This means that ittransforms like a vector. Compare it to our original transformation formula
forxμ.
x′μ=aμνxν
We may safely assume that all our derivatives with one index transform as a vector.
With this, lets work on the Euler-Lagrange equation to get it into covariant shape. Remember that
the fieldφis a Lorentz scalar.
∑k∂
∂xk(
∂L
∂(∂φ/∂xk))
+
∂
∂t(
∂L
∂(∂φ/∂t))
−
∂L
∂φ= 0
∑
k∂
∂xk(
∂L
∂(∂φ/∂xk))
+
∂
∂(ict)(
∂L
∂(∂φ/∂(ict)))
−
∂L
∂φ= 0
∂
∂xμ(
∂L
∂(∂φ/∂xμ))
−
∂L
∂φ= 0
This is theEuler-Lagrange equation for a single scalar field. Each term in this equation is a
Lorentz scalar, ifLis a scalar.
∂
∂xμ(
∂L
∂(∂φ/∂xμ))
−
∂L
∂φ= 0
Now we want tofind a reasonable Lagrangian for a scalar field. The Lagrangian depends on
theφand its derivatives. Itshould not depend explicitly on the coordinatesxμ, since that
would violate translation and/or rotation invariance. We also want tocome out with alinear wave
equationso that high powers of the field should not appear. The only Lagrangian we can choose
(up to unimportant constants) is
L=−
1
2
(
∂φ
∂xν∂φ
∂xν+μ^2 φ^2)
The one constantμsets the ratio of the two terms. The overall constant is not important except to
match theT−Vdefinition. Remember thatφis a function of the coordinates.
With this Lagrangian, the Euler-Lagrange equation is.
∂
∂xμ(
−
∂φ
∂xμ)
+μ^2 φ= 0∂
∂xμ∂
∂xμφ−μ^2 φ= 0✷φ−μ^2 φ= 0This is the known as theKlein-Gordon equation. It is a good relativistic equation for amassive
scalar field. It was also an early candidate for the relativistic equivalent of the Schr ̈odinger equation
for electrons because it basically has therelativistic analog of the energy relationinherent in