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Science Forum Index » Physics - Relativity Forum » Is there anything wrong with this canonical...
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| Jay R. Yablon... |
 Posted: Thu Jun 19, 2008 11:02 pm |
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I have linked a 2.5 page calculation for the commutators [x_k,p_0] at:
http://jayryablon.files.wordpress.com/2008/06/linear-mass-commutator-calculation.pdf
It is simple and to the point. Start with [O,m] = 0 for any operator O
including O=x_u. Take Dirac's equation to be true. Take the canonical
commutation relationship:
[x_i,p_j]=i hbar eta_ij i,j=1,2,3 (1)
to be true, where diag (eta_ij) = (1,1,1) are the space components of
the Minkowski metric tensor. From this, one may directly deduce that:
[x_k,p_0] psi = -i alpha_k psi (2)
where alpha_k = gamma^0 gamma_k, and the gamma^u are the Dirac matrices.
Equation (2), which is (2.7) in the linked file, suggests that the
[x_k,p_0] commutators times i=sqrt(-1) must be equal to the eigenvalues
of the Dirac alpha_k, which alpha_k, by the way, also sit along the 0k
components of the
-2i sigma_uv = [gamma_u, gamma_v] (3)
and that these [x_k,p_0] commutators can therefore only be considered in
relation to their associated eigenstate vectors.
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 [O,m^2]=0 and m=p^u.p_u cannot, by itself, reveal.
I would like some input on whether this is on the right track or whether
I am overlooking something.
Jay.
____________________________
Jay R. Yablon
Email: jyablon at (no spam) nycap.rr.com
co-moderator: sci.physics.foundations
Weblog: http://jayryablon.wordpress.com/
Web Site: http://home.nycap.rr.com/jry/FermionMass.htm |
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