Ah!

]]>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`

.

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.

]]>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.

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.

]]>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!

]]>here (still extremely rough)

]]>In which entry ?

]]>@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.

]]>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.

]]>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).

]]>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

]]>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.

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).

]]>Stubs for modular theory, Bisognano-Wichmann theorem and PCT theorem

]]>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 :-)

]]>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.

]]>Regarding the spin-statistics theorem, 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). ]]>

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?

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 ;-)

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).

]]>