Re: Is the canonical commutation relationship really just a low-mass approximation?




"Arnold Neumaier" <Arnold.Neumaier@xxxxxxxxxxxx> wrote in message news:4A07225B.4030204@xxxxxxxxxxxxxxx
Jay R. Yablon schrieb:

When apply we the canonical :

[x^i,p_j] = ih delta^i_j, i=1,2,3 (1)

is this really just an approximate relationship in the limiting case where a mass m-->0?

That is, is (1) really of the form:

[x^i,p_j] = ih delta^i_j + F(m)^i_j (1)

where F(m)^i_j --> 0 as m --> 0?

No. it is an exact relationship that is at the basis of quantum mechanics, valid for total position and momentum of any physical
system with nonzero total mass, even in the relativistic case and
in quantum field theory.

For systems of mass 0 and spin >1/2 , the position operator
is nonexistent, and therefore no associated CCR exists.
See Section S2g ''Particle positions and the position operator''
of my theoretical physics FAQ at
http://www.mat.univie.ac.at/~neum/physics-faq.txt


Arnold Neumaier


Does this also remain exact for *interacting* fermions, where there is a potential A^v employed in Dirac's equation and p^v-->p^v+eA^v?

Thanks,

Jay

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