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After sitting on this for days and hardly doing a thing, I added some applications to distribution and added a bit to the section on synthetic differential geometry. While I was dawdling, Andrew Stacey stepped in and added to some parts that needed expert attention -- thanks, Andrew.
There you go - an example of the n-lab working as it should! Perhaps we should write this up and post it ... somewhere.
I have started touching the entry distribution: added plenty of hyperlinks for keywords (some of them gray links to be created), and started to organize the material under “Traditional definition” into numbered definition/proposition/example environments.
I took the liberty of switching notation from $U$ for the base space (open subset of Euclidean ssapce) to $X$, since I think we really want to be speaking of distributions on smooth manifolds, the open subsets of Euclidean space just being particular examples. But I haven’t added anything regarding this generalization yet.
Is it wise that the entry distribution uses the notation $\mathcal{D}$ for the space of distributions, instead of the space of test funtions, as is common (originating with Schwartz, I suppose)? I like writing expicitly $C^\infty_c(X)$ for the test functions, not to hide behind notation what is easily made explicit, and on absolute grounds it would make sense to read $\mathcal{D}$ as short for “distributions”, but since this is in dangerous conflict with established convention, I’d rather we changed this.
I guess you’re right, but what a pity. Actually, your comment here should probably be incorporated into the article.
Okay, I have added a remark, here. Then I changed the notation on the page from $\mathcal{D}$ to $\mathcal{D}'$.
Following a suggestion that I made recently (here) I have removed from distribution the section “Multiplication of distributions”, which was introduced in rev 9 here.
I have replaced it with a pointer to the entry product of distributions, which I started recently.
The subsection which I removed started out with the words:
Distributions fail to address some uses to which physicists would like to put them (as in path integrals), since there is no general way to multiply distributions in a way that extends multiplication of functions .
Besides the fact that a pure math entry should say something about its topic before passing to discussion of physics, arguably the situation in physics is the opposite of what was suggested by these lines: It turns out that the theory of distributions supplemented by a proper analysis of their wave front sets serves to solve all the conceptual problems in perturbative quantum field theory: What are (or have been) widely-perceived as problems are not caused by an insufficient concept of distributions, but by simple technical errors in handling them. I have therefore written a paragraph on the actual situation in physics at “product of distributions” here. This also subsumes the pointer to Scharf’s textbook that appeared later in the section now removed.
Next in the subsection which I removed there was a paragraph sketching why a global product of distributions cannot exist, subject to some natural conditions. I have copied that paragraph over to “product of distributions”, here, essentially unmodified. But eventually this ought to be expanded a little more, to do the subject justice.
Finally there was a pointer to Colombeau algebras. This I turned into a stand-alone entry Colombeau algebra, see its own thread here.
This is very interesting to hear! I look forward to seeing the definition when it makes it into the entry, and maybe hearing a little more about the physical applications if they can be explained to a mathematician.
Thanks for the feedback. It is indeed very interesting. One of those “well known” facts that are not widely known.
I went and added some more detail on the construction of the product of distributions with compatible wave front sets: here. Ultimately a full proof requires spelling out a few more lemmas than I have done so far, but of course it is all in Hörmander’s book.
Apparently Hörmander had understood all this back in 1970 (here). Then Duistermaat-Hörmander wrote a famous (?) treatise using this theory, back in 1972 (here). In this book they use the theory to construct Feynman propagators for quantum fields on curved spacetime. One year later, in 1973, Epstein-Glaser show (here) that perturbative renormalization of quantum field theory is a well defined operation in terms of Hörmander’s microlocal analysis of distributions.
There have been some improvements since 1973, especially some streamlining of the tedious proof of the main theorem of perturbative renormalization and more generalization to fields on curved spacetime. But the mathematical foundation of perturbative quantum field theory in microlocal analysis of distributions was fully established by 1973.
