Lie algebra cohomology (fwd)
A.J. Tolland
I hope you won't delete this letter before reading it. It is not
a thoughtless flame of the sort you seem to have encountered. It's not
even strictly a defense of string theory. It does however contain a
response to some of the points raised in your paper.
I actually did read your paper. I have not really responded to it
yet, however. My comments on sci.physic.research should be taken in
context. If you read the entire post, which I've included at the end of
this note, you may notice that I was actually aiming the barb at Thomas
Larsson. Mr. Larsson has lately been espousing some ideas on s.p.r. about
gauge theory which look to me like a Lie algebraic version of numerology.
I'll admit that this was incivil and lousy form for scientific discussion
(and perhaps badly delivered), but I was tired of hearing him repeat
himself, and I took an opportunity to flame him.
[I owe you an apology for the incivility, Mr. Larsson. Sorry.]
I agree with you completely that string theory should not be
taught to high schoolers. They need to be learning why the sun shines,
and why putty tears and hardens. I don't see anything particularly wrong
with talking to science reporters, provided one doesn't make claims that
one can't back up. As for funding agencies... well, I wasn't aware that
string theorists got any funding... (sigh)
I should add that I don't share your notion that theoretical
physicists should be inspired by the common importance of fiber bundles
and the Dirac operator in phys and math. There's a historical argument
for this: the concepts relevant to developing physics have not often been
well-established already within mathematics. (It is called the Dirac
operator, after all...) However, there are also decent physical arguments
for doubting that gauge theories are the central structures in high energy
physics. First of all, we have fairly good indications that _any_ high
energy theory is forced to look like a gauge theory at low energy. This
pseudo-fact is reflected in the fact that gauge theory, for all of its
mathematical beauty, is an extremely arbitrary structure. It takes an
awful lot of data, and the consistent (i.e. anomaly free) theories tend to
be saved by weird cancellations. Trying to figure out a deeper theory by
studying mainly the Standard Model seems as inefficient as trying to guess
the laws of atomic physics from fluid dynamics. Since Distler seemed like
he wasn't going to bother responding to you, let me speculate that this is
what he meant when he called the Standard Model an "ugly hack".
The question of whether or not modern, non-perturbative studies of
string theory will lead to exciting mathematics is partly a matter of
taste. But let me say that I suspect that string theory is pointing
towards a new kind of geometry. The physical question here is "What does
a (possibly topologically non-trivial) spacetime look like to extended
objects?" It seems likely to me that even on the classical level, we are
likely be dealing with objects best realized as some deformations of fiber
bundles. (I should remark here that I suspect Lie group symmetries &
principal bundles would emerge from this framework only in a degenerate
limit.) I'm also willing to bet that the mathematical framework for the
quantum theory will be an even more interesting. I'll go out on a limb,
and guess that this framework, whatever it is, will bear approximately the
same relationship to algebraic geometry that functional analysis does to
linear algebra. I suppose you're annoyed by all the speculative
terminology here. Sorry. Ask me again when I become a graduate student.
Enough of this. I want to address what I see as another flaw in
your criticism of string theory. You complain of strings that "[a]ll that
exists at the moment is a divergent series that is conjectured to be an
asymptotic perturbation series for some as yet undefined non-perturbative
string theory." Let me remind you that quantum field theory itself is in
_exactly the same state_. Yes, people have proposed axioms, but these
axioms have zero contact with realistic theory (meaning Standard Model
class theories). For God's sake, there's a million dollar prize just to
prove the existence of a mass gap!
The situation is roughly as follows. We can think of a quantum
theory (of fields, strings, or whatever) as a map from the space of
physical parameters to the set of Hilbert spaces. I'm here cheerfully
ignoring a bunch of subtleties involving inequivalent representations of
commutation relations and the effects of radiative corrections on the
structure of the parameter space. But bear with me; we give this function
a set of physical parameters and it gives us back a state space and an
inner product with which to calculate correlation functions.
Unfortunately, we physicists have absolutely no idea how to calculate this
map in general. All we have at the moment is Feynman's functional
integral formalism, which gives us a recipe for computing correlation
functions directly. Unfortunately, this program really only works when we
have a small parameter to organize the superpositions with. This
perspective is unfortunately limited; we have to work very hard to see
non-perturbative phenomena. It's like standing with a torch at the edge
of an ocean of night. (Apologies to Newton for borrowing the ocean
metaphor.)
The perturbative string theory formalism, as I said, is
essentially the same situation. Old string theory, perturbative string
theory, the string theory of the 1980's, gives us a Feynman-style sum over
intermediate states recipe for computing correlation functions in the weak
coupling limit. To continue the ocean metaphor, until the early 1990's,
we were basically stuck on the shore, wondering what was out there at
larger coupling. We still don't really know, but there have been a few
discoveries: The five perturbative string theory shores may really
be shores of the same ocean. This ocean has other shores as well, such as
11D SUGRA and M(atrix) theory. There also appear to be a number of
shallow areas (BPS states) and a remarkable collection of islands
(non-commutative theory, the Little String Theories, etc,...).
At any rate, M-theory (or modern string theory, as I prefer) is
the study of this ocean.
Sorry if I've been lecturing or talking down to you. No offense
intended; I speak about this stuff as well as I'm able.
You said I few other things which probably should be responded to.
I've ranted for a while here, so I'll try to keep it brief.
First off, string theory is rather interesting and elegant. If
you put some serious effort into understanding the project you'll see what
I mean. I don't claim that the theory is formulated with nice math
concepts or even correct, but it exhibits many tantalizing (if enigmatic)
features. See for instance, the way that strings use their own length
scale to suppress the higher spin parts of their associated particle
spectrum. (Compare this to the phenomenal difficulties in studying higher
spin objects in conventional point particle theory.) Or the way that the
quantum theory breaks down if the background fields do not obey realistic
equations of motion. I think most of the physicists who stick with string
theory probably stick with it because of this weird consistency. At any
rate, this is what I mean when I claim that it "feels right."
2nd, in my 4 years at the U of C, I have _never_ heard a string
theorist make wild claims about the correctness of the theory. Maybe this
sort of thing happened a lot in the 80's. I don't know; it was before my
time. I have heard a number of string theory's critics accuse them of
this, but it seems to me that these critics are the only ones saying
anything of the sort. Perhaps string theorists merit the stereotype for
daring to think about such difficult topics? I don't feel that way
myself.
3rd, you claim that string theory predicts the cosmological
constant incorrectly. Please cite a source. I want to see if I trust the
calculation.
Finally, you accuse the majority of string theory researchers of
being fad-followers, unoriginal researchers. Then you say that faculty
positions are scarce. I might be wrong, but I was under the impression
that faculty positions are so hard to come by that you have no chance at
one if your research is not original and interesting. Perhaps my
experience here in Chicago has been limiting?
My apologies for the length of this note.
--A.J.
ps. Remember the 2nd paragraph? Here's the quoted message
---------- Forwarded message ----------
Date: Sun, 25 Feb 2001 21:23:51 -0600
From: A.J. Tolland <a...@hep.uchicago.edu>
Newsgroups: sci.physics.research
Subject: Re: Lie algebra cohomology
On Thu, 22 Feb 2001, Thomas Larsson wrote:
> I am also somewhat annoyed that he uses the phrase "not even wrong".
> I had saved that formulation for a rainy day.
Just so you know, "not even wrong" was one of Wolfgang Pauli's
signature phrases. I leave it up to you to decide whether Woit's use of
this phrase in discussing his own opinions smacks of arrogance.
--A.J.
A.J. Tolland
been shown to be a suitable definitiion for a quantum field theory.
Prof. Woit pointed out in an email that the continuum limit of
lattice gauge theory provides an adequate definition for one special class
of asymptotically free theories. Since I'm sure that a least of few of
s.p.r.'s readers were also thinking this, I'd like to clarify my argument.
Please note that I am not retracting it.
Lattice gauge theory does seem to an adequate definition for
asymptotically free theories. I want to suggest, however, that this is a
situation analagous to the one we have in some lower dimensional field
theories where perturbation theory captures enough information to define
the theory. (This is the case for phi-4 theory in D<4 and probably for
a lot of other things like the large N Gross-Neveau model.) What I'm
saying is that the path integral formalism is a calculational recipe which
just happens to work as a definition for a certain class of theories.
As far as I know, lattice gauge only works as a definition for
asymptotically free theories. I don't see how it might be made into a
useful definition for non-renormalizable interactions, a class which seems
to include gravity. You can claim avoid this problem by saying that the
fundamental theory will only include asymptotically free theories. This
may be true, but it seems unlikely to me: I just don't see why the
fundamental theory should be effectively free at high energy. In
particular, I don't think gravity is going to fall into this class of
interactions, because it seems to couple positively and universally to
energy. I suppose one could argue that particles at high energy ought to
be free; how else would they scatter? However, the problem of defining a
good notion of particle in QFT on curved spaces makes me suspect that
particle arguments are untrustworthy.
Well, I've been reduced to responding to my own posts. Madness
can't be far away. :)
--A.J.
Toby Bartels
And long time readers of this newsgroup know that
John Baez and his Acolytes like to make fun of string theory too.
But the string theorists on this group seem to be trying to make nice.
I'd like to give people an idea of where we nonstring theorists
get our ideas about how evil (tongue in cheek) string theory is.
A few weeks ago, Ed Witten gave a talk here at UC Riverside.
In it, he called string theory "the only theory of quantum gravity we have".
Asked after the talk about gr-qc/0005126, in which Ashtekar, Baez, and Krasnov
derive the Bekenstein Hawking formula using loop quantum gravity,
Witten replied that the paper was "wishful thinking" and
"made no calculation in any theory". Well, his assessment is correct!
However, the same thing can be said of any paper on M theory, can it not?
Quantum gravity students at UCR described Witten's attitude as "arrogance".
And can you blame them? He was under no obligation to discuss
alternatives to string theory; that string theory was the only option
was not the point of his talk; he didn't have to like loop quantum gravity.
But to specifically state that string theory was the only game in town,
to deride other approaches for failures that M theory shared,
smacked of arrogance.
The string theorists on this group don't seem to share that arrogance.
Unfortunately, they make less of an impression on the public than Witten does.
Witten's attitude doesn't justify broad attacks on M theory;
but I hope people can see why other quantum gravity theorists
have a tendency to be defensive.
-- Toby
to...@math.ucr.edu
A.J. Tolland
> OK, there seems to be no love lost between M theorists and Woit. And
> long time readers of this newsgroup know that John Baez and his
> Acolytes like to make fun of string theory too. But the string
> theorists on this group seem to be trying to make nice. I'd like to
> give people an idea of where we nonstring theorists get our ideas
> about how evil (tongue in cheek) string theory is.
Yes, but we string theory types tolerate the taunts of Baez et al
because we know that they are hard at work on their own theory. :) To my
mind, this makes their comments much more valuable than those of Woit,
whose major suggestion seems to have been "Stop working on string theory."
More generally, it seems to me that both theories are comically
incomplete. The string program knows very little about its theory outside
certain limits, and the canonical program, so far as I know, has the same
problem, just with different limits. There's a long tradition in physics
of borrowing concepts from one branch of theory for application in
another, so it puzzles me that there's not more interaction between the
two programs. (Of course, there are a few people who try to bridge the
gap. But they're mainly on the canonical side, and, if I may say so, I
think their critiques of the canonical program are rather less rigorous
than their critiques of the string program.)
Maybe the problem is partly one of language. String theory is
highly oriented towards Lagrangian/sum over states techniques, and its
mathematical language tends to be very geometrical. Canonical quantum
gravity, by contrast, uses primarily Hamiltonian techniques and its
language is heavily algebraic.
I see some signs that this may be changing. People are starting
to explore the nature of stringy geometry by looking at how normal notions
of spacetime geometry break down, studying non-commutativity, mirror
symmetry, and the like. There's obviously a lot of ground yet to be
covered, but since the natural language for discussing these topics is
algebra, perhaps eventually high powered algebra will work its way into
the basic string theory vocabulary. (If nothing else, I'm probably going
to be at Rutgers next year studying this stuff, and I sure as heck intend
to make it a part of _my_ vocabulary.)
As for Witten's comments, I really don't know what to say. I
wasn't under the impression that we had _any_ complete theory of quantum
gravity (except maybe the Einstein-Hilbert Lagrangian with a cutoff). I
personally find strings more interesting than the canonical program, but I
won't fault you guys for calling his remarks arrogant.
> he didn't have to like loop quantum gravity. But to specifically state
> that string theory was the only game in town, to deride other
> approaches for failures that M theory shared, smacked of arrogance.
