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I am starting a table
noncommutative geometry - contents
to be included as a “floating table of contents” in the relevant entries.
Clearly this is just a beginning. Zoran will have lots of items to add.
Is the Bohrification project a sign of something larger, that find the right setting and you can see noncommutative entities as really commutative? Or is there something unusual about that case which allows this to happen?
Good question. I don’t really know yet. And I think nobody does.
Two points to notice:
one insight of the Bohr topos-perspective is that it is not about non-commutativity but about non-associativity: the poset of commutative subalgebras of a suitable non-commutative algebra remembers really the Jordan algebra structure of the algebra.
This seems an exotic perspective these days, but this was what the founding fathers, notably Jordan himself, were actually motivated by. In as far as the observables themselves are concerned, one is really dealing with a Jordan algebra structure which is commutative, but not associative. The remaining commutator structure is what induces the Hamiltonian flows on the observables.
So a genuine non-commutative structure should actually be reflected not by a plain Bohr topos, but by a Bohr topos equipped with a notion of “Hamiltonian flows”. Two weeks ago Andreas Doering told me that he has figured out how this works.
(I have just edited at Jordan algebra a bit to reflect this.)
most, if not all, noncommutative algebras of observables in physics arise in fact as groupoid convolution algebras (say via geometric quantization of symplectic groupoids). This might make one wonder if one shouldn’t worry about the original groupoid more than about the algebra that it induces.
This is a general question which one can ask to noncommutative geometry: at least large chunks of it (at least in the context of C*-algebras) are but a dual formulation of geometric groupoids carrying 2-bundles/bundle gerbes. The non-commutativity of the algebra just so happens to encode both this “equivariance” and this “twisting” aspect. For instance I don’t think that Connes’ big textbook has many examples of non-commutative geometries that do not arise this way.
Thanks. Much to ponder on there.
I really do not get the point of claim that the dualities between algebra and geometry are Isbell duality. Isbell duality may be a heuristic language somebody may think of those, no objection, but as a theorem it is a triviality, while the identification of spaces with copresheaves is nontrivial. It makes no sense to say that a hard theorem is a special case of triviality. For example, Gelfand-Naimark duality relies on spectral theory of Banach algebra, not a trivial subject. Even in the case of sets the Gelfand-Naimark-Robbin theorem is true iff there are measurable cardinals, quite a property of a set theory. In any case, in the noncommutative geometry, the relation between spaces and algebras is essentially still called the spectral theory, and the functor is spectrum; this is older and more widely known than the little piece of categorical triviality called Isbell duality; it is not a special case of Isbell duality as the latter has no essential information on spectral theory in any of its variants. Reducing the spectra to copresheaves is nontrivial and not always neither possible nor essential.
I have nothing againts exploration of Isbell duality, just I think that emphasis is an overstatement, while the essential piece of the picture (spectral theory) is unfortunately consequently placed under the carpet.
I will add things to the entry, once I get there (I am being chased by other requests, related to the teaching etc.).
3: it is very interesting what you say about Hamiltonian flows; I hope we will have details in $n$Lab soon! I do not really like the pompous and mistifying metaphore “founding fathers”; people in foundations of QM since Jordan, one of the people from 1920-s and 1930-s considered Jordan algebras a lot; not that long time ago a Fields medal has been given for a work on those (Zelmanov). The fact that TQFT community does not study those is not the reflection of its status in the community studying foundations of QM, in my impression.
2: I don’t think Bohrification really reduces noncommutativity to commutativity in general. While it may be sufficient for QM and even for major aspects of C-star algebra theory, I think it has no power to capture the true width of noncommutative algebraic geometry. C-star algebras partly due involution are really much less noncommutative. Also in algebraic setup one has slightly noncommutative algebras, like PI-algebras, quantum deformations and alike; the fundamental importance is however with noncommutative algebras which are almost free, and which, in particular, have truly noncommutative smoothness properties and and infinite Gelfand-Kirillov dimension. The spectra of small, almost commutative, algebras can to some extent be measured using only ideals, like in commutative case, but the essence of spectra in noncommutative algebraic geometry requires essentially considering spectral points extracted from the whole category of modules, not only ideals. Gelfand-Naimark theorem is essentially about special kind od ideals leading to points, and in that sense it is very commutative; in special cases one can do this internally in presheaf categories. Now, of course, one can consider the categories of quasicoherent sheaves as an analogue, of this internal point of view. This is a legitimate point of view on Gabriel-Rosenberg theorem, for example…if stretched so far that it is a tautology and can be extended to noncommutative categories viewed as categories of qcoh sheaves (though they are not). So internalization in full generality is in strange categories replacing categories of qcoh sheaves; the latter are often constructed using descent theory from local categories of one-sided modules (which are not monoidal!), though there are cases when one gets them while representing some functors.
It makes no sense to say that a hard theorem is a special case of triviality.
I agree. Where do you see this claim? I like to link to Isbell duality because it explains well the fundamental role of the general mechanism.
Great that we agree! It was just my general feeling from several entries. Say, when one says at Isbell duality that Gelfand duality is an example of it, it is an impression that it implies the special case, rather than that a Gelfand duality may be heuristically viewed as a composition of Isbell duality and a nontrivial equivalence of categories.
I would also like to take your attention that one can also conceptually capture all those theorems and many more as reconstruction theorems, where Yoneda, Tannaka, Isbell, Makkai, Doplicher-Roberts, Giraud, Gelfand etc. all fit, while not really all are of spectral nature (though it may be stretched via thinking of generalized points etc.). Of course, reconstruction theorems do not necessarily identify which side is geometric, though they often do. This remark is more to stimulate more of our future work for synthesis of entries in $n$Lab.
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