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With Giuseppe Malavolta we decided to write a rather exhaustive and concrete dictionary between the Physics parlance and current Maths jargon (this dictionary should become Giuseppe's master thesis). our idea was that since there are all those smart guys digging and freezing and beaming under Geneva, it was quite a scandal we had no precise idea of what they were doing there. the dictionary will be completely classic Maths, with no nPOV in it (yet, a few shadows of it may happen to slide in..), but we decided the nLab could be a nice place where to develop it after having bumped into this nCafe' blog entry. World is really little.. :-)
you can access the work in progress on the dictionary from my area in the nLab (at the moment it only contains a rough table of contents)
we decided the nLab could be anice place wher to develop it
That would be a magnificent addition to the nLab!
you can access the work in progress on the dictionary from my area in the nLab
here!
an entry of the name standard model of particle physics had been requested by several other entries. So I created it with just a link to the page on domenic’s personal web.
I added a few links and references to the standard model page. An (online) dictionary of mathematical and physical language would be very valuable - so valuable indeed that it may well deserve to live in the public domain :-)
thanks! and for sure we'll move it to the public domain at a later stage. but at the beginning it will be so unstable and full of errors that a little more protective ambient will not harm. it's like a little toddler, we have to wait till it can walk before letting it free :-)
Yes, I agree, it is sometimes good to develop an entry in a quiet area to move it into the light of stage later on.
Thanks to Tim for the additions to the public domain entry. I have expanded it a little further now: standard model of particle physics.
I further expanded the section on “Variations and generalizations”.
Hm, now the entry has more on variations and generalizations than on the thing itself…
Giuseppe wrote a first draft of the introduction.
Keep at it! To facilitate talking about it, it would be nice to have some subtitles, and numbers or names for the propositions and definitions etc.
A professor once drove me wild by numbering his theorems like 5.2.4.2.6 and using this in his lectures, like "this follows immediatly from 5.2.4.2.6 and 7.3.5 dash 5". Since then I prefer to try to name propositions :-)
I am very embarrassed, because I noticed that I know references and proofs of the proposition you cited (irreps of the product of groups are products of irreps of the factors), and need a reference for, for compact groups only, so if you find one that will be of interest to me, too.
One question about the point 4:
4. A theory of field with global symmetries has no actually physical meaning...
"Global symmetry" has different meanings, and obviously I'm thinking about one that is not alluded to here, because I do not understand that statement :-)
Tim, thanks for feedback! :)
Global symmetries: I guess what Giuseppe meant there is that on a spacetime $X=\mathbf{R}^{1,3}$ one usually has a Lagrangian $\mathcal{L}$ which is manifestly invariant under the action of a gauge group $G$ on the space of fields. this action is unrealistic from a physical perspective since it corresponds to acting on fields at a point $x\in X$ always by the same element $g\in G$, where acting “locally” with an element $g(x)$ depending on $x$ would be more realistic. in other wors one moves from the symmetry group $G$ to the (much larger) symmetry group $maps(X\to G)$. usually the original lagrangian does not have this larger symmetries, and one has to suitably make it invariant, e.g, moving from derivations to $\mathfrak{g}$-valued connections.
Global symmetries, if I dredge my physics memory, means that the bundle on which the gauge fields are connections, is trivial (not merely trivialisable): $E = X \times G$ where $X$ is Minkowski space.
It’s better to explain what is meant, since non-specialists are part of the intended audience :-)
An example of what domenico explained is this: Let’s take a Dirac field $\psi$ with electric charge e, the free Dirac Lagrangian is then
$\mathcal{L} = \overline{\psi} (i \gamma^{\mu} \partial_{\mu} + m) \psi$which is manifestly = clearly invariant under the transformation $\psi \to e^{-ie \alpha}$ and $\overline{\psi} \to e^{ie \alpha}$ with $\alpha$ a constant. Physicists would call this transformation a global gauge transformation, because the element of the gauge group does not depend on the basepoint = point in Minkowski space. If you let $\alpha$ depend on the basepoint, they call it a local gauge transformation. But the bundle is trivial in both cases.
