nLab source field

Idea

In the path integral quantization formulation of quantum field theory the correlation functions (expectation values) are schematically path integrals of the form

$$\langle \phi(x_1) \cdots \phi(x_n)\rangle \coloneqq \frac{ \int [D\phi] \,\phi(x_1) \cdots \phi(x_n) \, \exp\left(i S\left(\phi\right)\right) }{ \int [D\phi]\,\exp\left(i S\left(\phi\right)\right) } \,.$$

Therefore, as for ordinary moments (and explicitly so under Wick rotation, if possible), there is generating functional for the correlators of the schematic form

$$\Psi(J) = \int [D \phi] \, \exp\left(i S\left(\phi\right) + i \int_X J \phi d\mu\right) \,.$$

Here in the exponent one may regard

$$S'(\phi,J) = S(\phi) + \int_X J \phi d\mu$$

as a new action functional defined on a larger space of fields that also contains the parameters $J$ as fields. In this context one calls $J$ a source field.

This is in the corresponding equations of motion of $S'$ $J$ will act like a source term. The Euler-Lagrange equations for the modified action are:

$$EL(S') = EL(S) + J = 0 \,.$$

Notably if $EL(S)$ is a homogeneous wave equation (as for a free field theory) then $J$ is the inhomogeneous term in such a wave equation which describes indeed a “source” of wave excitations.

Related concepts

holographic principle in quantum field theory

bulk field theory	boundary field theory
dimension $n+1$	dimension $n$
field	source
wave function	correlation function
space of quantum states	conformal blocks