quantum123 said:
Relativity theory is as guilty as Quantum Theory in reducing a physical situation into a mathematics - rubber sheet geometry (manifolds) vs Hilbert space state vectors.Time dilation vs collapsing the wave-particle duality.
The common folks out there especially those who play with car engines have a hard time swallowing all that.
I afraid my answers make situation with “the common folks out there especially those who play with car engines” even worse.
The non-relativistic QM as well as the Newtonian Mechanics may be roughly considered as the empty stage, platform. In order to give it the physical content one should introduce the interactions. If you consider that on the fundamental level, so far we know four and only four fundamental interactions and you use the presence of the phases in the description of the QM system (local gauge invariance). The simplest case is QED (minimal coupling) which require one phase:
U(1). That already show you why you need the complex numbers numerical system to comply with the experimental evidence. The detailed theoretical and experimental demonstration is given in the AB and A.Tonomura et al. papers. However, to describe weak (actually electroweak) interactions you need more phases:
U(2). Even before electroweak, W.Pauli and P.A.M. Dirac “spoiled” everything through introduction of spin. They demonstrated that the complex numbers are not enough. Later we saw that the electron carry not only the electric charge but also the weak charges. And now came E.Cartan and “spoiled” everything again. To describe QM system we need to use the unitary groups. But the unitary groups form the infinite chain. Why
U(2)? Which one out of Glashow prostitutes fit? Here E.Cartan gives us a hint: there is one finite chain - exceptional Lie groups. Why it is finite? You find that it is the underlined numerical system of the Cayley numbers responsible for that. And now you meet the remarkable mathematical result (the Hurwitz algebras and Hurwitz theorem).
I prefer to use a physical jargon instead of rigorous mathematical definitions. In contrast to infinite diversity of all possible algebras, the Hurwitz algebras are the number systems. In the literature they are called quadratic normal division algebras or, shortly, composition algebras. Below, I will describe conditions that they should satisfy, but now I present the result of the famous Hurwitz theorem: The dimensionalities of these algebras are respectively
1,
2,
4 and
8.
That’s all. There are no others. Most popular examples are:
1)1-dim algebra of real numbers;
2)2-dim algebra of complex numbers with real coefficients;
3)4-dim algebra of real quaternions (Hamilton numbers);
4)8-dim algebra of real octonions (Cayley numbers);
Roughly, the conditions they should satisfy are the following:
1) division - the equations a*x=b and x*a=b have unique
solution.
Translation to physics say: It is required repeatability for
the results of the physical measurements and their independence
from the experimentalist or interpreter subjective choice.
2) composition – N(x*y)=N(x)*N(y).
Translation to physics say: It is possible to construct
many-particles physical system without interaction (ensemble
of completely independent physical objects first and to
introduce the interaction at later stage).
The post #14 above contains small part of the relevant refs and wrong completely in each of its statements. QQM with the complex scalar product is
U(2) gauge field theory and the composition property assures that the consistent solution for the Fock second quantization procedure exists. OQM with the complex scalar product is
U(4) gauge field theory. CQM with the real scalar product is dispersion free field theory and thus allow the wave mechanical generalization of Newtonian Mechanics and the partial solution of the measurement problem (for the non-relativistic limit).
I worked on the problem of the tensor products in QQM with the complex scalar product during four years. I was convinced that the problem as well as the solution is trivial. Therefore I was convinced that I am simply stupid. Only S.L. Adler papers on the algebraic chromodynamics indicated to me where I was wrong and allows finding the solution. It turns out not so trivial. The solution allows defining the mathematical architecture of all of the physics according to W.R. Hamilton and E. Schrödinger conjecture.
Regards, Dany.