Ah, you’re right. In that case, could be think of the ’realistic’ Lagrangian as being less rigid than the naive one? Also, this comment,
this action is unrealistic from a physical perspective
seems unmotivated to me, and I have studied QFT (albeit a long time ago, and I may have forgotten why this is considered unrealistic). Perhaps it is like this (this is off the top of my head, and with a head cold, so it may be completely uninformed rubbish):
The symmetries of the $G$-bundle preserving a given connection exactly is just $G$(???Not sure about this bit), but the symmetries of the $G$-bundle preserving the connection up to something unphysical is the gauge group ($Map(X,G)$, say).
Sorry, I’m not sure what “rigid” in conjunction with “Lagrangian” means.
My interpretation of “unphysical” in this context (referring to my previous post) is this: Gauge transformations should encompass all transformations of the mathematical description of the system that are not observable, or, rephrasing: that are redundancies of the mathematical model. We expect that only the relative phase of interacting systems is observable, but not their absolute phases (think of interference patterns). Therefore we should postulate that a physically realistic theory has local gauge transformations, not only global gauge transformations. And this postulate leads to a different Lagrangian than the one I wrote down in my previous post, namly to that of QED.
rigid meaning the group of symmetries is smaller. But this was just intuiting, and probably waffle.
Gauge transformations should encompass all transformations of the mathematical description of the system that are not observable
ah good. This is what I thought. I was trying to come up with a non-circular definition above, but this is better than mine. So we then have the fact that fields (observable fields) correspond to points in $\mathcal{A}/\mathcal{G}$ where $\mathcal{A}$ is the space of connections and $\mathcal{G}$ (sometimes $= Map(X,G)$) is the gauge group. It’s coming back to me…
it seems just too much asking for “global” transformations symmetries.
But these global symmetries do play some role. That’s called the theory of superselection sectors. For instance the entire celebrated Doplicher-Roberts theorem is about global symmetries of relativistic field theories on Minkowski space.
It’s true that there is much less interesting information in these global symmetries than in gauge symmetries. But they are still there.
Guiseppe wrote:
…since in minkowski there are points causally unrelated, it seems just too much asking for “global” transformations symmetries.
I’m not sure I understand what is meant by “it seems just too much asking for”. We have gauge transformations where
a) the element of the gauge group does not depend on the basepoint (global) and
c) the element does depend on the basepoint.
I left b) out for the argument about causality: If $\mathcal{O}$ and $\mathcal{O}'$ are spacelike separated, then we should be able to prescribe some sort of gauge transformation on $\mathcal{O}$, and independently one on $\mathcal{O}'$, is that your point?
If so, then the postulate of local gauge invariance is stronger: Consider a timelike curve, then we allow gauge transformations where the element of the gauge group varies along the curve. I think that (danger: handwaving) you cannot infer this from causality alone, you need as additional assumption that the absolute phase of a (localized) excitation of the electromagnetic (quantum) field is not observable.
Urs said:
But these global symmetries do play some role. That’s called the theory of superselection sectors.
I have to dig farther into this, but off the top of my head AQFT says: The Hilbert space of the whole theory is a direct sum of coherent subspaces $H_k$ or superselection sectors. Within each $H_k$ one has the unrestricted superposition principle whereas phase relations between state vectors belonging to different sectors are meaningless. Measurements resp. observables map each sector to itself.
Translated to the context of Guiseppe’s draft that would correspond to quantum numbers: Each $H_k$ represents all possible states with prescribed quantum numbers.
The Hilbert space of the whole theory is a direct sum of coherent subspaces $H_k$ or superselection sectors
Yes, that’s the decomposition of the Hilbert space into irreps of the global group that is acting.
Translated to the context of Guiseppe’s draft that would correspond to quantum numbers:
Yes, quantum numbers are another name for irreps of some group.
I think that’s what’s going on. But let me know if you think I am mixed up.
See Halvorson from page 55 on and Halvorson from page 83 on. Notice that the group he calls the gauge group is clearly the global gauge group.
I definitly have to re-read that paper, but all I had in mind here was the dictionary “full set of quantum numbers” = “irrep of global gauge group” = “superselection sector”.
@Giuseppe: fine. I’ll add my comments in the form of query boxes there.
@anyone else: let me recall that The standard model: a dictionary for the mathematician is Giuseppe’s master thesis. He’s writing it under my supervision, but he’s writing it directly on the nLab (in my area), where I make my comments and corrections. Clearly, anyone interested can correct or comment: this way of developing thesis work is meant to be an attemp to explore the potentiality of the Lab in this direction.
this way of developing thesis work is meant to be an attemp to explore the potentiality of the Lab in this direction.