--------------------
Dang you! I had been saving that phrase for a rainy day! ;)
--A.J.
John Baez
>A few weeks ago, Ed Witten gave a talk here at UC Riverside.
>In it, he called string theory "the only theory of quantum gravity we have".
>Asked after the talk about gr-qc/0005126, in which Ashtekar, Baez, and Krasnov
>derive the Bekenstein Hawking formula using loop quantum gravity,
>Witten replied that the paper was "wishful thinking" and
>"made no calculation in any theory".
By the way, just so people can properly imagine this scene: I wasn't
there - I was off giving a talk in Milwaukee. I had scheduled the
talk months before and was disappointed to miss Witten's talk.
>Well, his assessment is correct!
I'll deal with *you* later.
A. J. Tolland wrote:
>More generally, it seems to me that both theories are comically
>incomplete. The string program knows very little about its theory outside
>certain limits, and the canonical program, so far as I know, has the same
>problem, just with different limits. There's a long tradition in physics
>of borrowing concepts from one branch of theory for application in
>another, so it puzzles me that there's not more interaction between the
>two programs.
The main problem is that each side has gotten used to its own shortcomings,
but is very sensitive to the shortcomings of the other.
For people working on loop quantum gravity, the idea of doing quantum
gravity in a background-dependent framework seems like a terrible
step back from the insights of general relativity. For people working
on string theory, the idea of being unable to compute dynamics seems
equally terrible. Of course, these are just two sides of the same problem:
nobody is good at understanding dynamics in background-free theories!
For people working on string theory, the loop quantum gravity crowd
seem woefully naive when they try to quantize gravity without matter.
To the loop quantum gravity people, the string theorists' approach
to including matter seems absurdly risky, given that until someone
figures out how to spontaneously break supersymmetry, it's predictions
are quite unphysical.
Supersymmetry: it seems people love it or hate it! String theory
is hard to love if you don't love supersymmetry. The string
theorists can precisely compute the entropy of a black hole...
but only in the supersymmetric context. They can prove confinement in
Yang-Mills theory... err, make that super-Yang-Mills. And so on. So
either supersymmetry is great, or it's a great example of how a false
assumption can simplify life.
Anyway, these are some reasons why it's tough for the two sides to
talk. But it's worthwhile to try.
By the way, speaking of partisan bickering, there may be an article
in the science section of the New York Times next Tuesday about string
theory and its doubters. I got a call from the reporter James Glanz
today, and he interviewed me about this. He had previously interviewed
Penrose, who is very outspoken in his opposition to string theory, and
he wanted a second opinion. I tried to say sensible things. I wonder
how they'll look after they've gone through the journalistic mill.
Ralph E. Frost
Toby Bartels <to...@math.ucr.edu> wrote in message
news:9829pn$c3q$1...@glue.ucr.edu...
[Moderator's note: Vast quantities of quoted text deleted. -TB]
As for Witten statement saying string theory is "the only theory of
quantum gravity we have", didn't Gellman say about the same thing in _The
Quark and the Jaguar_ where he said, in effect, if the string theorists
don't pull a rabbit out of their hat, everything else is up for grabs, too."
(Or something like that.)
As for a proponent of a theory being assertive about his position, why do
refer to that as "arrogant"? It seems to me you are just talking about
someone being "overly speculative". No?
--
Ralph E. Frost
Frost Low Energy Physics
http://www.dcwi.com/~refrost/index.htm Phase II
Imagine you needed to create knowledge. What one science education tool
would you use?
Aaron Bergman
<Pine.SGI.4.33.010307...@hep.uchicago.edu>,
"A.J. Tolland" <a...@hep.uchicago.edu> wrote:
> I see some signs that this may be changing. People are starting
> to explore the nature of stringy geometry by looking at how normal notions
> of spacetime geometry break down, studying non-commutativity, mirror
> symmetry, and the like.
Did you see Banks and Fischler's paper of a week or two ago?
> There's obviously a lot of ground yet to be
> covered, but since the natural language for discussing these topics is
> algebra, perhaps eventually high powered algebra will work its way into
> the basic string theory vocabulary. (If nothing else, I'm probably going
> to be at Rutgers next year studying this stuff, and I sure as heck intend
> to make it a part of _my_ vocabulary.)
Good luck with the ol' derived categories.
Aaron
--
Aaron Bergman
<http://www.princeton.edu/~abergman/>
Aaron Bergman
One more thing.
>A few weeks ago, Ed Witten gave a talk here at UC Riverside.
>In it, he called string theory "the only theory of quantum gravity we have".
>Asked after the talk about gr-qc/0005126, in which Ashtekar, Baez, and Krasnov
>derive the Bekenstein Hawking formula using loop quantum gravity,
>Witten replied that the paper was "wishful thinking" and
>"made no calculation in any theory". Well, his assessment is correct!
>However, the same thing can be said of any paper on M theory, can it not?
Actually, I would say sort of no. AdS/CFT almost qualifies as a
theory of quantum gravity in asymptotically Anti-de Sitter
backgrounds. The black hole entropy calculations (Strominger and
Vafa and so forth) are actual calculations done in a theory that might
not exist. I don't know what the situation is in cqg, so I won't
comment on it.
Aaron Bergman
I wouldn't say that. In fact, the people I hang out with around
the deparment were mostly amused by it.
>And long time readers of this newsgroup know that
>John Baez and his Acolytes like to make fun of string theory too.
I was on this group before I was a string theorist, so it doesn't
really bother me all that much. You should have heard some of the
questions I asked possible advisers as I was going around looking
for a project to do. The thing was that most of them actually had
considered the issues that string theorists are so often accused
of not considering. And most of them even had coherent responses.
As suchm I do get somewhat annoyed when the mocking seems to be more
directed as straw men rather than what actual string theorists think,
but it's not that big of a deal. For all the complaints of a lack of
background independence in string theory, I would think that the lack
of a classical limit is atleast as bad. Which is just to say that
both programs have their flaws, and neither is really at the point
where it can be called a an honest-to-Popper (or whatever) theory.
I'm not going to comment on other people's attitudes too much as,
well, they're their attitudes and not mine. I will say that
Ashtekar is giving a talk at the institute in May entitled
something like "Background Independent Quantum Geometry and Its
Applications", and I'm really curious as to how well it's going
to go. I've heard stories about Rovelli's talk a number of years
back, and supposedly that one didn't go over very well....
Toby Bartels
>Toby Bartels wrote:
>>Witten replied that the paper was "wishful thinking" and
>>"made no calculation in any theory". Well, his assessment is correct!
>>However, the same thing can be said of any paper on M theory, can it not?
>Actually, I would say sort of no.
Yes, you can say sort of no. But, when John was finished "dealing with me",
I'll have a fresh understanding of why you can say sort of no in his case too.
I'd rather give Witten the benefit of the doubt concerning
the simple truth of the statements he made about John's paper,
but then I have to give the same benefit of the doubt to anyone
wishing to make the same statements about the stringy
"calculations in a theory that might not exist".
In summary: You can have it either way, but you can't have it both ways.
-- Toby
to...@math.ucr.edu
A.J. Tolland
> For people working on string theory, the loop quantum gravity crowd
> seem woefully naive when they try to quantize gravity without matter.
Myself, I can wait on the matter. What weirds me out about loop
gravity is that I can't tell whether or not it's dealing with those higher
powers of the curvature that I've been ranting about lately... It seems to
me that that's where the real fun of quantum gravity is going to show up.
(Of course, one must accept the existence of a map taking "recalcitrant
nonlinear quantum theory" into "fun.")
Can you shed any light on this? I'd appreciate a reference.
--A.J.
John Baez
Aaron Bergman <aber...@Princeton.EDU> wrote:
> "A.J. Tolland" <a...@hep.uchicago.edu> wrote:
>> [...] perhaps eventually high powered algebra will work its way into
>> the basic string theory vocabulary. (If nothing else, I'm probably going
>> to be at Rutgers next year studying this stuff, and I sure as heck intend
>> to make it a part of _my_ vocabulary.)
>Good luck with the ol' derived categories.
Eh? So string theorists are using derived categories now?
That might give me an excuse to talk about them here! The
usual way of explaining them is (in my humble opinion) really
bad when it comes to conveying their "deep inner meaning".
They easily come across as a technical trick, but they're not.
They're really very nice.
To be relevant to this thread, I should include some partisan
bickering in this article. So:
My prediction: like a pyramid scheme, string theory will
collapse when it has used up all known branches of mathematics.
John Baez
Aaron Bergman <aber...@princeton.edu> wrote:
>In article <9829pn$c3q$1...@glue.ucr.edu>, Toby Bartels wrote:
>>[...] long time readers of this newsgroup know that
>>John Baez and his Acolytes like to make fun of string theory too.
>I was on this group before I was a string theorist, so it doesn't
>really bother me all that much. You should have heard some of the
>questions I asked possible advisers as I was going around looking
>for a project to do. The thing was that most of them actually had
>considered the issues that string theorists are so often accused
>of not considering. And most of them even had coherent responses.
Well, that's good. Most of my puny knowledge of string theory
comes from books, papers, and going to talks now and then, rather than
hanging out with string theorists and asking them lots of questions.
This is obviously not the best way to find out all the issues
some group of people has thought about... especially the problematic
issues where no neat answer has yet been found.
>I'm not going to comment on other people's attitudes too much as,
>well, they're their attitudes and not mine. I will say that
>Ashtekar is giving a talk at the institute in May entitled
>something like "Background Independent Quantum Geometry and Its
>Applications", and I'm really curious as to how well it's going
>to go.
How well it's going to go, eh? Sounds a bit like a gladiator
thrown into the lion pit... "How long will he last?" Seriously,
it takes guts to walk into a place full of top experts on theory X
and give a talk on theory not(X). I'm sure glad I'm not doing it!
>I've heard stories about Rovelli's talk a number of years
>back, and supposedly that one didn't go over very well....
Well, as usual with these political issues, it depends who you talk to.
The string theorists probably thought it sucked. The loop quantum
gravity people I know who attended those talks thought it was great -
"he really nailed those string theorists," etcetera. When people
are attached to a particular program, they usually see what they
expect to see. It's just like those after-debate "analyses" where the
Republican says the Republican won and the Democrat says the Democrat
won.
Doug Merritt
> Asked after the talk about gr-qc/0005126, in which Ashtekar, Baez, and Krasnov
> derive the Bekenstein Hawking formula using loop quantum gravity,
> Witten replied that the paper was "wishful thinking" and
> "made no calculation in any theory". Well, his assessment is correct!
Before Master Wizard Baez smites you, as he promised, and your smoldering
ruins are no longer able to inform us, would you please explain what you
meant when you agreed with his assessment?
(John, please withold the Smiting until he answers. :-)
> But to specifically state that string theory was the only game in town,
> to deride other approaches for failures that M theory shared,
> smacked of arrogance.
The history of physics (and every other field of human endeavor) is full
of instances like this; that's just human nature. Sometimes the arrogant
are right, and sometimes they're wrong. Arrogance is annoying and often
insulting, but isn't the *point*, of course.
And note that, since essentially everyone agrees that Witten is one hell
of a smart SOB, you could also take his arrogance as a sign of weakness.
If he can't dismiss alternate camps via purely intellectual means, and
has to resort to emotional means, then he must feel that his camp is
the weak one! (Or at least seriously threatened.)
> but I hope people can see why other quantum gravity theorists
> have a tendency to be defensive.
The minority has reason to be defensive, even if no one is arrogant, for
that matter. That goes with the territory. (Or the losing majority camp,
perhaps, in this case. Or both. Defensiveness is another part of the
human condition. Territoriality. Etc.)
Doug
--
Professional Wild-Eyed Visionary Member, Crusaders for a Better Tomorrow
A.J. Tolland
> In article <abergman-3077C1...@cnn.princeton.edu>,
> Aaron Bergman <aber...@Princeton.EDU> wrote:
> >Good luck with the ol' derived categories.
> Eh? So string theorists are using derived categories now?
Yep. Although not very many of them. The first thing that comes
to mind is hep-th/0011017, Michael Douglas's "D-Branes, Categories, and
N=1 Supersymmetry." I'll quote from the abstract: "We show that the
boundary conditions in topological open string field theory on Calabi-Yau
manifolds are objects in the derived category of coherent sheaves..."
Personally I'm looking forward to the "longer work with more
introductory discussion" he mentions in the introduction.
> That might give me an excuse to talk about them here! The
> usual way of explaining them is (in my humble opinion) really
> bad when it comes to conveying their "deep inner meaning".
> They easily come across as a technical trick, but they're not.
> They're really very nice.
I am _extremely_ interested in hearing this explanation.
