IVE4. An Equation with Interactions.

One can modify the complex-variable model to include interactions.  To do this, instead of a single set of , we use N sets, .  Then as in Eq. (IVD1-1) we set the linear operator equal to

(IVE4-1)  

where  is the operator of Eq. (E-1b), using the mth set of u,v.

The generators of the invariance group are now the sums of the generators for the individual sets of variables (Eqs. (IVE2-2) and (IVE2-3)).  To have an invariant theory,  must commute with those generators.  In particular, it must commute with the total momentum operators

(IVE4-2)  

In order to have the interaction transfer momentum from one “single-particle” state to another, and yet conserve total momentum, we must have

(IVE4-3)  

We can do this by making V a function of space-time-like variables, , that are conjugate to the ,

(IVE4-4)  

Then if

(IVE4-5)  

the constraints of Eq. (IVE4-3) are satisfied.

We require that the ’s transform like four-vectors under Lorentz transformations.  This can be achieved if their general form is

(IVE4-6)  

where I is a Lorentz invariant.  We set

(IVE4-7)  

and find that Eqs. (IVE4-4) are indeed satisfied.  To make Lorentz invariant, we can make V a function of .  We could also use the complex conjugates of the ’s.  Unfortunately, this interaction is not invariant under the internal symmetry group SU(n).

Note that the interaction term is invariant under exchanges of m and m’ variables, so O is invariant under the permutation group.  This implies that we could consider only solutions that are totally antisymmetric under exchange of sets of underlying variables.