I’m not sure how it will work out for Giuseppe, but I can already say that it is a great opportunity for me to check some things that I thought I already understood (since I dropped out of academia I don’t have the $\frac{obligation}{opportunity}$ to teach classes).
One organizational question: Is the master thesis supposed to contain some originally research of the author? (The German Diplomarbeit is/was supposed to, but it is usually accepted that the student writes a review of the topic if he/she did not come up with anything interesting).
If so, is there a problem if the final content is not attributable because it is not clear anymore who contributed what?
In Italy a master thesis can either contain original research or be a review. Giuseppe’s thesis will be a review.
Ok, good. I should add that (according to my personal, limited experience) usually the good students in Germany end up writing reviews instead of doing original research, because they go to the most prominent professors who assign tasks to them that are undoable.
I added my first comment :-)
In my experience, in Italy different research groups have different policies. Where I am, geometry and algebra master students are given review thesis, leaving original research for PhD students.
Thanks a lot for the comment! :-)
If Giuseppe’s thesis is in differential or algebraic geometry then my comment may be off topic, if the focus is on representation theory then the “rigorous” definition of quantum fields in the sense of AQFT should be ignored, I think. Quantum field = function from Minkowski space to an algebra (with a representation of the Poincare group) would suffice.
In my experience (which stems from several reasearch groups from three German universities) the professors would always assign some research project: To the good students (those they thought could succeed in academia) one that could lead to results that could be published, to the other ones easier ones (sometimes problems that they alredy solved themselves, so that the students were supposed to reconstruct what the professors already had done). I did my thesis in the research group of Franz Wegner in Heidelberg, and there writing a review thesis was considered to be the ultimate failure (I do not approve of this situation, but that was what I observed).
I hope this information will come in handy when Guiseppe notices some sceptical faces when telling German professors that his thesis was a review :-)
Well, let’s wait a bit how things develop. After all, a good deal of all original research springs from attempts to understand what people wrote up before you ;-)
Ok ;-)
I added the example of a neutral real (uncharged) scalar field from my comment as an example to the Wightman axioms (a little bit more detailed).
Tim,
can I maybe ask you for a favor? Do you feel like starting (a stub, maybe) on spin-statistics theorem?
I want to link to it from my book page, in the list of accomplishments of AQFT, but don’t feel I have the leisure to write something myself.
Just asking, since you wrote all these other nice entries on AQFT matters.
Ok, that is definitly on my list, the problem is: I do not know a simple explanation of it, all statements I know use some involved analysis using modular groups etc. Same for the PCT theorem. But I should be able to provide both a useful Idea and Reference paragraph. What is the deadline?
I know some simple explanations that I use with undergrads, but they may be overly simplified (though I try to not make them this way).
I’m all in for simple intuitive explanations, even for oversimplified ones: the question “can you see what is wrong about it?” can be more illumination than the best proofs, what I do not like is that physicists often “explain” things this way without pointing out that the explanation has weaknesses.
JB has written about it, too, “Spin, Statistics, CPT and All That Jazz”, can’t find the link right now.
What I meant was a mathematical proof using a set of axioms in the AQFT framework, “involved” means these use both some advanced mathematical machinery and other non-trivial results of AQFT. A self contained page seems out of reach.
Since the spin-statistics theorem that I would use comes from a paper of Daniele Guido and Roberto Longo (from a Festschrift for the 70th birthday of Hans-Jürgen Borchers), I took a look at the arXiv at their recent papers, and, who knows: Longo has published a joint paper with Edward Witten: An Algebraic Construction of Boundary Quantum Field Theory. I did not know that Witten is even remotly interested in AQFT :-)
Stubs for modular theory, Bisognano-Wichmann theorem and PCT theorem
Right. I think that was part of the problem with some of my early entries - they weren’t in the AQFT framework (which I’m not all that familiar with).
Ah no, I do not think that that is a problem. AQFT is only one little piece of the puzzle of creating a mathematical foundation for QFT (and String theory), see Mathematical Foundations of Quantum Field and Perturbative String Theory, and here the draft of the preface for an outline of Urs’ vision.
Credo:
There are physicists for whom mathematical physics is superfluous. There are physicists for whom mathematical physics tries to justify results obtained by the real physicists, and that’s all. And there are people like me who had an epiphany when reading the book “PCT, Spin, Statistics and all that”: So that’s what a QFT is, and I’ve been fascinated by AQFT since then. For me, mathematical physics provides the language that I need to think about physics.
