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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.
Did some proof reading.
Thanks, David!
I did some more work in the Sandbox, but not ready yet.
Have been further working on it, but still not done.
It finally dawned on me that it is pointless to try to squeeze this material into the “Idea”-section at field (physics). So I have declared it now a stand-alone chapter of geometry of physics. Now here:
Let me say though that I think what you’re doing recently (the Insights article at Physics Forum, the articles on S-matrix and pQFT, etc.) is just great. For me it’s some of the most useful exposition I’ve ever seen for this area – far better than the books by Witten et al., this really is Quantum Fields for Mathematicians. Keep it coming!
Thanks for the feedback, Todd!
When this chapter “A first idea of QFT” is closer to being stable (hopefully by tomorrow evening), I’d be very interested in hearing of your experience while reading through it, such as at which points the progression of the text seems unclear or maybe too unmotivated, or else maybe too pedantic or repetitive – whatever the case may be.
I have now made it to the description of the covariant phase space in section 7, and filled in full details on the Poisson bracket Lie $(p+1)$-algebra in section 5. (here)
Using this, next will finally be the discussion of the algebra of quantum observables in section 8. But not tonight.
I”m enjoying this too.
These examples may be combined: the mapping space $[\Sigma, \mathbf{\Omega}^n]$ is a kind of smooth classifying space for differential forms on $\Sigma$
Is ’on’ the right preposition here? Isn’t $[\Sigma, \mathbf{\Omega}^n]$ classifying $\Sigma$-parameterized differential $n$-forms?
Isn’t $[\Sigma, \mathbf{\Omega}^n]$ classifying $\Sigma$-parameterized differential $n$-forms?
What you are thinking of are the plots
$U \to [\Sigma, \mathbf{\Omega}^n]$which equivalently are differential $n$-forms on $U \times \Sigma$, which one may think of as $U$-parameterized differential forms on $\Sigma$ – since $U$ varies, while $\Sigma$ is fixed.
The subtlety here, which I did not want to get into in this “first idea”-chapter, is that the actual classifing smooth space of differential forms on $\Sigma$ should assign to $U$ not all the differential forms on $U \times \Sigma$, but just those which are plain functions with respect to $U$ (with no “legs” along $U$). This genuine classifying space of differential forms on $\Sigma$ is the concretification
$\sharp_1 [\Sigma, \mathbf{\Omega}^n] \,.$There is detailed discussion of this point at geometry of physics – differential forms, in the section Smooth moduli space of differential forms
I’m enjoying this too.
Thanks. If you have any comments (in particular critical comments) please do let me know.
I guess that’s a generalized element point of view, like we might have:
In $Set$, we say that $[A, 2]$ is the classifier of subsets of $A$, rather than just the set of subsets of $A$, because for any $X$, $X \to [A,2]$, picks out $X$-parameterized subsets of $A$.
$X \to [A,2]$, picks out $X$-parameterized subsets of $A$.
Exactly! But this is not how you said it in #10. In your analogy $\Sigma$ there corresponds to $A$ here.
Anyway, I suppose we perfectly agree on what’s going on.
I have now spelled out the example of the free real scalar field alongside the development of the theory. Currently it culminates in a detailed derivation of the “causal propagator”/”Peierls bracket”/”Pauli-Jordan distribution” from first principles – here.
I have been working on the next section: “Gauge symmetries”.
I imagine you’ll be editing offline, so I’ll note some typos here:
to give first good precise idea; allowed to be globally hyperbolic Lorentzian manifold; family of smooth functions on $\Sigma$ [should be $X$]; differential geoemtry; hence a field) [no opening bracket];
A stray ’s’ appears in the right hand diagram after
such that this system is compatible with the above projection maps, i.e. such that
Not sure I see this:
$\begin{aligned} \left(j^\infty_\Sigma(A)\right)^\ast(f_{\mu \nu}) & = F_{\mu \nu} \\ & = (d F)_{\mu \nu} \,. \end{aligned}$Why does $F_{\mu \nu} = (d F)_{\mu \nu}$?
That last one might have to be $(d f)_{\mu\nu}$
Thanks! All fixed now.
