From Correlation Functions to QFT Observables

In elementary quantum mechanics, an observable is represented by an operator, and predictions are probabilities or expectation values in a specified state. Quantum field theory keeps that logic, but its most useful calculational data are usually not isolated field values. They are correlation functions of fields or composite operators at several spacetime points.

Different correlators answer different physical questions:

Physical question	Correlator or derived object
Which excitations exist?	poles, thresholds, and spectral weights of two-point functions
How does a system respond to a source?	retarded correlator or susceptibility
What is a scattering amplitude?	on-shell residue of a connected time-ordered correlator
What can a Euclidean calculation measure?	long-distance decay and moments of Schwinger functions
How do symmetries constrain dynamics?	current correlators and Ward identities
Which fluctuations are present?	ordered, symmetrized, or connected correlators

The correlator is rarely the final experimental number by itself. One must still choose the state, ordering, contour, operator normalization, Fourier convention, and conversion rule appropriate to the observable.

The detailed path-integral construction is developed in Correlation Functions in Path Integrals, the pole structure is explained in Spectral Representation of Green Functions, response is derived in Green Functions and Response Preview, and scattering conventions are given in QFT Bridge: S-Matrix. Here those constructions are used to extract physical quantities.

Quantum-Mechanical Starting Point

Let $\lvert\Omega\rangle$ be a stationary state with energy $E_\Omega$. For Heisenberg operators $A(t)$ and $B(0)$, consider

$$C_{AB}(t) := \langle\Omega\rvert A(t)B(0) \lvert\Omega\rangle.$$

Insert a complete energy basis:

$$C_{AB}(t) = \sum_n e^{-i(E_n-E_\Omega)t/\hbar} \times \langle\Omega\rvert A\lvert n\rangle \langle n\rvert B\lvert\Omega\rangle.$$

This one formula already contains the central QFT logic:

• oscillation frequencies reveal energy differences;
• residues reveal matrix elements and selection rules;
• degeneracies and continua alter the spectral structure;
• the chosen operators decide which states have nonzero overlap.

The connected correlator removes factorized one-point data:

$$C_{AB}^{\mathrm c}(t) := \langle A(t)B(0)\rangle - \langle A\rangle\langle B\rangle.$$

For a nondegenerate ground state, this subtraction removes the $n=0$ vacuum contribution when $A$ and $B$ have nonzero expectation values. The smallest coupled gap sets the lowest nonzero spectral frequency. In real time, a finite discrete sum keeps all its oscillatory terms: large time alone does not isolate the lowest one. Lowest-gap dominance instead occurs at large positive Euclidean time, with the appropriate spectral assumptions.

At finite temperature or in a mixed stationary state, the spectral sum contains two energy labels and statistical weights. The principle remains the same: a correlator combines the spectrum with operator overlaps and state populations.

There Is No Single Correlation Function

For noncommuting operators, ordering is part of the definition:

Name	Schematic definition	Primary use
Wightman	$\langle A(x)B(y)\rangle$	fixed-order fluctuations and spectral positivity
Time ordered	$\langle\mathcal T A(x)B(y)\rangle$	perturbation theory, in-out amplitudes, source derivatives
Retarded	$-\frac{i}{\hbar}\theta(x^0-y^0)\langle[A(x),B(y)]\rangle$	causal response
Symmetrized	$\frac12\langle\{A(x),B(y)\}\rangle$	fluctuation and noise conventions
Euclidean	$\langle\mathcal T_\tau A(x_E)B(y_E)\rangle$	imaginary-time and lattice calculations
In-in	contour-ordered expectation in an initial state	real-time expectation values away from simple scattering

These objects can share a spectral density while differing in boundary values, support, thermal factors, and factors of $i$. Calling all of them “the propagator” or “the Green function” hides the information needed to use them.

Fermionic time ordering also contributes a minus sign for every odd permutation of fermionic operators. Thermal correlators obey periodic or antiperiodic imaginary-time conditions according to total fermion parity.

The QFT n-Point Hierarchy

For a scalar field and vacuum $\lvert0\rangle$, the fixed-order Wightman functions are

$$W_n(x_1,\ldots,x_n) := \langle0\rvert \phi(x_1)\cdots\phi(x_n) \lvert0\rangle,$$

while the time-ordered functions are

$$G_n(x_1,\ldots,x_n) := \langle0\rvert \mathcal T \left\{ \phi(x_1)\cdots\phi(x_n) \right\} \lvert0\rangle.$$

In a translation-invariant vacuum, an $n$-point function depends on coordinate differences rather than all positions independently. Lorentz symmetry, internal symmetries, locality, statistics, and conservation laws further constrain its tensor and analytic structure.

