IVF4. Vector Bosons.

We have given an implicit equation for vector bosons in Eq. (IVF2-3) but it is useful to have a more explicit equation for the boson field operator.  To give this, we apply Eq. (IVF2-3) to a single molecule, located at .  Then

(IVF4-1)   

where the last line holds because the molecule is invariant under global transformations.  This implies that the operator  is

(IVF4-2)  

where g  is a constant introduced so the boson state is properly normalized.  If we use the G of Eq. (IVF2-2)

(IVF4-3)  

then the wave function of the boson state generated by the gauge transformation is

(IVF4-4)  

The form of  in Eq. (IVF4-2) explains why we chose the interaction Lagrangian of Eq. (IVF1-2). 

The  operator is bilinear in the fermion operators and will therefore obey commutation (as opposed to anticommutation) relations.  This leads to symmetric statistics for vector boson states.