What a correlation function predicts

A correlation function records how outcomes at different positions or times vary together. For a free Gaussian field, knowing every two-point function determines all higher moments. With interactions, new connected correlations appear. In a relativistic theory with well-defined incoming and outgoing particles, those connected functions also lead to scattering amplitudes after several additional steps.

Start with the smallest calculation that shows the structure: an average of four field values.

Two points establish a length scale

For a zero-mean Euclidean field, define $G(x,y)=\langle\phi(x)\phi(y)\rangle$. The brackets mean an ensemble average, or the quantum expectation appropriate to the specified correlator. A positive value means that displacements of the same sign tend to occur together. Zero mean at each point does not force their product to average to zero.

On an infinite one-dimensional Euclidean line, the Gaussian action is

Coordinates use a chosen reference length, and the action is dimensionless. The mass parameter has inverse-length units. The covariance is the Green function of $-\partial_x^2+m^2$: away from the origin it solves the homogeneous equation, while integrating across the origin fixes the jump in its derivative and the factor $1/(2m)$.

The decay length is $\xi=1/m$. For $m=1$, the correlation falls from $0.500$ at zero separation to about $0.184$ at separation one and $0.0677$ at separation two. The figure shows how both the width and the height change when the mass doubles. These are ideal infinite-line values; finite-domain references in the experiment include their boundary conditions.

A generating function saves repeated integrals

For a finite set of field values, put their covariance in a matrix $C$. Let $J$ be a list of source strengths, one coupled to each field value. Define a normalized generating function $Z(J)=\langle\exp(J^{\mathsf T}\phi)\rangle$. A derivative with respect to one source brings down the corresponding field. Setting the sources back to zero therefore produces moments.

A Gaussian with positive quadratic kernel $K$ has $C=K^{-1}$. Complete the square in the exponent,

Shifting the integration variable leaves the normalization unchanged, so

This step explains the pattern behind Wick’s theorem. Differentiating the quadratic exponential four times can only group the four external fields into pairs. There are three possible pairings:

For one Gaussian variable with variance $0.25$, the fourth moment is $3(0.25)^2=0.1875$. The connected fourth moment subtracts these three pair products and is zero. In this Gaussian model, the covariance determines the fourth moment completely. The variance itself remains nonzero.

Interactions add connected information

Consider a relativistic real scalar field in three spatial dimensions, in natural units. Add an interaction density $\mathcal L_I=-g\phi^4/4!$ to its free Lagrangian. Here $g$ is dimensionless in four spacetime dimensions. The interaction weights configurations according to more than their quadratic amplitude.

Time evolution in the interaction picture is organized by the Dyson series. Its time-ordering operation puts the later-time operators first. Expand the exponential to first order in the coupling and evaluate averages in the free vacuum. For the connected four-point function, each external field can contract with one of the four fields at the interaction point. The $4!$ assignments cancel the factorial in the interaction:

Here $\Delta_F(x-y)=\langle0|T\phi(x)\phi(y)|0\rangle$ is the free time-ordered Minkowski propagator. It differs from the one-dimensional Euclidean function above in dimension, time convention and boundary prescription. The formula illustrates one connected interaction joining all four points.

What remains before we call it scattering

A four-point function includes propagation from the external positions to the collision region. To obtain an incoming-to-outgoing amplitude, Fourier transform the external coordinates, isolate stable single-particle poles, remove the external propagator factors and put the external momenta on their physical mass shells. The LSZ reduction formula formalizes this procedure and supplies the field-normalization factors.

At tree level in this scalar example, removing the four external propagators leaves $i\mathcal M=-ig$, or $\mathcal M=-g$ in that convention. A measured cross section also needs incident flux, final-state phase space and the correct counting of identical particles. A spatial covariance plot supplies none of those experimental ingredients by itself.

The point of the example is the connection: free two-point functions build the contractions, interactions produce connected terms, and their on-shell residues encode scattering. Loop corrections require regularization and parameter matching, which motivate the later chapters.

Correlation and influence answer different questions

Two locations can share a nonzero correlation without allowing a signal between them. In a relativistic local theory, spacelike local observables commute. A controllable response is governed by commutators and retarded propagation, not by the magnitude of a symmetric two-point function alone.

In the correlation Lab, first compare a sample estimate with its finite-lattice reference. Move the probes farther apart and predict the trend. The spin-chain quench Lab provides a separate real-time many-body setting: its Hamiltonian, observables and time evolution define a different experiment. Use it to ask how a change spreads through that specified chain, keeping its correlators separate from the Euclidean path samples.

Exercises

Calculate a mixed fourth moment

Two zero-mean Gaussian variables have $C_{11}=C_{22}=1$ and $C_{12}=1/2$. Find $\langle\phi_1^2\phi_2^2\rangle$.

Which result would reveal non-Gaussian behavior?

You measure a fourth moment that differs from the three Wick pair products. What should be checked before attributing it to an interaction?

Why keep external propagators at all?

If tree-level scalar scattering leaves a constant amplitude, why does its connected position-space four-point function depend on four positions?