> To be relevant to this thread, I should include some partisan
> bickering in this article. So:
>
> My prediction: like a pyramid scheme, string theory will
> collapse when it has used up all known branches of mathematics.
Nah, it'll just go into remission until someone invents
more math. :)
Immirizi parameter. Cough, cough.
More seriously: String theorists may get all excited about the
wondrous potential mathematics of stringy geometry, but you can't really
expect them to just invent new math when they could instead learn more
about the physics by exploring limits which can be described by known
mathematics.
--A.J.
Aaron Bergman
ba...@galaxy.ucr.edu (John Baez) wrote:
> In article <slrn9afes4....@cardinal0.Stanford.EDU>,
> Aaron Bergman <aber...@princeton.edu> wrote:
>
> >I'm not going to comment on other people's attitudes too much as,
> >well, they're their attitudes and not mine. I will say that
> >Ashtekar is giving a talk at the institute in May entitled
> >something like "Background Independent Quantum Geometry and Its
> >Applications", and I'm really curious as to how well it's going
> >to go.
>
> How well it's going to go, eh? Sounds a bit like a gladiator
> thrown into the lion pit... "How long will he last?"
Well, sure, but that's true for everyone here. Talks at Princeton
and the Institute can be quite fun. I say this having watched a
fair sized hole being torn in the paper that I'm currently
working on with my advisers and some other people.
> Seriously,
> it takes guts to walk into a place full of top experts on theory X
> and give a talk on theory not(X). I'm sure glad I'm not doing it!
I think it takes guts just to give a talk here.
> >I've heard stories about Rovelli's talk a number of years
> >back, and supposedly that one didn't go over very well....
>
> Well, as usual with these political issues, it depends who you talk to.
> The string theorists probably thought it sucked. The loop quantum
> gravity people I know who attended those talks thought it was great -
> "he really nailed those string theorists," etcetera. When people
> are attached to a particular program, they usually see what they
> expect to see.
Let's say, rather, that if he hoped to convince the string
theorists that there might be something to all this loopy stuff,
then he didn't succeed. And that's what talks are supposed to be
about, really. If all he wanted to do was "nail those string
theorists", it's no surprise it didn't go over well.
ba...@galaxy.ucr.edu
A.J. Tolland <a...@hep.uchicago.edu> wrote:
>What weirds me out about loop
>gravity is that I can't tell whether or not it's dealing with those higher
>powers of the curvature that I've been ranting about lately... It seems to
>me that that's where the real fun of quantum gravity is going to show up.
Fun. Hmm.
>(Of course, one must accept the existence of a map taking "recalcitrant
>nonlinear quantum theory" into "fun.")
Right.
>Can you shed any light on this? I'd appreciate a reference.
Well, here's all I can say. It's about spin foams rather than loop
quantum gravity per se. In the Barrett-Crane model the basic ingredient
is the amplitude for a 4-simplex with triangles labelled by spins (which
describe the areas of these triangles). In the limit of large spins
this amplitude is basically asymptotic to the exponential of the Regge
action - a discretized version of the good old Einstein-Hilbert action.
But for small spins there will be discrepancies. One should be able to
interpret these discrepancies as terms involving higher powers of R and
other curvature invariants - akin to the stringy corrections to the
Einstein-Hilbert action. However, I don't think anyone has actually
done this! Indeed, even in the case of 3d quantum gravity the rigorous
proof of the asymptotic formula came only in 1998:
Classical 6j-symbols and the tetrahedron
Justin Roberts
http://xxx.lanl.gov/abs/math-ph/9812013
though it was conjectured 30 years earlier:
G. Ponzano and T. Regge: in Bloch, F. (ed.),
Spectroscopic and Group Theoretical Methods in Physics,
Amsterdam: North-Holland 1968.
Of course, most of the delay was due to lack of interest in the
issue until recently! If we threw rigor to the winds and simply
wanted to get a plausible calculation, I'm sure somebody could start
with Barrett's ideas in the 4d case:
The asymptotics of an amplitude for the 4-simplex
John W. Barrett
http://xxx.lanl.gov/abs/gr-qc/9809032
and work out the higher-order terms. Personally, I am too
lazy to attempt this sort of calculation... it's not my sort of thing.
But if you think it's fun....
Aaron Bergman
>In article <abergman-3077C1...@cnn.princeton.edu>,
>Aaron Bergman <aber...@Princeton.EDU> wrote:
>> "A.J. Tolland" <a...@hep.uchicago.edu> wrote:
>>> [...] perhaps eventually high powered algebra will work its way into
>>> the basic string theory vocabulary. (If nothing else, I'm probably going
>>> to be at Rutgers next year studying this stuff, and I sure as heck intend
>>> to make it a part of _my_ vocabulary.)
>>Good luck with the ol' derived categories.
>Eh? So string theorists are using derived categories now?
I'd be surprised if there was a branch of math that doesn't have
some string theorist or another looking at it. I know next to
nothing (read: nothing) about derived categories, but, as I
understand it, Kontesevich said some deep stuff about mirror
symmetry using them, and, as mirror symmetry is deeply related to
(or just plain is) string theory on Calabi-Yaus, it sort of makes
sense that people would start thinking about 'em. People, of
course, meaning Douglas. [1]
>My prediction: like a pyramid scheme, string theory will
>collapse when it has used up all known branches of mathematics.
Hah. You just wait. We'll just talk the mathematicians into
inventing new mathematics.
Aaron
--
Aaron Bergman
<http://www.princeton.edu/~abergman/>
[1] Which isn't fair, of course, but there are at least one or two
people here who might find it amusing.
Toby Bartels
>Toby Bartels wrote:
>>Asked after the talk about gr-qc/0005126, where Ashtekar, Baez, and Krasnov
>>derive the Bekenstein Hawking formula using loop quantum gravity,
>>Witten replied that the paper was "wishful thinking" and
>>"made no calculation in any theory". Well, his assessment is correct!
>Before Master Wizard Baez smites you, as he promised, and your smoldering
>ruins are no longer able to inform us, would you please explain what you
>meant when you agreed with his assessment?
First, it's wishful thinking to suppose that the calculation
corresponds to anything in reality when we have
no experimental evidence linking loop quantum gravity to reality.
Of course, the same thing can be said of string theory.
Second, there was no specific theory that the calculation was in,
not even the Barrett Crane model (which didn't exist then).
Again, there is no M theory either, so far.
I don't find these assessments particularly objectionable.
Nevertheless, I've often charged that quantum gravity is
a branch of applied mathematics, not science,
and I believe that this is true of both spin foams and superstrings.
That doesn't mean that they're not valid avenues of research.
But you can hardly object to one avenue on these grounds
but not to the other!
-- Toby
to...@math.ucr.edu
Aaron Bergman
>
>Yes, you can say sort of no. But, when John was finished "dealing with me",
>I'll have a fresh understanding of why you can say sort of no in his case too.
>I'd rather give Witten the benefit of the doubt concerning
>the simple truth of the statements he made about John's paper,
>but then I have to give the same benefit of the doubt to anyone
>wishing to make the same statements about the stringy
>"calculations in a theory that might not exist".
>
>In summary: You can have it either way, but you can't have it both ways.
I'm not really sure what this is addressed to.
I would like to say, however, that it's difficult to
underestimate the importance of the black hole entropy
calculations in string theory. Starting with Strominger and Vafa, it
is a real calculation -- there are no heuristics involved. And,
it gets everything correct. People have gone and calculated grey
body factors and string theory gets them right, too. Many people
essentially began to believe in string theory because of these
calculations. It should also be mentioned that there are other
results that get the proportionalities correct for much more
generic black holes including Schwarzchild black holes.
Anyways, I just remembered a nice paper which probably has the
most balanced presentation I've seen: gr-qc/0011089.
Doug Merritt
>
> >>Witten replied that the paper was "wishful thinking" and
> >>"made no calculation in any theory". Well, his assessment is correct!
>
> corresponds to anything in reality when we have
> no experimental evidence linking loop quantum gravity to reality.
I don't understand. Experimental evidence is a rather different subject
than predictive calculation (although of course, ideally the one agrees
with the other. :-) Did you mean to say that loop quantum gravity cannot
make predictive calculations? Or that it does, but they disagree with
experiment, and that *that* was what Witten was complaining about?
> Nevertheless, I've often charged that quantum gravity is
> a branch of applied mathematics, not science,
> and I believe that this is true of both spin foams and superstrings.
Again, I don't understand. Is the phrase "mathematics is the queen of
science" lost to the current generation? :-) Surely (hard) science
*is* applied mathematics? Or have I misunderstood what academics mean
by those terms all these years???
Maybe in fact I *have* been confused. Maybe "applied mathematics" is commonly
used to describe only non-scientific real-world applications of math,
like measuring yarn when doing knitting, or calculating the probability
that a steel worker will die during building construction, but that
term ("applied mathematics") is never used to describe math applied to
science, such as data analysis of the data collection results of heavy ion
collisions?
Please straighten me out here, terminology-wise.
> That doesn't mean that they're not valid avenues of research.
> But you can hardly object to one avenue on these grounds
> but not to the other!
I would agree that alternate approaches, such as loops versus strings
etc, should be judged fairly and therefore by the same criteria. I'm
just a little up in the air, still, as to which criteria are fair to
either or both or neither, and whether you and/or Witten are clear on
that.
Also I'm still waiting curiously for your footnote about how John nailed
you on all this...if that's happened yet. :-) Perhaps he's procrastinating
because you did actually make a good point? (Albeit not one I fully understand
yet, given the above.)
P.S. to moderators: several of my posts in recent weeks were neither rejected
nor posted to the group. Should I assume they were rejected, or should I
assume they fell through the cracks, or what? I know moderator bandwidth
is a limited resource, so I'm not exactly complaining, but one can't
help but wonder when one hears nothing but crickets.
Doug
--
Professional Wild-Eyed Visionary Member, Crusaders for a Better Tomorrow
[Moderator's note: In general, meta-discussions about the workings of
sci.physics.research itself (like this last paragraph) are off-topic
for sci.physics.research (since they're not about physics). But this
question comes up from time to time, so I thought I'd answer it
publicly here.
Unless you post from a false e-mail address, every rejected post you
submit to sci.physics.research will result in a rejection e-mail
being sent to you. If you submit articles and they disappear
utterly, neither appearing in the newsgroup nor resulting in a
rejection e-mail, then something has gone wrong and you should let us
know about it by e-mailing <physics-rese...@ncar.ucar.edu>.
Note that the moderation process does involve unpredictable delays,
which can be several days in length. In addition, articles may take
time to propagate back to your local newsserver. So it can be up to
a week before you see a posted article (although it will usually be
less than that). If you submit an article and you haven't seen any
sign of it for a week, you might want to consider e-mailing us.
Also, note that the most reliable way to submit articles for posting
is to e-mail them directly to the submission address
<physics-...@ncar.ucar.edu>. A significant number of
newsservers are misconfigured in such a way that articles posted to
sci.physics.research via them do not reliably reach us. -TB]
Toby Bartels
>Toby Bartels wrote:
>>In summary: You can have it either way, but you can't have it both ways.
>I'm not really sure what this is addressed to.
I'm not sure either, because it took me so long to get back to you.
But it's at least in part addressed to Witten:
If Baez et al's calculations were in no particular theory,
then much of M theory is also in no particular theory.
Conversely, if M theory is a theory, then so is loop quantum gravity.
>Anyways, I just remembered a nice paper which probably has the
>most balanced presentation I've seen: gr-qc/0011089.
That was a nice read.
-- Toby
to...@math.ucr.edu
Toby Bartels
>Toby Bartels wrote:
>>First, it's wishful thinking to suppose that the calculation
>>corresponds to anything in reality when we have
>>no experimental evidence linking loop quantum gravity to reality.
>I don't understand. Experimental evidence is a rather different subject
>than predictive calculation (although of course, ideally the one agrees
>with the other. :-) Did you mean to say that loop quantum gravity cannot
>make predictive calculations? Or that it does, but they disagree with
>experiment, and that *that* was what Witten was complaining about?
There are no experiments. In a certain sense,
no form of quantum gravity makes any predictive calculations,
because the calculations made don't predict the results
of any experiments, proposed or actual.
Someday, we hope, they will.
>>Nevertheless, I've often charged that quantum gravity is
>>a branch of applied mathematics, not science,
>>and I believe that this is true of both spin foams and superstrings.
>Again, I don't understand. Is the phrase "mathematics is the queen of
>science" lost to the current generation? :-) Surely (hard) science
>*is* applied mathematics? Or have I misunderstood what academics mean
>by those terms all these years???
I like to say "Mathematics is the empty science.".