Strangely, it is not widely known, even though Scharf wrote two crystal clear textbooks (Scharf 95, Scharf 01)
hearing a little more about the physical applications if they can be explained to a mathematician.
That’s what I am preparing to write, a detailed exposition of the construction of perturbative quantum field theory for mathematicians. As you will have seen, so far I have been compiling bird’s eye overviews at causal perturbation theory and at locally covariant perturbative quantum field theory. You see the product of distributions in action in the entry Wick algebra. I’d be grateful for feedback. If you let me know which parts are (particularly) unclear, I’ll try to expand/improve.
ad Urs 5
Most of the literature I read in my life used $\mathcal{D}$ for the space of test functions and $\mathcal{D}'$ for the space of distributions/generalized functions. What is your classical source of having it differently ?
There may be another reason. On manifolds you distinguish the generalized functions and generalized distributions. So to get generalized functions you take the dual of test distributions (in the sense of 1-densities) and to get generalized distributions you take test functions. In R^n no difference. But the two have different functorial properties on manifolds as disussed in Guillemin, Sternberg, Geometric asymptotics.
Zoran, that’s just what I am saying in #5.
Sorry, I misunderstood.
14 seems to be a spam.
Urs, if you do not like $\mathcal{D}'$ you can proceed as you did with $\mathcal{D} = C^\infty_0$, namely people sometimes similarly write $C^{-\infty}_0$ for $\mathcal{D}'$.
At product of distributions the example for non-existence of a global product involves the Heaviside function. It would help me to see spelled out explicitly how this example fails the wave-front-sets condition.
Here is how the wave front condition is violated in this case:
the wave front set of $\delta \in \mathcal{D}'(\mathbb{R})$ is $WF(\delta) = \{( 0,k ) | k \in \mathbb{R}\setminus \{0\} \}$ (this example), i.e. it is concentrated over the origin, where it consist of all non-vanishing covectors.
So for a distribution $u$ to have a wave front set compatible with that of $\delta$ that wave front set must completely vanish at the origin. But by the Paley-Wiener-Schwartz theorem this means that there exists some bump function $b$ with $b(0) = 1$ such that $b \cdot u$ is a smooth function. This excludes in particular the Heaviside distribution.
But i am guessing that what you’d rather like to see is more of a reason for why this wave front condition is the right one for the product of distributions to make sense.
The idea is to define the product of distributions by first Fourier transforming locally (after multiplication with bump functions of small enough support), then forming convolution, then transforming back. In the process the convolution integral need not exist. The wave front condition is pretty directly a sufficient condition for this convolution integral to exist. Or rather, it is the necessary condition that this convolution integral exists for the two distributions and for all their derivatives, for this then ensures that on the resulting product the Leibniz rule works.
So let $u$ and $v$ be two distributions on $\mathbb{R}^1$. Their product $u \cdot v$ is supposed to be such that around every point $x \in \mathbb{R}$ there exists a neighbourhood $U_x$ and a bump function $b$ with $b|_{U_x} = 1$ such that
$\widehat{b^2 u \cdot v} = \widehat{b u} \star \widehat{ b v }$where $\widehat{(-)}$ denotes Fourier transform and $\star$ denotes convolution. Hence
$\widehat{b^2 u \cdot v}(\xi) = \int \widehat{b u}(k) \widehat{b v}(\xi - k) d k \,.$Now the Paley-Wiener-Schwartz theorem says that for compactly supported distributions such as $b u$ and $b v$ their Fourier transform grows at most exponentially, with power equal to the order of the distribution.
Hence for the above convolution integral alone to have a chance to exist, it is sufficient that either of the two Fourier transforms decays at least with power $-N-1$ in either of the two directions, where $N$ is the order of the other distribution. However, for this still to work also for all derivatives of the distributions, given that under Fourier transform each derivative increases the order by one, we need that in both of the two directions $k \to \pm \infty$ one of the two Fourier transform decays faster than any finite power. But this is the condition that $\pm k$ is not in the wave front set of one of the two distributions, hence is the compatibility conditions on wave front sets.
