Friday, July 1, 2016

A new definition of distance

I started this noncomutative series with the puzzle: what are quantum correlations higher than what we would normally expect? However I want to advocate that the real puzzle is the concept of distance or separation because the notion of distance plays no role whatsoever in the quantum mechanics reconstruction project. And if everything is quantum mechanical, where does the concept of distance comes from?

At first sight the problem seems pointless. Distance is something that we measure with a meter stick, meaning that between any two points we count the minimum numbers of meter sticks needed to connect the two points:

\(d(x,y) = Inf \{\int_{\gamma} ds |\gamma~is ~a~path~between~x~and~y\}\)

The study of the generalization of measure theory on pathological spaces led to a very different definition which is equivalent with the usual one in the standard case, but works under any circumstance:

\(d(x,y) = Sup \{|f(x)-f(y)|; || [D, f] || \leq 1\}\)

where \(f\) is a scalar value function subject to the constraint  that it does not vary too rapidly as controlled by the operator norm of the commutator \([D, f]\), and where \(D\) is the Dirac operator.

Moreover, there is a relationship between the Dirac operator and the infinitesimal line element \(ds\):

\(ds = 1/D\)

and the homotopy class of \(D\) represents the K-homology fundamental class of the space under consideration.

Alain Connes

I won't explain the technical details as I did not present the building blocks required, but coming back to physics, this new definition means that the notion of distance is spectral, and therefore it is fundamentally quantum mechanical in nature. As such the Tsirelson bound puzzle evaporates as everything (correlations from norm and distance from spectral information) comes from quantum mechanics and therefore quantum correlations do not cry out for an explanation as Bell put it.

There is much much more to noncommutative geometry both on the mathematical and the physical side. Noncommutative geometry is expressed in what is called a "spectral triple":

\((A, H, D)\)

where \(A\) is the algebra of coordinates on the geometric space, H is the Hilbert space of A and of the line element \(ds = D^{-1}\), and D is the Dirac operator. 

Dirac operator..., Hilbert space..., this is all physics. Is there a relationship between the noncommutative geometry and the Standard Model? Indeed it is and it is very deep. 

The symmetry group G of the Standard Model action together with the Einstein-Hilbert action is the semi-direct product of the group of gauge transformations with the diffeomorphism group. 

At this point Alain Connes asked two questions:
  1. Is there a space X such that Diff(X)  = G?
  2. Is there a simple action functional identical with SM+GR action when applied to X? 

For "normal" spaces the answer is negative, but recall that noncommutative geometry is about "pathological" spaces, and the answer in this case is positive. Connes calls this "Clothes for the SM beggar". I'll talk about this next time. Please stay tuned.


  1. Florin:

    "what are quantum correlations higher than what we would normally expect?"

    The answer is given in the book "The Emerging Quantum", chapter 7, "Disentangling Quantum Entanglement"

    The explanation consists in the interaction between the particles and the background electromagnetic field (named ZPF, zero-point field).

    Because the existence of this field is not taken into account the correlation between the entangled particles appear as mysterious.

    Quantum mechanics succeeds in describing Bell experiments because the Schrodinger's equation does not describe the particle alone, but the particle in interaction with the ZPF. The deduction of Schrodinger's equation is presented in the book at chapter 4.


    1. Andrei,

      After I am done talking about Connes' research I'll do a series on Bell theorem where hopefully I can show why it is impossible to achieve classical correlations above Bell limit.

  2. In a nutshell the Dirac operator defines a space of harmonic spinors that are the kernel and cokernel of the Dirac operators. These spinors are quaternions. The Dirac operator γ∂ is a map between half spin bundles

    γ∂:C^∞(S^+) → C^∞(S^-)

    One can then then look at a generalization of the Cauchy-Riemann theorem for quaternions. This derives Yang-Mills equations, and it defines the self-dual and anti-self-dual equations and by Donaldson's theorem the nature of differential structure of a four-manifold.