From: baez@galaxy.ucr.edu (John Baez) Subject: Re: Densitized Pseudo Twisted Forms Date: Wed, 21 Nov 2001 19:27:40 +0000 (UTC) Newsgroups: sci.physics.research Summary: Appropriate framework for forms on manifolds without orientation, metric In article <20011112.224414.1504569917.17853@empc07.ece.uiuc.edu>, Eric A. Forgy wrote: >I was originally claiming that fancy shmancy forms, a.k.a. densitized (D) >or pseudo (P) or twisted (T) or pseudo twisted (PT) forms were not REALLY >necessary to describe physics. Okay. Let me try the simplest example where we need them: electromagnetism on an unoriented spacetime. If you want to make my life difficult, you can argue that unoriented spacetimes are "unphysical" - e.g., we haven't actually sent any right-handed astronauts on long space journeys and had them come back left-handed, all their dextrose turned to levulose. But if you do this, I will be disappointed, because I think you know that nothing about Maxwell's equations really involves a "handedness" - all that "right-hand rule" crap is just the result of trying to use vectors in situations where vectors aren't really appropriate. As a result, it should be perfectly possible to formulate Maxwell's equations on a spacetime that's not equipped with an orientation. And it *is* - but not using differential forms. We need fancy-schmancy forms. You can probably guess the culprit: it's the Hodge star operator! We usually think of the Hodge star operator as a map from p-forms to (n-p)-forms, where n is the dimension of spacetime, but this only works when spacetime is equipped with a metric and orientation. If we have a metric but not an orientation, we can still define the Hodge star operator, but only as a map from p-forms to pseudo-(n-p)-forms. If we do this, ALL OF ELECTROMAGNETISM WORKS JUST AS WELL AS IT EVER DID. This strongly suggests that this is secretly what we "should have" been doing all along. We never really needed an orientation on spacetime; we were only using it as a crutch, because we were too lazy to understand pseudo-forms. Of course I put "should have" in quotes, because as long as we work on an oriented spacetime, it DOESN'T REALLY MATTER whether we use forms or pseudo-forms: there's a canonical isomorphism between the two, defined using the orientation. So, don't get me wrong: I'm not trying to persuade people to stop using forms and take up pseudo-forms, because in most applications we *are* working on an oriented spacetime, and then pseudo-forms are just an extra bother. I'm just saying that the usual treatment of electromagnetism makes use of a structure on spacetime which is not logically necessary - the orientation - and we can eliminate the need for this if we use pseudo-forms. The only people who should care about this are people who want to study physics on unoriented spacetimes *and* people who want to understand the foundations of geometry as thoroughly and carefully as possible. I count you among the latter. By the way, there are other theories, like the theory of the weak force, which really *do* make use of the orientation on spacetime. The weak force really *does* care about handedness. So it's nice to see how you can formulate electromagnetism without an orientation, but not the electroweak theory. >You and the Wizard nearly blasted me for not understanding why irreps of >GL(n) were so important :) Please, don't let me be a representative of how >low the education system has gone. I am not your typical student (thank >god!) :) I definitely learn things in an unorthodox manner. I think it was the Wizard, not I, who complained about you as an example of how low the educational system has sunk. This is just his usual irascibility, and you shouldn't take it too seriously. In fact, rather few physicists have taken the trouble to understand the complete classification of tensors and their kin using the representation theory of GL(n). But I think *you* would like to! >I don't expect many people to understand me, but I am spending a PAINFULLY >long time to understand the very basic geometrical concepts as clearly as >possible. It is a personal philosophy of mine that mathematics should not >be learned as a tool to study physics. I look at mathematics as speaking >directly to nature. This is my philosophy too. This is why I enjoy talking to you about this stuff. Most people rush through the geometry and miss out on a lot of fascinating subleties, LIKE THE IMPORTANCE OF DENSITIZED AND "PSEUDO" TENSORS. Ahem. You see, it's really incredibly cool how: representations of GL(n) correspond to different kinds of tensor-like gadgets in situations where spacetime is just a smooth n-manifold with no extra structure; representations of GL_0(n) correspond to different kinds of tensor-like gadges in situations where spacetime is a smooth n-manifold equipped with an orientation; representations of SL(n) correspond to different kinds of tensor-like gadgets in situations where spacetime is a smooth n-manifold equipped with an orientation and volume form; representations of O(n-1,1) correspond to different kinds of tensor-like gadgets in situations where spacetime is a smooth n-manifold equipped with a Lorentzian metric; representations of SO(n-1,1) correspond to different kinds of tensor-like gadgets in situations where spacetime is a smooth n-manifold equipped with a Lorentzian metric and orientation; and so on for various other sorts of groups and structures we like to put on spacetime. The more structure we put on spacetime, the smaller the relevant group gets, and the more kinds of tensor-like gadgets become "the same": two different representations of a big group can become equivalent when restricted to a smaller group! This is why, if you're always assuming spacetime is equipped with an orientation, you don't need to distinguish between p-forms and pseudo-p-forms: they correspond to different representations of GL(n), but equivalent representations of GL_0(n) (the subgroup of GL(n) that preserves the orientation). And this is also why some even more crass people don't distinguish between vectors and 1-forms: they're assuming spacetime is equipped with a metric, and while vectors and 1-forms correspond to different representations of GL(n), they correspond to equivalent representations of O(n-1,1) (the subgroup of GL(n) that preserves the metric). So you see, when you argue that "there's no real need for pseudo-forms - forms are all we need", you sound to me exactly like the people who argue that "there's no need for 1-forms - vector fields are all we need". Yes, they are equivalent as representations of a small group, but they're not as representations of a bigger group... so your attitude is fine when spacetime is equipped with lots of structure, but not when it's equipped with less! ============================================================================== From: baez@galaxy.ucr.edu (John Baez) Subject: Re: Densitized Pseudo Twisted Forms Date: Sun, 25 Nov 2001 02:39:10 +0000 (UTC) Newsgroups: sci.physics.research In article <20011112.224414.1504569917.17853@empc07.ece.uiuc.edu>, Eric A. Forgy wrote: >You and the Wizard nearly blasted me for not understanding why irreps of >GL(n) were so important :) And rightly so! By the way, I found out who figured out this stuff: Hermann Weyl and his student Alexander Weinstein, right around 1924. I happen to be reading a history of this business right now: Thomas Hawkins, Emergence of the Theory of Lie Groups, Springer, Berlin, 2000. First Weyl described how irreps of SL(n) correspond to various symmetry types of tensor, classified by Young diagram. Then he realized that GL(n) was in a way more fundamental, since it includes dilations and reflections. This is where the "densitized" and "pseudo" tensors come in - the density weight modifies how a tensor transforms under dilations, while the "pseudoness" throws in an extra minus sign for reflections. Weyl was the one who introduced the notion of "tensor density", and in 1921 he wrote: "By contrasting tensor and tensor-densities, it seems to me that we have rigorously grasped the difference between *quantity* and *intensity*, so far as the difference has a physical meaning". This was in his book "Space, Time, Matter" - which I highly recommend, by the way. It was in this book that the modern definition of "vector space" was first introduced! In his Ph.D. thesis published in 1922, Weinstein classified all the irreducible representations of GL(n). In 1925, Weyl wrote: "In place of the concept of a tensor that of a *tensor density* has arise; however, in the general sense that with the transition to a new coordinate system... multiplication by an *arbitrary* power of the transformation determinant occurs, the exponent being not necessarily 1 or even integral."