1.1 Symmetry as Guiding Principle for Particles
and Interactions
In ordinary life we observe symmetry of objects, like characteristic features of
geometrical forms, material objects or biological bodies. The concept is related to
the invariance of the object under definite transformations: One object is symmetric
if, after a transformation is applied, the result remains the same, i.e. it remains
“invariant”. But we also observe symmetry breaking, which is particularly of interest
when it is not a random effect but follows a definite pattern. In Fig. 1.1 we show the
three-span arch of the FermiLab entrance, near Chicago, which appears perfectly
symmetric when viewed from below, but has a calculated asymmetry from its other
views. Symmetry and symmetry breaking are very important concepts in the field of
elementary particle physics, however not referring to objects but to the fundamental
laws of physics.
We show here how symmetry has acted as a guiding principle for both the
existence of new particles and the formulation of interactions. One can claim that
“Symmetry dictates Interaction”, as stated by Yang. In Quantum Mechanics, the
symmetry is implemented by a unitary transformation b
U acting on states and
observables. If the dynamics, described by the Hamiltonian b
H , is invariant under
the transformation one has
b
H, b
U
h
i
¼ 0
ð1:1Þ
Under infinitesimal transformations generated by b
G ¼ b
G
{
, one obtains
immediately
Fig. 1.1 Symmetry Breaking
4
J. Bernabeu