($d F$ should have been $d A$ :-)
There is now some first actual content in the next section: Reduced phase space.
I am exploring ways to best draw the big picture regarding the quantization of gauge theories. Here is one attempt:
$\,$
$\,$
$\array{ \underline{\mathbf{\text{pre-quantum geometry}}} && \underline{\mathbf{\text{higher pre-quantum geometry}}} \\ \, \\ \left\{ \array{ \text{Lagrangian field theory with} \\ \text{implicit infinitesimal gauge transformations} } \right\} &\overset{ \text{explicate} \atop \text{gauge transformations} }{\longrightarrow}& \left\{ \array{ \text{dg-Lagrangian field theory with} \\ \text{explicit infinitesimal gauge transformations} \\ \text{ embodied by BRST complex } } \right\} \\ && \Big\downarrow{}^{\mathrlap{ \text{pass to} \atop \text{derived critical locus} }} \\ \Big\downarrow && \left\{ \array{ \text{dg-reduced phase space} \\ \text{ embodied by BV-BRST complex } } \right\} \\ && {}^{\mathllap{\simeq}}\Big\downarrow{}^{\mathrlap{\text{fix gauge} }} \\ \left\{ \array{ \text{ decategorified } \\ \text{ covariant } \\ \text{ reduced phase space } } \right\} &\underset{\text{pass to cohomology}}{\longleftarrow}& \left\{ \array{ \text{ dg-covariant} \\ \text{reduced phase space } } \right\} \\ && \Big\downarrow{}^{\mathrlap{ \array{ \text{ quantize } \\ \text{degreewise} } }} \\ \left\{ \array{ \text{gauge invariant} \\ \text{quantum observables} } \right\} &\underset{\text{pass to cohomology}}{\longleftarrow}& \left\{ \array{ \text{quantum} \\ \text{BV-BRST complex} } \right\} }$Here:
term | meaning |
---|---|
“phase space” | derived critical locus of Lagrangian equipped with Poisson bracket |
“reduced” | gauge transformations have been homotopy-quotiented out |
“covariant” | Cauchy surfaces exist degreewise |
I have now brought in the material which I had earlier written into separate entries, such as to have a skeleton for the complete notes (needs to be inter-connected and polished still, but should give an idea of the intended content): here.
There are now 19 sections. I don’t think I could do with less, but also I shouldn’t have much more.
I changed my strategy regarding introducing the required generalized geometry. Previously I had a lightning introduction to “functorial geometry” in the first section “Geometry”, but trying this out on readers like Arnold Nuemaier over in the PO-Insights discussion here I realized that this doesn’t work for the intended audience. So now I am instead trying to introduce the required geometry alongside as the corresponding physics gets introduced: first the “functorial geometry” in section 3 Fields then later the “higher geometry” in section 9 Gauge symmetries.
This still needs a bit more work, I suppose. But maybe one can see if it is going in the right direction now.
David C: I see that you made an edit yesterday evening. To check what you did, I tried to click on “see changes”, but that produced a “502” error. Now I saved another version of mine and then similarly tried to see your changes via the History link here:
https://ncatlab.org/nlab/revision/diff/geometry+of+physics+–+A+first+idea+of+quantum+field+theory/45
but again I get a 502 error.
I’ll report this to Adeel. Meanwhile: Is it easy for you to remember what you edited?
Yes, you had an extra curly bracket in
$D_\mu a_\nu^\alpha \;\coloneqq\; a^\alpha_{\nu,\mu} + \tfrac{1}{2} \gamma^{\alpha}{}_{\beta \gamma} a^\beta_{\mu} a^\gamma_{\nu} - (\mu \leftrightarrow \nu)$so it wasn’t compiling.
Oh dear, now I can’t see it compile here. It was the equation after
consider the functions on the jet bundle given by
[edit: silly me! I pasted in the original mistaken code.]
Thanks! I fixed it.
When you say “doesn’t compile” do you mean that the whole page failed, or just the equation? That’s why I keep checking in the Sandbox, to ensure that the page as a whole comes out readably after saving. On my end it alsways did.(?)
Just that equation.
I have started bringing in the example of the Dirac field (here). To that end I added to the section “Spacetime” discussion of spin, and to the section “Fields” discussion of supergeometry.