Quantum fields are operator-valued distributions. The rigorous objects are smeared:

$$\phi(f) := \int d^D x\, f(x)\phi(x),$$

and the correlation functions act as multilinear distributions on test functions. Coincident-point products such as $\phi^2(x)$ define composite operators only after regularization and renormalization.

Under suitable Wightman assumptions, the full hierarchy of vacuum correlation functions determines the Hilbert space, fields, vacuum, and symmetry action up to equivalence. This reconstruction statement explains why correlators can serve as primary data for a QFT. It does not imply that a few low-order correlators determine an arbitrary interacting theory, nor does it bypass gauge, positivity, or existence problems.

Sources, Full Correlators, and Connected Correlators

For a Lorentzian vacuum generating functional normalized by $Z[0]=1$,

$$Z[J] = \left\langle0\left| \mathcal T \exp\left[ \frac{i}{\hbar} \int d^D x\, J(x)\phi(x) \right] \right|0\right\rangle,$$

functional derivatives generate time-ordered products:

$$G_n(x_1,\ldots,x_n) = \left. \left( \frac{\hbar}{i} \right)^n \frac{\delta^n Z[J]} {\delta J(x_1)\cdots\delta J(x_n)} \right|_{J=0}.$$

Define the connected generator

$$W[J] := -i\hbar\log Z[J].$$

Derivatives of $W[J]$ generate connected correlators, with powers of $\hbar/i$ fixed by the derivative order and convention. Vacuum factors have already canceled in the normalized $Z$. Its logarithm removes products associated with disconnected external insertion sets.

Several related objects must be distinguished:

Object	What has been removed or reorganized	Typical role
Full correlator	nothing	complete expectation value
Connected correlator	disconnected factorized pieces	clustering, response, scattering
Amputated correlator	external propagator factors	approach to a vertex or amplitude
One-particle-irreducible vertex	pieces disconnected by cutting one internal line	effective action and self-energies

These are not interchangeable names for the same function. From Sources in QM to Generating Functionals in QFT develops the functional-derivative derivation, while From Evolution Operators to Time-Ordered Products explains its interaction-picture origin.

Two-Point Functions as Spectral Detectors

Set $\hbar=c=1$ for this section. For a Hermitian scalar operator in a positive-metric relativistic vacuum theory, subtract its vacuum mean. The nonlocal part of its momentum-space Feynman two-point function has a Källén–Lehmann representation of the form

$$\widetilde G_F(p) = \int_0^\infty d\mu^2\, \rho(\mu^2) \frac{i} {p^2-\mu^2+i0}.$$

Composite operators may need polynomial subtractions or local contact terms in addition to this spectral integral. If the operator overlaps a stable one-particle state of mass $m$, the spectral density contains an isolated contribution:

$$\rho(\mu^2) = Z_\phi \delta(\mu^2-m^2) + \rho_{\mathrm{cont}}(\mu^2).$$

The physical interpretations are:

• the pole location gives the stable particle mass;
• the residue $Z_\phi$ measures the overlap of the chosen field with that one-particle state;
• the continuum begins at multiparticle thresholds allowed by the quantum numbers;
• branch cuts encode continua rather than individual states;
• unstable resonances generally do not appear as positive delta-function contributions for asymptotic particles.

Pole locations and threshold positions can be physical, while residues depend on operator normalization and field choice. Two different interpolating operators with the same quantum numbers can reveal the same mass spectrum but have very different overlaps.

In gauge theories, propagators of gauge-dependent elementary fields need not have a positive spectral density and need not themselves be observables. Gauge-invariant composite operators, conserved currents, and on-shell quantities provide safer physical data.

Retarded Correlators as Response

Suppose an external source $f(y)$ couples to an operator $B(y)$ through

$$\delta H(t) = - \int d^d y\, f(y)B(y).$$

To first order, the induced change in $A(x)$ is

$$\delta\langle A(x)\rangle = \int d^D y\, \chi_{AB}^{R}(x-y) f(y),$$

where

$$\chi_{AB}^{R}(x-y) := \frac{i}{\hbar} \theta(x^0-y^0) \langle [A(x),B(y)] \rangle$$

for the source-sign convention used here. Hence $\chi^R=-G_R$ if $G_R$ uses the $-i/\hbar$ convention in the earlier table. This sign changes the response to a specified force and must be carried through the Fourier transform. The step function guarantees that a later perturbation cannot change an earlier expectation value.

