Re: please help my confusion about particles and irreps.
Igor Khavkine wrote:
Let me first agree with you. Yes, you can project out 2 extra
components, and restrict the space of functions using Dirac's
equation, and show that resulting functions afford an unitary
irreducible representation of the Poincare group, just as
Wigner's functions do. You can also apply the Foldy-Wouthuysen
transformation to Dirac's functions and arrive to the operators
of velocity and position that make certain physical sense.
Yes, you can do all that. My question is whether all these non-trivial
manipulations have any physical meaning? It seems to me that
we have started from something (solutions of Dirac's equation)
that didn't have much physical sense and then applied a series of
patches (like Foldy-Wouthuysen transformations) to put our creation
in a better shape.
Eugene, lets wrap up this subthread. Our discussion has already
fructured into too many pieces, to my preference.
Please note what happened here. As above, you disagreed with a
mathematical statement. The mathematical statement was that solutions of
the Dirac equation provide a unitary representation of the Poincare
group, whatever the mathematical steps necessary to get there. Your
objections could have been swept away by simply taking on the task of
verifying it yourself. Otherwise, undue burden is placed on whomever you
are corresponding with. A burden of which I am starting to get weary,
which means that sooner or later I'll stop contributing to this
discussion.
In the objections that remain in your message, it seems to me that your
original objections were motivated by your physical interpretation (or
lack of it) of the Dirac equation. If that is so, I recommend separating
your mathematical doubts from the physical ones. For the mathematics, I
think you should take upon yourself at least some of the rudimentary
tasks of verifying mathematical statements made in the discussion. As
for physics, simply because you lack a physical interpretation of a
physical theory does not imply that one exists. Therefore, I recommend
that instead of out flat denying existence of such an interpretation you
first inquire whether one exists or not.
For the case of the Dirac equation. It is immaterial whether it came
before or after Wigner's classification. Both have interpretations
associated with them. Complexity is not an issue here. You've already
presented some of the interpretations of Wigner's formalism, including
interpretation of the two components of the spinor through the
Stern-Gerlach experiment. Interpretation of the components of the Dirac
spinor come about in exactly the same way. They are given physical
meaning only in so far as the Dirac equation reduces to the Pauli
equation (Schroedinger + spin) in the non-relativistic limit,
In the relativistic case you need to keep 4 components in the 1-electron
wave function. These components have no observable physical meaning.
and by
extension to the Stern-Gerlach experiment. Same goes for Dirac wave
functions themselves. The use of the Dirac equation as opposed to any
other one that reduces to the Pauli equation is vindicated by the
correct prediction of single particle spectra (free particle, Hydrogen
atom, particle in external field). There is yet another advantage of the
Dirac formulation, it is a local covariant PDE over spacetime. All the
machinery that has been developed can be thrown at it to find solutions
and energy spectra in various conditions that can only be formulated
more abstractly in other settings.
Whatever your personal opinion of the Dirac equation and its
applications, you cannot deny that these applications exist. However, if
you have found alternatives, no-one is stopping you from using them. By
the same token, you cannot make people give up the Dirac equation if
they do not wish too, after all they are perfectly mathematically
justified in using it.
I think that arguing about such "details" as Dirac's equation we
are missing the *big picture*. I don't know what goals you are pursuing
by continuing this discussion, but my goals are as follows:
1. I want to convince you that I have a legitimate quantum theory of
electromagnetic interactions.
2. I want you to agree that this theory (RQD) is not equivalent to
the traditional QED. The two *big pictures* painted by these two
theories are different. There are places where the two theories
are very close to each other (e.g., the S-matrix calculations), but
there are other places with fundamental differences. For example,
the manifest covariance, that is central in QED, does not play any role
in RQD; RQD predicts instantaneous propagation of interactions
that is forbidden in the traditional approach.
3. I would like you to agree that if two physical theories predict
different results, then only one of them can be correct.
The question then is which way of doing things is better? By "better"
I mean mainly two aspects:
a) the ability to accurately predict/explain experiments
(e.g., the time evolution);
b) internal consistency and logic.
If this *big picture* approach is adopted, then our argument about
Dirac's equation is not that crucial. Dirac's equation belongs to
the traditional approach. I am not forcing you to give it up.
I am just telling that there is another way of doing things that does
not involve the Dirac's equation. For example, the (hyper)fine structure
of the hydrogen atom spectrum can be obtained from the Breit's
Hamiltonian as in sections 83 - 84 of Berestetskii, Livshits and
Pitaevskii book. This is just one place where our theories diverge.
Eugene.
.
