In quantum field theory, a quartic interaction or φ4 theory is a type of self-interaction of a scalar field. Other types of quartic interactions may be found under the topic of four-fermion interactions. A classical free scalar field φ {\displaystyle \varphi } satisfies the Klein–Gordon equation. If a scalar field is denoted φ {\displaystyle \varphi } , a quartic interaction is represented by adding an interaction energy term ( λ / 4 ! ) φ 4 {\displaystyle ({\lambda }/{4!})\varphi ^{4}} to the Lagrangian density. The coupling constant λ {\displaystyle \lambda } is dimensionless in 4-dimensional spacetime. This article uses the ( + − − − ) {\displaystyle (+---)} metric signature for Minkowski space.
Lagrangian for a massive, real scalar field The Lagrangian density for a massive, real scalar field with a quartic interaction is
L ( φ ) = 1 2 [ ∂ μ φ ∂ μ φ − m 2 φ 2 ] − λ 4 ! φ 4 . {\displaystyle {\mathcal {L}}(\varphi )={\frac {1}{2}}[\partial ^{\mu }\varphi \partial _{\mu }\varphi -m^{2}\varphi ^{2}]-{\frac {\lambda }{4!}}\varphi ^{4}.}
The first term between the brackets is the energy related to the four-momentum of the particle, the second term describes its restmass energy. This Lagrangian has a global Z2 symmetry mapping φ → − φ {\displaystyle \varphi \to -\varphi } .
Lagrangian for a complex scalar field The Lagrangian for a complex scalar field can be motivated as follows. For two scalar fields φ 1 {\displaystyle \varphi _{1}} and φ 2 {\displaystyle \varphi _{2}} the Lagrangian has the form
L ( φ 1 , φ 2 ) = 1 2 [ ∂ μ φ 1 ∂ μ φ 1 − m 2 φ 1 2 ] + 1 2 [ ∂ μ φ 2 ∂ μ φ 2 − m 2 φ 2 2 ] − 1 4 λ ( φ 1 2 + φ 2 2 ) 2 , {\displaystyle {\mathcal {L}}(\varphi _{1},\varphi _{2})={\frac {1}{2}}[\partial _{\mu }\varphi _{1}\partial ^{\mu }\varphi _{1}-m^{2}\varphi _{1}^{2}]+{\frac {1}{2}}[\partial _{\mu }\varphi _{2}\partial ^{\mu }\varphi _{2}-m^{2}\varphi _{2}^{2}]-{\frac {1}{4}}\lambda (\varphi _{1}^{2}+\varphi _{2}^{2})^{2},}
which can be written more concisely introducing a complex scalar field ϕ {\displaystyle \phi } defined as
ϕ ≡ 1 2 ( φ 1 + i φ 2 ) , {\displaystyle \phi \equiv {\frac {1}{\sqrt {2}}}(\varphi _{1}+i\varphi _{2}),}
ϕ ∗ ≡ 1 2 ( φ 1 − i φ 2 ) . {\displaystyle \phi ^{*}\equiv {\frac {1}{\sqrt {2}}}(\varphi _{1}-i\varphi _{2}).}
Expressed in terms of this complex scalar field, the above Lagrangian becomes
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