Monthly Archives: October 2007

Fictitious Snuark Vacuum II

In the previous post we introduced a fictitious snuark vacuum with the intention of using it to simplify calculations. In the snuark algebra, complex amplitudes show up only as a geometric result of an interaction between two mass vertices. We’d like to convert this sort of computation, which uses Pauli spin matrices, into calculations where we associate an amplitude with each mass vertex by itself. There are 36 mass vertices, 12 of which are trivial. We found that we would know the remaining 24 by symmetry if we could solve for just four of them: k_{+x,+y}, k_{+x,-y}, k_{-x,+y}, k_{-x,-y} . In this post, we find these numbers, and then test them by comparison with explicit calculations using Pauli matrices.

A Beautiful Theorem: When visiting a strange land, it’s always nice to take in the pretty sights. For our utilitarian calculations, we will use a beautiful result for products of primitive idempotents (density matrix states) of the Pauli algebra:

Let \vec{a}, \vec{b}, \vec{c} be three real unit vectors in 3 dimensions. They define the vertices of a spherical triangle. Let A, B, and C be the projection operators for Pauli spin in the \vec{a}, \vec{b}, \vec{c} directions. Then the projection operators are related by the equation:
General relationship between ABC and AC projection operators
where S_{ABC} is the oriented area of the spherical triangle defined by the three unit vectors.
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A Fictitious Snuark Vacuum State

The calculations for the NNLO and higher corrections to the snuark mass interaction are difficult. I’ve already got some sort of sign error. Before I go on, I’m going to write a computer program to do the higher level calculations.

Of course it is possible, and even natural, to write a Java program to do snuark QFT calculations. One can implement the Pauli algebra projection operators for spin in the +/- x, y, and z directions, and multiply away. One computes the phases by writing the product of projection operators, for example, \rho_x\;\rho_y\;\rho_z as a complex constant k times the final and initial projection operators: \rho_x\;\rho_y\;\rho_z = k \rho_x\;\rho_z , or in Pauli matrix form:
XYZ to k XZ projection operator reduction

In the same manner, arbitrarily long products of projection operators can be reduced to a complex multiple of the initial and final projection operators. If there is a forbidden transition somewhere in that product, then the complex multiple will be zero. For the above product, k = \sqrt{0.5}\exp(i\pi/4) .
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