CERN-THESIS-2012-153 26/07/2012 - CERN Document Server

2.2.1 Quantum Electrodynamics
Quantum electrodynamics (QED) refers to the electromagnetic interactions of leptons and photons. It is
characterized by renormalization and gauge invariance. The U(1) gauge invariance is in fact what brings
the Lagrangian for a free field to the Lagrangian of QED. The Dirac equation:
L = i ¯ ψγ µ ∂µψ − m ¯ ψψ, (2.1)
where ψ is the lepton vector field, is not invariant under a local gauge transformation such as:
ψ(x) → e iα(x) ψ(x), (2.2)
where α(x) depends on space and time in an arbitrary way. Because of Noether’s Theorem, invariance under
a transformation is relevant because it translates into a conservation law. The second term in Equation 2.1
is invariant under the transformation given in Equation 2.2, but the ∂µψ terms breaks the invariance. In
order to impose such invariance, an alternative derivative is introduced, Dµ, which transforms covariantly
under phase transformations, like ψ itself:
To form this covariant derivative, a vector field Aµ must be introduce.
where Aµ transforms as:
Dµψ → e iα(x) Dµψ. (2.3)
Dµ ≡ ∂µ − ieAµ, (2.4)
Aµ → Aµ + 1
2 ∂µα (2.5)
By construction, the unwanted term in the transformation of Equation 2.1 is canceled. Thus, the Lagrangian
is now:
L = ¯ ψ (iγ µ ∂µ − m) ψ + e ¯ ψγ µ ψAµ
By enforcing local gauge invariance, a vector field Aµ, a gauge field, is introduced. This gauge field couples
to the Dirac particle (with charge −e) in the same way as a photon field. If this is indeed the new physical
photon field, a term corresponding to its kinetic energy should be added to the Lagrangian. This kinetic
term must also be invariant under the transformation given in Equation 2.5, and thus can only involve the
5
(2.6)