Lecture 38.
We noted that the gauged phase symmetry of electromagnetism is a Lie group (reviewing Lie groups), specifically the abelian Lie group U(1). We then reviewed the 2 nonabelian Lie groups that we know, SO(3) and SU(2), and discussed the isospin symmetry of the strong interactions.
For SO(3) (rotations), we reviewed 1) the relationship between
rotations (the Lie group) and angular momentum generators (the Lie
algebra); 2) the ability to simultaneously diagonalize only one basis
generator, due to the angular momentum commutation relations; 3) the
creation of the tower of angular momentum eigenstates |j, m;SPMgt; for
fixed j -- a 2j+1-dimensional representation -- using the
step operators; and 3) the degeneracy of states within a
representation j.
We then defined the Lie group SU(2) -- rotations mixing 2 complex
fields. We noted that the spin-1/2 representation we found for SO(3)
is exactly the defining representation of SU(2). That is, the Pauli
generators 
are exactly a basis for the defining Lie
algebra, of 2-by-2 matrices whose exponentials are special, unitary
matrices. Thus all our representations of SO(3) are automatically
representations of SU(2). This is due to the 2-1 homomorphism
between the Lie groups SO(3) and SU(2).
While we will soon consider larger, more complicated nonabelian Lie groups, knowing SU(2) gives us a good start. This is because an SU(2) symmetry was, historically, the first noticed symmetry of the strong interactions; that is, of the forces that hold the nucleus together.
We then introduced isospin symmetry: the invariance of nuclear forces under rotations that mix the proton and neutron fields. This symmetry group, involving 2-dimensional complex rotations, is an SU(2) symmetry; it is called ``isospin'' in deference to SU(2)'s original role as the symmetry governing electron spin. It was initially imagined to be a symmetry of the strong interactions, violated only by weak and electromagnetic effects (we now know that it is only an approximate symmetry of the strong interactions as well, but quite a good one).
Consequences of isospin symmetry are that 1) all strong interaction
terms should be isospin invariant; and 2) particles should fall into
degenerate multiplets, the 2j+1-dimensional representations of
isospin SU(2). We noted the earliest known multiplets: the
proton-neutron isodoublet (with isospin 1/2), with masses degenerate
to within 
; and the pion isotriplet (with isospin 1), with
masses degenerate to within 
. We also noted that strong
interactions conserve electromagnetic charge Q as well, so that we
really have 2 independent quantum numbers, 
and Q, describing
every state. We noted that the linear combination 
, which is
of course also conserved, is the same for each state in an
isomultiplet. We can thus choose this quantum number (called the
``hypercharge'' Y), along with to be the 2 independent quantum
numbers carried by particles. We note that the isospin and hypercharge
symmetry found here emerge for initially unknown and seemingly
independent reasons; we will soon find that they are just aspects of a
larger symmetry group of the strong interactions.
-- KB