In field theory, response correlators include:

• current–current correlators for conductivity and polarization;
• density–density correlators for compressibility and structure;
• stress-tensor correlators for viscosity and energy transport;
• order-parameter correlators for susceptibilities near phase transitions.

The observable is obtained only after specifying tensor projections, source normalization, contact terms, and the order of limits. For example, uniform and static limits need not commute:

$$\lim_{\omega\to0} \lim_{\mathbf q\to0} \chi(\omega,\mathbf q) \ne \lim_{\mathbf q\to0} \lim_{\omega\to0} \chi(\omega,\mathbf q).$$

Ward identities from symmetries constrain these correlators. Contact terms are often required for the identity and cannot be discarded merely because they are local. Green Functions and Response Preview gives the canonical first-order derivation and convention checks.

Scattering from On-Shell Correlators

Time-ordered correlators are off-shell functions of their external momenta. The LSZ reduction procedure extracts scattering amplitudes by isolating stable one-particle poles and removing the external propagators.

For a scalar process, the structure is schematically

$$\mathcal M \sim \left[ \prod_i Z_i^{-1/2} \lim_{p_i^2\to m_i^2} \left( p_i^2-m_i^2 \right) \right] \times \widetilde G_{n,\mathrm c} (p_1,\ldots,p_n),$$

with momentum-conserving delta functions and convention-dependent factors separated from $\mathcal M$. The required input is the connected time-ordered correlator. Disconnected pieces describe independent propagation or spectator factors, not one connected scattering event.

LSZ requires identifiable asymptotic particle poles. Unstable particles, confinement, long-range gauge forces, massless infrared sectors, curved backgrounds, or thermal media can obstruct the elementary picture or require modified observables. Even when LSZ applies, the amplitude is not yet a cross section. One must square it, sum or average over quantum numbers, integrate over Lorentz-invariant phase space, and include flux and symmetry factors.

QFT Bridge: S-Matrix explains the normalization and asymptotic-state comparison. This page records why correlation functions are the object from which LSZ starts.

Euclidean Correlators and Lattice Observables

Let $O$ be a zero-momentum interpolating operator with its vacuum expectation subtracted. The correlator $C_E(\tau)=\langle O(\tau)O^\dagger(0)\rangle$ has spectral form

$$C_E(\tau) = \sum_n \left| \langle0\rvert O\lvert n\rangle \right|^2 e^{-(E_n-E_0)\tau} \qquad (\tau\gt0),$$

in natural units and finite volume. At large $\tau$, the lowest state with nonzero operator overlap dominates:

$$C_E(\tau) \sim \left| \langle0\rvert O\lvert1\rangle \right|^2 e^{-(E_1-E_0)\tau}.$$

This makes Euclidean decay a practical mass or energy-gap estimator. A lattice calculation commonly chooses several gauge-invariant interpolating operators, computes their correlation matrix, and separates low-lying exponentials through fits or variational methods.

Finite Temporal Extent Can Mimic a Smaller Gap

The preceding exponential is a vacuum result with unbounded Euclidean time. On a thermal circle of circumference $\beta$, a free mode of energy $E>0$ instead has the exact correlator derived in the free scalar thermal example:

$$C_\beta(\tau) =A\left[e^{-E\tau}+e^{-E(\beta-\tau)}\right], \qquad A=\frac{1}{2E(1-e^{-\beta E})}, \qquad 0\leq\tau\leq\beta.$$

This is the unit-mass harmonic-oscillator result, or one momentum mode of a free scalar field, in units $\hbar=c=k_B=1$. The two terms carry the same gap. Their relative weight is fixed by the thermal trace; the equivalent hyperbolic form is also given by Laine and Vuorinen, Eq. (3.38). No second excitation energy is needed to change the apparent logarithmic slope:

$$-\frac{d}{d\tau}\log C_\beta(\tau) =E\tanh\!\left[E\left(\frac\beta2-\tau\right)\right].$$

At $\tau=\beta/2$ this slope vanishes although $E$ is strictly positive. Thus taking later data at fixed $\beta$ does not reproduce the vacuum large-$\tau$ limit. To use the single-exponential slope on the first half of the circle, the backward-to-forward ratio $e^{-E(\beta-2\tau)}$ must be small. For an interacting correlator, that requirement must coexist with suppression of higher excitations. A usable fitting interval need not exist at the available temporal extent.

