Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
For purposes of linking, I had given an entry to decomposable differential form.
In more general ℕ-graded-commutative algebras than just that of differential forms, is there any established terminology for
0. elements that are sums of decomposables, i.e. sums of monomials in elements of degree 1?
What I’d really need is terminology for:
elements H of degree n+1 which split off at least one factor of degree 1, hence H=α⏟deg=1⋅β;
elements which are finite sums of these.
Is there anything?
just in order to be able to point to it, I created a stub for vector representation
Here is another stub: Albert algebra.
It would be nice to get a reference to clear up the number of (real) Albert algebras. John Baez's octonion paper, among other literature (including our Jordan algebra), takes it for granted that there is only one (which is true, over the complex numbers, but people are usually working over the real numbers). But John himself points out on a Wikipedia talk page that there are two (and that's what I followed).
We had a paragraph on split ocotnions buried in the entry composition algebra.
In order to be able to link to it, I have given that paragraph its own entry, now split octonions. But this deserves to be expanded of course.
Over in another thread, David Roberts asks for explanation of a bunch of terms in QFT (here).
Here I started a minimum of explanation for one item in the list: protection from quantum corrections.
at Bockstein homomorphism in the examples-section where it says
BnU(1)≃Bn+1ℤI have added the parenthetical remark
(which is true in ambient contexts such as ETop∞Grpd or Smooth∞Grpd)
Just to safe the reader from a common trap. Because it is not true in Top≃∞Grpd. The problem is that in all traditional literature the crucial distinction between Top and ETop∞Grpd (or similar) is often appealed to implicitly, but rarely explicitly. In Top≃∞Grpd we have instead BnU(1)≃K(U(1),n).
if you have been looking at the logs you will have seen me work on this for a few days already, so I should say what I am doing:
I am working on creating an entry twisted smooth cohomology in string theory . This is supposed to eventually serve as the set of notes for my lectures at the ESI Program K-Theory and Quantum Fields in the next weeks.
This should probably sit on my personal web, and I can move it there eventually. But for the moment I am developing it as an nLab entry because that saves me from prefixing every single wiki-link with
nLab:
Took a stab at a general formulation of Poisson summation formula, although the class of functions to which it is supposed to apply wasn’t nailed down (yet).
(Some of the ingredients of Tate’s thesis are currently on my mind.)
I have added the Fierz identities that give the S2-valued supercocycle in 5d here.
Added this briefly also at Fierz identity: here
This was long overdue: I have created a page lattice (disambiguation) and added corresponding warnings on terminology to the relevant entries.
In the course of this I have created stubs for lattice in a vector space, integral lattice and modular integral lattice.
wanted to record Borel’s theorem on Taylor series expansion, so created stubs for power series and Taylor series
At field (physics) I am beginning to write an actual introduction to the topic, now in a new section titled “A first idea of quantum fields”.
This means to introduce the concept with precise detail, but in a simple context (trivial and bosonic field bundles over Minkowski spacetime, perturbatively quantized) that allows to get a quick idea of the idea of the concept of (quantum) fields as such, without being distracted by other details.
So far I made it up to the derivation of the EOMs. Discussion of (deformation) quantization is to follow (maybe by tonight, depending on how much trouble I have with the trains) and I plan to sprinkle in the detailed example from scalar field in parallel with the abstract discussion.
started something at differential renormalization. More later
At dichotomy between nice objects and nice categories I added a quote from Deligne about allowing awful schemes gave a nice category of schemes. I can’t find the page I was thinking of where this dichotomy is also mentioned along with attribution of the general idea to Grothendieck. I wanted to add it there as well.
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
ηε≔diag(−1+iε,1+iε,…,1+iε)for ε>0∈ℝ.Then consider the ε-deformed Feynman propagator ΔF,ε,Λ with momentum cut off with scale Λ.
The question: does the limit satisfy
ΔF=limε→0Λ→∞ΔF,ε,Λin the Hörmander topology for tempered distributions with wave front set contained in that of the genuine Feynman propagator ΔF?
I have given Polchinski’s flow equation its own entry (spin-off from effective quantum field theory)
I have written out the rigorous formulation of renormalization group flow and running coupling constants (currently both redirect to the same entry).
This is an exegesis of Brunetti-Dütsch-Fredenhagen 09, section 4.2 and 5.1 with clues from Dütsch 18, section 3.5.3; but I have tried to disentangle, in the writeup, the general principle from the specific case where the flow is induced by scaling transformations. (The latter is going to be written out at Gell-Mann-Low renormalization cocycle, but don’t look at that entry just yet).
Accordingly I tried to streamline presentation and notation, to make it all clear at a glance what is going on… but of course this may be in the eye of the beholder.
for ease of linking I have given antibracket its own little entry (it used to just redirect to BV-BRST complex).
I had also given local antibracket an little entry of its own. Possibly these two should be merged…
I have split off Stückelberg-Petermann renormalization group from main theorem of perturbative renormalization, just so as to record references.
Both entries remain stubs for the time being (will get to filling in details soon.)
started some minimum at renormalizable interaction
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 subformula property.
created some minimum at scaling degree of a distribution
This is probably a request for Todd!
Over on colimits for categories of algebras there’s a corollary I really need right now, about Eilerberg-Moore categories being cocomplete, and the remark:
The hypotheses of the preceding corollary hold when C is a complete, cocomplete, cartesian closed category and C is the monad corresponding to a finitary algebraic theory.
That sounds like exactly what I want, but when I click on finitary algebraic theory I get taken to a page that doesn’t have the definition of “finitary algebraic theory”. I think I know what this means, so I could guess and stick it in, but I think I should let the expert do it.
