Extract connected fluctuations with a logarithm

Use the dimensionless Euclidean convention of a source plus Jx in the exponent, with ℏ=1. Define W as the logarithm of the normalized source function. Its first derivative is the tilted mean. Applying the quotient rule to the next derivative subtracts the product of means, giving the covariance. This subtraction is needed even if the underlying theory has no interactions, because a nonzero mean contributes a disconnected piece to the full two-point function.

For several sources the derivative indices identify the field positions or regulated coordinates. Mixed derivatives produce connected cross correlations. At zero mean the connected two-point function equals the full two-point function, but that equality is a special simplification rather than the definition. Higher derivatives require all set partitions, not just one subtraction. Declaring the mean and source point therefore makes a proposed connected-correlation formula much easier to audit.

If two independent regulated systems have separate actions and a product measure, their normalized source functions multiply. Their logarithms add, so cumulants of independent sums add and mixed connected correlations vanish. In diagrammatic language the partition function collects arbitrary products of connected components, while the logarithm retains one connected component at a time. Dividing by the zero-source partition function removes source-independent vacuum normalization without requiring the interacting vacuum to be identical to the free one.

For a centered Gaussian, W is exactly quadratic. Every cumulant above second order therefore vanishes, although even full moments remain nonzero through pairings. A nonzero full four-point function is thus not by itself evidence of an interaction. Conversely, a nonzero fourth cumulant signals departure from a Gaussian distribution in this finite setting; it does not directly identify a particular interaction Hamiltonian without additional information.

A connected diagram can fall apart when one internal propagator is cut. Such a diagram contributes to connected correlations but is not one-particle irreducible. The Legendre transform introduced next reorganizes connected functions into those irreducible building blocks. Confusing the logarithm with that additional transform would, for example, mistake an external-leg propagator factor for an elementary interaction vertex. Connectedness and amputation also answer different questions and should be checked separately.

The probability lab supplies exact finite moments. Reset and set success probability to 0.25 and trials to 4; predict mean 1 and variance 0.75. Reset and set probability to 0.5 and trials to 4; predict mean 2 and variance 1. The changing variance is a second cumulant of the binomial count. Playback only highlights exact bars; it does not resample data or generate field-theory diagrams, and the displayed mean alone is insufficient to infer higher cumulants.