A single hyperbolic cosine accounts for this boundary effect in the free example; Exercise 7 gives an exact discrete estimator. Interacting thermal correlators can include several transitions and constant contributions, so replacing an exponential fit by one hyperbolic cosine does not itself establish single-state dominance.

Finite volume makes the spectrum discrete and modifies thresholds. Finite lattice spacing modifies short-distance behavior. Taking the continuum and infinite-volume limits is part of the observable extraction, not an optional cleanup.

Euclidean correlators can also encode spectral and transport information, but recovering a real-time spectral density from noisy discrete data is ill-conditioned. From Euclidean Time to Euclidean QFT explains thermal circles, Matsubara frequencies, analytic continuation, and reflection positivity.

Symmetry Data and Ward Identities

Correlation functions make symmetry constraints calculational. If $J^\mu$ is a conserved current, differentiating a time-ordered correlator produces contact terms at operator insertions. Schematically,

$$\partial_\mu \langle \mathcal T J^\mu(x) O_1(x_1)\cdots O_n(x_n) \rangle = \sum_k \delta^{(D)}(x-x_k) \times \langle \mathcal T O_1\cdots\delta O_k\cdots O_n \rangle,$$

up to factors and signs fixed by the current and transformation conventions. These Ward identities relate different $n$-point functions and protect or constrain charges, masses, tensor structures, and renormalization.

Examples of physical information carried by symmetry correlators include:

• charge normalization from current matrix elements;
• polarization tensors from current two-point functions;
• energy density and transport from stress-tensor correlators;
• order parameters and susceptibilities from symmetry-breaking operators;
• anomaly coefficients from otherwise violated Ward identities.

The operator must be properly renormalized, and currents can mix with other operators allowed by the same symmetries. A formal insertion is not automatically a normalized observable.

From Correlator to Observable

The conversion can be summarized as follows:

Goal	Starting correlator	Required conversion
Stable mass or gap	two-point function	locate a pole or fit long Euclidean-time decay
Spectral density	Wightman, retarded, or Euclidean two-point data	choose spectral convention and perform boundary-value or inverse analysis
Linear susceptibility	retarded $AB$ correlator	specify source sign, tensor projection, contacts, and limits
Scattering amplitude	connected time-ordered $n$-point function	amputate, take on-shell residues, and normalize external states
Cross section or decay rate	on-shell amplitude	add phase space, flux, sums, averages, and symmetry factors
Static structure factor	density–density correlator	Fourier transform with experimental kinematics and normalization
Lattice energy level	Euclidean correlation matrix	control excited states, volume, spacing, and continuum extrapolation

This table is a map, not a set of interchangeable formulas. Each row uses a different ordering, state, limiting procedure, or normalization.

Fields Are Not Always Observables

An elementary field is often an interpolating operator rather than a directly measurable quantity. Its normalization can change under field redefinitions, and in a gauge theory it may be gauge dependent. Correlation functions of such fields remain useful intermediate objects, but physical claims should be attached to invariant consequences.

Further cautions include:

• composite operators require renormalization and can mix;
• off-shell Green functions can depend on gauge and scheme;
• pole masses, inclusive rates, conserved charges, and properly defined on-shell amplitudes have different invariance properties;
• infrared-safe experimental quantities may require sums over unresolved radiation;
• vacuum, thermal, in-out, and in-in correlators describe different states or measurement protocols;
• truncating the correlator hierarchy is an approximation whose error must be assessed.

The phrase “QFT observable” therefore includes both operator-algebra observables and experimentally inferred quantities. A calculated correlator becomes a prediction only after this conversion is specified.

Common Mistakes

• Calling every two-point function a propagator without specifying ordering and state.
• Treating a connected correlator, an amputated function, and a scattering amplitude as identical.
• Reading a field-strength residue as a universal observable independent of the interpolating operator.
• Identifying a gauge-dependent elementary-field correlator with a direct measurement.
• Using a time-ordered correlator where causal response requires a retarded commutator.
• Taking static and uniform limits without specifying their order.
• Extracting a cross section from $\lvert\mathcal M\rvert^2$ without flux, phase space, and symmetry factors.
• Treating an unstable resonance as an ordinary asymptotic one-particle state.
• Assuming Euclidean data can be continued to real time without uncertainty or additional input.
• Ignoring contact terms, operator mixing, and renormalization in current or composite-operator correlators.
• Forgetting that the state or contour changes the correlator being computed.

