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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