References: W. Greiner & J. Reinhardt, Field Quantization, Springer-Verlag (1996), Chapter 2, Section 2.4.
Arthur Jaffe, Lorentz transformations, rotations and boosts, online notes available (at time of writing, Sep 2016) here.
Continuing our examination of general Lorentz transformations, recall that a Lorentz transformation can be represented by a matrix which preserves the Minkowski length of all four-vectors . This leads to the condition
where is the flat-space Minkowski metric
It turns out that we can map any 4-vector to a Hermitian matrix defined as
[Recall that a Hermitian matrix is equal to the complex conjugate of its transpose:
Also note that Jaffe uses an unconventional notation for the Hermitian conjugate, as he uses a superscript * rather that a superscript . This can be confusing since usually a superscript * indicates just complex conjugate, without the transpose. I’ll use the more usual superscript for Hermitian conjugate here.]
Although we’re used to the scalar product of two vectors, it is also useful to define the scalar product of two matrices as
where ‘Tr’ means the trace of a matrix, which is the sum of its diagonal elements. Note that the scalar product of with itself is
The determinant of is
Thus is the Minkowski length squared.
From 3, we observe that we can write as a sum:
where the are four Hermitian matrices:
The last three are the Pauli spin matrices that we met when looking at spin- in quantum mechanics.
The are orthonormal under the scalar product operation, as we can verify by direct calculation. For example
The other products work out similarly, so we have
We can work out the inverse transformation to 3 by taking the scalar product of 12 with :
Now a few more theorems that will be useful later.
Irreducible Sets of Matrices
A set of matrices is called irreducible if the only matrix that commutes with every matrix in is the identity matrix (or a multiple of ). Any two of the three Pauli matrices , above form an irreducible set of Hermitian matrices. This can be shown by direct calculation, which Jaffe does in detail in his article. For example, if we define to be some arbitrary matrix
where are complex numbers, then
If is to commute with , we must therefore require and .
Similarly, for we have
so that requires and .
And for :
so that requires and , so (no conditions can be inferred for or ).
If we form a set containing and one of or , we see that and , so is a multiple of . If we form from and we again have , but we must have simultaneously and which can be true only if , so again is a multiple of .
A unitary matrix is one whose Hermitian conjugate is its inverse, so that . Some properties of unitary matrices are given on the Wikipedia page, so we’ll just use those without going through the proofs. First, a unitary matrix is normal, which means that (this actually follows from the condition ). Second, there is another unitary matrix which diagonalizes , that is
where is a diagonal, unitary matrix.
(The determinant can be complex, but has magnitude 1.)
From this it follows that and since is unitary and diagonal, each diagonal element of must satisfy . (Remember that could be a complex number.) That means that for some real number , so we can write
where is a diagonal hermitian matrix containing only real elements, non-zero along its diagonal: . As usual, the exponential of a matrix is interpreted in terms of its power series, so that
For a diagonal matrix with diagonal elements , the diagonal elements of are just .
From 34, we have
Now we also have, since
Therefore, from 37
where is another Hermitian matrix. In other words, we can always write a unitary matrix as the exponential of a Hermitian matrix.
In the case where is a matrix, we can write it in terms of the matrices above as
where the are real, since the diagonal elements of a Hermitian matrix must be real. This follows because the form an orthonormal basis for the Hermitian matrices. [For some reason, Jaffe refers to the as which is confusing since he has used as the diagonal elements of above, and they’re not the same thing.]
If , then
The second line follows because the determinant of a product of matrices is the product of the determinants, so we can rearrange the multiplication order. To evaluate the last line, we observe that for a diagonal matrix , using 37 and applying the result to each diagonal element
[By the way, the relation is actually true for any square matrix , and is a corollary of Jacobi’s formula.]
We can now use the cyclic property of the trace (another matrix algebra theroem) which says that for 3 matrices ,
This gives us
Finally, from 45 and the fact that the traces of the are all zero for , and , we have
Thus for some integer , but as all values of give the same original unitary matrix , we can choose so that and