nLab
source field

Context

Quantum field theory

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Contents

Idea

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

ϕ(x 1)ϕ(x n)[Dϕ]ϕ(x 1)ϕ(x n)exp(iS(ϕ))[Dϕ]exp(iS(ϕ)). \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

Ψ(J)=[Dϕ]exp(iS(ϕ)+i XJϕdμ). \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(ϕ,J)=S(ϕ)+ XJϕdμ 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 JJ as fields. In this context one calls JJ a source field.

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

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

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

in

n+1n+1 nn

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