nLab
source field

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Context

Quantum field theory

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

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.

holographic principle in quantum field theory

bulk field theoryboundary field theory
dimension n+1n+1dimension nn
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.