09 August 2006

A conventional question

I'm in the progress of writing up what I understand about tensors, defining them from scratch, using only intuition and Penrose's graphical notation. Eventually, perhaps I will write a version of my notes for Wikipedia, since their current article on the subject is laughably bad. I first read about them in this post by jao at physics musings; I had started reading Penrose's most recent book, The Road to Reality: A Complete Guide to the Laws of the Universe, which explores them in some depth. I am going to shamelessly reproduce jao's picture of such diagrams, so that you have some idea what I'm talking about:

Incidentally, I wonder what the history of such doodles really is. I hear talk of "einbeins", "zweibeins" and "dreibeins", lit. one-leg, two-leg, and three-leg, and if I knew more German, multi-legs ("mehrbeins"?), which sound like these tensorial pictures. Based on skimming the discussion here, it looks like einbeins are related, but not fully formulated. I wonder why someone would refer to an objects legs, though, unless it had legs.

Anyway, the notational question I wanted to ask was this:

We generally write "vectors" (as opposed to covectors) with raised indices, and covectors with lowered indices. This has physical significance: the Poincare group acts differently depending on whether the index is raised or lowered: on lowered indices, symmetries act by the adjoint, and so it's really a "dual" action, in the sense that it happens in the backwards order. So, although in some sense vectors and covectors are interchangeable, interpretations of diagrams are not.

Since vectors' indices are raised, Penrose proposes that a vector ought to have one "arm" (an edge coming out of the top), whereas a covector ought to have one leg. This makes sense, and closely matches how he thinks of contractions: contracting indices corresponds to drawing curves from the tops of the vectors to the bottoms of the covectors.

On the other hand, as soon as you start playing around with Penrose's diagrams — well, as soon as Josh H. started playing with them, when I introduced them to him over IM — you notice the connection between these diagrams and various ideas from quantum topology. In particular, diagrams like this look an awful lot like tangles.

This is actually no surprise. A n,m-tensor (one with n arms and m legs, so e.g. a vector is a 1,0-tensor), by definition, is a map from V tensored with itself m times to V tensored with itself n times. (By convention, V tensored with itself 0 times is the ground field — no, not a generic one-dimensional vector space, because I do in fact need the special number "1". This is so that there is a natural isomorphism between "V^0 tensor W" and W.)

But this, then, is a problem, because this commits me to reading my morphisms as going up. But my friends who study TQFTs think of their cobordisms as going down (see, for example, the many This Week's Finds starting at Week 73, in which Baez gives a mini course on n-categories).

So clearly one of us is right, and the other is wrong. Either we should think of vectors as having a head and a leg (more than half the time I catch myself drawing them this way anyway), or we should think of cobordisms, tangles, and their cousins as transforming the bottom of the page into the top of the page.

I'm leaning towards the latter, but only because there's one more, very established, case in which this matters. Diagrams of Minkowski space, and more generally of, for example, light cones in curved space, the positive time dimension is always drawn going up the page. And if our conventions are to have any sensible physical meaning, morphisms must correspond to forward time evolution.

So only typesetters and screen-renderers (and English language readers), who insist and putting (0,0) in the upper left corner of the page, have it backwards. Then again, mathematicians have known that for years.

But, of course, we do live in a democracy. If you tell me that infinitely more people and publications think time flows down, then I'll happily switch my conventions.

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