1Classical field theory
III Quantum Field Theory
1.2 Lorentz invariance
If we wish to construct relativistic field theories such that
x
and
t
are on an
equal footing, the Lagrangian should be invariant under Lorentz transformations
x
µ
7→ x
0µ
= Λ
µ
ν
x
ν
, where
Λ
µ
σ
η
στ
Λ
ν
τ
= η
µν
.
and η
µν
is the Minkowski metric given by
η
µν
=
1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 −1
.
Example. The transformation
Λ
µ
σ
=
1 0 0 0
0 1 0 0
0 0 cos θ −sin θ
0 0 sin θ cos θ
describes a rotation by an angle around the x-axis.
Example. The transformation
Λ
µ
σ
=
γ −γv 0 0
−γv γ 0 0
0 0 1 0
0 0 0 1
describes a boost by v along the x-axis.
The Lorentz transformations form a Lie group under matrix multiplication —
see III Symmetries, Field and Particles.
The Lorentz transformations have a representation on the fields. For a scalar
field, this is given by
φ(x) 7→ φ
0
(x) = φ(Λ
−1
x),
where the indices are suppressed. This is an active transformation — say
x
0
is the point at which, say, the field is a maximum. Then after applying the
Lorentz transformation, the position of the new maximum is Λ
x
0
. The field
itself actually moved.
Alternatively, we can use passive transformations, where we just relabel the
points. In this case, we have
φ(x) 7→ φ(Λ(x)).
However, this doesn’t really matter, since if Λ is a Lorentz transformation, then
so is Λ
−1
. So being invariant under active transformations is the same as being
invariant under passive transformations.
A Lorentz invariant theory should have equations of motion such that if
φ
(
x
)
is a solution, then so is
φ
(Λ
−1
x
). This can be achieved by requiring that the
action S is invariant under Lorentz transformations.
Example. In the Klein–Gordon field, we have
L =
1
2
∂
µ
φ∂
µ
φ −
1
2
m
2
φ
2
.
The Lorentz transformation is given by
φ(x) 7→ φ
0
(x) = φ(Λ
−1
x) = φ(y),
where
y
µ
= (Λ
−1
)
µ
ν
x
ν
.
We then check that
∂
µ
φ(x) 7→
∂
∂x
µ
(φ(Λ
−1
x))
=
∂
∂x
µ
(φ(y))
=
∂y
ν
∂x
µ
∂
∂y
ν
(φ(y))
= (Λ
−1
)
ν
µ
(∂
ν
φ)(y).
Since Λ
−1
is a Lorentz transformation, we have
∂
µ
φ∂
µ
φ = ∂
µ
φ
0
∂
µ
φ
0
.
In general, as long as we write everything in terms of tensors, we get Lorentz
invariant theories.
Symmetries play an important role in QFT. Different kinds of symmetries
include Lorentz symmetries, gauge symmetries, global symmetries and super-
symmetries (SUSY).
Example.
Protons and neutrons are made up of quarks. Each type of quark
comes in three flavors, which are called red, blue and green (these are arbitrary
names). If we swap around red and blue everywhere in the universe, then the
laws don’t change. This is known as a global symmetry.
However, in light of relativity, swapping around red and blue everywhere in
the universe might be a bit absurd, since the universe is so big. What if we only
do it locally? If we make the change differently at different points, the equations
don’t a priori remain invariant, unless we introduce a gauge boson. More of this
will be explored in the AQFT course.