is obtained by including amass term,
(
̄h^2 ∂μ∂μ−m^2 c^2
)
φ(x) = 0 (21.44)This equation for a massive scalar field may be solved by Fourrier analysis, and we have
φk(x) =eik·x ̄h^2 kμkμ+m^2 c^2 = 0 (21.45)Sincepμ= ̄hkμ, we see that this relation gives the energy - momentum relation for a relativistic
particle (or wave) with massm. A further Lorentz-invariant generalization of the scalarfield wave
equation is obtained by adding an arbitrary functionV′(φ),
̄h^2 ∂μ∂μφ−m^2 c^2 φ−V′(φ) = 0 (21.46)The latter may be derived via the action principle from the invariant Lagrangian density,
L=−1
2
̄h^2 ∂μφ∂μφ−1
2
m^2 c^2 φ^2 −V(φ) (21.47)The Higgs particle, for example, is an elementary scalar particle, which is described by a real scalar
fieldφ, with a quartic potentialV(φ). Of course, the Higgs particle couples to many other particles,
such as electrons, quarks and neutrinos, and this description will require additional fields in the
Lagrangian density.
21.8 Relativistic invariance of Maxwell equations
The prime example of a vector field under Lorentz transformations is provided by Maxwell the-
ory of electro-magnetism. A general (covariant) vector field Vμ(x) is a collection of 4 fields
V 0 (x),V 1 (x),V 2 (x),V 3 (x) which behave as follows under a Lorentz transformation Λ,
Vμ(x) →Vμ′(x′) = ΛμνVν(x) (21.48)and analogously for a contravariant vector field.
21.8.1 The gauge field and field strength
In electro-magnetism, we encounter two vector fields, the gauge potentialAμ= (−Φ/c,A), and the
electric current densityjμ= (ρc,j), where Φ is the electric potential,Athe usual 3-dimensional
vector potential,ρthe charge density, andjthe 3-dimensional electric current density. Consistency
of Maxwell’s equations requires the current density to be conserved,
∂μjμ=
∂ρ
∂t+∇·j= 0 (21.49)Maxwell’s equations are invariant under gauge transformations on the vector potential,
Aμ→A′μ=Aμ+∂μθ (21.50)