Okay, an electron or positron has spin in some direction defined by a unit vector. The operator for spin 1/2 in the Dirac algebra in the direction is:

The projection operator for the state with spin +1/2 in the direction is therefore:

To distinguish between electron and positron I need a “charge” operator. Peskin & Schroeder use , and the projection operator for this is

Therefore, the complete solution for the density operator (qubit) representation of an electron / positron is the product of these two (commuting) projection operators:

where the P&S convention is that the + sign gives the electron and - sign gives the positron. To translate this into the usual (inelegant) spinor notation, one chooses a representation for the operators, and computes the matrix equivalent:

The spinor is any non zero column of the above matrix. For the particular example that P&S slaves over, and the resulting spinor is (1,0,1,0), which is what one obtains as a special case above (times 1/2).

Pick up your copy of Peskin and Schroeder and look up the section early in the book where they derive the constant spinors in the chiral representation. The derivation takes a half dozen pages and involves assumptions about spinors as representatives of the Lorentz or Poincare symmetry. At the end, a spinor pops out, but it was a hell of a lot of work. This is the way particle theory is done because it is based on the assumption that symmetry is fundamental.

The way I do particle physics is from the operator point of view. For the Dirac algebra, it is the operators themselves, as elements of a Clifford algebra that are foundational, and so the theory is directly related to the vectors of the Clifford algebra, .

In the operator language, symmetry, and spinors, are just mathematical conveniences, it is the operators that are fundamental. And these operators are geometric (i.e. Hestenes’ Geometric Algebra) in nature. In this language, the problems which are so messy in the symmetry language become quite trivial in the operator language. For example, the wikipedia entry for spinors uses operator theory. In one page it derives the spinor for an electron or positron in an arbitrary orientation. About 1/6th of P&S’s efforts for more results.

Standard physics makes the assumption that relativity is perfect (a symmetry principle) and derives everything from this, at great effort and with oodles of undetermined constants (as arise in any attempt to use symmetry to solve a math problem). Operator physics makes the assumption that the geometry of spacetime is the Dirac algebra and gets the same results trivially.

Now let’s try and unify the particles. From the symmetry point of view, you have a damned near infinite number of possible choices of symmetry, which particles are present, and how they interact. From the operator point of view, the only thing you can do is generalize the Dirac algebra to a more general Clifford algebra. Which one of these techniques makes sense to you? How many vacua can 5000 particlel theorists explore in 30 years given that there are to look through? Operator theory is very restrictive, and that means that you have a hope in Hell of solving the problem.

The reason that obsolete techniques become obsolete in physics, in the face of the very strong sociological pressures towards conservatism, is because graduate students don’t like to solve problems the hard way. Symmetry is the hard way and is doomed.

The foundations of physics are shrouded in mystery, symmetry is a useful tool for this. Biology is based on chemistry and all that, the foundations are very practical and are well understood at an intuitive level. If anything, the biologists should be over giving assistance to the physicists.