References

• G. Källén, “On the Definition of the Renormalization Constants in Quantum Electrodynamics,” Helvetica Physica Acta 25, 417–434, 1952.
• R. Kubo, “Statistical-Mechanical Theory of Irreversible Processes. I,” Journal of the Physical Society of Japan 12, 570–586, 1957, doi:10.1143/JPSJ.12.570.
• M. Laine and A. Vuorinen, Basics of Thermal Field Theory, Lecture Notes in Physics 925, Springer, 2016; updated lecture notes arXiv:1701.01554, Eqs. (3.38) and (8.33).
• H. Lehmann, “Über Eigenschaften von Ausbreitungsfunktionen und Renormierungskonstanten quantisierter Felder,” Il Nuovo Cimento 11, 342–357, 1954, doi:10.1007/BF02783624.
• H. Lehmann, K. Symanzik, and W. Zimmermann, “On the Formulation of Quantized Field Theories,” Il Nuovo Cimento 1, 205–225, 1955, doi:10.1007/BF02731765.
• K. Osterwalder and R. Schrader, “Axioms for Euclidean Green’s Functions,” Communications in Mathematical Physics 31, 83–112, 1973, doi:10.1007/BF01645738.
• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014.
• R. F. Streater and A. S. Wightman, PCT, Spin and Statistics, and All That, Princeton University Press, 2000.
• S. Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995.

Exercises

1. Connected spectral sum

Let $\lvert0\rangle$ be a nondegenerate ground state and $A=A^\dagger$. Show that

$$C_A^{\mathrm c}(t) := \langle0\rvert A(t)A(0)\lvert0\rangle - \langle0\rvert A\lvert0\rangle^2$$

contains no $n=0$ term in its spectral sum. What determines its smallest nonzero oscillation frequency? Does it have to decay in real time?

Solution

Insert the energy identity:

$$\langle0\rvert A(t)A(0)\lvert0\rangle = \sum_n e^{-i(E_n-E_0)t/\hbar} \left| \langle0\rvert A\lvert n\rangle \right|^2.$$

The $n=0$ term is

$$\left| \langle0\rvert A\lvert0\rangle \right|^2,$$

which is exactly removed by the connected subtraction. Therefore

$$C_A^{\mathrm c}(t) = \sum_{n\ne0} e^{-i(E_n-E_0)t/\hbar} \left| \langle0\rvert A\lvert n\rangle \right|^2.$$

The slowest nonconstant frequency is set by the smallest gap $E_n-E_0$ for which $\langle0\rvert A\lvert n\rangle$ is nonzero. The lowest energy in the full spectrum is irrelevant if the operator cannot couple to it because of symmetry or other selection rules. A finite sum of such phases does not generally decay; its recurrences and beat times are distinct from the inverse of the lowest gap.

2. Same mass, different residues

Two scalar interpolating operators $O_1$ and $O_2$ carry the same quantum numbers and both overlap a stable one-particle state $\lvert p\rangle$ of mass $m$. Explain why their two-point functions have a pole at the same $p^2=m^2$ but can have different residues.

Solution

The spectral resolution of each two-point function contains the same physical intermediate state. Its energy–momentum relation fixes the pole location:

$$p^2=m^2.$$

The coefficient is the squared operator-state overlap:

$$Z_i \propto \left| \langle0\rvert O_i(0)\lvert p\rangle \right|^2.$$

Because $O_1$ and $O_2$ can be normalized differently or contain different mixtures of local structures, $Z_1$ and $Z_2$ need not agree. The mass is a property of the state; the residue also records how efficiently the chosen operator creates that state from the vacuum.

3. Why the logarithm selects connected pairs

For a normalized generating functional $Z[J]$, show directly that the second derivative of $\log Z[J]$ at $J=0$ subtracts the product of one-point functions.

Solution

Differentiate:

$$\frac{\delta^2\log Z} {\delta J(x)\delta J(y)} = \frac{1}{Z} \frac{\delta^2 Z} {\delta J(x)\delta J(y)} - \frac{1}{Z^2} \frac{\delta Z}{\delta J(x)} \frac{\delta Z}{\delta J(y)}.$$

At $J=0$, source derivatives of $Z$ insert fields, with convention-dependent factors of $i/\hbar$. Since $Z[0]=1$, the expression is proportional to

$$\langle \mathcal T\phi(x)\phi(y) \rangle - \langle\phi(x)\rangle \langle\phi(y)\rangle.$$

This is the connected two-point function. Higher derivatives of $\log Z$ similarly subtract every disconnected partition of the insertion set.

