nLab
phi^n interaction

Contents

Context

Fields and quanta

Algebraic Quantum Field Theory

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

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Contents

Idea

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

L int=ϕ ndvol Σ \mathbf{L}_{int} \;=\; \phi^n \, dvol_\Sigma

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

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

S int =ΣΦ(x)Φ(x)Φ(x)Φ(x)nfactorsdvol Σ(x) =Σ:Φ(x)Φ(x)Φ(x)Φ(x)nfactors:dvol Σ(X), \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 nn 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 ϕ k\phi^k as above, up to order nn, is often called simply “Φ n\Phi^n-theory”.

Examples

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

References

The weak adiabatic limit for mass-less Φ 4\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

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