Dear Paul,
thank you very much for your support. It is very important to me to have feedbacks.
You are right about the analogy with the Klein's idea. I briefly mention this aspect and the dualism with extra dimensional theories on a paper which I have submitted to PRD few days ago. I will post it on arXiv as soon as possible.
An extract from the introduction:
"Nevertheless it must be
noted that Klein's original proposal was to use 5D field theory
in an attempt to interpret Quantum Mechanics (QM). In
his famous paper Quantentheorie und funfdimensionale Relativitatstheorie
\cite{Klein:1926tv} he noticed that Periodic Boundary Conditions
(PBCs) at the ends of a compact XD provide an analogy with the Bohr-Sommerfeld
quantization condition - in particular he used this hypothetical cyclic XD to interpret the quantization of the electric charge. Similarly, it is well known that the solution of the
mass spectrum of an XD field theory is performed by
a mathematical procedure which turns out to be parallel to the one used for the semi-classical determination
of the energy spectrum of simple Schrodinger problems. Examples are
the analogies between the resolution of: the mass spectrum of a
Kaluza-Klein (KK) theory and the quantization of a ``particle'' in an infinite well
potential \cite{Randall:1999vf}; the mass spectrum for an XD theory
with mass or kinetic lower dimensional terms at the boundaries of a compact XD and the Schrodinger problem with a Dirac delta potentials \cite{Dvali:2001gm,Carena:2002me};
the mass spectrum with soft-walls or dilatons and the semi-classical quantization of the harmonic oscillator \cite{Karch:2006pv}.
The KK mass spectrum in a given compact XD background is in fact fixed by imposing
consistent Boundary Conditions (BCs), whereas the evolution along
the XD is described by bulk Equations of Motion (EoMs) which play the role
of the Schrodinger equation of the problem \cite{Dvali:2001gm,Carena:2002me}. We may also note that an XD sector contributes to other typical quantum phenomena such
as the anomalous magnetic momentum, the lamb shift, the Casimir effect
and so on \cite{Hong:2009zw,Brodsky:2008tk,Hosotani1983193}."
What I am actually trying to do is to show that there is a matching between classical fields with intrinsic periodicities and ordinary quantum field theory. I have found that classical fields can be described in Hilbert space, with the Schrodinger equation and commutation relations, the space-time evolution is described by the ordinary Path Integral. The parallelism is surprising and goes from elementary aspects of QM mechanics to more evolved aspects of modern physics. For instance, the correspondence between classical fields (dual to XD fields) and quantum fields is of the same kind of the AdS/CFT correspondence. The theory resemble also string theory....
There is a lot to say and this seems to be only the beginning of a long story...
Best regards,
Donatello
