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)

=0

and therefore:

[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 Jordan product!!!

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 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.

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Solving Hilbert’s sixth problem

Commutative vs. non-commutative geometry

Last time I introduced the fundamental relationship of (Hamiltonian) mechanics:

ρ₁₂= ρ⊗θ +    θ⊗ρ

θ₁₂ = θ⊗θ + x ρ⊗ρ

This looks very abstract, and nothing like the usual formulation but we will show that this leads to classical and quantum mechanics.

However, we want to take a closer look first at the relationship and notice a very interesting analogy. Suppose x = -1.Where have we seen a relation like this before? How about complex number multiplication?

z = a+ib, w = p+iq

z*w = ap - bq + i (aq + bz)

Im(zw) = Im(z) Re(w) + Re(z) Im(w)

Re(zw) = Re(z) Re(w)  - Im(z) Im(w)

Since x=-1 corresponds to quantum mechanics, it is no wonder that quantum mechanics is expressed best over complex numbers!

In general the products ρ and θ can be proven to be skew-symmetric and symmetric but I am going to skip the proof. What I want to concentrate today is the concrete realizations of the two products which is a very interesting story with unexpected mathematical links.

It turns out that there are two such realizations, one based on state space, and one based on Hilbert space. However, those realizations tell a much larger story, that of commutative and non-commutative geometry.

In mathematics there is a nice duality between geometry and algebra. The easiest way to see this is to consider the ordinary 2D plane and old fashion Euclidean geometry in a plane. Then add a coordinate system and express all lines and circles as algebraic relationships. For example a circle with center at position (a, b) and radius r obeys (x-a)^2+ (y-b)^2 = r^2. And then all usual geometric theorems can be expressed algebraically.

This duality cuts very deep across many mathematical concepts and structures, and in particular generalizes to non-commutative geometries where the notions of point and lines are not well defined. In non-commutative geometry at core, the very notion of distance is modified from an infimum (the shortest distance between two points is a line which minimizes the distance) to a supremum.

There is a dictionary of correspondence of mathematical structures between commutative and non-commutative realm (http://arxiv.org/pdf/math/0408416v1.pdf):

Commutative	Noncommutative
measure space	von Neumann algebra
locally compact space	C∗- algebra
vector bundle	finite projective module
complex variable	operator on a Hilbert space
real variable	sefadjoint operator infinitesimal compact operator
range of a function	spectrum of an operator
K-theory	K-theory
vector field	derivation
integral	trace
closed de Rham current	cyclic cocycle
de Rham complex	Hochschild homology
de Rham cohomology	cyclic homology
Chern character	Chern-Connes character
Chern-Weil theory	noncommutative Chern-Weil theory
elliptic operator	K-cycle
spin Riemannian manifold	spectral triple
index theorem	local index formula
group, Lie algebra	Hopf algebra, quantum group
Symmetry	action of Hopf algebra

To this list on the commutative side I will add: phase space and to the noncommutative side I will add Hilbert space. I will also discuss only quantum mechanics (meaning the elliptic composability case of x=-1)

Here are the realizations of the two products

Commutative	Noncommutative
Product ρ (skew symmetric)
A ρ B=A 2/ ħ(sin (ħ/2)∇) B Moyal bracket	A ρ B = i/ħ (AB-BA)   commutator
Product θ (symmetric)
A θ B=A (cos (ħ/2)∇) B	A θ B = 1/2(AB+BA) Jordan product

Where ∇ is the Poisson bracket (where the first partial differential acts on the left and the second partial differential acts on the right).

The particular choice of realization depends on the problem under consideration and the ease of the formalism to solve it, but both realizations give the same final answer. Let us play a bit with the noncommutative side products to derive an (trivial in this formalism) identity (but which is absolutely essential).

We will use [,] as a notation for the commutator, and {,} for the anti-commutator (Jordan product). Then consider this:

[A,[B,C]] – [[A,B],C] and {A,{B,C}} – {{A,B},C}

those are called associators because they measure the lack of associativity in the products [], {}.

[A,[B,C]] – [[A,B],C] = [A, BC-CB] – [AB-BA, C]=

ABC-ACB-BCA+CBA   - ABC+BAC+CAB-CBA = BAC+CBA – ACB-BCA

{A,{B,C}} – {{A,B},C} = {A, BC+CB} – {AB+BA, C}=

ABC+ACB+BCA+CBA   - ABC-BAC-CAB-CBA = -(BAC+CBA – ACB-BCA)

In fact the two associators are proportional, and this is the consistency relationship between dynamic and ontology!!! It is this relationship that transforms the two products into a mechanic (quantum or classical)! What this shows is that the two products can be combined into an associative product. Why is this important? Because it allows for the introduction of probabilities into the formalism and the notion of a state/phase space which is needed if we have to make experimental predictions, doing physics and not pure math. In fact, the table above should be extended as follows:

Commutative	Noncommutative
Product ρ (skew symmetric)
A ρ B=A 2/ ħ(sin (ħ/2)∇) B Moyal bracket	A ρ B = i/ħ (AB-BA)   commutator
Product θ (symmetric)
A θ B=A (cos (ħ/2)∇) B	A θ B = 1/2(AB+BA) Jordan product
Associative product
f⋆g = f θ g + i ħ/2 f ρ g the start product	AB = AB (matrix multiplication) Ordinary complex number multiplication

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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

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Solving Hilbert’s sixth problem

Invariances of the laws of nature

How is special theory of relativity derived? One starts with the invariance of the laws of nature to changing inertial reference frames. From this you get either the Lorentz transformation or the Galilean transformation, and use the second postulate, that of the constant value of the speed of light, to select between the two choices.

