Distributions and Green functions

A test function is smooth and compactly supported. A distribution is a continuous linear functional on the space of these probes, with continuity understood in the test-function topology. Every locally integrable function defines a distribution by integration against a probe. The delta at a instead returns the probe's value at a. Writing δ(x−a) is convenient notation for this functional; it does not create a finite-valued function that vanishes everywhere except at one point.

A family of narrow, normalized smooth bumps can approach delta distributionally: every fixed smooth probe sees its weighted average approach the probe's value at the center. The bumps need not converge to an ordinary integrable function with area one. This distinction resolves the apparent paradox of an infinitely narrow source. In quantum field theory even a field may be operator-valued distributional data, so smearing against probes is mathematically more basic than assigning an operator independently at every spacetime point.

Integration by parts suggests the definition of a distributional derivative: apply the original distribution to the derivative of the test function and insert a minus sign. Compact support removes boundary terms. This extends ordinary differentiation and permits derivatives of jumps. The Heaviside step has derivative delta because the integral of the probe's derivative over the positive half-line is minus its value at zero. The arbitrary assigned value of the step at the single point zero is irrelevant to its associated distribution.

A distribution can be multiplied by a smooth function because the smooth function times a test function is still a valid test function. A product of two arbitrary distributions has no such general definition. Expressions such as δ² require extra structure or a specified regularization and limiting procedure, and different procedures may disagree. Renormalization is not simply permission to treat singular objects as ordinary numbers; it supplies controlled prescriptions and consistency conditions for the products arising in particular theories.

For a linear differential operator L, a fundamental solution satisfies LG=δ in a translation-invariant full-space setting. Convolving G with a suitable source then constructs a solution, when the convolution exists. On a bounded domain, a Green kernel generally depends separately on the observation and source positions and must satisfy the selected boundary conditions. Adding any homogeneous solution leaves the source equation unchanged, which is why the equation LG=δ alone rarely selects a unique physical response.

For time-dependent problems, retarded and advanced Green functions answer different questions. The retarded solution has no response before the source acts. A Feynman propagator uses a time-ordering prescription appropriate to quantum correlation functions; it is not identical to a classical retarded response. In frequency space these distinctions often appear as pole prescriptions. They carry physical information that is lost if one merely writes an algebraic reciprocal of the differential operator's symbol.