Now there is need to harmonize some notation a bit better. But not today.
Next I should try to sort out that issue with distributions as linear maps in the Cahiers topos (here).
At geometry of physics – A first idea of quantum field theory
For $\mathfrak{g} = \mathfrak{su}(2)$ this is a field history for the gauge field of the strong nuclear force in quantum chromodynamics.
This is a typo, no?
Also: “ant-symmetry”, “so smal”
In your “Remark 2.22. (two-component spinor notation)”, you write
denoted $(\xi^{\dagger \dot a})_{\dot a = 1}^2$
which at first I thought was a power, but then I realised you meant $\dot a=1,2$. This might be less confusing as the notation is already rather dense. Similarly for the left-handed spinor.
In “Definition 2.28. (adiabatic switching)”, you have
the vctor space space $C^\infty(\Sigma)\langle g \rangle$ spanned by a formal variable $g$
which apart from the typo was slightly confusing in the sense that you are, immediately before, talking about a cutoff function $g_{sw}$, so at first I guessed that $g$ here might be such a function, but I guess it is the gauge coupling? Might be worth disambiguating, or sign-posting here.
an infinitesimally thickened Cartesian space $\mathbb{R}^n \times Spec(A)$ is represented by a commutative algebra $C^\infty(\mathbb{R}^n) \otimes Spec(A) \in \mathbb{R} Alg$ which
that second $Spec(A)$ should be just $A$, I think.
for each infinitesimally thickened Cartesian space $\matbb{R}^n \times Spec(A)$
\matbb
rather than \mathbb
More: “the 1-dimensional even vector sspace”, “an super Cartesian space”
super-commutative algebra $C^\infty(\mathbb{R}^n) \otimes Spec(A) \in \mathbb{R} Alg$
again, $A$, not $Spec(A)$, I think.
That’s down to the start of the section “Field variations”. More later, I hope.
Thanks for catching all this! All fixed now.
In “Definition 2.28. (adiabatic switching)”, […] slightly confusing
I have tried to improve the wording to clarify this. Now it reads like so:
For a causally closed subset $\mathcal{O} \subset \Sigma$ of spacetime say that an adiabatic switching function or infrared cutoff function for $\mathcal{O}$ is a smooth function $g_{sw}$ of compact support (a bump function) whose restriction to some neighbourhood $U$ of $\mathcal{O}$ is the constant function with value $1$:
$Cutoffs(\mathcal{O}) \;\coloneqq\; \left\{ g_{sw} \in C^\infty_c(\Sigma) \;\vert\; \underset{ {U \supset \mathcal{O}} \atop { \text{neighbourhood} } }{\exists} \left( g_{sw}\vert_U = 1 \right) \right\} \,.$Often we consider the vector space space $C^\infty(\Sigma)\langle g \rangle$ spanned by a formal variable $g$ (the coupling constant) under multiplication with smooth functions, and consider as adiabatic switching functions the corresponding images in this space,
$\array{ C_c^\infty(\Sigma) &\overset{\simeq}{\longrightarrow}& C_c^\infty(X)\langle g\rangle }$which are thus bump functions constant over a neighbourhood $U$ of $\mathcal{O}$ not on 1 but on the formal parameter $g$:
$g_{sw}\vert_U = g \,$In this sense we may think of the adiabatic switching as being the spacetime-depependent coupling “constant”.
Aha, that’s interesting. Both my guesses were wrong.
More:
Their variational derivative uniquely decomposes as 1) the Euler-Lagrange derivative $\delta_{EL}\mathbf{EL}$ which is
is (or defines) a prequantum [Lagrangian field theory]]
and just before “Here $\star_\eta$ denotes the Hodge star operator of Minkowski spacetime.”
one sees an equation where it is given as $\wedge_\eta \star$. Also the sentence following seems to be truncated. Perhaps you were going to expand the example, or just meant to make a comment saying more general Lie algebras were possible.
such that $\delta EL$ is proportional to the variational derivative of the fields (but not their derivatives[…]
missing underscore.
More later.
Thanks again!! All fixed now.
I have now considerably expanded the exposition of the underlying geometry.
