At the quark level we have motivated the existence of a symmetry called isospin. The world is very nearly invariant at the level of strong interactions under the continuous (unitary) transformation that mixes the up and down type quarks. Clearly, when we consider electromagnetic and weak interactions, in which the up and down quarks couple differently, this is not the case, but lets forget this at present.
Turning to the nucleons, and forgetting about weak and electromagnetic interactions, with identical spins and nearly identical masses it is a very good approximation to say that the world would be invariant under continuous, unitary transformations that mix the proton and neutron. Another way of saying this is that the strong Hamiltonian (Lagrangian) is invariant under isospin transformations.
We group the proton and the neutron together into the fundamental
representation of the SU(2), and define the isospin
nucleon field, N,
Let us now consider the 's in the context of isospin.
There are three different charge states ,
They fit simply into a
Fo our purposes, we will only be considering the pion in the context of
nucleons, i.e. for nuclear physics purposes, and so it is simpler for us to
put the pions into the adjoint representation of SU(2), which is accomplished
by contracting the index on the
field with the index of the pauli
matrices, which defines the
field with one up and one down index,
In terms of states