Mathematics is a sort of degenerate science.
Science is about organising explanations of reality
in accordance with observation and experiment.
It uses mathematics to do so, so mathematics underlies all science.
But there is no actual science going on in the study of pure mathematics,
because there is no connection to an explanation of reality,
and there may be no actual science going on in a study of applied mathematics
if there is no accordance with observation and experiment.
I like to think of the stereotype of the ancient Greeks
studying physics philosophically without the benefit of experiment.
When we study quantum gravity, we are like that.
We study the method that appeals to us philosophically,
and use our methods of philosophy (high level mathematics)
to take us far beyond the region of our actual knowledge.
We hope that this will get us the right answer, and it may.
But it's no more science than the ancient Greek philosophising.
Deemokritos was right about the atoms
and perhaps for sound philosophical reasons,
but he wasn't doing science because
there was no way for an unbiased observer in ancient Greece
to know that he was right.
We are in the same position now and will *still* be
even if both M theory and spin foams come to complete fruition;
assuming the theories don't merge,
there will still be argument and no way to resolve it.
>Maybe in fact I *have* been confused. Maybe "applied mathematics" is commonly
>used to describe only non-scientific real-world applications of math,
>like measuring yarn when doing knitting, or calculating the probability
>that a steel worker will die during building construction, but that
>term ("applied mathematics") is never used to describe math applied to
>science, such as data analysis of the data collection results of heavy ion
>collisions?
I would apply the term to either case,
but the term "science" only to the latter.
In this case, there is no data, nor any proposal for any.
>Also I'm still waiting curiously for your footnote about how John nailed
>you on all this...if that's happened yet. :-) Perhaps he's procrastinating
>because you did actually make a good point? (Albeit not one I fully understand
>yet, given the above.)
I think that it was an empty threat;
still, I'll give that procrastination idea a spin ^_^.
-- Toby
to...@math.ucr.edu
Greg Kuperberg
Toby Bartels <to...@math.ucr.edu> wrote:
>I'm not sure either, because it took me so long to get back to you.
>But it's at least in part addressed to Witten: If Baez et al's
>calculations were in no particular theory, then much of M theory is
>also in no particular theory.
I don't know string theory or "loop quantum gravity" in any detail,
but it seems really foolish to lecture Ed Witten about his own work in a
public forum. It's not just that he's a genius. It's not just that he
reputedly understands string theory and M-theory better than anyone else.
He also, by reputation, has a supreme command of mathematics and physics
related to string theory, including the blind alleys and the alternatives.
Isn't it possible that his apparent bias for string theory over loop
quantum gravity is actually astute intuition? And if so, wouldn't it be
better to seek wisdom from string theorists privately than to "debate"
them publicly?
--
/\ Greg Kuperberg (UC Davis)
/ \
\ / Visit the Math ArXiv Front at http://front.math.ucdavis.edu/
\/ * All the math that's fit to e-print *
Charles Francis
<dougm...@earthlink.net> writes
>> Nevertheless, I've often charged that quantum gravity is
>> a branch of applied mathematics, not science,
>> and I believe that this is true of both spin foams and superstrings.
How can it be applied mathematics if it is not applied to science? If I
were to flatter, I would call these notions pure maths. But as I see in
papers on these subjects nothing of the standards of exposition and
rigour that I would demand of pure maths, I don't know whether I can
describe them as anything at all.
>Again, I don't understand. Is the phrase "mathematics is the queen of
>science" lost to the current generation? :-) Surely (hard) science
>*is* applied mathematics? Or have I misunderstood what academics mean
>by those terms all these years???
As I have a Master of Arts in maths, not a MSc, I hesitate as to whether
the description of mathematics as the queen of science should be treated
as much more than flamboyant rhetoric. Nonetheless we can make sense of
the words by thinking of the epistemelogical meanings. Art actually
means "skill", and is in its original meaning much closer to craft than
to the modern idea that it is to do with aesthetics. Science means
knowledge. Mathematics (i.e. pure math) is pure knowledge because we
know it to be true by tautology. I think that does justify the title
"queen of science". In modern terminology "science" has come to mean
something more like "natural philosophy", which means love of the study
of nature. Physics itself literally means nature. Maths has its own
existence quite divorced from nature, so maths is not physics, it is not
natural philosophy, and so I think it is not a science in the modern
sense of the word. In that sense applied maths is a tool of science
without being a science.
>Maybe in fact I *have* been confused. Maybe "applied mathematics" is commonly
>used to describe only non-scientific real-world applications of math,
>like measuring yarn when doing knitting, or calculating the probability
>that a steel worker will die during building construction, but that
>term ("applied mathematics") is never used to describe math applied to
>science, such as data analysis of the data collection results of heavy ion
>collisions?
Applied mathematics is the application of math to any field including
science, but not statistics which for strange reasons is called
applicable maths. At least this is so at Cambridge where we have the
Department of Pure Maths and Mathematical Statistics, and the Department
of Applied Maths and Theoretical Physics.
--
Charles Francis
Toby Bartels
>Toby Bartels wrote:
>>I'm not sure either, because it took me so long to get back to you.
>>But it's at least in part addressed to Witten: If Baez et al's
>>calculations were in no particular theory, then much of M theory is
>>also in no particular theory.
>I don't know string theory or "loop quantum gravity" in any detail,
>but it seems really foolish to lecture Ed Witten about his own work in a
>public forum. It's not just that he's a genius. It's not just that he
>reputedly understands string theory and M-theory better than anyone else.
>He also, by reputation, has a supreme command of mathematics and physics
>related to string theory, including the blind alleys and the alternatives.
>Isn't it possible that his apparent bias for string theory over loop
>quantum gravity is actually astute intuition? And if so, wouldn't it be
>better to seek wisdom from string theorists privately than to "debate"
>them publicly?
I think it'd be pretty silly of me to assume that
Witten's predilection for string theory over loop QG
is due to astute intuition just because he's so damn smart.
And I *don't* know that he has a command of the alternatives to string theory;
why should he, when they're not his major field of study?
Also, I didn't literally mean that my remarks were addressed to Witten;
there's no reason to think that he reads this newsgroup;
that was shorthand for the remarks' being inspired by what he said,
a shorthand that I didn't think would cause any confusion.
Sorry that it did.
-- Toby
to...@math.ucr.edu
[Moderator's note: Participants in this thread are hereby admonished
to stick to the physics. Discussions of Witten's mental acuity and
related subjects are not about physics. -TB]
Toby Bartels
>Doug Merritt wrote:
>>Toby Bartels wrote:
>>>Nevertheless, I've often charged that quantum gravity is
>>>a branch of applied mathematics, not science,
>>>and I believe that this is true of both spin foams and superstrings.
>How can it be applied mathematics if it is not applied to science? If I
>were to flatter, I would call these notions pure maths. But as I see in
>papers on these subjects nothing of the standards of exposition and
>rigour that I would demand of pure maths, I don't know whether I can
>describe them as anything at all.
Call it nonrigorous pure math, then;
today's standards of rigour are only a couple centuries old.
I call it "applied" because it isn't developed for its own sake
but is *meant* to be applied to another endeavour.
When you do applied math, it's irrelevant whether
there's a science to apply it to or not;
if you were taught a completely ficitious science as a joke
and then worked on developing the math for it,
then you'd do the same thing as if the science were real.
So, I think applied math is still applied math
even if there's no science to apply it to yet,
even if, as in this case, we're just waiting for the day
when we can propose tests to put it to.
>Applied mathematics is the application of math to any field including
>science, but not statistics which for strange reasons is called
>applicable maths.
Never heard that term here in the USA.
-- Toby
to...@math.ucr.edu
Charles Francis
<gr...@math.ucdavis.edu> writes
> it seems really foolish to lecture Ed Witten about his own work in a
>public forum. It's not just that he's a genius. It's not just that he
>reputedly understands string theory and M-theory better than anyone else.
>He also, by reputation, has a supreme command of mathematics and physics
>related to string theory, including the blind alleys and the alternatives.
>Isn't it possible that his apparent bias for string theory over loop
>quantum gravity is actually astute intuition?
You have produced a nice circular argument. "Ed Witten is a genius
because he understands string theory better than anyone else, so string
theory must be right because Ed Witten is a genius and approves of it".
> And if so, wouldn't it be
>better to seek wisdom from string theorists privately than to "debate"
>them publicly?
Not really. First one should understand the arguments which led to
string theory in the first place. If they don't hold up, then the whole
edifice of string theory is without foundation and can be expected to
collapse.
Currently it seems that neither string theorists nor quantum loop people
acknowledge the validity of finite qed (see thread Quantum Field Theory
without Divergences or Renormalization, II; e.g. John Baez dismisses
this as a "trick", making jokes but not answering the physical issue).
To my way of thinking finite qed does remove the motivation for string
theory, and I could not describe anyone, however competent and well
regarded, as a genius if they build mathematical models of physics
without first getting the foundations right.
--
Charles Francis
Bill Hobba
As a person with a degree in applied math I would like to think so; but
clearly it is not. One of the characteristics of science is that it is (at
least in principle) based on experimental evidence. Mathematics is a
collection of tools (if your an applied mathematician) or a collection of
theorems with no connection to reality other than its 'beauty' if you are a
pure mathematician. Either way it is not based on experiment.
Of course Quantum Gravity has not been subjected to experiment YET. However
are you sure it never will be?
I doubt that such a fundamental theory (now that's an interesting word - is
math a theory? - I do not think so) will forever be impervious to the
experimentalists.
It is interesting that this question is covered early on in Feynmans
lectures and the rare person that asks me about this is referred there.
Bill
Charles Francis
writes
>Doug Merritt wrote:
>
>>Toby Bartels wrote:
>>> gr-qc/0005126, where Ashtekar, Baez, and Krasnov
>>>derive the Bekenstein Hawking formula using loop quantum gravity,
>First, it's wishful thinking to suppose that the calculation
>corresponds to anything in reality when we have
>no experimental evidence linking loop quantum gravity to reality.
>Of course, the same thing can be said of string theory.
>Second, there was no specific theory that the calculation was in,
>not even the Barrett Crane model (which didn't exist then).
>Again, there is no M theory either, so far.
>
>I don't find these assessments particularly objectionable.
>Nevertheless, I've often charged that quantum gravity is
>a branch of applied mathematics, not science,
>and I believe that this is true of both spin foams and superstrings.
>That doesn't mean that they're not valid avenues of research.
>But you can hardly object to one avenue on these grounds
>but not to the other!
On the other hand you can (and it sounds as if we both do) object to
both avenues on these grounds, i.e.
1) that neither has any experimental relation to physics.
2) That in many cases calculations are carried out without first
rigorously defining a model to which the calculations apply.
To my way of thinking 1) suggests that neither even qualify either as
applied mathematics or physics and 2) suggests that neither qualify
either as pure mathematics or science.
As such I am not sure that either are valid as avenues to research.
Research should be carried out in a rigorous and disciplined manner, not
by speculating on theories which do not satisfy fundamental criteria of
science. I do not believe that any progress in this field can be made
without first entering into the "arguments" over the interpretation of
quantum mechanics, formulating quantum mechanics in far more rigorous
manner than that found in general textbooks, and resolving the question
of what quantum mechanics really has to say about space-time. Once this
has been done, I don't believe anyone would have a great deal of
interest in researching either spin foams or string theory.
--
Charles Francis
Doug B Sweetser
Hello Charles:
You claimed:
>Once this has been done, I don't believe anyone would have a great
>deal of interest in researching either spin foams or string theory.
I think there is empirical evidence against this :-) The kind of
people who can understand and do research on spin foams or string
theory are a pretty sophisticated lot. I have no problem with them
grabbing all the attention and research dollars. This is research, so
it is not reasonable to mandate the logic be solid. That happens much
later.
What is more important is to make sure and refine ones own questions,
and not worry about other people. Why does causality behave
differently for quantum mechanics than for classical mechanics? An
excellent question! This is one that bothered Einstein the last half
of his life. He hoped for a solution like general relativity, that
changed the logic, but hardly changed the experimental results at all
(the strong field stuff was worked out mostly after his death). If
this is your focus, I wish you the best of luck.
Convincing anyone to follow your efforts is quite another thing.
Since research is exploratory, people have their own sense of where to
go next, and it is darn hard to convince anyone otherwise. Here is a
part of my own story. I wrote a professor at York, an expert in my
obscure area, about some things I had developed, such as a way to
write the Maxwell equations. He wrote me a nice reply, detailing
exactly why it was not possible to write out the Maxwell equations as
I had claimed. I thanked him for his reasoning, but showed him in two
lines of ASCII that he was in error. The busy man had to go on
holiday, but said he would look into it. As far as I can tell, he has
done nothing about this, probably because he is off playing with
octonions in some manner. That is where is gut sense leads him.