Now section 1. Geometry is, in principle, a self-contained (if terse) exposition of differential geometry over Cartesian spaces, and section 3. Fields develops from that in a similarly pretty self-contained way diffeological spaces, smooth sets, formal smooth sets, super smooth sets; all with some key examples and facts, following the demand of the development of the field theory.
The first three sections are meant to go live on PhysicsForums-Insights in a few days.
Further down the line I have been adding some basics on distributions as the multilinear observables on the super smooth set of field histories in 7. Observables – Polynomial observables
When further editing 7. Observables – General observables I noticed that for $E \to \Sigma$ a supergeometric field bundle (such as for the Dirac field), then the tight identification of the mapping super smooth set
$\left[ \Gamma_\Sigma(E), \mathbb{C} \right]$“of observables” seems to requires a bit more work. It is easy to see that smooth functions on the diffeological space of sections of the odd-shifted field bundle induce generalized elements “at odd stage” in this super smooth set, but maybe I need to show that this construction exhausts that space.
Is this open for editing, or are you working on the latest version offline?
Typos:
$k$ should be $\alpha$.
Should be $n_3$.
$X$ should be $\mathbb{R}$
should be \mathbb{n}
Why the ’C’ in ’CAlg’?
C^\infty8\mathbb{R}^{n_1}
not so inclined my ignore this
may.
Out of interest, does each session of your course cover about 1 section?
Thanks for the list of typos! Fixed now.
I am not following the script linearly. I mean to optimize the text for reading.
Now on PhysicsForums-Insights: A first Idea of Quantum Field Theory.
The chapters will be appearing incrementally. The first one is at
The second chapter is now at
Meanwhile I have considerably boosted the chapters 7. Observables and 8. Phase space, using the all-important proposition 2.1 (and its enhancement) and lemma 2.5 from
Covariant phase space, constraints, gauge and the Peierls formula,
Int. J. Mod. Phys. A, 29, 1430009 (2014) (arXiv:1402.1282)
which now appear as theorem 7.26 and theorem 8.7.
It’s pretty neat how this part of the story comes together. Thanks to Igor.
finally chapter 9. Propagators is taking shape…
Until now I could still save the entry A first idea of quantum field theory by repeatedly trying. After one or two dozen of attempts it would suddenly save (instead of giving the 502 error)
But apparently this has reached a limit, today I haven’t been able to save my latest version, even after many attempts.
Now I had the idea that I might try to split off each chapter as an include-file, save these separately (which works fine), and then include them all. Accordingly there is now a list of entries created titled
…
But this trick turns out not to have any effect: the main file A first idea of quantum field theory, which is now mainly a list of include-commands, still produces errors when saving.
And how will these correspond to geometry of physics – A first idea of quantum field theory?
That’s the entry I am talking about. Or was talking about, will have to abandon that project now.
Check out the source of that entry. It ends now with the lines
[[!include A first idea of quantum field theory -- Geometry]]
[[!include A first idea of quantum field theory -- Spacetime]]
[[!include A first idea of quantum field theory -- Fields]]
[[!include A first idea of quantum field theory -- Field variations]]
[[!include A first idea of quantum field theory -- Lagrangians]]
[[!include A first idea of quantum field theory -- Symmetries]]
[[!include A first idea of quantum field theory -- Observables]]
[[!include A first idea of quantum field theory -- Phase space]]
The goal would be to expand this list by the lines
[[!include A first idea of quantum field theory -- Propagators]]
[[!include A first idea of quantum field theory -- Gauge symmetries]]
and make it save. But it doesn’t work.
Hah, I may have found a second-order trick that works around the bug:
I first save the main entry with its full include-list while the entries-to-be-included are still empty! When that is saved, then I fill the content into the entries-to-be-included.
Too bad, that doesn’t help either. While now the material saves, it still produces 502 errors when displaying.
Okay, so that’s the end of it then.
Oh I see, ’A first idea of quantum field theory’ was included as a redirect at ’geometry of physics – A first idea of quantum field theory’.
What to do now?
What to do now?
What we should have done years ago: Somebody needs to set up a non-broken wiki and write a converter that turns our existing database of entries into whatever the new format is.
