This Week's Finds in Mathematical Physics (Week 158)
Robert C. Helling
> Lo and behold, this is what the Standard Model + gravity Lagrangian
> looks like: gravity is non-renormalizable, but very weak!
>
> In short, if spacetime is not infinitely subdivisible, I don't
> see why a nonrenormalizable theory must arise as the effective
> field theory associated to some more fundamental renormalizable
> theory.
O, fair enough. But this is not entirely different from the particle
physics point of view. Your theory with minimal distance scale oder
quantum geometry or whatever is nothing but the ULTIMATE REGULATOR I have
been using string theory as an example of. It might not be best to think
about it in these field theory terms but it would be a possible
description.
As others have pointed out, string theory is very much like such a minimal
distance scale theory. Our geometrically inspired pictures of strings
propagating in some background manifold are strictly only valid if all
distances and scales of the manifold are large compared to the length of
the string. If you come to the sub-stringy regime, the world look pretty
much different because for example strings (and branes and stuff) can wrap
around circles and do similar things.
As of today, our only control of this regime is in terms of conformal
field theory. Only in corners of the moduli space (interpreted as the
space time manifold becoming large) there exists a geometric (sigma
model) picture at all. After that, there is only CFT.
You can turn this around and use CFT to define what you mean by quantum
geometry. Michael Douglas has thought about this a lot and summarized some
of his thoughts in
hep-th/9901146 [abs, src, ps, other] :
Title: Two Lectures on D-Geometry and Noncommutative Geometry
Authors: Michael R. Douglas
Comments: LaTeX, 26 pp. Added reference
unfortunately, Mike's papers are often not too clear to read but they are
all very much worth it.
A very simple example of that is M(atrix)-Theory. I might elaborate on
this in some later posting.
But there is one (practical) problem with all non-renormalizable
effective theories: I assume you would expect the manifold picture to
break down somewhere around the Planck scale. For the time being, let us
ignore the details of your master theory and assume a simpler
regularization: We cut off all momenta at m_pl (and let me add that what
follows does not depend on how we regularize, we only have to assume that
the manifold+QFT picture is correct up to the Planck scale)
The problem is now that all the non-renormalizable couplings receive
perturbative loop-corrections at the order of the cut-off, i.e. m_pl or
some (possibly negative) power of it. If you are going to end up with
couplings at our energy range that are neither gigantic (we would have
noted those) nor un detectably small, i.e all of more or less unit
strength, then there is a need for an enormous amount of fine-tuning just
before the Planck scale. Your master theory would have to provide enormous
numbers fine tuned to many digits so that after flowing to the IR (our
regime) we would end up with reasonable couplings.
So the real question seems to be: Why do we notice gravity at all, given
that all numbers in the ultimate theory are of order 1? This is of course
a difficult question but it is probably due to the fact that gravity is
always attractive and cannot be shielded. But its weakness is probably due
to the fact that it is non-renormalizable.
> In the long run, the solution is to get more mathematicians
> interested in supergravity. They will chew away on those signs
> and complain bitterly until one day they will come up with an
> formalism that takes care of them automatically without any fuss.
> With any luck, lots of grungy calculations that involve "miraculous
> cancellations" will be replaced by proofs involving ideas from
> geometry - or supergeometry.
Superspace is some kind of a good start. There, supersymmetry can be kept
manifest at all stages of the calculation and the "miraculous
cancellations" are not miraculous at all but more or less obvious. The
problem is that it only really works for D=4 N=1. There is a version for
N=2, called harmonic superspace and propagated mainly but the Dubna people
but there again the formalism is horrible. For N=4 there is no superspace.
People have tried very hard to find it but it turned out that the
formalism always implied the _classical_ equations of motion which is not
a good start for a quantum theory.
Again, for higher dimensions, there is no good version of superspace even
though people have tried hard to find it. An example what this leads to is
the recent preprint
hep-th/0010167 [abs, src, ps, other] :
Title: Supersymmetric higher-derivative actions in ten and eleven
dimensions, the associated superalgebras and their formulation in
superspace
Authors: Kasper Peeters, Pierre Vanhove, Anders Westerberg
If you ask for a book on what is possible, the one by the Stony Brook
people
SUPERSPACE OR ONE THOUSAND AND ONE LESSONS IN SUPERSYMMETRY.
By S.J. Gates (MIT, LNS), M.T. Grisaru (Brandeis U.), M. Rocek (SUNY,
Stony Brook), W. Siegel (UC, Berkeley). 1983.
Reading, Usa: Benjamin/cummings ( 1983) 548 P. ( Frontiers In Physics,
58).
is the best on the market. Nevertheless, it is quite a project to work
thru it. Their equation (5.2.81) is the one I talked about in an earlier
posting (figuring out what +h.c. or better -h.c. is supposed to mean).
Still this is not very mathematical and/or geometric. But those authors
have done a lot of the famous susy calculations and this book contains
their combined knowledge on the subject. But their formalism looks more
like some effective rather than an elegant tool.