Science has had a tradition that if a no-name like myself slaps a
learned Englishman in the face with a fish, that woke some people up
[actually, it usually takes a few attempts at fish-slapping, since
most bright folks know how to duck]. Anyway, I will keep going
forward, next trying to figure out curves. The point is to be true to
your own vision, and if it ever pans out, great, but enjoy the journey
in the mean time. Understanding anything profoundly about Nature is
wonderful, even if it is partially an illusion.
Charles Francis
et...@ncar.UCAR.EDU> writes
>You claimed:
>>Once this has been done, I don't believe anyone would have a great
>>deal of interest in researching either spin foams or string theory.
>I think there is empirical evidence against this :-) The kind of
>people who can understand and do research on spin foams or string
>theory are a pretty sophisticated lot. I have no problem with them
>grabbing all the attention and research dollars. This is research, so
>it is not reasonable to mandate the logic be solid. That happens much
>later.
The point is it is already much later in the history of field theory,
and the logic is still not rock solid. This needs to be fixed before
using it to justify more speculation.
>What is more important is to make sure and refine ones own questions,
>and not worry about other people. Why does causality behave
>differently for quantum mechanics than for classical mechanics? An
>excellent question! This is one that bothered Einstein the last half
>of his life. He hoped for a solution like general relativity,
The strange thing is there already was a solution, but he never seemed
to understand quantum mechanics. But then he was well into his forties
when its mathematical structure was discovered.
>Convincing anyone to follow your efforts is quite another thing.
It is indeed.
--
Charles Francis
John Baez
Toby Bartels <to...@math.ucr.edu> wrote:
>Doug Merritt wrote:
>>I don't understand. Experimental evidence is a rather different subject
>>than predictive calculation (although of course, ideally the one agrees
>>with the other. :-) Did you mean to say that loop quantum gravity cannot
>>make predictive calculations? Or that it does, but they disagree with
>>experiment, and that *that* was what Witten was complaining about?
>There are no experiments. In a certain sense,
>no form of quantum gravity makes any predictive calculations,
>because the calculations made don't predict the results
>of any experiments, proposed or actual.
This isn't quite true. There are a lot of experiments
in favor of classical general relativity and quantum field
theory on Minkowski spacetime; any theory of quantum gravity
should at least be able to explain these. That's harder than
it might appear. In fact, I don't think any theory of quantum
gravity has fully met this challenge yet.
There are also other tests that I think any would-be theory
of quantum gravity should pass.
For example, it should be able to explain the results of
semiclassical calculations like the Bekenstein-Hawking formula
for black hole entropy - or else say why these calculations are
wrong.
It should also have something to say about the singularities
associated with black holes and the big bang. It may eliminate
these singularities or it may not, but it should make *some*
predictions about them.
I think getting theories of quantum gravity to meet these tests
should keep us busy for a while - at least if we want them to be
reasonably elegant.
>I like to say "Mathematics is the empty science.".
That's probably why you like the empty set so much.
>>Also I'm still waiting curiously for your footnote about how John nailed
>>you on all this...if that's happened yet. :-) Perhaps he's procrastinating
>>because you did actually make a good point?
>I think that it was an empty threat;
>still, I'll give that procrastination idea a spin ^_^.
You agreed with Witten that a paper of mine "made no calculation in
any theory" - and added that same thing could be said about any paper
on M-theory.
I decided that before blasting you with a thunderbolt, I should figure
out what you meant.
I sort of understand what you mean now. But I still can't tell whether
you have actually read that paper of mine. Nothing suggests that you
have. You haven't made any specific comments about anything in the
paper, for example.
If so, I won't waste a perfectly good thunderbolt on you now. I'll
just lock you in the dungeon someday and force you to read that paper!
It makes a set of explicit assumptions and then uses mathematics to
calculate the entropy of a black hole based on these assumptions.
The math is probably right, so the real fun starts when we examine the
underlying assumptions.
Aaron Bergman
ba...@galaxy.ucr.edu (John Baez) wrote:
> It should also have something to say about the singularities
> associated with black holes and the big bang. It may eliminate
> these singularities or it may not, but it should make *some*
> predictions about them.
Horowitz and mumble in '95 have some argument that a theory of
quantum gravity should, in fact, not eliminate all singularities.
Kevin A. Scaldeferri
Charles Francis <cha...@clef.demon.co.uk> wrote:
>In article <GB8DA...@world.std.com>, Doug B Sweetser <uunet!world!swe
>et...@ncar.UCAR.EDU> writes
>>You [Charles Francis] claimed:
>>>Once this has been done, I don't believe anyone would have a great
>>>deal of interest in researching either spin foams or string theory.
>>I think there is empirical evidence against this :-) The kind of
>>people who can understand and do research on spin foams or string
>>theory are a pretty sophisticated lot. I have no problem with them
>>grabbing all the attention and research dollars. This is research, so
>>it is not reasonable to mandate the logic be solid. That happens much
>>later.
>The point is it is already much later in the history of field theory,
>and the logic is still not rock solid. This needs to be fixed before
>using it to justify more speculation.
It's not really much later in the history of field theory in my
opinion. Remember that it took hundreds of years before calculus was
given a rigorous formulation. The mathematical underpinnings of
classical mechanics also took quite some time to develop (from a
couple hundred years to a couple thousand years depending on where you
want to start counting.)
--
======================================================================
Kevin Scaldeferri Calif. Institute of Technology
The INTJ's Prayer:
Lord keep me open to others' ideas, WRONG though they may be.
Paul D. Shocklee
> In article <9afvg8$5ss$1...@glue.ucr.edu>,
> ba...@galaxy.ucr.edu (John Baez) wrote:
> > It should also have something to say about the singularities
> > associated with black holes and the big bang. It may eliminate
> > these singularities or it may not, but it should make *some*
> > predictions about them.
> Horowitz and mumble in '95 have some argument that a theory of
> quantum gravity should, in fact, not eliminate all singularities.
Where mumble = Rob Myers:
The Value of Singularities
gr-qc/9503062
http://xxx.lanl.gov/abs/gr-qc/9503062
--
Paul Shocklee
Graduate Student, Department of Physics, Princeton University
Researcher, Science Institute, Dunhaga 3, 107 Reykjavik, Iceland
Phone: +354-525-4429
Charles Francis
writes
>In article <99v640$186$1...@glue.ucr.edu>,
>Toby Bartels <to...@math.ucr.edu> wrote:
>
>>Doug Merritt wrote:
>
> There are a lot of experiments
>in favor of classical general relativity and quantum field
>theory on Minkowski spacetime; any theory of quantum gravity
>should at least be able to explain these. That's harder than
>it might appear. In fact, I don't think any theory of quantum
>gravity has fully met this challenge yet.
>
>There are also other tests that I think any would-be theory
>of quantum gravity should pass.
>
>For example, it should be able to explain the results of
>semiclassical calculations like the Bekenstein-Hawking formula
>for black hole entropy - or else say why these calculations are
>wrong.
>
>It should also have something to say about the singularities
>associated with black holes and the big bang. It may eliminate
>these singularities or it may not, but it should make *some*
>predictions about them.
>
>I think getting theories of quantum gravity to meet these tests
>should keep us busy for a while - at least if we want them to be
>reasonably elegant.
I don't really see the problem. You have just been talking of
"microlocal" reference frames in another thread. But really the
Minkowsky reference frames we actually use in ordinary physics are
local tangent spaces which are true to the limits of experimental
accuracy, and we can formulate qfts within these local tangent spaces.
In other words we say that qfts on Minkowsky space are local
approximations which hold to the limits of experimental accuracy, in the
same sense that the Minkowsky space itself is a local approximation
valid to the limit of experimental accuracy.
Then all that we need do is study the physical measurements which give
us the connection between these tangent spaces. It should be simple to
see that semiclassical calculations like Bekenstein-Hawking stand up. At
a singularity the field theoretic description of particle
creation/annihilation may be thought still to hold, although obviously
at a singularity the space in which the qft is defined could not be
described as a tangent space.
I have not understood the response of hard working physicists to this
proposal. Is it "its so easy that it can't be true"? Of course that was
said of Darwin's evolutionary theory. Or is it "it doesn't make sense".
If not, why not?
>If so, I won't waste a perfectly good thunderbolt on you now.
I heard recently on sci.physics that there is a free supply of
thunderbolts. Anyone can do it. You just have to wave your hand at
approaching the speed of light in the earth's magnetic field.
--
Charles Francis
Steve Carlip
> There are also other tests that I think any would-be theory
> of quantum gravity should pass.
> For example, it should be able to explain the results of
> semiclassical calculations like the Bekenstein-Hawking formula
> for black hole entropy - or else say why these calculations are
> wrong.
> It should also have something to say about the singularities
> associated with black holes and the big bang. It may eliminate
> these singularities or it may not, but it should make *some*
> predictions about them.
> I think getting theories of quantum gravity to meet these tests
> should keep us busy for a while - at least if we want them to be
> reasonably elegant.
Let me add a few:
1. A theory of quantum gravity ought to explain why the
cosmological constant is not huge. (It may not ``solve'' the
cosmological constant problem, but should at least explain
why quantum corrections don't necessarily lead to a conflict
with observation.)
2. A theory of quantum gravity should give a concrete resolution
to the black hole information paradox.
3. Some still fairly hand-waving arguments predict that cumulative
quantum gravitational effects should have implications for
gamma ray bursts (frequency-dependent dispersion relations,
birefringence) and for noise in gravitational wave detectors, at
levels that may be observable in the not-too-distant future. A
quantum theory of gravity ought to make concrete predictions,
either that these effects will be seen (with detailed signatures) or
that they will not be. Note that even a negative prediction of
something potentially observable is a prediction.
4. Some versions of quantum gravity predict CP and CPT violations
at levels that may soon be observable. Again, a theory ought to
make concrete predictions, either that these effects will be seen
(with detailed signatures) or that they will not be.
5. A quantum theory of gravity *might* make other predictions,
especially in cosmology---spatial topology, initial state from
which fluctuations in inflationary cosmology evolve, etc. While
these are not necessary, in the sense that a theory that made no
predictions of this sort might still be acceptable, they offer tests
for some theories.
Steve Carlip
Ilja Schmelzer
> There are a lot of experiments
> in favor of classical general relativity and quantum field
> theory on Minkowski spacetime; any theory of quantum gravity
> should at least be able to explain these. That's harder than
> it might appear. In fact, I don't think any theory of quantum
> gravity has fully met this challenge yet.
Really? What about some (more or less arbitrary) explicit
regularization of the GR Lagrangian in a preferred Minkowski
background frame (a la Feynman/Deser)?
I know very well that there are other important criteria like beauty,
explanatory power, predictive power and so on, and based on these
criteria this particular theory isn't nice. But IMHO it fully mets
the above challenge.
> There are also other tests that I think any would-be theory
> of quantum gravity should pass.
> For example, it should be able to explain the results of
> semiclassical calculations like the Bekenstein-Hawking formula
> for black hole entropy - or else say why these calculations are
> wrong.
Hm, if we obtain the same basic formulas for semiclassical fields,
what's the problem with "explanation of calculations"?
> It should also have something to say about the singularities
> associated with black holes and the big bang. It may eliminate
> these singularities or it may not, but it should make *some*
> predictions about them.
In the theory mentioned above will be no black hole singularity.
> I think getting theories of quantum gravity to meet these tests
> should keep us busy for a while - at least if we want them to be
> reasonably elegant.
Is there a difference between "reasonably elegant" and
"in agreement with your personal metaphysical beliefs"?
> I'll just lock you in the dungeon someday and force you to read that
> paper!
I would like to do the same with you about gr-qc/0001101, to argue
about elegance (as far as this is possible and different from "in
agreement with your personal metaphysical beliefs").
Ilja
--
I. Schmelzer, <il...@ilja-schmelzer.net>, http://ilja-schmelzer.net
Steve Carlip
> ba...@galaxy.ucr.edu (John Baez) writes:
>> There are a lot of experiments
>> in favor of classical general relativity and quantum field
>> theory on Minkowski spacetime; any theory of quantum gravity
>> should at least be able to explain these. That's harder than
>> it might appear. In fact, I don't think any theory of quantum
>> gravity has fully met this challenge yet.
> Really? What about some (more or less arbitrary) explicit
> regularization of the GR Lagrangian in a preferred Minkowski
> background frame (a la Feynman/Deser)?
Since no consistent explicit regularization is known, I think John's
point stands.
