Chapter 48

An Introduction to the

Standard Model

The theory of fundamental particles and their non-gravitational interactions is
encapsulated by an extremely successful quantum field theory known as the
Standard Model. This quantum field theory is determined by a particular set
of quantum fields and a particular Hamiltonian, which we will outline in this
chapter. It is an interacting quantum field theory, not solvable by the methods
we have seen so far. In the non-interacting approximation, it includes just the
sorts of free quantum field theories that we have studied in earlier chapters.
This chapter gives only a very brief sketch of the definition of the theory, for
details one needs to consult a conventional particle physics textbook.
After outlining the basic structures of the Standard Model, we will indi-
cate the major issues that it does not address, issues that one might hope will
someday find a resolution through a better understanding of the mathematical
structures that underlie this particular example of a quantum field theory.

48.1 Non-Abelian gauge fields

The Standard Model includes gauge fields for aU(1)×SU(2)×SU(3) gauge
group, with Hamiltonian given by the sum 46.10

hY M=

∑^3

j=1

1

2

∫

R^3

trj(|Ej|^2 +|Bj|^2 )d^3 x (48.1)

TheEjandBjtake values in the Lie algebrasu(1),su(2),su(3) forj= 1, 2 , 3
respectively. trj indicates a choice of an adjoint-invariant inner product for
each of these Lie algebras, which could be defined in terms of the trace in some
representation. Such an invariant inner product is unique up to a choice of
normalization, and this introduces three parameters into the theory, which we
will callg 1 ,g 2 ,g 3.