Is that good or bad, is it a strength or a weakness? I don’t know, but I strive to learn more about AQFT not because I think it is the ultimate theory and solution for everything, a TOE or the most important subject that anyone could think about right now, but because it helps me to understand a little bit better what QFT is about. And that is part of the spirit of the nLab: Use a mathematically precise language for whatever you say, not as an empty excercise in rigour, but as a way to a better understanding.
an empty excercise in rigour
Certainly exercises in rigour are worth doing to the point that you’re confident that you could make something completely rigorous given enough time. Physics has not yet reached this point, so I disagree that such exercises in rigour are superfluous
I disagree that such exercises in rigour are superfluous.
Yes, thought so :-)
This depends of course on the subjective perspective. Jaques Distler had some interesting remarks about rigour in physics on his blog, his motive was his discussion of Rehren’s “counterproof” of the AdS/CFT correspondence. This discussion is only the tip of the iceberg of a cultural clash of the AQFT and the string communities, and part of the clash is the difference of perspectives with regard to rigour.
The whole AQFT project has been characterized by other physicists as having “contributed less than $\epsilon$ to particle physics”. Why? Because for these scientists the primary objective of QFT is/was to calculate numbers that can be compared with experiments, like cross sections, and AQFT can “only prove general theorems” but cannot calculate any such numbers (yet?). And some of those general theorems were even known before AQFT was invented, like the spin-statistics theorem and the PCT theorem (although “theorem” should be understood in the sense of theoretical physics).
Longo has published a joint paper with Edward Witten:
Yes, and apparently, from the looks of it, what happened was that Longo had insights on boudary CFT and Witten noticed that this serves to put some of his decade-old papers on string backgrounds on more solid footing.
@Tim: You make a very impassioned (and sensible) argument. I will have to add it to my already enormous list of things to get a handle on.
Regarding rigor (rigour, for Anglophiles) in physics, I agree to a degree, but I think that there are parts of physics that will defy all attempts to make them rigorous. I do not believe - and I would bet a large percentage of physicists would agree - that we will ultimately fail in our attempt to fully axiomatize physics. I truly do not think it is possible. It doesn’t mean we shouldn’t make it as rigorous as possible - we should. But it means that there will always be fuzzy, gray areas.
In which entry ?
here (still extremely rough)
After a very long pause, with Giuseppe we have taken up again the standard model project. A very introductory but fairly complete account on mass, spin, helicity has now been added here.
Infinite thanks to Andrew for the latex to itex conversion!
Infinite thanks to Andrew for the latex to itex conversion!
Ah, interesting. Is the whole page a conversion from LaTeX?
By the way, a bunch of formulas don’t display as math. Either one sees the dollar signs displayed, or they appear in “code-typesetting” or whatever that is called.
Urs, that section that Domenico links to is converted from LaTeX.
I’ll have to look through the history to see how much Domenico has changed after the conversion, but the errors that I saw on a quick scan through just now were due to two things:
Syntax like S_\mathrm{A}
. This is okay in LaTeX, but is dangerous. It’s much better to write S_{\mathrm{A}}
. The handling of _
and ^
in LaTeX is a bit special and though they seem like macros, they aren’t. But my script treats them as if they were macros so misses this special behaviour.
Nested mathematics and text. The command $\text{$x$}$
is invalid in iTeX. I’m not sure how best to handle this in my script, though, I’ll have to think about this.
But my script treats them as if they were macros
Not your script but iTeX itself, right? Your script shouldn’t have to do anything to that, a priori.
So both problems are limitations of iTeX that your script merely doesn’t (yet) have a way to fix.
No, it is my script. My script tries to emulate TeX and expand all macros. Those that are known to be iTeX macros seem to get left alone, but actually they get expanded to \noexpand\macroname
which avoids being expanded and gets passed on to the output routine. Subscripts and superscripts need to be expanded because what might seem to be a reasonable subscript to TeX might not be to iTeX and vice versa. So in my script, ^
is active, takes one argument, and expands to \^{#1}
(where \^
further expands to ^
but with catcode 12 (other) so no further expansion takes place). However, this isn’t actually how TeX does its superscripts (but it is if you load the mathtools package) so if you write a^\mathcal{A}
which is legal in both TeX and iTeX, then my script interprets that as a^{\mathcal{}}A
.
Ah!
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