The kind soul who wrote the converter from Instiki to Wordpress for me did so in two evenings, and it has all the functionality. This shows how easy this actually is for an expert (and how weird it is that after so many years Instiki is still so severely broken.)
With so many computer-science-affine contributors here it would seem somebody should know somebody who would enjoy spending two evenings with migrating us to another working wiki platform?
Given that we are hosted at Carnegie Mellon University, funded by the HoTT MURI grant, they must want nLab to function well. Could some extra funding be found for a software upgrade?
There are many possibilities and ideas, but somebody needs to take some action. If you have some free energy, you could try to follow up that suggestion by emailing, say, Steve Awodey.
Hi Urs, could you possibly try to reproduce the 502 error on saving, with an as accurate as possible time when it happens? if you have the time, could you also try triggering it in the Sandbox, again letting me know the timestamp when you do so? E.g if you are able to find something that is on the border between saving and not, and leave what does save in the sandbox, that would be very useful.
Hi Richard,
luckily I had managed to make the error appear on loading instead of on saving (by saving small separate files and then “!include”-ing them in the main file). So if it is about reproducing the error message, just call geometry of physics – A first idea of quantum field theory and this will give the error.
Thanks a million for looking into this!!
Hi Urs, thanks! I am interested in reproducing both errors, though I suspect that both have the same cause (that the page cannot be loaded in time). In particular, it may be (depending upon which order things are carried out) that your earlier changes did in fact save, just that the new page could not be displayed. Probably the 502 error is the result of some timeout. Not finished debugging yet, though.
Hi Richard,
thanks. The saving error is reproduced by copying the content of the !include-files into a single file (the first half dozen or so should suffice to trigger the error).
Indeed, for some time this would show an error but eventually save anyway. I always checked for this. But lately this wasn’t the case anymore.
Generally, a curious aspect of this bug is its time-dependency. For several weeks I proceeded by submitting my edits a dozen or two dozen times in a row, always getting 502 until at some random point suddenly it would work.
As the file grew larger, the number of times I had to submit before it “got through” increased.
This could be consistent with your suggestion of a timeout. I am hoping you’ll find the cause!
Hi Urs, thanks for the additional details.
You have probably seen the email I sent to Adeel, but for any others interested: yes, I think this issue is actually very simple. I believe that there is a request timeout of 30 seconds currently in place on the nLab, a request here meaning when you either try to save the page or load it (one is a POST or PUT, the other a GET, but the timeout seems to be the same for all). It is very usual for web applications to have a timeout in place, but 30 seconds is not reasonable for the nLab, given that pages take quite some time to process. As a quick fix, we can simply increase this timeout. It is just a question of locating exactly where it is set in the codebase/server config; I have asked Adeel to help clarify this, as the nLab’s unicorn config (the nLab is running as nginx + unicorn on top of Instiki) is not very clear to me.
Of course the longer term goal would be to bring down the page rendering time, but this should hopefully gets things at least working for now.
Thanks, this sounds good! Could one just grep for “30”?
Hehe, yes, tried that :-). The timeout is set to 30 in one of the unicorn config files, and I suspect that the nLab is taking the value from there, but the way in which the unicorn config for the nLab is specified is a bit confusing to me, and I am not exactly sure where it is actually taking the values from. This is what I have asked Adeel to clarify, since he is familiar with the unicorn config.
Oh, I see. But if you’d just increased the values where you found them? At worst the nLab would not be affected while some other application (??) would get a longer timeout threshold.
I was a bit wary because unicorn would need to be restarted, and this is obviously a bit dangerous (i.e. it will bring down the nLab if something goes wrong), so I preferred to have a second pair of eyes on it.
But happily Adeel has now changed it from 30 to 60 seconds, and the problematic page now loads (on my machine at least). Let us know if the phenomenon re-appears.
For future reference, in case anybody remembers, in addition to the unicorn timeout (now 60 seconds), there are also some nginx timeouts, which are currently 3 minutes for one of them, possibly 90 seconds for the other, I do not remember just now. Should we ever increase the unicorn timeout beyond 90 seconds, we would also need to increase the nginx ones.
