Playground 13

Generating Functional and Source Trick

Introduce a source J(x), differentiate the generating functional to pull down field insertions, and watch correlators and diagrams emerge from one master object.

Animation Controls

Current build step

1. Add a source J(x)

Source strength0.70

Derivative order2

Interaction strength0.55

Master object

Z[J]

Insertions

Correlator

2-point

Source Trick Build

Differentiate once, insert once

Every functional derivative with respect to the source brings down one field insertion from the exponential.

\frac{\delta Z[J]}{\delta J(x)} \sim \langle \phi(x) \rangle

Master correlator machine

Higher derivatives generate higher-point correlators without rebuilding the path integral from scratch each time.

\frac{\delta^{2} Z[J]}{\delta J(x_1)\cdots \delta J(x_2)}

Correlators from one object

Setting the source back to zero after differentiating leaves exactly the field insertions we want inside the expectation value.

\left.\frac{\delta^{2} Z[J]}{\delta J(x_1)\cdots\delta J(x_2)}\right|_{J=0} \to \langle T\phi(x_1)\cdots\phi(x_2)\rangle

Diagram expansion

Propagator

Vertex

Series weight

0.69

diagram terms become organized rather than mysterious