nLab phi^n interaction



Fields and quanta

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mesonslight mesons:
pion (udu d)
ρ-meson (udu d)
ω-meson (udu d)
ϕ-meson (ss¯s \bar s),
kaon, K*-meson (usu s, dsd s)
eta-meson (uu+dd+ssu u + d d + s s)

charmed heavy mesons:
D-meson (uc u c, dcd c, scs c)
J/ψ-meson (cc¯c \bar c)
bottom heavy mesons:
B-meson (qbq b)
ϒ-meson (bb¯b \bar b)
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neutron (udd)(u d d)

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Algebraic Quantum Field Theory

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field theory:

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interacting field quantization



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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”.


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.


An introduction to Φ 4\Phi^4 theory could be found in lecture 13 of

  • Sourav Chatterjee, Introduction to Quantum Field Theory for Mathematicians, pdf

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 7, 2024 at 18:16:42. See the history of this page for a list of all contributions to it.