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Hi, what comes to the mind of a physicist, when he hears words like filtered ring and associated graded? What do these guys describe? What are basic/typical/illuminating examples in physics? Of course also mathematicians may answer from their perspective :)

Edit: Information on my background: My "primeval" motivation to understand filtered rings/graded rings, comes from mathematics: They are the basics for D-Module theory. Though I would also like to see these rings from a different perspective, therefor I asked the question. My knowledge in physics is a bit limited, I only attended some undergraduate courses (quantum mechanics, electro dynamics, classical mechanics). Though more sophisticated answers are fine, I don't need to understand every detail, I just want to get the flavor.

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It would help if you gave some indication of your own background. –  Qiaochu Yuan Jan 4 '10 at 8:11
A filtered ring and its associated graded ring should be the same for a physicist as for a mathematician. –  Chandan Singh Dalawat Jan 4 '10 at 8:26
I agree with you Chandan, but the definition of a good or illuminating example might be different, depending on whether you are talking to a mathematician or a physicist. In any case the question should be retagged, and perhaps some more information about the OP's background would be useful (as suggested by Qiaochu). –  Grétar Amazeen Jan 4 '10 at 8:43
Well, I'm not sure (otherwise I would not have asked this question), but I think for a physicist, a filtered ring describes a "quantum system" and the associated graded describes something like "the corresponding classical system". A mathematician on the other hand, will maybe not think in these terms. –  Jan Weidner Jan 4 '10 at 8:47
I gave this question a less peculiar title. –  Reid Barton Jan 4 '10 at 16:52

2 Answers 2

up vote 18 down vote accepted

It is debatable that a physicist would use those very words, and if they did one would hope their meaning would be the same as for a mathematician, since it means that they are trying to speak the same language.

Having said, and coming from a Physics background, when I first learnt about filtered objects and associated graded objects, I immediately recognised the following examples from Physics. They all have to do with quantisation/classical limit in one way or another.

  1. The Clifford algebra is filtered and its associated graded algebra is the exterior algebra. Under the "classical limit" map which takes the Clifford algebra to the exterior algebra, the first nonzero term in the graded commutator of two elements defines a Poisson structure on the exterior algebra. You can then view the Clifford algebra as the quantisation of this Poisson superalgebra. In Physics the exterior algebra is the "phase space" for free fermions and Clifford modules (=representations of the Clifford algebra) are Hilbert spaces for quantized fermions. Things get more interesting when the underlying vector space is infinite-dimensional, since not all Clifford modules are physically equivalent. (The relevant buzzword is Bogoliubov transformations; although you would not guess this from the wikipedia page.)

  2. The algebra of differential operators on $\mathbb{R}^n$, say, is also filtered and the associated graded algebra is the algebra of functions on $T^*\mathbb{R}^n \cong \mathbb{R}^{2n}$ which are polynomial in the fiber coordinates (=the "momenta"). Again the first nonzero term in the commutator of two differential operators defines the standard Poisson bracket on $T^*\mathbb{R}^n$ and one can view the algebra of differential operators as a quantisation of this Poisson algebra. In Physics, this corresponds to quantising $n$ free bosons.

In both cases there is no unique section to the map taking a filtered algebra to the associated graded algebra, but one has to make a choice. There are number of more or less standard ones: Weyl ordering for the bosons, complete skewsymmetrisation for the fermions,...

By the way, this (and a lot more) is explained in the fantastic paper Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras by Kostant and Sternberg.

Edit (inspired by Mariano's answer)

Kontsevich's deformation quantization is not just of interest to physicists, but has a quantum field theoretical reformulation due to Cattaneo and Felder. It is basically the perturbative computation of the path integral of the Poisson sigma model. (This is analogous to how the perturbative evaluation of the path integral of Chern--Simons theory gives the Vassiliev invariants of (framed) knots.)

The picture that seems to be emerging is that indeed quantisation (be it deformation or path-integral or what have you) of a classical physical system gives rise to a filtered object, filtered by powers of $\hbar$.

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Thank you, I found the first example and the last lines of your edit especially helpful :) –  Jan Weidner Jan 4 '10 at 21:28

The deformation quantization construction, famously carried into full generality by Kontsevich, is the construction, from the datum of a manifold $M$ and a Poisson structure $\{\mathord-,\mathord-\}$ on it, of a generally non-commutative associative algebra $\mathcal A$ which is endowed with a filtration such that the associated graded algebra (which is naturally a Poisson algebra) $\mathrm{gr}\mathcal A$ is isomorphic to the algebra of smooth functions on $M$ and its Poisson structure. This should be an example of interest to physicists! (Also, this includes José's example 1, as the Weyl algebra is a deformation quantization of the ring of (polynomial) functions on the plane, and more or less example 2)

In general, a filtered object $X$ is an objects with a chosen aproximation, given by its associated graded object $\mathrm{gr}\\,X$, which ideally reflects some of the interesing propoerties of $X$ but which is, at the same time, simpler than $X$. Indeed, one can in most cases think of elements of $\mathrm{gr}\\,X$ as «elements of $X$ up to details».

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Filtered algebras give deformations of their associated gradeds which are C^* invariant. In particular they have the feature that the family is constant up to isomorphism once you remove h=0. They also have the feature that the deformation extends (trivially) to all values of the Planck constant. These are not features of general deformation quantizations. (In fact in great generality filtered objects are the same as objects over the affine line which are equivariant with respect to the multiplicative group.) –  David Ben-Zvi Jan 4 '10 at 21:03
Thanks for your answer, and thanks for inspiring José Figueroa-O'Farrill! –  Jan Weidner Jan 4 '10 at 21:27

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