4. Causality and analyticity of retarded response

Let

$$\chi^R(t) = \theta(t)F(t),$$

where $F(t)$ is locally integrable and at most polynomially growing for large $t$. Explain why $\chi^R(t)$ gives no response before a source is applied and why $\chi^R(\omega)$ is analytic for $\operatorname{Im}\omega\gt0$.

Solution

The step function makes

$$\chi^R(t)=0 \qquad \text{for }t\lt0.$$

Thus a source at time $t'$ contributes only to measurements at $t\gt t'$. For complex frequency,

$$\chi^R(\omega) = \int_0^\infty dt\, e^{i\omega t}F(t).$$

If $\operatorname{Im}\omega\gt0$, then

$$\left| e^{i\omega t} \right| = e^{-\operatorname{Im}\omega\,t},$$

which supplies exponential damping. Under the stated growth condition, the integral converges uniformly on compact subsets of the upper half-plane and defines an analytic function there. Retarded support and upper-half-plane analyticity are two forms of the same causal structure.

5. No scattering from a free four-point function

For a free scalar vacuum, Wick’s theorem gives the four-point function as a sum of products of two-point functions. Why does LSZ yield no connected $2\to2$ scattering amplitude?

Solution

The free four-point function is

$$\langle \mathcal T\phi_1\phi_2\phi_3\phi_4 \rangle = G_{12}G_{34} + G_{13}G_{24} + G_{14}G_{23}.$$

Every term factorizes into two disconnected propagators. Therefore the connected four-point function vanishes:

$$G_{4,\mathrm c} = 0.$$

LSZ amputates the connected correlator for one connected scattering event, so

$$\mathcal M_{2\to2} = 0.$$

The disconnected terms describe free propagation and identity contributions to the $S$-matrix, not an interaction-induced collision.

6. Euclidean effective energy

Suppose

$$C_E(\tau) = A_0e^{-E_0\tau} \left( 1+r e^{-\Delta\tau} \right),$$

with $A_0\gt0$, $r\gt0$, and $\Delta\gt0$. Define

$$E_{\mathrm{eff}}(\tau) := - \frac{d}{d\tau} \log C_E(\tau).$$

Find $E_{\mathrm{eff}}(\tau)$ and its large-$\tau$ limit.

Solution

Differentiate the logarithm:

$$\begin{aligned} E_{\mathrm{eff}}(\tau) &= E_0 + \frac{ \Delta r e^{-\Delta\tau} }{ 1+r e^{-\Delta\tau} }. \end{aligned}$$

As $\tau\to\infty$,

$$r e^{-\Delta\tau} \to 0,$$

so

$$E_{\mathrm{eff}}(\tau) \to E_0.$$

The approach is exponential with gap $\Delta$. In a real lattice analysis, additional states, noise growth, finite temporal extent, and correlated fit uncertainties complicate this simple plateau picture.

7. A gap estimator on the thermal circle

For the free-mode correlator above, let $a_\tau>0$ be the sampling interval and use three points with $a_\tau\leq\tau\leq\beta-a_\tau$. Show that

$$E_{\cosh}(\tau) =\frac{1}{a_\tau}\operatorname{arcosh} \left[\frac{C_\beta(\tau-a_\tau)+C_\beta(\tau+a_\tau)}{2C_\beta(\tau)}\right] =E.$$

For $E=1$, $\beta=6$, and $a_\tau=0.2$ in consistent natural units, compare this estimator with the logarithmic slope at $\tau=1$ and $\tau=3$. Explain why a noisy estimate whose arcosh argument is below one is not evidence for an imaginary excitation energy.

Solution

Write $C_\beta(\tau)=2Ae^{-E\beta/2}\cosh[E(\beta/2-\tau)]$. The addition identity gives

$$C_\beta(\tau-a_\tau)+C_\beta(\tau+a_\tau) =2\cosh(Ea_\tau)\,C_\beta(\tau).$$

The principal real arcosh therefore returns $Ea_\tau$, since $E a_\tau>0$. The estimator gives $E_{\cosh}=1$ at both points. The logarithmic slopes are respectively $\tanh2\simeq0.96403$ and $0$.

For exact data in this model the argument is $\cosh0.2\simeq1.02007$. Statistical errors, subtraction errors, or failure of the assumed correlator model can take an estimated ratio below one; then this real-valued diagnostic is undefined. Assess the data covariance and fit model instead of interpreting a complex arcosh as a physical gap. Agreement with a single-cosh formula checks the assumed spectral form, not the absence of all finite-spacing or finite-volume effects.