Yay! The page loads!. A million thanks!!
By the way, Urs, you may prefer to change back your includes now to having the content on the page, so that the user sees something happening. Or, if you prefer, we could look into changing the behaviour of rendering of includes, if people wish. Currently they are ’pre-processed’, i.e. each include finishes processing before the page loads. It looks fairly simple to change things so that includes process on the fly like normal, but we’d need to test it carefully locally first. Let us know if we should look into this.
Good idea.
Right now I am enjoying the split into smaller separate field. Because even though the big file presumeably saves now, it still means that it needs more than half a minute to do so, while the smaller files save more quickly. Also the big file I have to edit locally in an editor, because my small webbook stalls when trying to edit big files in the nLab edit window; while the small files I can edit here, which is to be preferred.
Thanks to Richard’s admin work, finally we may get back to the content of the entry.
I am finalizing the polishing of chapter 11 “Reduced phase space”. Please let me know how it reads.
Sorry to interrupt, I will try to avoid posting here after this so that you can get back to the content.
Thanks for your thoughts, I agree with your plan, then I will put this down as something to implement. I will look at the unicode issue first, though.
(Adeel actually made the change, so he deserves the credit, but thanks! :-))-
Now I am finalizing chapter 12. Gauge fixing. (This construction lifts the remaining obstruction to quantization by causal perturbation thory, which is finally the topic from chapter 13 on…)
In editing I briefly create or touch various related entries, without however having time at the moment to do these justice as stand-alone entries (I’ll try to come back to this later). For instance here I edited wave polarization.
finally I have finalized chapter 13. Quantization
now I have finalized chapter 14. Free quantum fields
Now I am finalizing chapter 15. Interacting quantum fields
This is essentially the material that I have been compiling into the entry S-matrix in the section “In causal perturbation theory”.
Dear Urs,
(I am a/the student in your MQFT lecture that often asks questions that are quite of the same content until I realize it. Here are likely another questions of the same nature.)
Thanks for your comment.
I assume you are referring to this paragraph?
Looking at it, i see that I used the symbol “$\Psi$” twice, where I should have used two different symbols. I have changed it now, let me know if this clarifies the issue.
(By the way, the source code in your question looks okay, probably what you need to do to make it display properly is to check the box “Markdown+Itex” below the edit pane.)
(Another by the way: when we talked about discussion in “the forum” I was thinking of PhysicsForums, such as here. It’s fine with me either way, but over at PhysicsForums there are more participants interested in quantum field theory, and hence more chance for you to find a useful exchange. )
Dear Urs,
Yes, I’ve just found out the proper way to display the maths.
Thank you. I will switch to the PhysicsForum and continue there. Also, I forgot to tell you that if it’s more convenient for you to answer my written questions verbally when we meet, please do so and I then could totally post your answers to the forum for the benefit of the participants there (and of mine, as this method could serve as a way for me to check whether I have really got the answers.)
Thanks, great. I’ll try to react to questions in any form, but there might be delays. If you volunteer to type up answers that you extract from what I say verbally, that would be magnificent! Thanks.
That would be my pleasure too!
I have sprayed a few new questions on the PhysicsForum. For the above question, I would not have raised it if from the start you had used the current notations. The old notation left an impression on me that the “even-degree-to-even-degree” plot $\Psi_{(-)}: \mathbb{R}^{n} \to \Gamma_\Sigma(E_{even})$ is obtained from “mixed-degree” plot $\Phi_{(-)}: \mathbb{R}^{n|1} \to \Gamma_\Sigma(E_{odd})$ by multiplying $\Phi_{(-)}$ with some odd-degree element $\theta$ (in $\mathbb{R}^{0|1}$?). If this impression is true, then the “multiplying” procedure has not been clear to me.
(By the way, is there any general strategy to obtain a link to a specific paragraph in a nLab’s article? So far, my method is to search for a hyper-mentioning of the Def/Prop./Theorem to extract its address.)
