Hyperbolic Quantum Mechanics

In a recent post Lubos Motl stated:

     There only exist two types of theories in all of physics:

    

          1. Classical physics

          2. Quantum mechanics

          3. There is simply not third way.

     I added the third option in order to emphasize that it doesn't exist.

This is correct (gee, I am agreeing with Lubos, is everything all right?), but (as advertised last time) today I want to dig in deeper in option 3 and prove why this is so. In the process we will better understand quantum mechanics and explore a brand new (and unfortunately sterile) landscape of functional analysis.  

Quantum mechanics shares with classical physics a key property: the laws of nature do not change when we consider additional degrees of freedom. For example, the tensor product of two Hilbert spaces is still a Hilbert space and the composed system does not become classical, quantum-classical, or something else. This invariance of the laws of nature under composition is best expressed in the category formalism and in there we have three classes of solutions:

1. Elliptic composition (quantum mechanics)
2. Parabolic composition (classical physics)
3. Hyperbolic composition (a hypothetical hyperbolic quantum mechanics)

For quantum mechanics von Neumann has unified Heisenberg's matrix formulation and Schrodinger's approach in the Hilbert space approach, and we will see that the functional analysis has a categorical origin and there is a mirror categorical formalism for hyperbolic quantum mechanics as well. We will see that this is unphysical for obvious reasons and option 3 while nice as a mathematical curiosity has no physical usefulness. But it is helpful to better understand the Hilbert space formulation of quantum mechanics.

Let me start with some elementary mathematical preliminaries. In quantum mechanics one uses complex numbers and they originate from the fundamental composition relationships of the two quantum mechanics products: the commutator and the Jordan product. In hyperbolic quantum mechanics one uses split-complex numbers

In split complex numbers the imaginary unit is \(j\) with \(j^2 = +1\) but \(j \ne 1\). In matrix representation, \(j\) is the 2x2 matrix  with zero on the diagonal and 1 on the off-diagonal. Split complex numbers have polar representations similar with the complex numbers, but the role of the sines and cosines is replaced by hyperbolic sines and hyperbolic cosines. Very important, the fundamental theorem of algebra does not hold in split complex numbers and this has very important physical consequences for hyperbolic quantum mechanics.

Formally, hyperbolic quantum mechanics is obtained by replacing \(\sqrt{-1}\) with \(j\) in the commutation relations. However, there are no Hilbert spaces in hyperbolic quantum mechanics!!!

So let us see the corresponding formulation of hyperbolic quantum mechanics.

Starting with de Broglie's ideas, in a hypothetical universe obeying hyperbolic quantum mechanics one would attach to a particle not a wave, but a scale transformation \(e^{jkr}\) and the scale transformation caries a given momenta in accordance with the usual de Broglie relation (in hyperbolic quantum mechanics one still have the same Planck constant): \(p = \hbar k\)

Continuing with matrix mechanics, Heisenberg approach carries forward identically, but here we hit the first roadblock: the matrices cannot be always diagonalized because the fundamental theorem of algebra does not hold in this case.

Continuing on Schrodinger equation, it's hyperbolic analog is:

\(+\frac{\hbar^2}{2m}\frac{d^2}{d x^2} \psi (x) + V(x) = E \psi (x) \)

To really understand what is going on in the hyperbolic case, we need to investigate the functional analysis of split-complex numbers. The best starting point are the metric spaces, and not the more abstract setting of topology. Key in a metric space is the triangle inequality. If you follow any standard functional analysis book you see that all metric spaces over complex numbers ultimately prove their triangle inequality starting from the triangle inequality of complex numbers. Moreover, this arises out of the trigonometric identity:

\({cos}^2 x + {sin}^2 x = 1\)

But in the hyperbolic case, one has this identity:

\({cosh}^2 x - {sinh}^2 x = 1\)

and in each of the four quadrants of the split-complex numbers, a reversed triangle inequality holds. If you cross the diagonals all bets are off, but it turns out that one can successfully introduce a vast functional analysis landscape for split complex numbers just as rich as the usual functional analysis. The reason for this is that ultimately the fundamental Hahn-Banach theorem holds in the split-complex case as well. To coin a name for the mirror analysis of split complex numbers, we will place in front the prefix para. As such in the hyperbolic case we will have a para-Hilbert space which is a very different beast than the usual Hilbert space.

There is a translation dictionary for all definitions and proofs in para-analysis:

Elliptic	Hyperbolic
Triangle inequality	Reversed triangle inequality
Sup	Inf
Bounded	Unbounded
Complete	Incomplete

A sequence \(x_n\) in a metric space X = (X,d) is said to be para-Cauchy if for every \(\epsilon \gt 0\) there is an \(N = N(\epsilon)\) such that:  \(d(x_m,x_n) \gt \epsilon \) for every \(m,n\gt N\) The space X is said to be para-incomplete if every para-Cauchy sequence in X diverges.

A para-Hilbert space is an indefinite para-inner product space which is para-incomplete.

The para-Hilbert space is non-Hausdorff, but the key showstopper is that given a point x, and a convex set (which would correspond to a state space) there is not a unique perpendicular to the set. As such a hyperbolic GNS construction is not possible! This means that we lack positivity and hence we cannot define a physical state.

Invariance under composition demands 3 solutions: elliptic, parabolic, or hyperbolic. We can define a state only in the elliptic and parabolic cases (quantum and classical physics). 
By overwhelming experimental evidence, nature is quantum mechanical, no exceptions allowed.