phi^n interaction

**standard model of particle physics**

**matter field fermions** (spinors, Dirac fields)

1st | 2nd | 3d |
---|---|---|

up? | charm | top |

down? | strange? | bottom |

**hadron** (bound states of the above quarks)

**minimally extended supersymmetric standard model**

bosinos:

**dark matter candidates**

**Exotica**

**algebraic quantum field theory** (perturbative, on curved spacetimes, homotopical)

**quantum mechanical system**, **quantum probability**

**interacting field quantization**

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”.

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.

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)

See also

- Wikipedia,
*Quartic interaction*

Last revised on January 13, 2018 at 08:59:51. See the history of this page for a list of all contributions to it.