But are the laws of nature invariant only to changes in inertial reference frames? How about a trivial invariance: the laws of nature do not change during time evolution?

And how about another: the laws of nature do not change if we partition in our mind a larger system into smaller ones (the composability principle, or the invariance of the laws of Nature under tensor composition)?

Those statements are completely obvious, but their mathematical consequences are far from trivial. We need one more ingredient of a technical nature: a continuity property: if we represent the state of a physical system with a point in a state/phase/configuration space manifold we want to be able to compute derivatives, meaning we should be able to define a tangent plane (after all we will recover the Hamiltonian formalism and the cotangent bundle).

So now we are ready to begin the journey of deriving the two products we talked about last time. It will turn out that the symmetric product describes the ontology and the skew-symmetric one the dynamic. Their compatibility condition (which will be derived too) will help recover the quantum and classical mechanics. All this will come out of the two new invariance laws stated above!!! If you want to follow along in a technical paper (and peek into the future post contents) use http://arxiv.org/abs/1303.3935 as guidance. Let us begin…

First we consider a set of (abstract) unspecified operations {o} which include local laws of nature. As concrete examples of such operations in quantum mechanics, we think of the commutator understood as a product, and the Jordan product. At this point we do not assume any properties for those products, or even that they should exist. How can we mathematically state that the set {o} remains invariant under time evolution? The idea is that of an isomorphism: “o“ at time t must be isomorphic with “o” at time t+delta t because the isomorphism preserves the algebraic relationships. If T is a time translation operator (T A(t) = A(t+ delta t)) the isomorphism can be written as:

T [G o H] = [T(G) o T(H)]

Or equivalently:

(G  o H)(t +delta t) = G (t +delta t) o H(t+delta t)

This is the only consequence we can derive from the invariance of the laws of nature under time evolution.

Now consider infinitesimal time evolutions and use the ability to derivate. This introduces a tangent plane at “t” and a (particular) vector field associated with (a particular) time evolution.

If T_epsilon is a particular infinitesimal (time) translation operator we have:

T_epsilon [G(t) o H(t)] = [T_epsilon G(t) o T_epsilon H(t)]

If rho is one of the products in the set {o} corresponding to the time translation transformation T_epsilon, we can construct the following product between a distinguished f and any g:

f ρ g = [T_epsilon g – fg]/ ε in the limit of ε -> 0 (we pick this as the definition)

equivalently:

f (I + ε ρ) g = T_epsilon g

(Here T_epsilon and “f” are not arbitrary, but depend on each other. “f” in general corresponds to a particular Hamiltonian, and ρ corresponds to the Poisson bracket in classical mechanics and the commutator in quantum mechanics.)

T_epsilon can be uniquely represented as f (I + ε ρ) because if f (I + ε ρ) = T_epsilon = r (I + ε η), in the zero order: f=r. In turn this means that f (I + ε ρ) = T_epsilon = f (I + ε η) and therefore: ρ = η.

We generalize the product ρ for all f’s and g’s by repeating the argument for all conceivable dynamics. To make sure the domains of f and g are identical and well behaved, in case of pathologies, we can restrict the domain of g’s to the span of all possible f’s.

From the invariance of the laws of Nature under time evolution we have:

f (I + ε ρ) (g o h) = T_epsilon (g o h) = (T_epsilon g) o (T_epsilon h) =

= [f (I + ε ρ) g] o [f (I + ε ρ) h]

In first order in epsilon we have a left Leibnitz identity:

f ρ (g o h) = (f ρ g) o h + g o (f ρ h)

for any product “o” in the set {o} and for any f, g, h.

Similarly one derives a right Leibnitz identity:

(g o h) ρ f = (g ρ f) o h + g o (h ρ f)

The Leibnitz identities turn out to be of critical importance as we will see in subsequent posts.

In summary, the fact that the laws of nature do not change during time evolution along with differentiation demands the existence of left and right Leibnitz identities for an abstract product ρ.

As an important note, we do not know at this time that ρ is necessarily skew-symmetric. If we assume this from the very beginning the whole derivation is much shorter.

Next time we’ll show that dynamic alone is not enough and we will need a supporting product which in the end will show that it corresponds to observables (ontology). Then we’ll establish the most general relations between dynamic and ontology.

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