Trade the source for the mean field

For a finite positive Euclidean measure, use W(J)=log Z normalized at zero source and ℏ=1. Define the mean φ as the derivative of W. Where its covariance Hessian is positive definite, solve this relation for J as a function of φ. Define Γ as J dotted into φ minus W. Differentiating Γ cancels the terms containing the derivative of J, leaving its first derivative equal to J in this Euclidean sign convention.

A global Legendre-Fenchel transform takes a supremum over sources and remains meaningful when an ordinary inverse is unavailable. It need not be differentiable everywhere. Bounded variables restrict the attainable mean-field range, while thermodynamic limits can produce flat portions or nondifferentiable behavior. At finite volume a positive measure often smooths phenomena that become sharp only after taking an infinite-volume limit. The choice and order of these limits matter when interpreting spontaneous symmetry breaking.

Differentiate the relation between the source and mean once more. The Hessian of Γ is the matrix inverse of the Hessian of W, hence the inverse connected covariance in the declared units. In a field regulator this is an inverse kernel with an explicit integration measure and boundary conditions. A raw second moment containing a mean-squared contribution is not the object to invert. Null response directions require restriction or a more careful transform, just as quadratic zero modes did earlier.

Diagrammatically the transform removes the reducible chains that connect irreducible pieces through a single full propagator. Derivatives of Γ supply the proper vertices; connected correlations are reconstructed by joining these vertices in trees with full propagators. The exact Γ already includes loop information. Integrating it again as though it were a fresh bare action and keeping all loops would double count that information rather than reproduce the original quantum theory.

For the real-time convention ZM=exp(iWM) with source plus Jφ, the commonly used effective action is ΓM=WM−∫Jφ. Its derivative is minus J. This sign differs from the positive Euclidean Legendre transform above, and the second derivative of WM itself contains a factor i times the connected correlator. Copying only the name Γ between conventions does not preserve these derivative formulas. Define the convention whenever comparing references.

The optimization lab illustrates a local iterative issue rather than constructing Γ. Reset and set step size to 0.5 and initial position to 1, then Step once; predict position 0.5. Reset and set step size to 2.5 and initial position to 1, then Step once; predict −1.5, an increased magnitude. The objective is fixed at x²/2, so these runs test step stability. They do not invert a covariance, evaluate a Legendre transform, or establish convergence of a quantum effective-action truncation.