The Lorentz Group

There are 2 real variables for each u and each v for a total of 8 per i.  Thus there are a total of 8n real variables.  If we switch to variables in which O is diagonal, it will be a quadratic form with half the coefficients +1 and half the coefficients –1.  Thus the O of Eq. (IVE1-1b) has a symmetry group with (8n)(8n –1)/2 generators.  For now, we are only interested in those generators that correspond to the inhomogeneous Lorentz group.  They must satisfy the commutation relations

(IVE2-1a)  

(IVE2-1b)  

(IVE2-1c)  

The generators of the group SL(2), homomorphic to the homogeneous Lorentz group, are
(IVE2-2a)  

where b is summed from 1 to 2 and i from 1 to n.  And the generators of translations are

(IVE2-2b)  

We can also give the Lorentz transformations macroscopically.  The homogeneous Lorentz transformation L(A) corresponding to the 2x2 matrix A from SL(2) has the effect (we drop the subscript i here)

(IVE2-3a)  

while the effect of translations is

(IVE2-3b)  

Unitary Transformations SU(n).

It is useful to introduce an SU(n) “internal” symmetry, where is the set of all nxn unitary matrices with determinant 1.  If the kets  transforms as the n representation of SU(n) and the kets  as the  representation, then  (summed on j) is invariant under SU(n).  In order to make the generators of the inhomogeneous Lorentz group SU(n) invariants, we suppose the variables with b=1 transform as the n representation and the variables with b=2 transform as the  representation.  The SU(n) transformation properties of all the variables and their derivatives are then

(IVE2-4)  

So we see that the operator of Eq. (IVE1-1) and the generators of the inhomogeneous Lorentz group are invariants under SU(n).