> But in the short run, there's a big problem even getting to the
> point where mathematicians can *understand* supergravity. If
> they can't even see those grungy calculations written down somewhere,
> how are they going to simplify them?
To my mind, another problem of this susy stuff is that it is a bit out of
fashion. Not many people do these kinds of calculations any more. Most of
the ones that are both worth doing and doable are done now more than 15
years ago by the real experts. What is surviving are mainly their results
like non-renormalization theorems. For people outside phenomenology, it
doesn't matter anymore what exactly the form and coefficients of some
higher loop scattering amplitudes are.
Modern uses of susy center more around things like the BPS property.
Nowadays, susy is used to have a control on what kinds of things are
protected from receiving corrections from quantization and to make
statements that are robust even at strong coupling.
Susy is some kind of a trick to make a quantum theory behave classical at
least as some properties are concerned. And there the consequences of
susy are well understood and even have some geometrical/mathematical
expressions. Let me mention two examples: If you what are the requirements
that string theory sugra on some space
R^n x M (M being a compact manifold)
has some susy remaining. The anser is that M has to be a Calabi-Yau
manifold. Now, you can use your knowledge about CY's to determine
properties of the IR physics in this setup. This is susy at work.
Another example is Seiberg-Witten theory: There one is in the situation of
having N=2 susy. General arguments tell you that this implies the (low
energy, i.e. not more than two derivatives) (effective) action to be given
entirely in terms of a holomorphic (better: meromorphic) function F on
some Riemann surface. This is an implication of susy (to be used without
horrible indices and/or sign errors). Then S&W go on and argue that the
Riemann surface has to be a sphere with three punctures with the action of
some (S-duality) discrete symmetry group on it that exchanges the
punctures. One puncture corresponds to a semi-classical weak-coupling
regime where perturbation theory is valid. There some of the old component
calculations (that you can look up and don't have to go thru yourself [*])
tell you what the singularity structure of F is around that puncture. Then
they use the symmetry and deduce the structure at the other singularities.
Finally, they employ complex calculus to extract the complete function by
the knowledge of its singularities.
This is how you can use susy without this terrible mess we have been
ranting about.
footnote [*]: Of course, it is always good to go thru calculations
yourself in case the others have made errors. But in this case, there is
an important difference: Usually it pays also to go thru others'
calculations because you learn their tools of the trade and you can employ
them yourself at future problems. But I doubt pretty much that this is the
case for the old susy component calculations: As I said, you will have a
hard time to find one that isn't done but can be done with some effort and
is worth doing it. But this is my personal opinion and it might be a bit
extreme tonight.
The point I am trying to make is that even inventing a much more elegant
formalism by staring at the old calculations for a very long time probably
doesn't enable you to do anything interesting that hasn't been done
before.
> To be blunt, I find this disgusting. Squirrelling away facts to gain
> advantage over others is exactly the opposite of true scholarship.
> A true scholar makes himself indispensable by explaining things so
> clearly that they become obvious to everybody.
I totally agree. But I think this is also very common among
mathematicians: The point is not that you write paper that cannot be
checked or found to be correct. It is more, that you just state your
results and not how (I mean heuristic) you obtained them. I have the
impression that this is considered to be considered good style in the math
community: Present a very short and elegant proof even if it hides
completely how you got the idea that this theorem might be true in the
first place.
Those index gymnastic papers are often like one way encryption: For
example it is relatively easy to check that some expression is
supersymmetric (i.e. a Lagrangian). But you need many more tools to adjust
all the possible coefficients to make it the case. Again, you can write a
very short elegant and even self contained paper even if your original
calculations filled hundreds of pages.
> Write an article
> explaining supergravity and all those sign conventions, and be sure,
> you will be cited - and respected!
I will not because I myself have never done almost all of those
calculations and myself I am missing the tools. And even more important:
As I said, I doubt too many people are interested in this. The ones that
are are the old guys. They all have their drawer or have just done all
this themselves. And younger people are not interested anymore because all
this is out of fashion. This is probably not worth spending a life-time
on.
Robert
--
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Robert C. Helling Institut fuer Physik
Humboldt-Universitaet zu Berlin
print "Just another Fon +49 30 2093 7964
stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling
John Baez
>John Baez wrote:
>> In the long run, the solution is to get more mathematicians
>> interested in supergravity. They will chew away on those signs
>> and complain bitterly until one day they will come up with an
>> formalism that takes care of them automatically without any fuss.
>> With any luck, lots of grungy calculations that involve "miraculous
>> cancellations" will be replaced by proofs involving ideas from
>> geometry - or supergeometry.
>Superspace is some kind of a good start. There, supersymmetry can be kept
>manifest at all stages of the calculation and the "miraculous
>cancellations" are not miraculous at all but more or less obvious.
Right, it's definitely a good start, and if the current superspace
formalism gave a good geometrical framework for everything that's going
on in supersymmetric physics, I would be happy. But alas, as you know
infinitely better than I, it does not!
