A Poincaré group element gp in the group Gp is defined by a sequence of parameters {pi}, the number of which, as we have already said, represents the dimension of the group. The matrix dg (g = e) is built from the quantities {dpi}. The above mapping is therefore of the type:
(81)
In other words, to a set of scalars dpi we associate an equal number of scalars dpi'. Duality consists of postulating the invariance of a scalar, according to:
(82)
where n is the dimension of the group (ten, for the Poincaré group). The scalars Ji represent the components of the momentum, in the same number.
We will decide to decompose this momentum J into two objects. The first will be an antisymmetric matrix M of size (4,4), thus having six components, and the second a "four-vector" P, a matrix of size (4,1):
(83)
(84) J = { M , p , E} = { M , P } We will write the scalar product in the form:
(85)
Tr meaning "trace of", and we will also have:
(86)
a linear form whose invariance ensures the duality.
with:
(87) (87b)
(87c)
but GG = 1 therefore this equals:
(88)
Identify the terms in y (89)
That is to say:
(90)
----> Here again follow details of matrix calculations. If you wish, clicking here you can go directly to the result.
In the trace, we can perform a cyclic permutation of the terms.
(90a)
(90b)
(90c)
the second term of the right-hand side is equal to the product of a row matrix by a column matrix.
This is equal to the trace of the reversed product (below, schematically, the product of a row matrix by a column matrix):
(90d)
In this trace, I can perform a cyclic permutation:
(90e)
Hence:
(90f)
(90g)
Here we will apply again the theorem on the traces of matrices which are the product of another matrix by a symmetric matrix.
Any matrix can be symmetrized or antisymmetrized. Moreover, the trace of the product of a matrix by a symmetric matrix is zero.
(90h)
I can apply this to the matrix (90i) since we take the trace
(90j)
(90k) = sym ( ) + antisym ( )
but:
(90l)
hence
(90m) (90n)
(90o)
(90p)
and:
(90q)
finally:
(90r)
By grouping and changing the primes on one side I obtain my group action: