Solving Hilbert’s sixth problem

The fundamental relationship between ontology and dynamic

Last time we have derived the Leibnitz identity which is the root cause for information conservation in nature (from this one can derive unitarity in quantum mechanics for example). How does Leibnitz identity hold under tensor composition?

The quantum reconstruction argument is categorical, meaning it is naturally expressed in terms of category theory but whenever possible we will stress a more physical point of view. So now let us introduce a composability category U(⊗,R, ρ,…) where ⊗ is the tensor product which combines two physical systems A and B into a larger system: A⊗B. As an example, A⊗B could be a hydrogen atom, where A is the electron, and B is the proton.

R represents the real number field and we pick R over any other mathematical field because we want to be able to compute probabilities in the usual way. ρ,… belong to our unspecified set of local operations {o}. This composability category has as the identity element the chosen field R (U⊗R = R⊗U = U) and elements of R will be understood as arbitrary constant functions.

Now we can see how ρ acts on a constant function 1. Using the Leibnitz identity:

f ρ 1 = f ρ (1 x 1) = (f ρ 1) x 1 + 1 x (f ρ 1) = 2x (f ρ 1)

and so we have in general that f ρ1 = 0 for any function f. (This is very natural, we just stated in a fancy abstract way that the derivation of a constant function is zero)

Using the tensor product and using the unit of the composability category we have:

(f ⊗1) ρ₁₂ (g ⊗1) = (f ρ g) ⊗1 = (f ρ g)

where ρ₁₂ is the bipartite product ρ_.

The bipartite (and in general the n-partite) products must be build out of the available products in {o}. If in our universe of discourse the collection {o} of the available products contains only the product ρ we have:

.   (f ⊗1) ρ₁₂ (g ⊗1) = (f ρ g) ⊗(1 ρ 1) = 0 because (1 ρ 1) = 0

Hence the ρ product is trivial if it exists by itself. There must exists at least another product θ to have an interesting domain. If ρ corresponds to the dynamic θ corresponds to ontology (observables).

Suppose that {o} contains only ρ and θ. The bipartite products ρ₁₂ and θ₁₂ must be constructed out of ρ and θ. The most general way for this is as follows:

(f₁ ⊗ f₂)ρ₁₂(g₁ ⊗ g₂) = a(f₁ρ g₁) ⊗ (f₂ρ g₂) +b(f₁ρ g₁) ⊗ (f₂θ g₂) + c(f₁θ g₁) ⊗ (f₂ρ g₂) + d(f₁θ g₁) ⊗ (f₂θ g₂)

(f₁ ⊗ f₂) θ ₁₂(g₁ ⊗ g₂) = x(f₁ρ g₁) ⊗ (f₂ρ g₂) +y(f₁ρ g₁) ⊗ (f₂θ g₂) + z(f₁θ g₁) ⊗ (f₂ρ g₂) + w(f₁θ g₁) ⊗ (f₂θ g₂)

In shorthand notation:

Rho_12 = a rho_1 rho_2 + b rho_1 theta_2 + c theta_1 rho_2 + d theta_1 theta_2

Theta_12 = x rho_1 rho_2 + y rho_1 theta_2 + z theta_1 rho_2 + w theta_1 theta_2

Now the goal is to determine the values of the parameters a,b,c,d,x,y,z,w. Since the relationship is general, we can pick f₁  =f₂ = 1 and we can use the identity: 1 rho g = 0;

Then in the first relationship only c and d terms survive. If we normalize Theta such that (1 theta 1) = 1 this demands c=1 and d=0. Similarly z=0, w = 1. Using the same trick with g₁  = g ₂ = 1 demands b=1 y=0

So we must have:

(f₁ ⊗ f₂)ρ₁₂(g₁ ⊗ g₂) = a(f₁ρ g₁) ⊗ (f₂ρ g₂) +(f₁ρ g₁) ⊗ (f₂θ g₂) + (f₁θ g₁) ⊗ (f₂ρ g₂)

(f₁ ⊗ f₂) θ ₁₂(g₁ ⊗ g₂) = x(f₁ρ g₁) ⊗ (f₂ρ g₂) + (f₁θ g₁) ⊗ (f₂θ g₂)

The a term can be eliminated by applying the Leibnitz identity on itself on the bipartite products ρ₁₂. Therefore the fundamental composability relation becomes:

ρ₁₂ = ρ⊗θ + θ⊗ρ

θ ₁₂ = θ⊗θ + xρ⊗ρ

where x could be normalized to be +1, 0, -1.

The three possible parameters -1, 0, 1 corresponds to “fixed points” in a categorical (composability) theory and they correspond to (this remains to be shown):

-quantum mechanics (elliptical composability)

-classical mechanics (parabolic composability)

-split-complex (hyperbolic) quantum mechanics (hyperbolic composability)

If Theta represents the algebra of observables and Alice and Bob form a bipartite system (EPR pair), x=0 means that the observables are separable. x != 0 means that the observables are affected by the dynamics and the system can be entangled!!!

We can normalize x to be +1,0,-1, but if we do preserve the dimensions, when it is not zero, x = +/- ħ²/4.

Invariance of the laws of nature under composability demands that x remains the same or, equivalently that the Plank constant is the same for all quantum systems!

Now for some references.

The core ideas were developed by Emile Grgin and Aage Petersen at Yeshiva University in the 70s (Aage Petersen was Bohr’s assistant and Bohr had the hunch that classical and quantum mechanics share core features beyond the correspondence principle)

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1103900192

The idea that composability demands the invariance of the Plank constant was developed by Sahoo, a colleague of Grgin: http://arxiv.org/abs/quant-ph/0301044

Expanding on Grgin’s original idea I wrote: http://arxiv.org/abs/1303.3935 which was uploaded on the archive only 11 days before a similar result by Anton Kapustin of Caltech: http://arxiv.org/abs/1303.6917 Kapustin’s paper had the same old Grgin paper for inspiration and is written in the category theory formulation. We worked independently and the papers are about 80% identical in content notwithstanding that they look very different. Eliminating (correctly and conclusively) the hyperbolic composability class was done in http://arxiv.org/abs/1311.6461 and his lead to a major generalization of functional analysis (I’ll cover this in subsequent posts). There are still more unpublished results.

If the reader is interested into an excellent reference for classical and quantum mechanics (which includes the Lie-Jordan algebraic formulation of quantum mechanics) I highly recommend Nicolas Landsman’s book: http://www.amazon.com/Mathematical-Classical-Mechanics-Monographs-Mathematics/dp/038798318X

Tweet