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, quantumclassical, 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:
 Elliptic composition (quantum mechanics)
 Parabolic composition (classical physics)
 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 splitcomplex 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 offdiagonal. 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): \(p = \hbar k\)
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 splitcomplex 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 splitcomplex 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 HahnBanach theorem holds in the splitcomplex 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 paraHilbert space which is a very different beast than the usual Hilbert space.
There is a translation dictionary for all definitions and proofs in paraanalysis:
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 paraCauchy 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 paraincomplete if every paraCauchy sequence in X diverges.
A paraHilbert space is an indeﬁnite parainner product space which is paraincomplete.
The paraHilbert space is nonHausdorff, 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.
By overwhelming experimental evidence, nature is quantum mechanical, no exceptions allowed.
I think there may be circumstances where hyperbolic structure exists. It must be remembered that relativity is hyperbolic, and we must grapple with that. The Cartan decomposition of a group G with the quotient H = G/K has the algebraic generators g = h + k with
ReplyDelete[h, h] ⊂ h, [h, k] ⊂ k, [k, k] ⊂ h.
The decomposition can be taken according to g = g^0 + g^+ + g^ so that g^0 is Euclidean, g^+ is compact and g^ is noncompact. We may then write g^+ = h + k and it in Hermitian symmetric spaces we can have g^ = h + ik. In this sense we have g^ as the complexification of of g^+. Since g^ is uniquely determined by g^+ we can have in this case a duality where the noncompact construction (hyperbolic) is isomorphic to the elliptic construction.
An case of this is SL(2n,C)/SU(2n). For n = 1 we have that the quotient SL(2,C)/SU(2) that is dual to SL(2,C)/SU(1,1), which is evident by SL(2,C) ~ SU(1,1)xSU(2). I think in these cases such as E8(8)/SO(16), the N = 3 and E7(7)/SU(8) for 8 fold entanglements ~ black holes in supergravity.
The important thing to remember is this duality, which constrains the hyperbolic theory to be identical to the elliptic theory. So in some sense you are right, or that the third option either does not exist or in the case of Hermitian symmetric spaces is just a copy of an elliptic case.
LC
Lawrence, SL(2,C) is another number system allowed under elliptic composability. This corresponds to another decomposition of the d'Alembertian and is provable to be equivalent with Dirac's equation. This generalization of the number system corresponds to conservation of a current probability density.
DeleteGroups enter QM as different number systems, but they have nothing to do with invariance under composition. You encounter Lie algebras, not Lie groups in QM.