Example: Gromov-Hausdorff distances and the Heisenberg group, PART 1

This post continues the previous one “Quantum physics and the Gromov-Hausdorff distance“.

Let me take an example. We are in the following Heisenberg group (this is really a physicist Heisenberg group): the semidirect product G = \mathbb{R}^{2} \times S^{1}. Elements of the group have the form

X = (x, e^{2\pi iz}) with x \in \mathbb{R}^{2} and z \in \mathbb{R} / \mathbb{Z}

(think that z \in [0,1] with the identification o = 1).
The group operation is given by:

X \cdot Y = (x,e^{2\pi i z}) \cdot (y, e^{2 \pi i u}) = (x+y, e^{2 \pi i (u+z+ \frac{1}{2} \omega(x,y))})

where \omega: \mathbb{R}^{2} \times \mathbb{R}^{2} \rightarrow \mathbb{R} is the “symplectic form” (or area form)

\omega(x,y) = x_{1} y_{2} - x_{2} y_{1}

Remark 1. The pair (\mathbb{R}^{2}, \omega) is a symplectic (linear) space. As we well know, the Hamiltonian mechanics lives in such spaces, therefore we may think about (\mathbb{R}^{2}, \omega) as being a “classical”, or “large scale” (see further the justification, in PART 2) phase space of a mechanical system with one degree of freedom.

The group G is generated by the subset D = \mathbb{R}^{2} \times \left\{ 1 \right\}, more precisely any element of G can be expressed as a product of four elements of D (in a non-unique way).

What is the geometry of G, seen as generated by D? In order to understand this, we may put an euclidean norm on D (identified with \mathbb{R}^{2}):

\| (x, 1) \| = \| x\|, where \|x\|^{2} = x_{1}^{2} + x_{2}^{2} for example.

Then we define “horizontal strings” and their “length”: a string w = X_{1} ... X_{n} of elements of G is horizontal if for any two successive elements of the string, say X_{i}, X_{i+1} we have

X_{i}^{-1} \cdot X_{i+1} \in D, where X^{-1} denotes the inverse of X with respect to the group operation. Also, we have to ask that X_{1} \in D.

The length of the horizontal string w = X_{1} ... X_{n} is defined as:

l(w) = \|X_{1}\| + \| X_{1}^{-1} \cdot X_{2}\| + .... + \|X_{n-1}^{-1} \cdot X_{n}\|. The source of the string w is the neutral element s(w) = E = (0,1) and the target of the string is t(w) = X_{1}\cdot ... \cdot X_{n}.

OK, then let us define the “group norm” of an element of G, which is an extension of the norm defined on D. A formula for this would be:

\| X\| = \, inf \left\{ l(w) \mbox{ : } t(w) = X \right\}.

Small technicality: it is not clear to me if this definition is really good as it is, but we may improve it by the following procedure coming from the definition of the Hausdorff measure. Let us introduce the “finesse” of a horizontal string, given by

fin(w) = \max \left\{ \|X_{1}\| , \| X_{1}^{-1} \cdot X_{2}\| , ... , \|X_{n-1}^{-1} \cdot X_{n}\| \right\}

and then define, for any \varepsilon > 0, the quantity:

\| X\|_{\varepsilon} = \, inf \left\{ l(w) \mbox{ : } t(w) = X \mbox{ and } fin(w) < \varepsilon \right\}.

The correct definition of the group norm is then

\| X\| = \, sup \left\{\| X\|_{\varepsilon} \mbox{ : } \varepsilon > 0 \right\}.

With words, that means: for a given “scale” $\varepsilon > 0$, take discrete paths from E to X, made by “small” (norm smaller than \varepsilon) horizontal increments, and then take the infimum of the length of such curves. You get \| X\|_{\varepsilon}. Go with \varepsilon to o and get the norm \| X\|_{\varepsilon}.

Up to some normalization, the bigger is the norm of an element of G, the bigger is the infimal length of a horizontal curve which expresses it, therefore the group norm gives a quantitative estimate concerning how the group element is generated.

In disguise, this norm is nothing but a sub-riemannian distance!

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