Scalar Field Theory with InteractionΒΆ

This section introduces the self-interaction theory of scalar field. As usual, the interaction is implemented as an additional interaction term \(\mathcal{L}_{\text{int}}\) in Lagrangian. The interaction discussed here is assumed to satisfy the following condition

  • It is the self-interaction, i.e. \(\mathcal{L}_{\text{int}} = \mathcal{L}_{\text{int}}[\varphi]\)
  • It has the form of polynomials.

Similarly, the construction starts from the generating functional. The generating functional of an interaction theory is

\[\begin{split}W[J] &= \int\mathcal{D}\varphi\exp\{\mathrm iS\} = \int\mathcal{D}\varphi\exp\{\mathrm i\!\!\int\!\!\mathcal{L}\! + \!\mathcal{L}_{\text{int}}[\varphi]\}\\ &= \int\mathcal{D}\varphi\exp\{\mathrm i\!\!\int\!\!\mathcal{L}_{\text{int}}[\varphi]\}\exp\{\mathrm i\!\!\int\!\!\mathcal{L}\}\end{split}\]

Next, we expand the first exponential (the one with interaction) according to definition

\[W[J] = \int\mathcal{D}\varphi\sum_{n=0}^\infty\frac{1}{n!}\left[\mathrm i\!\!\int\!\!\mathcal{L}_{\text{int}}[\varphi]\}\right]^n\exp\{\mathrm i\!\!\int\!\!\mathcal{L}\}\]

Now, recall our first identity

\[\frac{1}{\mathrm{i}}\frac{\delta}{\delta J(x)}W_0[J] = \int\mathcal{D}\varphi\ \varphi(x)\exp\{\mathrm{i}S\}\]

Therefore, each \(\varphi(x)\) in \(\mathcal{L}_{\text{int}}[\varphi]\) can be replaced with \(\delta/\mathrm{i}\delta J(x)\), as long as \(\mathcal{L}_{\text{int}}[\varphi]\) is a pure polynomial of \(\varphi\). In this case, we can write

\[\begin{split}W[J] = \int\mathcal{D}\varphi\sum_{n=0}^\infty\left[\mathrm i\!\!\int\!\!\mathcal{L}_{\text{int}}\!\!\left[\frac{1}{\mathrm{i}}\!\frac{\delta}{\delta J}\right]\}\right]^n\exp\{\mathrm i\!\!\int\!\!\mathcal{L}\}\\ = \sum_{n=0}^\infty\frac{1}{n!}\left[\mathrm i\!\!\int\!\!\mathcal{L}_{\text{int}}\!\!\left[\frac{1}{\mathrm{i}}\!\frac{\delta}{\delta J}\right]\}\right]^n\int\mathcal{D}\varphi\exp\{\mathrm i\!\!\int\!\!\mathcal{L}\}\end{split}\]

where expression \(\mathcal{L}_{\text{int}}\!\!\left[\delta/\mathrm{i}\delta J\right]\) represents the the expression where all \(\varphi\) in \(\mathcal{L}_{\text{int}}\) is replaced with \(\delta/\mathrm{i}\delta J(x)\). The terms after the replacement no longer contain \(\varphi\) and thus can be moved out of the path integral. And we find now that the path integral gives the generating functional \(W_0[J]\) of free field. Transform back into the exponential form and we get

\[W[J] = \exp\{-\mathrm{i}\int\mathcal{L}_{\text{int}}\!\!\left[\frac{1}{\mathrm{i}}\!\frac{\delta}{\delta J}\right]\} W_0[J]\]

which is the result of second stage of our construction. The interaction of other kinds of fields (except for vector field in gauge theories) can be performed through similar manner.