Re: Reality Conditions

On Fri, 28 Nov 2003, island wrote:

> Dirac admitted that it can't be said the negative energy operators of
> his dynamic quantum field solutions in quantum space represent
> positrons, because it would make the dynamic relations "all wrong", ...

Fortunately Dirac also realized the correct statement, which allowed him
to predict the existence of the positron. ;-) If he had not realized that
point, his Nobel prize would be uncertain. Fortunately he did - and the
prediction of the antiparticles was one of the most amazing predictions of
quantum physics; most other major discoveries were postdictions because
experimentalists were very fast.

        http://www.nobel.se/physics/laureates/1933/

Dirac sort of understood Pauli's exclusion principle. The number of
electrons in a state must be either 0 or 1. Nature always wants to
minimize the energy; in other words, the physical vacuum must be the
lowest energy eigenstate. In the case of his relativistic equation, we
must fill all the negative-energy states with 1 electron (there can't be
more electrons in a single state), because adding an electron with
*negative* energy *lowers* the energy.

Once we fill all these negative energy states - the Dirac sea - we obtain
the physical vacuum and we may renormalize its energy so that we can call
it "zero" (the additive shift has no physical consequences in a
non-gravitational theory, and we know that the vacuum energy even in
relativity must be essentially zero). The excitations of the physical
vacuum are either positive-energy particles (electrons), or "holes" in the
Dirac sea of negative-energy solutions. These "missing electrons with
negative energy" obviously have *positive* energy and *positive* charge,
and they are identified with positrons - new particles.

If one started with an equation for positrons, one would have obtained the
same result - the electrons would appear as the holes in the Dirac sea of
negative-energy states of the positron. There are electrons, and there are
positrons. There is a symmetry between the creation operators and the
annihilation operators for fermions. Unfortunately, this symmetry does not
exist for bosons (because they don't use the anticommutator {a,b}=ab+ba,
which is symmetric, but rather the commutator [a,b]=ab-ba, which is
antisymmetric), and the concept of the "Dirac sea" does not work for
bosons. In fact, there can be 0,1,2,3... infinity bosons in a single
state, and we could not fill the "bosonic sea". The general principle that
the vacuum is the lowest energy state still holds, of course.

There was a huge confusion among the leading physicists in the 30s whether
the second quantization should have also been applied to fermions - almost
no one properly understood that the Dirac sea *was* equivalent to the
second quantization in the fermionic case.

> since classical dynamics breaks the association between real positive
> and negative energy solutions based on the reality condition which

I am not quite sure which reality condition you are talking about. Dirac
fields must *always* be complex in 3+1 dimensions. In fact, all *charged*
particles must be described by complex fields because the electromagnetic
U(1) symmetry acts as a rotation of the overall phase. A reality
projection for Dirac's spinors is possible - the Dirac spinor is reduced
to something called the Majorana spinor (which is important for neutrino
physics) - but once again, it can't be done for charged particles.

I apologize but the last 6 paragraphs of your post made no sense to me,
and therefore I can't reply to them.
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