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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.
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