Mathematical duality

Let me set the scene. A brooding overcast morning in post-industrial Manchester. A Mancunian café underneath an old railway arch. Two large lattés. On my side of the table a physics book for mathematicians, on Le Fox’s the Economist. I opened my book on at a chapter on the basics of quantum mechanics. It stated:

“A description of physical reality is made in terms of two set of objects: observables and states. A set of obsevables A, B,… will be denoted by A and states w, x, y,… by W. Each state assigns to each observable its probability distribution on the real line. This pairing (mean value) defines a duality between A and W.”

I just love how physicists like to use phrases like physical reality… It reminds me of Arnold’s classic book Mathematical Methods in Classical Mechanics, which I seem to remember states in no uncertain terms

Definition: The universe is a four dimensional affine space.

Which was perhaps inspired by Wittgenstein’s

1 The world is all that is the case
1.1 The world is the totality of facts, not of things. etc.

In any case, I started to wonder about the notion of duality. In mathematics at least, duality seems to be reasonably well defined. The modern idea I suppose started with the idea of a vector space having a dual which might have been somehow inspired by duals of platonic solids through the reflexive property of taking duals (at least for finite dimensional spaces). But then there’s also Poincaré duality for compact oriented manifolds which says that the kth and (n-k)th Betti numbers are equal (for coefficients in Z2, orientability isn’t required).

The picture of foolhardiness, I then tried to give Le Fox some idea of what Poincaré duality might mean through a couple of badly explained examples messily scribbled on a napkin – decomposing a sphere into a 0-cell (point) and a 2-cell (plane) and a torus into a 0-cell, two 1-cells and a 2-cell.

As her eyes slowly glazed over, I tried to think of a more concrete way of thinking of duality. Suppose that I have a bag containing ten balls. I then take four balls out of the bag. If I know I have four balls in my hand then I know that there are six balls in the bag. Knowledge of how many balls are outside the bag is therefore somehow equivalent to knowledge of how many balls are in the bag as long as we know the total number of balls. It seemed to me that these two statements were therefore in some sense dual to each other. In the back of my mind I had Poincaré duality which could be rephrased along the lines of “Knowledge of k-cells is equivalent to knowledge of (n-k)-cells as long as the space is compact and oriented…”. Here it seems that knowing that there are ten balls in total and knowing that the manifold is compact play the same role, setting up the bridge between the two dual statements in each case.

For finite dimensional vector spaces, the notion of the dual space can be formulated in a similar way. Given a set with the structure of a finite dimensional vector space, the space of linear functionals can be formed. This is non-canonically isomorphic to the original space, just by mapping a given basis to its dual basis. On the other hand, given the dual space, I can construct the original space again by applying the dualising process. It’s one of those well known facts that a finite dimensional vector space is canonically isomorphic to its double dual (reflexivity).

On one level, the analogy with the balls fits quite nicely into the first half of this: knowledge of the dual is “the same as” knowledge of the space. On another, it doesn’t really work so well – how do I get a notion of ‘canonical isomorphism’ in this setup?

I still haven’t thought of an example… any ideas?

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