Derive quantum equations with their contact terms

Use a finite Euclidean integral with weight exp(−S(x)+Jx), normalized by its convergent partition function and with ℏ=1. Differentiate an insertion f times that weight and integrate over the declared domain. If the boundary term vanishes, the average derivative of f equals the average of f times S′−J. For the full real line, decay of the product at both ends is sufficient. A restricted interval or a rapidly growing insertion can invalidate that step.

In several dimensions there is one identity for each coordinate. The field-theory notation follows by replacing a regulated coordinate derivative with a functional derivative and retaining the regulator’s measure convention. A nontrivial measure can contribute its own logarithmic derivative; it should not be treated as a flat product silently. Gauge fixing, constraints, and possible anomalous Jacobians require particular care before a formal shift of every field variable is assumed harmless.

Choosing f to be another coordinate gives a constant derivative, so the two-point identity has a nonzero right-hand side. In continuum language, differentiating an inserted field produces a delta function when the varied field point coincides with that insertion. These contact terms ensure that a Gaussian two-point kernel is the inverse of the quadratic action operator. Dropping them would incorrectly turn the propagator equation into a homogeneous classical equation everywhere.

At real time with weight exp(iS) the same integration-by-parts calculation introduces a factor i. Its sign must be derived from that exponent rather than copied from the Euclidean identity. Time ordering also makes coincident-point distributions important. Renormalized composite operators can require local counterterms, so products of continuum fields at the same point are not automatically finite quantities merely because their regulated identity was exact.

A quartic action couples an equation for the second moment to the fourth moment, whose equation involves the sixth, and so on. A Gaussian closure replaces the fourth moment by three times the variance squared in a centered state. This can provide a useful estimate, but it discards non-Gaussian cumulants and is not the exact Schwinger–Dyson hierarchy. Solving the closed equation to many digits tests only that approximation’s internal algebra; it does not bound its error against the original integral.

The root-finding lab provides a narrow comparison. Reset and set target c to 1 and upper bound U to 2, then Step once; predict midpoint 1 and zero bracket bound because the midpoint is an exact root. Reset and set c to 2 and U to 2, then Step once; predict midpoint 1.5 and bound 0.5. These certified brackets apply to x²−c. They do not certify a chosen hierarchy truncation or solve the quartic moment closure discussed here.