The multiplying-by-$\theta$-business comes from using the natural bijection between maps of the form
$\mathbb{R}^{0\vert 1} \longrightarrow [ \Gamma_{\Sigma}(E), \mathbb{C} ]$with maps of the form
$\mathbb{R}^{0\vert 1} \times \Gamma_\Sigma(E) \longrightarrow \mathbb{C}$By pullback, these need to take the canonical even coordinate $c$ on $\mathbb{C}$ to an even coordinate on the Cartesian product $\mathbb{R}^{0\vert 1} \times \Gamma_\Sigma(E)$.
Now if $E = \Sigma \times S_{odd}$, then the odd coordinates of $\Gamma_\Sigma(E)$ are the $\mathbf{\Psi}^ \alpha(x)$, regarded in odd degree. But on the Cartesian product with $\mathbb{R}^{0\vert 1}$ we may multiply these with the canonical odd coordinate $\theta$ on $\mathbb{R}^{0 \vert 1}$ to get the even element $\theta \mathbf{\Psi}^\alpha(x)$:
$c \mapsto \theta \mathbf{\Psi}^\alpha(x)$This is known as “superfield-expansion”.
I have now added a further example to this effect, here.
By the way, is there any general strategy to obtain a link to a specific paragraph in a nLab’s article? So far, my method is to search for a hyper-mentioning of the Def/Prop./Theorem to extract its address.
Unfortunately there is no good way to do this.
What you can do is go to the source of the page by replacing “…/show/…” in the URL with “…/source/…”, then find your paragraph there and see if it is equipped with an anchor of the form
#AnchorName
That’s how I pointed to a paragraph in message #69 above. But it’s a bit tedious…
I have sprayed a few new questions on the PhysicsForum.
Thanks. I see one question, here. Did you send more?
Hmm your multiplying-by-$\theta$-business above seems to be on the $Obs$ industry instead of the $\Gamma_\Sigma(E)$ industry I was asking. But the same business goes on for the latter? My guess is an “even-degree-to-even-degree” plot
$\Psi_{(-)}: \mathbb{R}^n \rightarrow \Gamma_\Sigma(E_{even})$(or rather
$\Psi_{(-)}: \mathbb{R}^{n+p+1} \rightarrow E_{even}$) that dually takes an even coordinate $\psi^a \in C^\infty (E_{even})$ to an even coordinate $x\in C^\infty( \mathbb{R}^{n+p+1})$ corresponds to an “odd-degree-to-odd-degree” plot
$\Phi_{(-)}: \mathbb{R}^{n|1} \to \Gamma_\Sigma(E_{odd})$(or rather
$\Phi_{(-)}: \mathbb{R}^{p+n+1|1} \to E_{odd}$that takes an odd coordinate $(\psi^a)^\ast \in C^\infty(E_{odd} )$ (which is dual to $\psi^a$) to an odd coordinate $\theta x \in C^\infty(\mathbb{R}^{p+n+1|1} )$. This could justify denoting $\Phi_{(-)}$ by $\theta \Psi_{(-)}$ (which is the reverse direction of the old notation.)
(Thank you. The tedious method is good enough for me, as I was only troubled by not knowing a general way to get the link. No, those two were all I did send. I will be posting more and more questions tonight and the coming days.)
I am now preparing chapter 16. Renormalization from the material that I have been compiling in the entry renormalization.
What is still missing is example computations. I have to see if I find time for that.
For the time being I’ll declare A first idea of quantum field theory done.
I did not manage to write the last intended chapter 17. on QED, and without that a bunch of examples are missing. I hope to get back to this, but for the moment I need to focus on other tasks.
Also, there are little remaining gaps and probably mistakes. I’ll keep an eye on it and will try to incrementally optimize things. But for the moment I need a break.
I hope you get to teach the course again. It’s a great luxury being able to improve a course incrementally from year to year. I hate to think of the number of once-delivered courses I ran from the early days.
Can you get back to research now?
Yes, so I need to get some backlog of referee reports and my contribution to “New Spaces…” out of the way, then I am free.
I am on academic leave at NYUAD now, for about two years, with no obligations besides research.
Great! Will you be out there on campus much of the time?
Saadiyat island. The view from my window is across the desert to the Persian Gulf.
Have just been discussing with my colleague Dan Grady here the issue of boosting the pull-push quantization via stable homotopy types to pQFT. Maybe there is a way…
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