Daily Archives: November 24, 2007

Bound States as Density Matrices

Review: Previously, we showed that non Hermitian density matrices arise naturally as the result of using density matrices to define symmetry operations on density matrices, and that these sorts of things might be used in defining Feynman diagrams modeling bound states. We begin with a proton modeled as a set of three valence quarks held together by gluons:
Proton as bound state of three quarks and gluons and sea quark

We cut the gluon lines and look only at the valence quarks (as it is these that determine the quantum numbers of the bound state). This gives us a simplified model:
Proton from quarks ignoring gluons and particle / antiparticle creation
To model this from density matrices requires a simple sort of Feynman propagator, one that corresponds to a non Hermitian density matrix:
Example non Hermitian propagator / density matrix

The corresponding non Hermitian density matrix is a product of two pure Hermitian density matrices. Let us assign to the colors red, green, and blue, the density matrices for spin in the +x, +y, and +z directions, respectively. Then, for example, the above propagator becomes:
Z to X non Hermitian density matrix.
In the remainder of this post we take Feynman diagrams like this, and assemble them into density matrices that represent the bound state. (They will be 3×3 matrices of Feynman diagrams or, equivalently, 3×3 matrices of non Hermitian density matrices.)
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