Oblique projections on metric spaces

New paper out there:

M. Polettini, Oblique projections on metric spaces, arXiv:1711.04672

This one falls into a line of research dedicated to algebraic methods in graph theory, which eventually turn out to be useful for the analysis of thermodynamic systems on a network, though in this paper I decided to keep the physical motivation to a minimum and concentrate on the math, which is just plain linear algebra. One of the reasons for keeping this paper so abstract and simple is that, though I see several connections to other parts of physics and math, I’m not yet in the position to make these observations precise enough. Nevertheless, I think the paper is deep and sound enough to be kicked out as it is.

However, here I can be more courageous and attempt a few connections that I did not dare in the main body of the paper, and which might hopefully make the interpretation a bit more straightforward.

Projection operators are crucial in physics (consider e.g. the measurement process in Quantum Mechanics). Most often we assume orthogonal basis states: in this case everything is trivial and my paper does not matter, because I only have something to say about projections on oblique subspaces. Nonorthogonal states are a bit more rare in Quantum Mechanics, though they do have a role (one that I will try to explore in more detail soon…): for example coherent states are nonorthogonal (and over-complete), molecular orbitals as well, and I would guess there might be a number of similar situations in Quantum Computations, where one keeps moving from Bell states to pure states.

So what does my result say? Let’s take one step back: in my previous paper […], among some other novel considerations, I re-derived the known fact that, given a projection P and its complement Q=I-P, and letting P*, Q* denote their adjoints, the operators P*P and Q*Q have the same spectrum, as regards eigenvalues larger than 1. They can also have eigenvalues 0 and 1 in their spectrum, coming with different multiplicities (possibly none).

What do these operators P*P and Q*Q do? It’s more easily explained with this simple example:

projections.png

The two solid nonorthogonal axis are the directions where we project, and the dashed ones are the two directions orthogonal to them. P*P first projects along one of the orthogonal solid axis, and then projects along the orthogonal to the other axis, and viceversa. This gives rise to two maps. These two maps have the same eigenvalues > 1. In fact, I can prove something stronger, that physicists might call a “supersymmetry” (and indeed, as I briefly discuss at the end of the the paper, might have more than a faint connection to that supersymmetry): there exists a “supercharge” D such that

P*P – I = DD* and QQ*-I=D*D

Now in this latest paper I considered the same kind of construction when the vector space is also provided with a metric G that is not the usual scalar product. Somehow it becomes natural to define P’= √G-1 P √G and Q’=√G-1 P √G, and again construct the operators P’*P’ and Q’*Q’. Then the same results as above hold.

In the paper I discuss some applications to electrical circuit analysis and in the future I plan to go more in depth in the analysis of nonequilibrium systems. At the mathematical level, it is very tempting to notice that the transformation P’= √G-1 P √G looks very much like a gauge transformation, something analogous to √g(x)-1x √g(x) in differential geometry, and the application to graph theory I give in my paper hints at this connection to exterior derivatives and Hodge theory, though at the moment I fail to see the bigger picture.

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