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]xρ =0
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] ρ = ((A ρ A) ρ B) ρ A – (A ρ A) ρ (B ρ A) = ((0) ρ B) ρ A – (0) ρ (B ρ A)
[A θ A, B, A]θ =((A θ A) θ B) θ A – (A θ A) θ (B θ A) = 0
which shows that θ is a
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)
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
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. Jordan
In the classical case (x=0) there are no
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 split complex numbers.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 lack of positivity. In other words, we are not guaranteed to have positive probability predictions, and we cannot define probabilities!!! 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.
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 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.