Here are some basic problems such a regularized theory would have
to be able to address before it could be said to give a good classical
limit:
--The cut-off theory must still have some version of the Hamiltonian
and momentum constraints of GR, since they're required for a
self-consistent coupling to a divergence-free stress energy tensor.
What operator ordering gives a consistent closed algebra? Are
there anomalies? What observables commute with the constraints?
Note that they will necessarily be nonlocal; how do you reconstruct
a local picture of geometry in the classical limit?
--Does the regularization break unitarity? If not, how, specifically,
does it avoid doing so? If so, are you sure the breaking doesn't
affect low energy observables?
--The self-interaction of gravity doesn't go away at large distances, so
you can't use noninteracting ``free field'' states as asymptotic states.
What are the asymptotic states, and how do you interpret the S
matrix?
--How do you normalize states? Specifically, how do you make the
inner product of two states that satisfy the constraints finite? The
physical interpretation, and in particular the classical limit, will
depend on your answer.
--You're defining the theory by expanding around a flat background.
Is this consistent? Is flat spacetime a minimum of the full effective
action? (Think about trying to do the same thing in the much
simpler theory of QCD.)
--Do the results depend on your choice of gauge-fixing? Note that
off-shell quantities, including the effective action, will, generically,
and that it's at least unclear what happens on shell when you are
dealing with nonlocal observables (as you must, since observables
have to commute with the constraints).
--A related question: gauge-fixing involves, among other things, a
choice of time slicing. Does the evolution from a fixed initial
slice to a fixed time slice depend on the choice of intermediate
slicing. (Note that it does for even a scalar field on a flat spacetime
of dimension greater than two.) If it does, how do you decide what
the right choice is?
--If you answer the preceding question by picking a preferred slicing,
here's a follow-up. If you have a preferred time slicing with a time
coordinate T, you can find the classical Hamiltonian by solving the
Hamiltonian constraint for the momentum p_T conjugate to T.
How do you make the result into a self-adjoint operator? Some of
the problems you're likely to run into:
* With some choices of T, p_T involves the square root of an
operator that's not necessarily positive. Even if it is positive,
you can usually get a unique square root only through spectral
decomposition. How do you find an operator ordering that
makes the result self-adjoint?
* For most other choices of T, p_T is defined only implicitly through
a differential equation. Again, how do you make such an implicitly
defined operator self-adjoint?
Steve Carlip
Charles Francis
<ke...@cco.caltech.edu> writes
>>The point is it is already much later in the history of field theory,
>>and the logic is still not rock solid. This needs to be fixed before
>>using it to justify more speculation.
>
>It's not really much later in the history of field theory in my
>opinion. Remember that it took hundreds of years before calculus was
>given a rigorous formulation. The mathematical underpinnings of
>classical mechanics also took quite some time to develop (from a
>couple hundred years to a couple thousand years depending on where you
>want to start counting.)
Good point. It was much more than a hundred of years from Newton to
Weirstrauss et al, but one should probably start to count the years of
the development of the calculus with Archimedes.
Perhaps I am just impatient, but I think the time is right for the
rigorous construction of axiomatic quantum field theory, and I still
think we cannot hope to build more eleaborate models like string theory
if we cannot get the foundations right.
My experience working in the foundations has been that it is rather like
building a house of cards. If one gets the expression of something even
slightly wrong on one page, it takes another five or ten pages of
rigorous mathematics before the model actually breaks down.
--
Charles Francis
Ilja Schmelzer
> > Really? What about some (more or less arbitrary) explicit
> > regularization of the GR Lagrangian in a preferred Minkowski
> > background frame (a la Feynman/Deser)?
>
> Since no consistent explicit regularization is known, I think John's
> point stands.
Fine. Let's try to see where the problems occur. Note that we don't
care in any way about the beauty of the resulting theory, only if it
gives the relevant limits.
We fix an explicit preferred frame, define a regular grid on space, on
a large enough cube with periodic boundary conditions (or a torus).
We discretize the variables of the gravitational field and the SM. We
also discretize the Lagrangian. We also destroy all symmetries and
degeneracies of the Lagrangian in this step. We are now back to
quantization of a classical theory with finite number of steps of
freedom and a non-degenerated Lagrange formalism. So let's use some
standard normal ordering to define the quantum theory.
Again, I know sufficient reasons for not liking this theory and do not
propose it, the only question is about the predictions in the relevant
limits.
> --The cut-off theory must still have some version of the Hamiltonian
> and momentum constraints of GR, since they're required for a
> self-consistent coupling to a divergence-free stress energy tensor.
> What operator ordering gives a consistent closed algebra?
Diff symmetry as well as gauge symmetry are clearly broken. But the
formulation in the Lagrange formalism is quite consistent, at least I
don't understand what may be inconsistent.
> Are there anomalies? What observables commute with the constraints?
There are anomalies, we have no constraints and nothing commutes.
> Note that they will necessarily be nonlocal; how do you reconstruct
> a local picture of geometry in the classical limit?
No, I have local operators g_ij(x_k,t), A_i(x_k,t) and so on from the
start.
> --Does the regularization break unitarity? If not, how, specifically,
> does it avoid doing so? If so, are you sure the breaking doesn't
> affect low energy observables?
No, we formulate the theory in the "big" space, and do not introduce a
factorization of the "non-physical" steps of freedom. They are
considered to be physical.
I'm not completely sure that the symmetry breaking doesn't affect low
energy observables. That's possibly the weak point, having in mind
fermion doubling.
> --The self-interaction of gravity doesn't go away at large distances, so
> you can't use noninteracting ``free field'' states as asymptotic states.
> What are the asymptotic states, and how do you interpret the S
> matrix?
I don't see that I need them for the definition of the theory. S
matrices may be derived later as usual in classical quantum gravity.
This will be as subtle and complex as quantum field theory for large
crystals. But nobody claims that classical quantum theory is undefined
because we have problems computing large crystals.
> --How do you normalize states?
As usual in classical quantum theory.
> Specifically, how do you make the
> inner product of two states that satisfy the constraints finite?
I have no constraints.
> --You're defining the theory by expanding around a flat background.
Not exactly expanding around. Its lattice theory on this background,
not pertubation theory.
> --Do the results depend on your choice of gauge-fixing?
Yep. I use de facto an explicit gauge and don't care about others.
> --A related question: gauge-fixing involves, among other things, a
> choice of time slicing. Does the evolution from a fixed initial
> slice to a fixed time slice depend on the choice of intermediate
> slicing.
Yep. I use an explicit preferred frame and don't care about others.
> --If you answer the preceding question by picking a preferred slicing,
> here's a follow-up. If you have a preferred time slicing with a time
> coordinate T, you can find the classical Hamiltonian by solving the
> Hamiltonian constraint for the momentum p_T conjugate to T.
I don't solve the Hamiltonian constraint. Instead, I start from a
pertubed classical Lagrangian which breaks the symmetry, therefore the
Hamiltonian is not a constraint.
> How do you make the result into a self-adjoint operator?
Using standard theory about quantization in classical quantum theory
(finite number of steps of freedom, no field theory subtleties). If
you like to preserve positivity, use anti-normal ordering instead of
normal ordering or Weyl quantization.
Steve Carlip
> Steve Carlip <sjca...@ucdavis.edu> writes:
>> > Really? What about some (more or less arbitrary) explicit
>> > regularization of the GR Lagrangian in a preferred Minkowski
>> > background frame (a la Feynman/Deser)?
>> Since no consistent explicit regularization is known, I think John's
>> point stands.
> Fine. Let's try to see where the problems occur.
> We fix an explicit preferred frame, define a regular grid on space, on
> a large enough cube with periodic boundary conditions (or a torus).
> We discretize the variables of the gravitational field and the SM. We
> also discretize the Lagrangian. We also destroy all symmetries and
> degeneracies of the Lagrangian in this step. We are now back to
> quantization of a classical theory with finite number of steps of
> freedom and a non-degenerated Lagrange formalism.
First question: does this have the right continuum limit? You've defined
a theory, but is it gravity? There's been lots of work on quantum general
relativity on a lattice, but very little evidence that you get back classical
GR as a limit. (A possible exception is recent work by Ambjorn, Loll,
et al., but their model is rather different from the one you describe.)
In particular:
> Diff symmetry as well as gauge symmetry are clearly broken. [...]
> I have no constraints. [...]
> I have local operators g_ij(x_k,t), A_i(x_k,t) and so on from the
> start.
Then you have too many degrees of freedom---six per point rather
than two. To have any hope of getting a good classical limit, where
``good'' means ``in agreement with simple observation,'' you have to
get rid of the extra degrees of freedom. The only way I know to do
this is to reintroduce constraints. Then all of my questions stand.
Furthermore, since you have broken diffeomorphism invariance, you
no longer have Noether's theorem for translations, and thus have no
conserved stress-energy tensor. Again, to have any hope of agreeing
with observation, you need to reintroduce some symmetry.
> As usual in classical quantum theory.
In other words, you write your wave functions as functions of the
g_ij(x_k) and require that some integral of |\psi|^2 over the g_ij is
equal to one, right? But this is geometrically wrong. As Wheeler
and his collaborators pointed out, the metric g_ij at a fixed time t
contains information not only about the three-geometry, but about
the time---a slice with a specified g_ij will fit into a four-manifold
only in a few restricted ways that determine the time at which the
space has that geometry. So your normalization implicitly includes
an integral over time; if you require your wave functions to be
normalized this way, you won't get a good classical limit.
Steve Carlip
Ilja Schmelzer
> First question: does this have the right continuum limit? You've
> defined a theory, but is it gravity? There's been lots of work on
> quantum general relativity on a lattice, but very little evidence
> that you get back classical GR as a limit. (A possible exception is
> recent work by Ambjorn, Loll, et al., but their model is rather
> different from the one you describe.)
Of course, there are some problems to obtain a classical limit. Some
of them I know, and I have avoided them in this approach. First of
all, its highly problematic to quantize gravity in a way that
preserves relativistic symmetry, in a background-free way. In
gr-qc/0001101, App. D I argue that this is impossible, and use
incompatibility with some classical limit (Schrödinger theory with
Newtonian potential) as the main argument. According to this
argument, dynamical triangulations cannot give the classical limit.
That's not what I have proposed.
If we have partial differential equations, we use computers to obtain
solutions. I have no special experience in numerical simulation of
general relativity, but they are second order equations, and in
harmonic gauge we have a well-defined Cauchy formulation. Not much
reason to doubt that it is possible to simulate these equations on a
computer.
Now, I agree that we discretize here a theory on flat background,
which is already different from GR because it does not allow
nontrivial topology. It will not cover the part inside the BH horizon
and so on. Therefore, I guess, there may be also remaining
differences near the horizon which will not vanish for arbitrary small
h. In this sense, I'm sure that it doesn't give GR, but gives a
metric theory of gravity on flat background which, in some regime (~
sufficiently small cosmological constants or graviton mass) gives the
Einstein equations.
For example, assume it gives in the limit my GET. We have to make the
cosmlogical constants X,Y sufficiently small, that's all we need to
fit observation.
>> Diff symmetry as well as gauge symmetry are clearly broken. [...]
>> I have no constraints. [...]
>> I have local operators g_ij(x_k,t), A_i(x_k,t) and so on from the
>> start.
> Then you have too many degrees of freedom---six per point rather
> than two. To have any hope of getting a good classical limit, where
> ``good'' means ``in agreement with simple observation,'' you have to
> get rid of the extra degrees of freedom.
No, I havn't. Its sufficient if these extra degrees of freedom do not
interact with the others in the classical limit. Or interact in a
sufficiently weak way (via some small cosmological constants).
> Furthermore, since you have broken diffeomorphism invariance, you no
> longer have Noether's theorem for translations, and thus have no
> conserved stress-energy tensor. Again, to have any hope of agreeing
> with observation, you need to reintroduce some symmetry.
I have used a flat background, and a regular rectangular grid for
discretization. So, we don't have to destroy classical translational
symmetry, and the classical Noether theorem gives classical
energy-momentum conservation.
> In other words, you write your wave functions as functions of the
> g_ij(x_k) and require that some integral of |\psi|^2 over the g_ij
> is equal to one, right?
Yep.
> But this is geometrically wrong. As Wheeler and his collaborators
> pointed out, the metric g_ij at a fixed time t contains information
> not only about the three-geometry, but about the time---a slice with
> a specified g_ij will fit into a four-manifold only in a few
> restricted ways that determine the time at which the space has that
> geometry. So your normalization implicitly includes an integral
> over time;
It seems this confuses GR time and quantum time. Of course GR time is
part of the g_ij(x_k). But its unrelated to quantum time, which is
absolute and unrelated to GR time in this approach. We have fixed the
gauge (during the step I have described as "We also destroy all
symmetries and degeneracies of the Lagrangian". Ok, let's add "except
simple translational symmetry".)
