Blackforest said:
I explain: the way to give a precise position to the different coordinates of the vector satisfying a given fundamental form "seems" to be entirely the result of an arbitrary choice. Why? What does it represent (in relationship with the geometry or with a special property for this matrix: orthogonality...)? Is there any geometrical signification in this way of doing?
Dear Blackforest;
There is another, deeper, way of looking at the Pauli spin matrices. It's called "Clifford algebra" or "geometric algebra", and perhaps that is what you are looking for.
The Clifford algebra for the Pauli spin matrices is called "complexified CL(2,0)" by the mathematicians.
The algebra amounts to supposing that we wish to create an "algebra" that defines the geometry of 3-dimensional space. In mathematics, an algebra is a collection of symbols that satisfy certain relationships. In the case of the Pauli spin matrices or complexified CL(2,0), the symbols and their relationship consists of the following:
\hat{1} = 1 = Scalar number. This can be multiplied by any complex constant, and the result is still a member of the algebra.
\sigma_x = Vector for the x direction. This is not a 2x2 matrix, as in the Pauli spin matrices, but instead is to be thought of as a vector of unit length in the x direction. Any multiple of this vector by a complex number, for example (3-i)\sigma_x is still a member of the algebra.
The same applies to y and z giving \sigma_y and \sigma_z as members of the algebra, along with all their multiples.
In addition to what's been described, the algebra also contains all possible sums of complex multiples of the above four elements. So a typical element of the algebra might be:
(2-i)\hat{1} + 7i\sigma_x + 22\sigma_y - \sigma_z
So far we've defined addition for the algebra and multiplication by complex numbers. An algebra also needs a rule for multiplication. In this case, the rule will give the products of the vectors as follows:
\sigma_x^2 = \sigma_y^2 = \sigma_z^2 = \hat{1}^2 = \hat{1}
\hat{1} \sigma_n = \sigma_n for n = x,y,z
\sigma_x \sigma_y = -\sigma_y \sigma_x = i \sigma_z
\sigma_y \sigma_z = -\sigma_z \sigma_y = i \sigma_x
\sigma_z \sigma_x = -\sigma_x \sigma_z = i \sigma_y
If you try these rules out with the Pauli spin matrices you will discover that the Pauli spin matrices also satisfy these rules. That means that the Pauli spin matrices are a "representation" of the algebra.
But you do not have to pick a representation in order to make calculations with the algebra.
In this way, the association of a vector with a matrix becomes instead an association of a vector with a member of the algebra. For example, if u_x, u_y, u_z are complex numbers, then you can associate
(u_x,u_y,u_z) <=> u_x\sigma_x + u_y\sigma_y + u+z\sigma_z
In a lot of ways, it's easier to make calculations with Clifford algebra than with the Pauli spin matrices. There is a relationship between the Clifford algebra as defined above, and the spinors or vectors used with the Pauli spin matrices.
The relationship is subtle and beautiful, but is probably beyond the scope of this thread: It involves special elements of the Clifford algebra that are called "primitive idempotents", which means that when you square them, they don't change, that is, that \iota^2 = \iota, and they're particularly "primitive".
Among the Clifford algebra CL(2,0), an example of a primitive idempotent is:
(\hat{1} + \sigma_z)/2
Among the Pauli spin matrices, an example of a primitive idempotent (that corresponds to the above Clifford algebra element) is:
\left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}\right).
If you take this matrix above, and multiply an arbitrary 2x2 matrix by it, what you get is a matrix that is nonzero only in the left column. That left column is a spinor. The same thing happens with the Clifford algebraic idempotent, but it is harder to see. These idempotents are used as "projection operators" and they reduce the full algebra (either of 2x2 matrices or the CL(2,0)) down to a subalgebra that is half the size.
The arbitrary choice in the spinor amounts to the arbitrary choice in the selection of the idempotent used to create the spinor. In the above examples, the idempotent chosen was the one that corresponds to spin+1/2 in the z-direction.
The z-direction was arbitrarily chosen. Perhaps this is the arbitrary choice you are thinking of. The choice of the z-coordinate causes the spinor that corresponds to spin aligned in the z-direction to be particularly simple.
This all comes about because the idempotent is a way of selecting a "subalgebra" of the Clifford algebra. It is the choice of this idempotent (in this case, the one aligned in the +z direction) that is arbitrary.
One can use the idempotents themselves as replacements for the spinors, but this is not generally done in physics. I suppose the reason that it is not is because you end up with "unphysical degrees of freedom". That is, a complex 2x2 matrix or a complexified CL(2,0) Clifford algebra (where idempotents live) has 8 degrees of freedom. A spinor only has 4. Thus by using spinors instead of idempotents, one eliminates 4 unphysical degrees of freedom.
Having done this, the spinor still has two unphysical degrees of freedom. These correspond to multiplication by an arbitrary complex number. One of these can be removed by requiring that the spinor be "normalized". The other is left as an arbitrary complex phase.
Carl
