The two concepts are similar, but in density matrices, you need only make a single arbitrary choice of the vacuum state, which amounts to a choice of a unit vector (i.e., in spin-1/2, one must choose a unit vector).vanesch said:Yes, so my "arbitrary choices of representatives" comes down to your arbitrary choice of the "help-vacuum" for the sum. The superposition principle simply says that you can make all thinkable combinations of rays by adding together all kinds of representatives of the original rays. You do that with arbitrary help vacuum choices.
In order to set up a system of representatitives for the rays, you must make an arbitrary phase choice for each of the rays. To do this, you first have to make a choice of representation of the Pauli algebra. Then, for each possible unit vector, you must choose a particular normalized eigenvector. Choosing the eigenvector amounts to making a choice of a real number between 0 and 2 pi. But you have to do this for each unit vector, which is equivalent to making a choice of a real valued function on the unit sphere. What a mess.
The thing to note about the above geometrization procedure is that it gives a natural geometric meaning to spin-1/2 particles for the Pauli algebra, for spin-1/2 X spin-1/2 particles for the Dirac algebra, and for (spin-1/2)^3 particles for an extension, etc. That is, the natural quantum numbers look like [tex](\pm 1, \pm 1, \pm 1, ...)[/tex] where the number of dimensions is roughly half the number of spacetime dimensions. One of the spin-1/2s is the usual spin, the others are presumably various internal symmetries.vanesch said:Really, this only looks to me like a "change in representation" between equivalent concepts. I admit of course not knowing this "addition rule" and I have to admit it looks elegant. As you say, sometimes a more elegant formulation of equivalent concepts can get rid of useless complications like useless phase factors or gauges or things like that, sure. It can also serve as a source for inspiration.
However, it still remains equivalent. A bit as trying to solve fundamental problems in classical physics by going from a force formulation to a Lagrangian formulation. You don't change any foundational problems that way. You can at most find suggestions for new principles.
But it does not give a geometrization for scalars. By the way, David Hestenes says pretty much the same thing in his geometrization of the Dirac equation spinors. That is, he says that the Pauli spinors are what you get when you take a Dirac spinor and ignore the antiparticles, and that the Schroedinger equation is what you get when you take the Pauli equation and ignore spin down (or spin whatever). Here's a reference:
Consistency in the Formulation of the Dirac, Pauli and Schroedinger Theories
D. Hestenes (& R. Gurtler), J. Math. Phys. 16, 573-583 (1975).
... Consistency with the Dirac theory is shown to imply that the Schroedinger equation describes not a spinless particle as universally assumed, but a particle in a spin eigenstate. The bearing of spin on the interpretation of the Schroedinger theory discussed. ...
In other words, a primary result of geometrizing the density matrix formalism and assuming that it should be the foundation of QM is that it suggests that all truly elementary particles should be spin-1/2 in density matrix form, that is, that the elementary particles and their interactions should be composed purely of Dirac bilinears. This amounts to the requirement that the standard model's "elementary" particles be composite in such a way that the true elementary particles are never created nor destroyed, but that their creation and annihilation operators appear together in equal numbers in each term in a Lagrangian.
The standard model uses a scalar Higgs particle in giving mass to the elementary fermions and this is incompatible with the above balancing of creation and annihilation operators. To fix it, you have to replace the Higgs with a Dirac bilinear, which also implies that we should look at square roots of masses instead of masses.
Since the elementary particles are point particles, making them composite in this manner requires that one solve the quantum mechanics problem for how one combines multiple particles that can interact with each other in the usual way that spinors do (i.e. (1+cos(theta))/2), but with a potential energy between them that reduces their internal wave function (i.e. the relative displacements of the particles) to be what one gets with an infinite potential well of infinitesimal spatial extent. This makes the bound state appear as a point particle, and the matching of every annihilation with a creation is what you expect if you suppose that the subparticles interact with energies on the order of the Plank mass (and where the low mass of their bound states is due to their extremely deep binding energy). Once you have solved this problem, read the following paper, and pay close attention to the way that matrices are used:
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