## 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

**ρ**1 = 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)

**ρ**(g ⊗1) = (f_{12}**ρ**g) ⊗1 = (f**ρ**g)
where

**ρ**is the bipartite product_{12 }**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)

**ρ**(g ⊗1) = (f

_{12}**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__The bipartite products

**ρ**and**θ.****ρ**and

_{12}**θ**must be constructed out of

_{12}**ρ**and

**θ.**The most general way for this is as follows:

(f

_{1}⊗ f_{2})**ρ**_{12}(g_{1}⊗ g_{2}) = a(f_{1}**ρ**g_{1}) ⊗ (f_{2}**ρ**g_{2}) +b(f_{1}**ρ**g_{1}) ⊗ (f_{2}**θ**g_{2}) + c(f_{1}**θ**g_{1}) ⊗ (f_{2}**ρ**g_{2}) + d(f_{1}**θ**g_{1}) ⊗ (f_{2}**θ**g_{2})
(f

_{1}⊗ f_{2})**θ**_{ 12}(g_{1}⊗ g_{2}) = x(f_{1}**ρ**g_{1}) ⊗ (f_{2}**ρ**g_{2}) +y(f_{1}**ρ**g_{1}) ⊗ (f_{2}**θ**g_{2}) + z(f_{1}**θ**g_{1}) ⊗ (f_{2}**ρ**g_{2}) + w(f_{1}**θ**g_{1}) ⊗ (f_{2}**θ**g_{2})
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 f

_{1 }=f_{2 }= 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 g_{1 }= g_{ 2 }= 1 demands b=1 y=0
So we must
have:

(f

_{1}⊗ f_{2})**ρ**_{12}(g_{1}⊗ g_{2}) = a(f_{1}**ρ**g_{1}) ⊗ (f_{2}**ρ**g_{2}) +(f_{1}**ρ**g_{1}) ⊗ (f_{2}**θ**g_{2}) + (f_{1}**θ**g_{1}) ⊗ (f_{2}**ρ**g_{2})
(f

_{1}⊗ f_{2})**θ**_{ 12}(g_{1}⊗ g_{2}) = x(f_{1}**ρ**g_{1}) ⊗ (f_{2}**ρ**g_{2}) + (f_{1}**θ**g_{1}) ⊗ (f_{2}**θ**g_{2})
The

**a**term can be eliminated by applying the Leibnitz identity on itself on the bipartite products**ρ**_{12}. Therefore__the fundamental composability relation becomes:__**ρ**

_{12}=

**ρ**⊗

**θ**+

**θ**⊗

**ρ**

**θ**

_{ 12}=

**θ**⊗

**θ**+ x

**ρ**⊗

**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),

**means that the**__x=0____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 =
+/- ħ

^{2}/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)

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

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