# nLab source field

Contents

### Context

#### Quantum field theory

functorial quantum field theory

# Contents

## 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.

holographic principle in quantum field theory

bulk field theoryboundary field theory
dimension $n+1$dimension $n$
fieldsource
wave functioncorrelation function
space of quantum statesconformal blocks

Created on July 2, 2013 at 23:59:33. See the history of this page for a list of all contributions to it.