So either 1) people must be missing some important ingredient or 2)
theories like 11d supergravity are freaks of nature with no elegant
explanation, understandable only through laborious calculation. Option
2) is simply too disgusting to be believable: I'm no believer in
supersymmetric physics, but at least *as mathematics* there's something
cool about it, so there must be some conceptual explanation of how
a lot of the calculations work - not just index-juggling.
>To my mind, another problem of this susy stuff is that it is a bit out of
>fashion.
That's good! I avoid working on stuff that's too fashionable, because
when something is in fashion, everyone feels the need to work in a
great rush, to beat everyone else. That makes it hard to take
the time to understand things properly. I like everything to be
completely obvious when I'm done with something - but that takes
time.
Theoretical physicists move along with fashions quite quickly,
so they get a lot done, but often leave a big mess behind in
their rush to get on to the next big thing. Mathematical
physicists are like scavengers: they come along when the
theoretical physicists think the picnic is over, and then
they pick the bones clean. When it's all nice and cleaned
up, the pure mathematicians come along and generalize the
heck out of everything, completely ignoring the original
context. This is nice, because then those generalizations
will be sitting there, nicely packaged, when the theoretical
physicists wander back into the vicinity again!
For example, K-theory is fashionable now among string theorists,
but it was developed by pure mathematicians as part of a long
tradition of algebraic topology, which started in part with
Poincare - who was a mathematical physicist. A few years
from now the string theorists will all be off doing something
else, leaving K-theory behind in the dust. But the subject
will continue its development, perhaps enriched by the string
theorists' brief visit.
>Usually it pays also to go thru others'
>calculations because you learn their tools of the trade and you can employ
>them yourself at future problems. But I doubt pretty much that this is the
>case for the old susy component calculations: As I said, you will have a
>hard time to find one that isn't done but can be done with some effort and
>is worth doing it.
Luckily, I don't want to do any new calculations: I would be satisfied
if I could understand the old ones! But even understanding the old
stuff seems to require some new math, not just the superspace formalism.
I'm focussing on 11d supergravity because I have a hunch it really should
be something very simple and pretty by the time it's properly formulated.
As you know - and are probably sick of hearing! - I hope that two of the
ingredients missing from the usual approach are 1) octonions and 2)
connections on n-gerbes. But who knows? It could require something
completely different.
Anyway, this is just one of my many hobbies - I have lots of old bones
I like to gnaw on, and occasionally one cracks.
Penny55113
What I want is simple. I want undergraduate analysis for anti-commuting
variables. Things like the mean value theorem, a good fundamential theorem of
calculus etc. Then, we could do analysis of the supergravity equations similar
to what we have done with Yang-Mills.
I have a few ideas along these lines. There was some work on the
integration issues in the eighties.
So far, the antisymmetric theory seems dominated by algebraic theories.
pennysmith
John Baez
Penny55113 <penny...@aol.com> wrote:
>What I want is simple. I want undergraduate analysis for anti-commuting
>variables. Things like the mean value theorem, a good fundamental theorem of
>calculus etc.
Have you looked at Bryce DeWitt's book "Supermanifolds", published
by Cambridge U. Press in 1992? It does a bunch of basic supercalculus.
It may not fully satisfy you - it's a bit klunky in some ways - but it's
worth a look.
Also there is Feliz Berezin's book "Introduction to Superanalysis",
published by D. Reidel (the super-expensive ripoff press) in 1987.
I must admit that I haven't looked close at this one, even though
Berezin is the one of the folks who invented this stuff.
>Then, we could do analysis of the supergravity equations similar
>to what we have done with Yang-Mills.
Hmm - isn't the big problem the difficulty with finding a formulation
of higher-dimensional supergravity theories? I.e. one in which
supersymmetry is manifest even "off-shell" - before imposing the
equations of motion?
Mind you, I don't understand this stuff at all. I'm just parrotting
what people have told me.
By the way, Yvonne Choquet-Bruhat has proved local existence and
uniqueness theorems for the Lorentzian version of various supergravity
theories. But I bet you're talking more about the Riemannian version,
which could have cool relationships to topology.
> I have a few ideas along these lines. There was some work on the
>integration issues in the eighties.
> So far, the antisymmetric theory seems dominated by algebraic theories.
Well, analysis for odd variables seems inherently a lot more algebraic
than for even ones, since they are nilpotent, so all power series
converge, and all functions are analytic. Whoopee! This is just
what algebraists wish analysis was like.
squ...@my-deja.com
penny...@aol.com (Penny55113) wrote:
> dear john,
> What I want is simple. I want undergraduate analysis for anti-
> commuting variables.
You might be intrested in de Witt's "Supermanifolds". He is using
approach in a sense along these lines. I.e. he constructs a specific
algebra of supernumbers, rather than working with only "supervariables"
which is a non-commutative geometry style approach.
Best regards, squark.
Sent via Deja.com http://www.deja.com/
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