> if you require your wave functions to be normalized this way, you
> won't get a good classical limit.
Hm. I agree I have no complete proof. The point of this discussion
for me is to understand what I have to prove.
In my understanding we have several limits: the large distance limit
regularization -> GR which is purely classical (at least I see no need
to consider here quantum questions). And the limit h-> 0 between QG
and the regularized theory. The regularized theory is, by
construction of the regularization, a classical theory with finite
number of steps of freedom and Lagrange formalism in a classical
framework with absolute time.
Let's, for clarity, introduce a third intermediate step: a continuous
theory on a flat background with Lagrange formulation and broken diff
symmetry (fixed gauge). Here we can choose GET as an example.
So, we have GR <----- GET <--- classical discretization <------ QG.
X,Y-> 0 h-> 0 hbar-> 0
So, which part is the problematic one? The first is fine, AFAIU the
situation with empirical evidence in gravity. Note that this first
steps already covers all the questions related with symmetry, gauge,
conservation laws, constraints and the number of steps of freedom.
The second is the everyday job of programmers, not always easy but
usually for different reasons (we need accurate and stable
discretizations for computers, not simply discretizations which give
the correct limit). Fermion doubling is an interesting point here,
which shows that there may be unexpected problems. But to handle
symmetrical tensor fields is the everyday job in mechanics. Thus, the
context is quite classical, not very extravagant.
The last is simply classical QM.
Miguel Carrion
Charles Francis <cha...@clef.demon.co.uk> wrote:
>In article <9agkif$5...@gap.cco.caltech.edu>, Kevin A. Scaldeferri
><ke...@cco.caltech.edu> writes
>>>The point is it is already much later in the history of field theory,
>>>and the logic is still not rock solid. This needs to be fixed before
>>>using it to justify more speculation.
>>It's not really much later in the history of field theory in my
>>opinion. Remember that it took hundreds of years before calculus was
>>given a rigorous formulation. The mathematical underpinnings of
>>classical mechanics also took quite some time to develop (from a
>>couple hundred years to a couple thousand years depending on where you
>>want to start counting.)
>Good point. It was much more than a hundred of years from Newton to
>Weierstrass et al, but one should probably start to count the years of
>the development of the calculus with Archimedes.
In my opinion, it is true that it is already "much later", and the
comparison of the times involved in rigorizing different theories cannot
be done so simply. Science has been progresing at an almost exponential
rate since the Scientific Revolution of the 1600's, and maybe since the
Renaissance, whether you measure it in terms of the number of papers
published, number of journals, number of active scientists... This means
that seventy years starting in 1930 is much, much longer than 200 years
starting in 1660 (for classical mechanics).
There is a good reason why Quantum Field Theory has developed very
quickly but with relative disregard for rigor, and that is that it has
been driven by the necessity to explain a plethora of experimental
results, like the zoo of particles in the 1950's. For much of the
history of QFT, experiment was ahead of theory. In 1971-1973 this was
reversed, but instead of going back and rigorising the quantum theory
that then was known to work, thirty years have been spent on all kinds
of theories with no experimental support or even hints. (Even the
neutrino mass can be included in the Standard Model)
As for calculus, as much as I am a fan of Archimedes, I think the
starting point must be somewhere around Cavalieri's time (early 1600's),
due to the historical discontinuity between, say, the destruction of the
Library of Alexandria around 400, and the mathematical research of the
Renaissance.
Although available in the original Greek or in Arabic or Latin
translations, Eudoxus and Archimedes' exhaustion method was apparently
too hard for Renaissance mathematicians to use, and therefore the method
of "indivisibles" was devised. Also, Archimedes' "Method to derive
geometrical solutions from mechanics" was only disovered in a Palimpsest
around 1900, so it unfortunately had no impact on the development of
modern science or the rigorization of either calculus or mechanics.
Regards,
Miguel Carrion
--
}---------------------------------------------------------------------
| homepage: <http://www.math.ucr.edu/~miguel/>
| International Association of Physics Students, IAPS
| url: <http://www.iaphys.org/> e-mail: ia...@nikhef.nl
Toby Bartels
>Toby Bartels wrote:
>>I don't find these assessments particularly objectionable.
>>Nevertheless, I've often charged that quantum gravity is
>>a branch of applied mathematics, not science,
>>and I believe that this is true of both spin foams and superstrings.
>>That doesn't mean that they're not valid avenues of research.
>>But you can hardly object to one avenue on these grounds
>>but not to the other!
>On the other hand you can (and it sounds as if we both do) object to
>both avenues on these grounds, i.e.
>1) that neither has any experimental relation to physics.
>2) That in many cases calculations are carried out without first
>rigorously defining a model to which the calculations apply.
No, I do not object to either as an avenue of research.
I object to double standards, and I object to calling them "science"
(in the modern sense of the term, meaning natural philosophy).
First of all, even the purest mathematics in the halls of academe
is not beset with perfect rigour, not in the research stages.
Yes, mathematicians try to prove things, but they try to understand first;
they try to get a grasp of the subject, use that intuition to get conjectures,
and *then* prove the conjectures rigorously, if they can.
And if they can't do that ... well then, they publish the conjectures.
I can't speak of string theorists from personal experience,
but JB has talked about proving theorems with hardcore analysis before
in applications to loop quantum gravity, so this side is not forgotten.
But this step (rigorous proof) is not strictly necessary at any time;
we have to do it sometime before we can say that we're *done*,
but if we're still getting the lay of the mathematical land,
well, we can just put it off if it's too hard.
That's what graduate students are for, or so I'm told.
IOW, we do *not* have to secure the foundations first;
we have to secure the foundations *eventually*.
Secondly, both of these avenues of research have the potential
to be applicable to science in the future, after science catches up.
I don't consider reproduction of the Bekenstein Hawking formula
to be a scientific test, since Bekenstein Hawking is entirely theoretical.
But it may be subjectable to an experimental test at some later date.
From another direction, there is no sign that I've heard of
of either approach predicting a reasonable value for the cosmological constant.
If there were such a sign, I'd have to reevaluate my position that
we weren't doing physics, because that is experimentally measurable.
This is why I've used the term "applied"; it is potentially applicable,
and intended to be applied. There's just nothing to apply it to yet;
there's no science yet; but there is math and that has its value.
I also object to string theory on philosophical grounds,
but I suspect that it is salvageable and is moving in the right direction,
so that's OK too.
-- Toby
to...@math.ucr.edu
Toby Bartels
>You agreed with Witten that a paper of mine "made no calculation in
>any theory" - and added that same thing could be said about any paper
>on M-theory.
>I decided that before blasting you with a thunderbolt, I should figure
>out what you meant.
>I sort of understand what you mean now. But I still can't tell whether
>you have actually read that paper of mine. Nothing suggests that you
>have. You haven't made any specific comments about anything in the
>paper, for example.
I read it. At least I think that I read it.
I found a paper on gr-qc by Ashtekar, Baez, & Krasnov
which calculated the entropy of a wide class of nonrotating black holes,
reproducing the Bekenstein Hawking formula for all of them
with a single free (Immirzi) parameter across the board.
That should be it, right?
>If so, I won't waste a perfectly good thunderbolt on you now. I'll
>just lock you in the dungeon someday and force you to read that paper!
>It makes a set of explicit assumptions and then uses mathematics to
>calculate the entropy of a black hole based on these assumptions.
>The math is probably right, so the real fun starts when we examine the
>underlying assumptions.
The assumptions don't seem to be derived from a *theory*
but instead from the beginning *attempts* at a theory.
This is a fine distinction, and you can argue that Witten
put the distinction on the wrong side of the line.
That's a valid arguemnt; but I was being generous with Witten's comment.
I thought that it could be justified, but only by saying that
a theory is not a theory yet until you have it in your hands,
that until then you're merely getting ideas for a theory you seek.
Then the same accusations[*] can be levelled against M theory.
If you want to be more broad with your definition of
"calculation in any theory", then I won't argue;
I'm just trying to give Witten the benefit of the doubt.
[*]Not that they're very damning accusations,
since there's nothing wrong with seeking a theory you don't have yet.
-- Toby
to...@math.ucr.edu
Dushan Mitrovich
Toby Bartels <to...@math.ucr.edu> wrote:
`
` The assumptions don't seem to be derived from a *theory*
` but instead from the beginning *attempts* at a theory.
` This is a fine distinction, and you can argue that Witten
` put the distinction on the wrong side of the line.
` That's a valid arguemnt; but I was being generous with Witten's comment.
` I thought that it could be justified, but only by saying that
` a theory is not a theory yet until you have it in your hands,
` that until then you're merely getting ideas for a theory you seek.
The first sentence above puzzled me: deriving 'assumptions' from a theory??
I thought a theory is based on assumptions. Or do you have something else
in mind than what I understood?
Also, could you define 'have it in your hands'? Maybe your answer to this
will shed light on the above.
- Dushan
dus...@abq.com
Steve Carlip
>>> Diff symmetry as well as gauge symmetry are clearly broken. [...]
>>> I have no constraints. [...]
>>> I have local operators g_ij(x_k,t), A_i(x_k,t) and so on from the
>>> start.
>> Then you have too many degrees of freedom---six per point rather
>> than two. To have any hope of getting a good classical limit, where
>> ``good'' means ``in agreement with simple observation,'' you have to
>> get rid of the extra degrees of freedom.
> No, I havn't. Its sufficient if these extra degrees of freedom do not
> interact with the others in the classical limit. Or interact in a
> sufficiently weak way (via some small cosmological constants).
Agreed---it's sufficient to have the extra degrees of freedom decouple.
Note that they have to decouple from everything, including the ``physical''
metric degrees of freedom, since otherwise you'd get violations of the
strong equivalence principle.
Tthere are two ways I know to make such decoupling occur. One is
to have the extra degrees of freedom acquire large masses; the other is
to have constraints. Do you have another?
Now, if you're obtaining your discrete equations of motion by simply
discretizing general relativity on a lattice, then in fact you *do* have
constraints. Four of the ten field equations involve no time derivatives
in canonical form. (They have first time derivatives of the metric, but
these only appear in combinations that become the undifferentiated
canonical momenta.) Unless your discretization somehow introduces
new time derivatives, these equations will continue to be constraints.
Thus my questions still stand. There are basic features of the
quantization of a constrained system (observables commute with
constraints, wave functions are annihilated by constraints) that
are independent of the details of the system being quantized.
Moreover, to get a consistent classical limit, you have to find an
operator ordering in which the algebra of constraints closes.
>> In other words, you write your wave functions as functions of the
>> g_ij(x_k) and require that some integral of |\psi|^2 over the g_ij
>> is equal to one, right?
> Yep.
>> But this is geometrically wrong. As Wheeler and his collaborators
>> pointed out, the metric g_ij at a fixed time t contains information
>> not only about the three-geometry, but about the time---a slice with
>> a specified g_ij will fit into a four-manifold only in a few
>> restricted ways that determine the time at which the space has that
>> geometry. So your normalization implicitly includes an integral
>> over time;
> It seems this confuses GR time and quantum time. Of course GR time
> is part of the g_ij(x_k). But its unrelated to quantum time, which is
> absolute and unrelated to GR time in this approach.
Is your quantum evolution described as evolution in this ``quantum time''?
If so, how do recover the classical limit, in which the evolution equations
describe evolution in what you call ``GR time''? (I think you can't really
mean that the two are ``unrelated.'')
> In my understanding we have several limits: the large distance limit
> regularization -> GR which is purely classical (at least I see no need
> to consider here quantum questions). And the limit h-> 0 between
> QG and the regularized theory.
You can't separate the two limits. That's one of the reasons quantizing
gravity is so hard. Quantization puts certain constraints on the theory
that are at least well hidden in the classical theory. The basic problem
is that the obvious ``observables'' in the classical theory (discretized
or not) are not quantum observables---a quantum observable must
commute with the constraints, and it is *extremely* difficult to find
any such quantities at all.
Steve Carlip
John Baez
Toby Bartels <to...@math.ucr.edu> wrote:
>John Baez wrote:
>>I sort of understand what you mean now. But I still can't tell whether
>>you have actually read that paper of mine.
>I read it. At least I think that I read it.
>I found a paper on gr-qc by Ashtekar, Baez, & Krasnov
>which calculated the entropy of a wide class of nonrotating black holes,
>reproducing the Bekenstein Hawking formula for all of them
>with a single free (Immirzi) parameter across the board.
>That should be it, right?
