MadSci Network: Physics

Re: VRD: first ,second and nth quantisation

Date: Tue May 30 11:35:12 2000
Posted By: Ronald Fisch, Physics, Washington University
Area of science: Physics
ID: 957302549.Ph

First quantization refers primarily to the quantum mechanics
of a single particle in some classical potential.  Many complex problems
can be approximately reduced to problems of this form.  The usual way
to solve such a problem is to find the eigenvectors and eigenvalues of
the Hamiltonian function (the Schroedinger equation), which represents
the motion of the single particle in the potential.  While the time
dependence of the Schroedinger equation is linear, the eigenfunctions
and eigenvalues are not linear functions of the potential.  It is
sometimes helpful to think of first quantization as the quantization of
angular momentum in integer multiples of the fundamental unit h-bar.

Second quantization occurs in quantum field theory.  It is the
quantization of particle number.  For electrons, protons or any other
charged particles, quantization of particle number is essentially the
same as quantization of the electric charge.  But it is also true
that other particles, such as photons (the modes of the electromagnetic
field), which have no electric charge, are quantized.  Quantization of
particle numbers is not merely a result of charge conservation.  Dirac
argued that quantization of the electrical charge is required if
magnetic charges exist.  Such magnetic charges occur in some popular
theories of the early universe.  However, despite major efforts to find
them, experimental evidence for magnetic charges remains elusive.
Other people have suggested that the reason why particle numbers are
quantized is that particles are "solitons", localized solutions of some
nonlinear field theory.

I am not sure what you mean by "nth quantization".  Some people believe
that in order to find a fully self-consistent theory of quantum gravity
we should quantize the fluctuations in the geometry of space and time.
As far as I know, this has not been done successfully yet.

I do not really understand what you mean when you say that "first
quantization was a mistake".  It is an approximation which is often
useful.  In a single particle problem we do not need to worry about the
quantization of particle numbers.

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