Solving Hilbert’s sixth problem
Commutative vs. noncommutative geometry
Last time I introduced the fundamental relationship of (Hamiltonian)
mechanics:
ρ_{12}= ρ⊗θ + θ⊗ρ
θ_{12} = 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
skewsymmetric 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 noncommutative 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 (xa)^2+ (yb)^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 noncommutative geometries
where the notions of point and lines are not well defined. In noncommutative
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 noncommutative 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

Ktheory

Ktheory

vector ﬁeld

derivation

integral

trace

closed de Rham current

cyclic cocycle

de Rham complex

Hochschild homology

de Rham cohomology

cyclic homology

Chern character

ChernConnes character

ChernWeil theory

noncommutative ChernWeil theory

elliptic operator

Kcycle

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 (ABBA)
commutator

Product θ (symmetric)


A θ B=A (cos (ħ/2)∇) B

A θ B = 1/2(AB+BA)

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 anticommutator (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, BCCB] – [ABBA, C]=
ABCACBBCA+CBA  ABC+BAC +CABCBA
= BAC +CBA
– ACBBCA
{A,{B,C}} – {{A,B},C} = {A, BC+CB} – {AB+BA, C}=
ABC+ACB+BCA+CBA  ABCBAC CABCBA
= (BAC +CBA
– ACBBCA)
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 (ABBA)
commutator

Product θ (symmetric)


A θ B=A (cos (ħ/2)∇) B

A θ B = 1/2(AB+BA)

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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