19 research outputs found
Unambiguous quantization from the maximum classical correspondence that is self-consistent: the slightly stronger canonical commutation rule Dirac missed
Dirac's identification of the quantum analog of the Poisson bracket with the
commutator is reviewed, as is the threat of self-inconsistent overdetermination
of the quantization of classical dynamical variables which drove him to
restrict the assumption of correspondence between quantum and classical Poisson
brackets to embrace only the Cartesian components of the phase space vector.
Dirac's canonical commutation rule fails to determine the order of noncommuting
factors within quantized classical dynamical variables, but does imply the
quantum/classical correspondence of Poisson brackets between any linear
function of phase space and the sum of an arbitrary function of only
configuration space with one of only momentum space. Since every linear
function of phase space is itself such a sum, it is worth checking whether the
assumption of quantum/classical correspondence of Poisson brackets for all such
sums is still self-consistent. Not only is that so, but this slightly stronger
canonical commutation rule also unambiguously determines the order of
noncommuting factors within quantized dynamical variables in accord with the
1925 Born-Jordan quantization surmise, thus replicating the results of the
Hamiltonian path integral, a fact first realized by E. H. Kerner. Born-Jordan
quantization validates the generalized Ehrenfest theorem, but has no inverse,
which disallows the disturbing features of the poorly physically motivated
invertible Weyl quantization, i.e., its unique deterministic classical "shadow
world" which can manifest negative densities in phase space.Comment: 12 pages, Final publication in Foundations of Physics; available
online at http://www.springerlink.com/content/k827666834140322