Daily Archives: December 16, 2007

Regge Trajectories and Koide’s formula

As many of you know, I’m a proponent of the density operator formalism of quantum mechanics as opposed to the usual state vector / spinor formalism. The basic idea with density operator formalism is that quantum states should be represented by a density operator, or density matrix, rather than a state vector. This sort of idea is similar to the S-matrix theory that was part of the foundations of string theory, and I thought it would be worth exploring the similarities and differences.

The idea behind the state vector / spinor formalism is that quantum states are represented by state vectors or spinors. These sorts of objects are linear. In the state vector / spinor formalism, the spin state of an electron is typically represented by a Pauli spinor, a 2 element complex vector. Such a quantum state corresponds to spin-1/2 in some direction, \vec{v} = (v_x,v_y,v_z) . The operator for spin in this direction is typically written out as a 2×2 matrix using the Pauli spin matrices. One takes the dot product of the spin direction with the vector of Pauli spin matrices, giving:
Pauli spin operator in v direction

The spinor representing spin-1/2 in the v direction is a two element vector which is an eigenvector of the above matrix with eigenvalue +1. It is easy to see that for almost any 3 dimensional unit vector \vec{v} , such an eigenvector (in ket form) is given by:
Pauli spinor for spin 1/2 in direction v
The above formula gives a valid ket for any spin direction except (0,0,-1), but it isn’t normalized. To normalize it, divide by \sqrt{2+2v_z} . Higher spin states than spin-1/2, say spin-n/2, need spinors of size (n+1)x1 in size. So spin 3/2 will be 4×1 vectors, spin 2 will be 5×1, etc, but we won’t discuss these much.
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