Fwd: notes 129

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Subject: Fwd: notes 129
Date: Tue, 14 Apr 2009 14:51:56 EDT

Let me give some comments.

In eq. (26) of 129(7) there seem to be typos in the indexes. You have given the correct form in eq. (14) of 129(9).

The current density j0 of the dirac spinor is the usual fermion current density. Are there physical interpretations of the other densities j1, j2, j3? As far as I know there are not. j3 could be interpreted as a difference of spin densities.

In 129(8) you wrote eq. (23) as an operator equation but applied to nothing. This seems not to be meaningful, although I remember that it is written this way in books. Probably psi bar has to be placed to the right, but then the multiplication of eqs. (22,23) gives other terms. Does it have to do with the fact that quantum mechanics is constructed as a linear operator theory? Then you can omit the nonlinear terms and probably arrive at the same result as from eqs. (22,23).

Finally in one of the earlier notes 129 you introduced the 4-spinor by “rewriting” a 2×2 tetrad. Although both types of objects are isomorphic, they operate on different vector spaces. The tetrad connects the base manifold with the tangent space, while a 4-spinor consists of objects being all of the same type, it is normally not interpreted as a transformation. Can you give some mathematical arguments why it is allowed to equate both types of objects?

Horst

Many thanks for going through the notes as usual. This is always very helpful.

1) Eq. (26) of 129(7) and the third term of Eq. (14) of 129(9) are identical. 2) The current densities certainly have physical meaning in relativistic quantum mechanics, but the probability density is the one used in the Schroedinger equation of non-relativistic quantum mechanics 3) Eq. (23) of note 129(8) operates to the left on the adjoint spinor wavefunction, as discussed by L. H. Ryder in “Quantum Field Theory” (Cambridge University Press, 2nd ed., 1996). Then Eq. (26) is the expectation value of the quantity in brackets, leading self consistently to Eq. (31). Multiply eq. (22) from the left by eq. (23), and we algebra of type (a + b)(a – b) = a squared + b squared. 4) As in ECE papers 4 and 36 and in several other ECE papers, some referred and published in “Foundations of Physics Letters”, the ECE wave equation is always one in a tetrad wavefunction, which is arranged in SU(2) rep space in a 2 x 2 matrix. The Pauli matrices are also tetrads. Each element of the tetrad matrix obeys the wave equation independently. When arranged in a column four vector each element obeys the wave equation independently, QED. This is the wave representation of the Dirac equation, and the Klein Gordon equation. Finally factorize the d’Alembertian to give the Dirac equation.

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