That’s the idea. I don’t see what I just sketched stated as a theorem in Hörmander’s book, though. It must be implicit in his proof of theorem 8.2.4 (pullback of distributions) but it seems to require work to make this explicit. I see it stated without proof in various reviews (for instance in arXiv:1404.1778 item iv) on top of p. 12). It seems like a proof might be explicit in the book
but I have not seen (enough of) that book yet.
Actually #17 is what I wanted. So the problem is not with the initial product $H^n(x)$, but that the product $n H^{n-1}(x) \delta(x)$ in the “chain rule” is not valid. What is the wave front set of $H$ exactly?
According to example 15 in arXiv:1404.1778 the wave front set of the Heaviside function $H$ is also $\{ (0,k) \vert k \neq 0 \}$. (Hm, but does the proof offered there really show this?) This would mean that also $H^n$ is not admissible in itself.
And that makes sense in view of the example considered, because the derivative is $H' = \delta$, so if $H^{n+1}$ were admissible, then $H^n \delta$ would be.
Shouldn’t there be a disambiguation
There was such, right after the table of contents. I have moved it to the top of the entry.
Thanks. What’s an example of a distribution whose wave front set is nonempty but not all of $k\neq 0$?
Also, it looks from a superficial glance at the definitions as though we could consider the wave front set to be a subset of the 1-dimensional projective space over the cotangent space. Is that right?
Mike, what do you mean by
1-dimensional projective space over the cotangent space.
do you mean a $\mathbb{P}^1$-bundle over $T^*X$?
I meant apply the functor $P^1 : Vect \to Top$ fiberwise to $T^*X$.
But P^1 is not the functor, it is P, and the dimension comes from the dimension of the vector space, no? Note also, that the cone is not necessarily symmetric around the origin, which means that even taking the projective bundle of the cotangent bundle won’t work.
Oh, sorry, yes, I meant $P$. I was thinking of the fact that the points of projective space are 1-dimensional subspaces, but I forgot that when you consider higher dimensional subspaces, for some reason it’s called a “Grassmanian” instead of a projective space.
I also missed that the “conic sets” are only stable under multiplication by positive scalars. But in that case it seems even easier: the wave front set should be a subspace of the unit sphere bundle of the cotangent bundle.
Right, the wave front set is what they call “conal” or “conic”. I agree that thinking of it as a subspace of the unit co-sphere bundle would be more elegant.
What’s an example of a distribution whose wave front set is nonempty but not all of $k\neq 0$?
The Heaviside distribution on higher dimensional space has wave fronts being orthogonal to the hyperplane at which the function jumps.
More generally, the wave front set of a characteristic function of a subset $U \subset \mathbb{R}^n$ with smooth boundary $\partial U \subset \mathbb{R}^n$ (hence the distribution which integrates its argument over $U$) is the conormal bundle of $\partial U \subset \mathbb{R}^n$. (See around figure 3 in arXiv:1404.1778. Apparently this is how wave front sets are used in computer image recognition.)
Then the delta-distribution $\delta(x-y)$ regarded as a generalized function of two arguments has wave front set of the form $(x,(k,-k))$ (see at Wick algebra here). More sophisticated such examples are listed at propagators - table.
But how about an example on $\mathbb{R}^1$? I haven’t checked (am on vacation now…), but I suspect the reason that the Heaviside function on $\mathbb{R}^1$ has wave front at the origin pointing in both directions may be attributed to the graph of the Heaviside function having two kinks, one to the left and one to the right of zero. If that is right, then we should get a wave front set pointing in only one of the two directions by smoothing out one of these two kinks. So I am guessing that a distribution like $\phi \mapsto \int_0^\infty \sqrt{x} \phi(x) \, dx$ should have wave front at the origin pointing only to the left.
Re #28: great; I added some remarks about this to wave front set.
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