## Solving Hilbert’s Sixth Problem

### Tying it all together for quantum mechanics

Using commutators and anti-commutators we have seen last
time the relationship between the two products. The remarkable fact is that
this relationship can be derived from composability (see http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1103900192
for the lengthy proof) and in general you get:

[A, B, C]

_{θ}**+ ħ**^{2 }/4[A, B, C]**=0**_{xρ}
where [A, B, C] = (AB)C – A(BC) is the associator

with x=+1, 0, -1

In the quantum mechanics case (x=-1), since

**ρ**is skew-symmetric:
[A

**ρ**A, B, A]**= ((A**_{ρ}**ρ**A)**ρ**B)**ρ**A – (A**ρ**A)**ρ**(B**ρ**A) = ((0)**ρ**B)**ρ**A – (0)**ρ**(B**ρ**A)
=0

and therefore:

[A

**θ**A, B, A]**=((A**_{θ }**θ**A)**θ**B)**θ**A – (A**θ**A)**θ**(B**θ**A) = 0
Since

**ρ**is skew-symmetric and obeys the Leibniz identity:
A

**ρ**(B**ρ**C) = (A**ρ**B)**ρ**C + B**ρ**(A**ρ**C) it is easy to show that it obeys the Jacobi identity:
A

**ρ**(B**ρ**C) = (A**ρ**B)**ρ**C + B**ρ**(A**ρ**C) = -C**ρ**(A**ρ**B) - B**ρ**(C**ρ**A)
and so:

A

**ρ**(B**ρ**C) + B**ρ**(C**ρ**A) + C**ρ**(A**ρ**B) = 0 [Jacobi identity]
Hence

__ρ____is a Lie algebra.__

__In turn the__

__Jordan__

__and Lie algebra give rise to a C* algebra and we obtain quantum mechanics in the algebraic formalism. The standard Hilbert space formulation is recovered by the GNS theorem/construction.__
In the classical case (x=0) there are no Jordan
algebras, and in this case one has the regular function multiplication and the
Poisson bracket as realizations of the products

**θ**and**ρ**.
What can we say about the third case, the hyperbolic
composability x=+1?

In this case we are lead to a hypothetical quantum mechanics
over

**The interesting part is that in this number system, the functional analysis is completely changed because the norm triangle inequality which is the foundation of most of the results in functional analysis is replaced by a reversed triangle inequality (http://arxiv.org/pdf/1311.6461v2.pdf ). The key difference however between complex quantum mechanics (parabolic composability) and split-complex quantum mechanics (hyperbolic composability) is the**__split complex numbers.__**. In other words, we are not guaranteed to have positive probability predictions, and**__lack of positivity__**Hyperbolic composability violates one of the principles of nature introduced in prior posts: positivity. Mathematically hyperbolic quantum mechanics is just as rich and interesting as ordinary quantum mechanics, but it cannot correspond to anything in nature. Only parabolic composability (classical mechanics) and elliptic composability (quantum mechanics) can describe nature.**__we cannot define probabilities!!!__
But how can we tell classical and quantum mechanics apart?
Simple: by experimental evidence in the form of violations of Bell inequalities. In classical mechanics, x=0 which means that the ontology always factorizes
neatly into system A and system B, but because x=-1 in quantum mechanics, this
factorization is no longer possible, and this is known as

Tweet

**entanglement due to the superposition of the wavefunction**. It is superposition which allows for higher correlations than what one can expect from any local realistic model.Tweet