Source field

In theoretical physics, a source field is a background field ${\displaystyle J}$ coupled to the original field ${\displaystyle \phi }$ as

${\displaystyle S_{\text{source}}=J\phi }$.

This term appears in the action in Richard Feynman's path integral formulation and responsible for the theory interactions. In Julian Schwinger's formulation the source is responsible for creating or destroying (detecting) particles. In a collision reaction a source could be other particles in the collision.^[1] Therefore, the source appears in the vacuum amplitude acting from both sides on the Green's function correlator of the theory.

Schwinger's source theory stems from Schwinger's quantum action principle and can be related to the path integral formulation as the variation with respect to the source per se ${\displaystyle \delta J}$ corresponds to the field ${\displaystyle \phi }$, i.e.^[2]

${\displaystyle \delta J=\int {\mathcal {D}}\phi ~e^{-i\int dt~J(t)\phi (t)}}$.

Also, a source acts effectively^[3] in a region of the spacetime. As one sees in the examples below, the source field appears on the right-hand side of the equations of motion (usually second-order partial differential equations) for ${\displaystyle \phi }$. When the field ${\displaystyle \phi }$ is the electromagnetic potential or the metric tensor, the source field is the electric current or the stress–energy tensor, respectively.^[4][5]

In terms of the statistical and non-relativistic applications, Schwinger's source formulation plays crucial rules in understanding many non-equilibrium systems.^[6][7] Source theory is theoretically significant as it needs neither divergence regularizations nor renormalization.^[1]