Physical Foundations of Cosmology

4.1 Basics 137

Problem 4.2Verify the transformation law forFμν.(HintTo simplify the calcu-
lation, justify and use the following commutation rule:D ̃μU=UDμ.)

Thus, starting from a simple idea, we have achieved a significant result. Namely,
the interactions between fermions and gauge fields, as well as the simplest possible
Lagrangian for the gauge fields, were completely determined by the requirement
of gauge invariance. We would like to stress once more that the gauge fields are
massless because a mass term would spoil the gauge invariance.
There is an important difference betweenU( 1 )andU(N)groups. All elements
of theU( 1 )group (complex numbers) commute with each other (Abelian group),
while in the case of theU(N)group the elements do not generally commute (non-
Abelian group). This has an important consequence. TheU( 1 )gauge field has no
self-coupling and interacts only with fermions (the last term in (4.13) vanishes when
N=1), or, in other words, this field does not carry the group charge (photons are
electrically neutral). The non-AbelianU(N)fields do carry group charges and the
last term in (4.13) induces their self-interaction.

Problem 4.3ConsiderNcomplex scalar fields instead of fermions and find the
interaction terms in this case. Draw corresponding diagrams including those de-
scribing the self-interaction of the gauge fields.

To find the minimum number of compensating fields needed to ensure gauge
invariance, we have to count the number of generators of theU(N)group or, in
other words, the number of independent elements of anN×Nunitary matrix. Any
unitary matrix can be written as

U=exp(iH), (4.15)

whereHis anHermitianmatrix (H=H†).

Problem 4.4Verify that the number of independentrealnumbers characterizing
N×N Hermitian matrix is equal toN^2.

In turn, an HermitianHcan always be decomposed into a linear superposition
ofN^2 independentbasis matrices, one of which is the unit matrix

H=θ 1 +

N∑^2 − 1

C= 1

θCTC=θ 1 +θCTC, (4.16)

whereTCare traceless matrices andθCare real numbers; hence

U=eiθexp

(

iθCTC

)

. (4.17)

The first multiplier corresponds to theU( 1 )Abelian subgroup ofU(N)and the
second term belongs to theSU(N)subgroup consisting of all unitary matrixes