substantially before that region can suggest the wrong scale.

In actual simulations, going to larger time also loses statistical precision. “Just wait longer” therefore encounters a measurement problem. A stable-looking fit over a narrow time window does not settle it.

Try changing the overlap in the correlator lab while keeping both energies fixed. The physical model has not changed, but the apparent plateau can change dramatically.

Improve the probe, not the spectrum

Use several operators with the same desired quantum numbers:

$$C_{ij}(t)= \langle\Omega|\widetilde O_i^\dagger e^{-tH} \widetilde O_j|\Omega\rangle.$$

For coefficients $v_i$, the combination $\sum_i v_i\widetilde O_i$ can have a better overlap with a chosen state than any single basis operator.

The generalized eigenvalue problem

$$C(t)v_n=\lambda_n(t,t_0)C(t_0)v_n$$

organizes such combinations. When the relevant basis and asymptotic assumptions hold, the leading dependence of a principal correlator is approximately $e^{-E_n(t-t_0)}$, with contamination from states outside the retained basis. Blossier and collaborators analyze these corrections.

Smearing and an enlarged operator basis can improve overlap. They do not alter the physical energies of the underlying theory when used as measurement choices. A finite basis still cannot prove that every possible low-energy state has been detected.

Why variational estimates need careful interpretation

For a normalized state $\psi$ orthogonal to a unique vacuum,

$$\langle\psi,H\psi\rangle\geq \inf\bigl(\operatorname{spec}(H)\setminus\{0\}\bigr).$$

A trial state therefore produces an upper bound on the bottom of the excited spectrum, assuming the relevant domain and orthogonality conditions. Finding a trial energy near a positive value does not provide the lower bound needed for a mass-gap proof.

This direction of the inequality is easy to reverse informally. A successful approximation to a low excitation is valuable spectroscopy, but “I found no lower state” is not the same as “no lower state exists.”

For the full gap, one needs an estimate that controls the appropriate entire vacuum-orthogonal space, or an equivalent sufficiently complete class of physical observables. The Proof Map keeps this coverage question visible.

Exercises

1. Remove the lighter state

In the two-state example, set $A=0$. What happens to the large-time limit of the effective mass?

Hint. Take the limit only after setting the weight to zero.

Solution. The result is $E_2$. Taking $t\to\infty$ first at each fixed $A>0$ gives $E_1$, so the limiting operations are not interchangeable at this boundary.

2. What does a trial state prove?

A normalized, vacuum-orthogonal trial state has mean energy $2$. Does this prove that the physical gap is at least $2$?

Hint. Apply the variational inequality.

Solution. It proves only that the bottom of the excited spectrum is at most $2$. The gap could be smaller, or zero. A positive lower bound requires a different argument.

Continue at your own depth

Spectral extraction with an operator basis develops this lesson without adding a prerequisite to the next lesson.

Read further

• Blossier and collaborators, generalized eigenvalue methods.
• Athenodorou and Teper, operator channels and glueball spectroscopy.

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