Oh, whoops! - as usual, I actually need a multi-sorted generalization. But still it would be nice to have this clarified.
spelled out the statement of the quantum master Ward identity and proof of its equivalence to the quantum master equation (before renormalization): here
(still need to expand/polish the former entry itself)
I have created little entries electron propagator and photon propagator, for the moment mostly so that one may link to these terms: at the moment they contain pointers to other entries with technical details, and both include the example-for-inclusion-entry Feynman diagrams in causal perturbation theory – summary
started something at vacuum stability (spin-off from S-matrix)
I had occasion to create minimal entries for basic combinatorial concepts: factorial, multinomial coefficient
I created locally compact groupoid with an attempt at a very general definition, that will be refined to connect with the notion in the literature. In particular, if one has a system of Haar measures on the source (or target) fibres, then this will most likely place further constraints on the topological structure.
I created locally proper map and filled it with some basic properties. I linked to it from proper map, locally compact space, direct image with compact support.
needed to be able to point to connected graph, so I created some minimum
I have given multigraph its own little entry, so that I can point to it (from discussion of Feynman diagrams).
I have given finite graph its own minimum entry, just so as to be able to point to it
we didn’t have face
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.
Someone called Hammad Rana has created the stub Surreal geometry and the more substantial (but… odd) Surreal space. The latter claims to look at vector spaces over the surreal numbers and relate them to other things.
I created a new page distributivity of products and colimits, where I recorded what I learned after asking this question: http://nforum.mathforge.org/discussion/6255/commutativity-of-homotopy-sifted-colimits-and-products-in-categories-other-than-sets-or-spaces/
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 definition
remove all vague mentioning of application in physics and instead add a pointer to Wick algebra and microcausal functional, which I will create shortly.
started an entry interacting vacuum with some pointers. For instance to
idiogravity and quantum manifold have been started with no content.
started a little entry electron-photon interaction. In the process I also touched phi^n interaction.
I have started spelling out details at quantum master equation, following the rigorous derivation in causal perturbation theory due to Fredenhagen-Rejzner 11b, Rejzner 11.
So far I have added some backgound infrastructure and then the proof of this theorem ((Rejzner 11, (5.35) - (5.38)):
Consider an adiabatically switched non-point-interaction action functional in the form of a regular polynomial observable
Sint∈PolyObs(EBV-BRST)reg[[ℏ]],Then the following are equivalent:
The quantum master equation (QME)
12{S′+Sint,S′+Sint}𝒯+iℏΔBV(S′+Sint)=0holds on regular polynomial observables.
The perturbative S-matrix on regular polynomial observables is BV-closed
{−S′,𝒮(Sint)}=0.Moreover, if these equivalent conditions hold, then the interacting quantum BV-differential is equal, up to a sign, to the sum of the time-ordered antibracket with the total action functional S′+Sint and iℏ times the BV-operator:
ℛ∘{−S′,(−)}∘ℛ−1=−({S′+Sint,(−)}𝒯+iℏΔBV)I am starting to write up at BV-operator an account of the rigorous derivation/construction of the BV-operator and the BV quantum master equation in causal perturbation theory, due to Fredenhagen-Rejzner.
As a first step, the statement and proof of the BV-operator arising as the difference of the plain and time-ordered BV-differential in free field theoy is now here.
after typing “not a free field theory” for the third time, I decided to create a quick entry interacting field theory. Added a minimum of text and cross-linked a little.
just out of a whim, I expanded a little the text at Fermat curve
started a bare minimum at Møller operator.
Added a link to Informal Notes from the Harvard Fargues-Fontaine Curve seminar at Fargues-Fontaine curve since people like Lurie and Gaitsgory are apt to explain things in a manner that appeals to people at the nlab.
Mike just started explicit mathematics. I added a bit more flesh, but it’s still quite stubby.
Epstein zeta function, just recording the definition and the two classical references by Kronecker and Siegel. Nominally, this is the page 13133 of the nLab ;)
I am starting Wick algebra. So far I have an Idea-section, references, and a discussion of the finite dimensional case, showing how the traditional “normal ordered Wick product” is the Moyal star product of an almost-Kähler vector space.
I finally gave time-ordered product its own entry (it used to just redirect to Dyson formula). Still a stub.
at S-matrix and elsewhere is reference to the “causal order“-relation, the relation saying that for a pair (S1,S2) of subsets of a spacetime, S1 does not intersect the past of S2, or equivalently that S2 does not intersect the future of S1.
(Following a suggestion by Arnold Neumaier, a neat suggestive notation for this is S1∨∧S2, which I have been implementing now at S-matrix.)
I am starting to give this concept its own entry, currently titled “causal order”; but what’s good terminology?
This relation ∨∧ is not really an ordering, since it is not transitive. It would seem tempting to say “causal relation”, but googling for this term shows that has an different established meaning.
added some actual text to the Idea-section at renormalization scheme.
Created t-norm.
I have created a table-for-inclusion
and included it into the relevant entries
I have added to star product some basic facts, and their proofs, for the case of star products induced from constant rank-2 tensors ω on Euclidean spaces: the definition, proof of the associativity, proof that shifts of ω by symmetric contributions are algebra isomorphisms.
for ease of linking, I gave the concept of “field observables” its own entry.
I gave Dickey bracket its own entry (just a brief Idea-section and references)
(the term “Dickey bracket” used to redirect to conserved current, where however it was mentioned only in the references. Now it should be easier to discern what the pointer is pointing to. Of course the entry remains a stub nonetheless.)