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I am confused with the idea section to functorial analysis. It has two parts, first which talks abstractly aboutusing functor of points point of view in functional analysis and avoiding topological vector spaces. Some ideas come to mind like some classes of TVS can indeed be viewed as generalized spaces obtained say as pro objects from a more narrow class. So all OK, now let us see examples. Then the second part roughly says that it is to avoid inequalities of analysis and it introduces the work with partially defined function. Yes, e.g. sheaves (which are part of functorial philosophy) descibe things which are defined only on some subdomains (functional analysis needs more general families than those which glue as a sheaf; for example bounded functions do not make a sheaf as something what is bounded on an infinite cover is not necessarily a cover). Of course, the inequalities of analysis can not be replaced, they give more strength, and some things crucially depend on this. Then it comes a statement that mathematicians look at QFT as not well defined business as they do not understand that physicist work formally with partially defined functions. Well, for example, usually there are no proofs that the things do not depend on various choices like a choice of a renormalization scheme. Most of the deformation quantization for example is better defined precisely because the series involved are formal, but the fact that they are well defined does not mean that the specialization for Planck constant is a real number indeed work out well defined amplitudes on a physical space.
It is not that I do not like the idea of separating the formal story from the analysis, but the claims should be more humble here. Second there is a subject algebraic analysis of Michio Sato and Masaki Kashiwara, which has been claiming something very alike since Sato’s vision around 1960and had a great success and it is about using D-modules in a way which will separate analytic questions from geometric ones. It is hard to see a substantial difference in idea section between algebaric analysis and functorial analysis, except that the first flag has a widely realized bulk of work.
I should also point out the Morris-Pareigis formal scheme, where kind of an extended Yoneda lemma on the “site” of is used where one represents by topological rings. It is precisely some idea where the formal distributions and the formal topologies are related to ind-objects in the categorical sense (formal schemes are a subclass of ind objects on ).
There is another subtlety of great importance to the quantum mechanics. Namely the distinction of the symmetric operator and a self-adjoint operator. The symmetric formally satisfies the required relations but it does not suffice for the purposes of quantum mechanics. For self-adjoint the domain needs to be dense and the domain of the adjoint has to be inside. For symmetric operators one can often find self-adjoint extensions. The extensions are not necessarily unique, and the study of the self-adjoint extension is not only of a theoretical interest but in fact is a very improtant practical methods, for example in finding new integrable systems.
See http://en.wikipedia.org/wiki/Extensions_of_symmetric_operators, http://en.wikipedia.org/wiki/Self-adjoint_operator. Wikipedia says
In quantum mechanics, observables correspond to self-adjoint operators. By Stone’s theorem, self-adjoint operators are precisely the infinitesimal generators of unitary groups of time evolution operators. However, many physical problems are formulated as a time-evolution equation involving differential operators for which the Hamiltonian is only symmetric. In such cases, either the Hamiltonian is essentially self-adjoint, in which case the physical problem has unique solutions or one attempts to find self-adjoint extensions of the Hamiltonian corresponding to different types of boundary conditions or conditions at infinity.
Hi Zoran,
maybe we can fine-tune the phrasing. The technical details referred to are those of section 3 of the book.
One point there is: there is a direct formulation of distributions as certain linear morphisms on sheaves, that is equivalent to the traditional definition, but possibly a bit better suited to the way they are actually used.
Second there is the observation that it makes sense to consider presheaves with values in the category-of-sets-with-partially-defined-functions-between-them and that this gives a good way to integrate the above observation into the general sheaf story.
Third there is, I think, the reflection of actual conversation with other mathematicians, who rejected following a discussion of functionals in physics as long as the domain of definition has not been specified. That section 3 means to make the point that it is fully formally justified to work with the algebraic characterization of the functional and leave the determination of its domain of definition for some later point.
That’s anyway my understanding. Frederic might want to expand on this. But yes, to the extent that this is not clear, let’s try to refine the formulation in the entry.
So you are saying that the sheaf of all Schwarz distributions, possibly of a generalized sort, can be defined in a different way. That is far easier statement which would not need a claim of creation of a separate direction of mathematics. While this can work just fine, one can not get far with operator analysis in general, however, if neglecting domains, as many standard constructions in operator theory soon stop making sense. Hyperfunctions of Sato are more general than Schwarz distributions and are usually defined in solely cohomological terms. There one uses analyticity which is much closer in properties to the formal calculus and algebraic geometry than the smooth classes.
Once, as a graduate student I wanted to do sheaf theoretically certain things which referred to measure theory. My advisor pointed to me: sheaf theory and measure theory are not very compatible. Smooth works better and analytic much better.
Hmm, def. 3.1.4 (in the book) of a distribution (in a more general sense than Schwarz) does have a serious condition on a domain. On the other hand, even if you do not study conditions on a domain, and consider partial functions, one still distinguishes partial functions which agree on the common domain, but whose domain is different, so one needs to define the domain in most examples, implicitly at least.
Rosenberg had in his 1988 Stockholm work on Q-categories sheaves with values in functional spaces as the main motivation, not the sheaves of sets. That manuscript is full of cumbersome notation on and calculations with partial functors which he said one should not be scared with, and is useful for such uses, but in his later treatments he did not return to such notions much (there are few exceptions of domain related calculations like changing the domain of a faithful functor to redefine it in a universal way to a coreflection, this is mentioned in the appendix on Q-categories in Kontsevich/Rosenberg paper which we looked at).
We have near duplicates functorial analysis and categorical analysis.
Does this have much to do with what Shapira speaks of here
I hope that this very sketchy panorama of almost fifty years of Algebraic Analysis (perhaps one should now better call it “Functorial Analysis”) will have convinced you of the importance of the theory and of the fact that Masaki plays the main role in it since the early seventies.
Zoran in #1 is pointing to this connection too.
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