The Field Equations
The commutator of 2 co-variant derivatives of a complex vector using 1.8 for the affine connection gives
2.1
Where
By adding 
to both sides of equation 2.1 and equating the LHS to zero, and lowering
2.2
2.3
Contracting equation 2.3 by setting 
gives
2.4
In a geodesic frame, this reduces to
2.4a
and since 
, it can be shown that 
is imaginary. Contracting equation 2.2 by letting 
and 
gives
2.5
The vector 
can be eliminated from 2.4 and 2.5 by using the relations (See Appendix A)
where 
are NxN matrices where 
and replace 
with its matrix equivalent to give
2.6
2.7
Eliminating the symmetric connection from equations 2.6 & 2.7 gives
2.8
For 
, the euclidean metric, the matrices are found to satisfy the following, (see section 3 for the calculation of N)
where I is the 6x6 unit matrix. A solution for the 
i=1,2,3 matrices is
where the matrices 
are given by, see reference [2]
These are the matrices for Spin 1 particles, thus the matrix 
is the spin wavefunction for spin 1 bosons