Contents

Contents

Idea

In quantum field theory of the scalar field $\Phi$, the canonical local interaction term is a Lagrangian density of the form

$\mathbf{L}_{int} \;=\; \phi^n \, dvol_\Sigma$

(with notation as at A first idea of quantum field theory).

For $g_{sw} \in C^\infty_{cp}(\Sigma)$ any bump function on spacetime, the corresponding adiabatically switched local observable is

\begin{aligned} S_{int} & = \underset{\Sigma}{\int} \underset{ n \, \text{factors} }{ \underbrace{ \mathbf{\Phi}(x) \cdot \mathbf{\Phi}(x) \cdots \mathbf{\Phi}(x) \cdot \mathbf{\Phi}(x) } } \, dvol_\Sigma(x) \\ & = \underset{\Sigma}{\int} : \underset{ n \, \text{factors} }{ \underbrace{ \mathbf{\Phi}(x) \mathbf{\Phi}(x) \cdots \mathbf{\Phi}(x) \mathbf{\Phi}(x) } } : \, dvol_\Sigma(X) \end{aligned} \,,

where in the first line we have the integral over a pointwise product (this def.) of $n$ field observables (this def.), which in the second line we write equivalently as a normal ordered product, by the discusssion at Wick algebra (this def.).

The interacting field theory with Lagrangian density that of the free scalar field plus interactions of the form $\phi^k$ as above, up to order $n$, is often called simply “$\Phi^n$-theory”.

Examples

The mass term of the free scalar field is a $\Phi^2$-interaction.

The Higgs field involves a quadratic and quartic interaction of this form.

The potential for the inflaton field in chaotic cosmic inflation is a $\Phi^2$-interaction.

References

The weak adiabatic limit for mass-less $\Phi^4$ theory was established in

• P. Blanchard and R. Seneor, Green’s functions for theories with massless particles (in perturbation theory), Ann. Inst. H. Poincaré Sec. A 23 (2), 147–209 (1975) (Numdam)