I have been looking over the following three links for the Foldy-Wouthuysen transformation from the Dirac-Pauli to the Newton-Wigner representation of Dirac’s equation:
The first shows the calculation itself of this transformation:
The second, an excellent and lucid exposition of the physics (why this is of interest), is to be found at:
The third, dealing with Zitterbewegung motion and the velocity operator in the Dirac-Pauli representation, is at:
What I would like to discuss, for the purpose of getting your reactions as to whether I am on a sensible track, is the possibility that a mechanism for generating fermion mass may be hidden in all of this.
I say this in particular because in the Dirac-Pauli representation, the velocity operator is given by:
where , see reference III. Further, the eigenvalues of this velocity operator constrain the velocity of the Fermion of be the speed of light, see reference II in the middle of page 3. This means that the fermion must be massless and luminous, in the Dirac-Pauli representation. Why this is so, has long been a mystery, and is thought not to make any sense, for obvious reasons.
Now, transform into the Newton-Wigner representation via Foldy-Wouthuysen. The velocity operator in Newton-Wigner now takes the classical form:
where is the position operator. But even more importantly, Newton-Wigner permits a range of eigenvalues less than the speed of light, and so, the fermions permitted by Newton-Wigner are massless and sub-luminous.
Following this to its logical conclusion, this seems to suggest that somewhere hidden in the Foldy-Wouthuysen transformation, we have gone from a fermion which is massless and luminous, to one which has a finite, non-zero rest mass and travels at sub-luminous velocity. It seems, then, that it would be important to specifically trace how the velocity operator (1) of the Dirac-Pauli representation with eigenvalues transforms into the velocity operator (2) of Newton-Wigner which allows a continuous, sub-luminous velocity spectrum, and at the same time, to trace through how the rest mass goes from necessarily zero (with decoupled chiral components), to non-zero with chiral couplings.
By doing so, perhaps one would find a mechanism for generating fermion masses.
One contrast to make here: think about how vector boson masses are generated. One starts with a Lagrangian in which the boson mass term is omitted entirely. Then, via a well-knows technique, one breaks the symmetry and reveals a boson mass. Perhaps the mystery of luminous velocity eigenvalues in the Dirac-Pauli representation is telling us a similar thing: Start out with a Dirac-Pauli Lagrangian in which the mass of the fermion is zero, i.e., without a mass term. Then, the +/- c velocity eigenvalues make sense. Transform that into the Newton-Wigner representation. Somewhere along the line, a mass must appear, because a subliminous velocity appears.
I will, of course, try to pinpoint how this all happens, if it does indeed happen. But I would for now like some reactions as to the tree up which I am barking.