Physical Foundations of Cosmology

4.1 Basics 135

Therefore, this diagram describes electron–positron annihilation with the emission
of a photon. Because the photon is its own antiparticle, we do not need an arrow on its
line. More complicated processes can be described by simply combining primitive
vertices. For instance, Figure 4.1(c) is responsible for the Coulomb repulsion of
two electrons.
The replacement of all particles by antiparticles (charge conjugationC) corre-
sponds to the reversal of all arrows on the diagrams. Lagrangian (4.7) is invariant
with respect to charge conjugation.

Problem 4.1Consider a complex scalar fieldφwith Lagrangian

L=^12 (∂μφ∗∂μφ−m^2 φ∗φ). (4.8)

How should this Lagrangian be generalized to becomelocallygauge-invariant?
Write down the interaction terms and draw the corresponding vertices.

4.1.2 Non-Abelian gauge theories

The gauge transformations we have considered so far can be thought of as a mul-
tiplication ofψby 1×1 unitary matricesU≡exp(−iθ),satisfyingU†U= 1.
The group of all such matrices is calledU( 1 ). The localU( 1 )gauge invariance of
electrodynamics was realized a long time ago. However, the importance of such
symmetry was not fully appreciated until 1954, when Yang and Mills extended it
toSU( 2 )local gauge transformations. This symmetry was later used to construct
the electroweak theory.
The transformations generated byN×Nunitary matricesUare calledU(N)
gauge transformations. Generalizing fromU( 1 )gauge transformations is very
straightforward. Let us considerNfree Dirac fields with equal masses. Then the
Lagrangian is

L=

∑N

a= 1

(

iψ ̄aγμ∂μψa−mψ ̄aψa

)

=iψ ̄γμ∂μψ−mψψ ̄ , (4.9)

wherea= 1 ,...,N and in the second equality we have introduced the matrix
notation

ψ=

⎛

⎝

ψ^1

···

ψN

⎞

⎠, ψ ̄=

(

ψ ̄ 1 ,···,ψ ̄N

)

.

One should not forget that every element of these matrices is in its turn a four-
component Dirac spinor. The fieldsψahave the same spins and masses, and
therefore differ only by charges (for instance, in quantum chromodynamics these