started some minimum at *convolution product of distributions*

there had been no references at *Hilbert space*, I have added the following, focusing on the origin and application in quantum mechanics:

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John von Neumann,

Mathematische Grundlagen der Quantenmechanik. (German) Mathematical Foundations of Quantum Mechanics. Berlin, Germany: Springer Verlag, 1932.George Mackey,

The Mathematical Foundations of Quamtum MechanicsA Lecture-note Volume, ser. The mathematical physics monograph series. Princeton university, 1963E. Prugovecki,

Quantum mechanics in Hilbert Space. Academic Press, 1971.

(brushed up the definition, added four basic classes of examples)

]]>I found the section-outline of the entry *distribution* was a bit of a mess. So I have now edited it (just the secion structure, nothing else yet):

a) There are now two subsections for “Operations on distributions”,

b) in “Related concepts” I re-titled “Variants” into “Currents” (for that’s what the text is about) and gave “Hyperfunctions and Coulombeau distributions” its own subsection title.

c) split up the References into “General” and “On Coulombeau functions”.

(I hope that this message is regarded as boring and non-controversial.)

]]>For ease of linking to from various entries, and in order to have all the relevant material in one place, I am creating an entry

Presently this contains

an Idea-section,

some preliminaries to set the scene,

the statement and proof for the case of compactly supported distributions, taken from what I had just writted into the entry

*compactly supported distribution*,the informal statement for general distributions, so far just with a pointer to Kock-Reyes 04,

a section “Applications”, so far with

some comments on the relevance in pQFT;

some vague pointer to Lawvere-Kock’s generalization to a more general theory of “extensive quantity”

both of which deserve to be expanded.

Eventually I want to have more details on the page, but I’ll leave it at that for the time being. Please feel invited to join in.

I’ll go now and add pointers to this page from “distribution” and from other pages that mention the fact.

]]>I have added to *locally convex topological vector space* the standard alternative characterization of continuity of linear functionals by a bound for one of the seminorms: here

(proof and/or more canonical reference should still be added).

]]>I created a stub entry for *Hörmander topology*, just to record some references.

The following seems to be waiting for somebody to answer it:

Consider the deformed Minkowski metric

$\eta_\epsilon \coloneqq diag( -1 + i \epsilon, 1+ i \epsilon , \dots , 1 + i \epsilon )$for $\epsilon \gt 0 \in \mathbb{R}$.Then consider the $\epsilon$-deformed Feynman propagator $\Delta_{F,\epsilon,\Lambda}$ with momentum cut off with scale $\Lambda$.

The question: does the limit satisfy

$\Delta_{F} \;=\; \underset{ {\epsilon \to 0} \atop {\Lambda \to \infty} }{\lim} \Delta_{F,\epsilon,\Lambda}$in the Hörmander topology for tempered distributions with wave front set contained in that of the genuine Feynman propagator $\Delta_F$?

]]>have created *extension of distributions* with the statement of the characterization of the space of *point-extensions* of distributions of finite degree of divergence: here

This space is what gets identified as the space of renormalization freedom (counter-terms) in the formalization of perturbative renormalization of QFT in the approach of “causal perturbation theory”. Accordingly, the references for the theorem, as far as I am aware, are from the mathematical physics literature, going back to Epstein-Glaser 73. But the statement as such stands independently of its application to QFT, is fairly elementary and clearly of interest in itself. If anyone knows reference in the pure mathematics literature (earlier or independent or with more general statements that easily reduce to this one), please let me know.

]]>created some minimum at *scaling degree of a distribution*

I have created an entry-for-inclusion titled

which is one “Summary box” that means to give a lightning but accurate summary of the origin and meaning of Feynman diagrams in the rigorous description via causal perturbation theory.

This makes use of a set of nicely done slides in Brouder 10; a citation is contained.

I am meaning to include this as a Summary-box into relevant entries, such as *Feynman diagram* and *renormalization*.

I gave *product of distributions* its own entry. For the moment it just points to the definition in Hörmander’s book.

This should eventually supercede the section “Multiplication of distributions” at *distributions*, which I find suboptimal: that section starts very vaguely referring to physics as if the issue only appears there, and it keeps being very vague, with its three sub-subsections being little more than a pointer to one reference by Colombeau.

I suggest to

remove that whole subsection at

*distribution*and leave just a pointer to*product of distributions*move the mentioning of Colombeau’s reference to

*product of distributions*and say how it relates to Hörmander’s definitionremove all vague mentioning of application in physics and instead add a pointer to

*Wick algebra*and*microcausal functional*, which I will create shortly.

created *pullback of a distribution*, just for completeness

Added to *Hadamard distribution* the standard expression for the standard choice on Minkowski spacetime, as well as statement and proof of its contour integral representation (here)

Gave some content to *causal propagator*: an Idea-section and the main formulas for the causal propagator on Minkowski spacetime.

I want to be adding some details to *Cauchy principal value*. What’s a good reference? Say for the proof that up to addition of a delta-distribution, $f(x) = pv\left( \frac{1}{x}\right)$ is the unique distributional solution to $x f = 1$?

I gave the definition of *symbol order* its own entry (an estimate on the decay of the principal symbol of a (psedo-)differential operator that enters the assumptions of the propagation of singularities theorem).

Maybe there is a better name for this? The literature refers to it mostly only in formal notation as “$q \in S^m_{\rho, \delta}(X)$”.

]]>I gave the definition of *properly supported pseudo-differential operator* its own entry.

A while back I had started an overview table *propagators - table*.

Now I see that a nice table in this spirit, but larger with much more information, has been produced by some M. B. Kocic. I have added pointer to this pdf in a few places.

]]>I gave the entry *wave front set* more of an Idea-section, and I added pointer to Hörmander’s book.

added to *advanced and retarded propagator* statement and proof of the expression

I have given *generalized function* its own little entry (it used to be just a redirect to *distribution*) with some expository words on how to think of and read distributions as “generalized functions”.

This question will just show my ignorance, please bear with me:

For $P \colon C^\infty(X) \to C^\infty(X)$ a differential operator (self-adjoint say), there is an evident linear map

$DistribSolutions(P) \longrightarrow \left(C^\infty(X)/im(P)\right)^\ast$from distributions $u$ for which $P u = 0$ to linear duals on the cokernel of $P$.

When is this surjective?

(I’d be happy to add various qualifiers if necessary, say compactly supported distributions, or whatever it takes.)

]]>This used to be a side-remark at *wave front set*, I have given it its own entry: *Paley-Wiener-Schwartz theorem*.

just for completeness and for ease of linking, I gave *product of distributions with a smooth function* its own entry

I keep adding basic material to *Fourier transform* and copying paragraphs over to the respective stand-alone entries, where missing.

Now I added the example of the Fourier transform of the delta-distribution here and copied it over to *Dirac distribution – Fourier transform*.