It has been awhile since my last blog entry, but if you want to check out some my recent wanderings through physicsland, check out sci.physics.foundations, relativity, and research.
Here, I would like to show a rather simple calculation, which may cast a different light on how one needs to think about the canonical commutation relationship . I would very much like your comments in helping me sort this through. You may download this in pdf form at https://jayryablon.files.wordpress.com/2008/06/linear-mass-commutator-calculation.pdf.
I. A Known Square Mass Commutation Calculation
Consider a particle of mass m as a single particle system. Consider canonical coordinates , and that at least the space coordinates are operators. If we require that the mass m must commute with all operators, then we must have , and by easy extension, . It is well known that the commutation condition , taken together with the on-shell mass relationship and the single-particle canonical commutation relationship , where is the Minkowski tensor, leads inexorably to the commutation relationship:
where is the particle velocity (in c=1 units) along the kth coordinate. I leave the detailed calculation as an exercise for the reader not familiar with this calculation, and refer also to the sci.physics.research thread at http://www.physicsforums.com/archive/index.php/t-142092.html or http://groups.google.com/group/sci.physics.research/browse_frm/thread/d78cbfecf703ff6a.
I would ask for your comments on the following calculation, which is totally analogous to the calculation that leads to (1.1), but which is done using the linear mass m rather than the square mass , and using the Dirac equation written as , in lieu of what is, in essence, the Klein Gordon equation that leads to (1.1).
2. Maybe New?? Linear Mass Commutation Calculation
Start with Dirac’s equation written as:
Continue to use the canonical commutator . Multiply (2.1) from the left by noting that to write:
This separates into:
Now, use the canonical relation to commute the space (k) equation, thus:
In the final line, we use Dirac’s equation written as , and specifically, the component equation .
If we require that , which is (2.2), then (2.5) reduces easily to:
Finally, multiply from the left by , and employ and to write:
If we contrast (2.7) to (1.1) written as , we see that the velocity has been replaced by the Dirac operator , that is, .
Here are my first set of questions:
1) Is the calculation leading to (2.7) correct, and is (2.7) a correct result, or have I missed something along the way?
2) If (2.7) is correct, has anyone seen this result before? If so where?
3) Now use the plane wave so that we can work with the Dirac spinors , and rewrite (2.7) as:
The upper member of (3.1) is an eigenvalue equation. Reading out this equation, I would say that the commutators are the eigenvalues of the Dirac matrices, which are:
and , (3.2)
in the respective Pauli/Dirac and Weyl representations, and that the u are the eigenvectors associated with these eigenvalues . Am I wrong? If not, how would one interpret this result? Maybe the commutators can be discussed in the abstract, but it seems to me that the commutators can only be discussed as the eigenvalues of the matrices with respect to the eigenstate vectors u. This, it seems, would put canonical commutation into a somewhat different perspective than is usual.
Just as Dirac’s equation reveals some features that cannot be seen strictly from the Klein Gordon equation, the calculation here seems to reveal some features about the canonical commutators that the usual calculation based on and cannot, by itself, reveal.
I’d appreciate your thoughts on this, before I proceed downstream from here.