There are two papers that meet this description. There's the
short summary:
gr-qc/9710007
Quantum Geometry and Black Hole Entropy
Phys. Rev. Lett. 80 (1998) 904-907
and then there's the long paper that fills in all the details:
gr-qc/0005126
Quantum Geometry of Isolated Horizons and Black Hole Entropy
Adv. Theor. Math. Phys. 4 (2001) 1-94
>The assumptions don't seem to be derived from a *theory*
>but instead from the beginning *attempts* at a theory.
I guess this all depends on what you demand from a "theory".
Personally I'm pretty relaxed about what I call a theory.
As far as I'm concerned, any sort of framework where you start
with a bunch of assumptions and can use them to (perhaps
approximately) calculate answers to (potentially) physical
questions deserves this name. This seems to mesh with general
physics practice.
But I think we can agree not to argue about the definition
of "theory". To me, it's more interesting to ponder the specific
pros and cons of the above calculation.
The good news is, it's mathematically rigorous. This is
pretty rare in physics. Compare for example the computation
of the Lamb shift in QED. It's impressively accurate, but nobody
knows any rigorous consistent formulation of QED!
The bad news is, it's based on a theory with some holes that
still need to be filled in. For example, in loop quantum gravity
nobody can agree on the right formula for the Hamiltonian
constraint. Also, the theory involves a mysterious new constant,
the Immirzi parameter, which we cannot yet determine on theoretical
grounds.
The good news is, our calculation only uses a weak assumption
about the Hamiltonian constraint, which holds for all known
proposals about what it might be. Also, our calculation can
be used to pick out a unique "best" value of the Immirzi parameter.
The bad news is, the whole goal of trying to find a correct
Hamiltonian constraint for loop quantum gravity might be doomed
to failure.
The good news is, Martin Bojowald's latest work suggests otherwise!
And so on. Clearly it's all a bit tentative and a bit of a mess.
However, this is the rule rather than the exception in physics.
I'd rather get in there and try something than keep my hands clean
and watch other people quantize gravity.
Ilja Schmelzer
>>>> Diff symmetry as well as gauge symmetry are clearly broken. [...]
>>>> I have no constraints. [...]
>>>> I have local operators g_ij(x_k,t), A_i(x_k,t) and so on from the
>>>> start.
>>> Then you have too many degrees of freedom---six per point rather
>>> than two. To have any hope of getting a good classical limit, where
>>> ``good'' means ``in agreement with simple observation,'' you have to
>>> get rid of the extra degrees of freedom.
>> No, I havn't. Its sufficient if these extra degrees of freedom do not
>> interact with the others in the classical limit. Or interact in a
>> sufficiently weak way (via some small cosmological constants).
> Agreed---it's sufficient to have the extra degrees of freedom decouple.
> Note that they have to decouple from everything, including the ``physical''
> metric degrees of freedom, since otherwise you'd get violations of the
> strong equivalence principle.
If they decouple from matter and have a weak coupling with the metric
they look like dark matter. Not really a problem with observation
today.
> There are two ways I know to make such decoupling occur. One is to
> have the extra degrees of freedom acquire large masses; the other is
> to have constraints. Do you have another?
Yep. The additional degrees of freedom are essentially the preferred
background. A very specific set of degrees of freedom. If we couple
them like in GET (gr-qc/0001101) (or RTG - hep-th/9711147) with the
metric, we have nor constraints, nor large mass. The background
coordinates are harmonic (thus, massless if we handle them as
"fields"), or the graviton may be interpreted as massive (in RTG), but
with very small mass.
> Now, if you're obtaining your discrete equations of motion by simply
> discretizing general relativity on a lattice, then in fact you *do* have
> constraints. Four of the ten field equations involve no time derivatives
> in canonical form. (They have first time derivatives of the metric, but
> these only appear in combinations that become the undifferentiated
> canonical momenta.) Unless your discretization somehow introduces
> new time derivatives, these equations will continue to be constraints.
>
> Thus my questions still stand. There are basic features of the
> quantization of a constrained system (observables commute with
> constraints, wave functions are annihilated by constraints) that
> are independent of the details of the system being quantized.
> Moreover, to get a consistent classical limit, you have to find an
> operator ordering in which the algebra of constraints closes.
This depends on your interpretation what are the "true" steps of
freedom. If you believe that the constraints appear only because you
use inappropriate variables to describe the system, and the "true"
configuration space is the factor space, then you indeed have to
follow this scheme.
But if you, instead, believe that the constraints are only an artefact
of the classical large distance approximation, and the steps of
freedom you use to describe the physics are already the true steps of
freedom, then you don't have to follow this scheme. Instead, you add
some regularizing term which breaks the symmetry related with the
constraints and have no longer any constraints.
This last way is usually ignored because it destroys the symmetries we
like to preserve. I agree with these feelings, and my currently
preferred way of quantizing gravity does not remove all constraints
because of these reasons. But this is not the question discussed here
- which is if there appear problems in the classical large scale limit
with observation.
>> It seems this confuses GR time and quantum time. Of course GR time
>> is part of the g_ij(x_k). But its unrelated to quantum time, which
>> is absolute and unrelated to GR time in this approach.
> Is your quantum evolution described as evolution in this ``quantum
> time''? If so, how do recover the classical limit, in which the
> evolution equations describe evolution in what you call ``GR time''?
> (I think you can't really mean that the two are ``unrelated.'')
In the classical limit the evolution equations describe evolution in
some (i.e. harmonic) coordinate condition. Thus, quantum time becomes
harmonic time, and remains unrelated to GR proper time.
>> In my understanding we have several limits: the large distance limit
>> regularization -> GR which is purely classical (at least I see no need
>> to consider here quantum questions). And the limit h-> 0 between
>> QG and the regularized theory.
> You can't separate the two limits. That's one of the reasons quantizing
> gravity is so hard. Quantization puts certain constraints on the theory
> that are at least well hidden in the classical theory. The basic problem
> is that the obvious ``observables'' in the classical theory (discretized
> or not) are not quantum observables---a quantum observable must
> commute with the constraints, and it is *extremely* difficult to find
> any such quantities at all.
But this is not the problem in a theory without constraints. Which is
the way of quantization discussed here.
Toby Bartels
John Baez wrote:
>Toby Bartels wrote:
>>I read it. At least I think that I read it.
>>I found a paper on gr-qc by Ashtekar, Baez, & Krasnov
>>which calculated the entropy of a wide class of nonrotating black holes,
>>reproducing the Bekenstein Hawking formula for all of them
>>with a single free (Immirzi) parameter across the board.
>>That should be it, right?
>There are two papers that meet this description. There's the
>short summary: gr-qc/9710007
>and then there's the long paper that fills in all the details: gr-qc/0005126.
The short summary was also written by Korichi.
I read it too, but that's not the one that I meant.
>>The assumptions don't seem to be derived from a *theory*
>>but instead from the beginning *attempts* at a theory.
>I guess this all depends on what you demand from a "theory".
>Personally I'm pretty relaxed about what I call a theory.
>As far as I'm concerned, any sort of framework where you start
>with a bunch of assumptions and can use them to (perhaps
>approximately) calculate answers to (potentially) physical
>questions deserves this name. This seems to mesh with general
>physics practice.
Sure, you can do that, and then M theory gets to be a theory too.
That's fine, but that's apparently not what Witten was doing.
Then M theory doesn't get to be a theory. That was my whole point.
>But I think we can agree not to argue about the definition
>of "theory". To me, it's more interesting to ponder the specific
>pros and cons of the above calculation.
OK.
-- Toby
to...@math.ucr.edu
John Baez
Toby Bartels <to...@math.ucr.edu> wrote:
John Baez wrote:
>>There are two papers that meet this description. There's the
>>short summary: gr-qc/9710007
>>and then there's the long paper that fills in all the details: gr-qc/0005126.
>The short summary was also written by Korichi.
Whoops! Very naughty of me to leave out a coauthor like that!
Let's get his name right, too: it's Alejandro Corichi.
The short summary article on black hole entropy from loop quantum
gravity contained a description of the classical phase space
for nonrotating event horizons, followed by a quantization of
this phase space and an entropy calculation. It was written by
Ashtekar, Baez, Corichi and Krasnov. The details appear in two
long papers. First there's a paper on classical stuff by Ashtekar,
Corichi and Krasnov:
gr-qc/9905089
Isolated Horizons: the Classical Phase Space
Adv. Theor. Math. Phys. 3 (2000) 419-478
Then there's a paper on quantum stuff and entropy by
Ashtekar, Baez and Krasnov - that's gr-qc/0005126.
>>Personally I'm pretty relaxed about what I call a theory.
>>As far as I'm concerned, any sort of framework where you start
>>with a bunch of assumptions and can use them to (perhaps
>>approximately) calculate answers to (potentially) physical
>>questions deserves this name. This seems to mesh with general
>>physics practice.
>Sure, you can do that, and then M theory gets to be a theory too.
>That's fine, but that's apparently not what Witten was doing.
>Then M theory doesn't get to be a theory. That was my whole point.
Right. We agree on everything here.
Steve Carlip
If I understand you correctly, you're proposing to modify the original
Lagrangian in a way that eliminates the constraints. For example, the
Hamiltonian constraint occurs in the Lagrangian the combination NH,
where N is the lapse functions, essentially 1/sqrt{g^{00}}. To remove
this as a constraint, you need to add some term that involves a time
derivative of N, then redo the canonical analysis to find the momentum
conjugate to N, etc.
Let me again stress that putting the theory on a lattice is not enough,
even though it seems to break diffeomorphism invariance; there are
still four sets of lattice equations that involve no time derivatives of
the canonical variables. To eliminate the constraints, you have to add
new time derivatives by hand.
Then you have a new set of questions:
1. The modified field equations are no longer those of GR. For example,
if you add a term \lambda dN/dt to remove the Hamiltonian constraint,
you get an equation of motion H = -d\lambda/dt rather than H=0.
Since the initial value of \lambda is arbitrary (that's what happens
when you remove constraints---you get new fields with arbitrary
initial values), it certainly doesn't generally decouple from the usual
gravitational fields. So how do you recover GR?
You might object that I chose the wrong way to eliminate the
constraints. Maybe, but you then need to show that there's a ``right''
way---that there is some way to modify the classical Lagrangian
that leads to an unconstrained system but in which the eight new
phase space degrees of freedom, which can now have arbitrary initial
values, still decouple well enough from the four standard ``graviton''
degrees of freedom to leave the classical theory intact.
2. If you eliminate the constraints, you eliminate the corresponding
conservation laws. You said earlier that you proposed to keep some
``translational'' invariance to protect energy conservation. But then
there will be a corresponding constraint.
In short, what you're suggesting is not a way of quantizing GR, but a way
of quantizing a classical theory that is very different from GR (for one
thing, because its space of solutions is much, much larger than that of GR).
Before you can talk about quantization, you have to show that this classical
theory really reduces to GR in some limit, and that all the extra degrees
of freedom don't muck up the classical predictions.
Steve Carlip
Ralph Hartley
particular comment set something off.
Steve Carlip wrote:
> 2. If you eliminate the constraints, you eliminate the corresponding
> conservation laws. You said earlier that you proposed to keep some
> ``translational'' invariance to protect energy conservation. But then
> there will be a corresponding constraint.
One needs to be careful in rejecting a theory just because it isn't
invariant with respect to some symmetry. The symmetries of a fine
scale theory can be very different from those of its large scale
limit.
Most physicists are familiar with "spontaneous symmetry breaking"
where the microscopic theory has some symmetry which is not observed
macroscopically. The underlying equations have a symmetry, but the
solutions that can actually be observed do not.
I think it is also important to remember that the reverse can also
happen; macroscopic observations display a symmetry that is not
present in a microscopic description.
You don't have to look far for an example, in fact you are probably
looking at one right now - the glass screen of your CRT.
At the atomic level, glass is not at all uniform or isotropic. The
neighborhood of an individual atom has nearly as much regular
structure as in a perfect crystal. If we were interested in the path
of an electron confined to that neighborhood, assuming translation or
rotation invariance would be a very bad approximation. Some symmetry
remains, even in a crystal, but this usually a discrete subgroup of
the normal symmetries of space.
Looked at macroscopically, however, glass is amazingly uniform, and
almost perfectly isotropic. What happened? Because glass is a
disordered state, all the local violations of translational and
rotational symmetry canceled out. If you shine a light trough the
glass, at least if its wavelength is much longer than the correlation
length, it will act just like a uniform continuum.
It is usually assumed (I don't know if there is any proof) that
materials with disordered states will also have lower energy
crystalline states, but in the bulk limit the relaxation time may be
infinite (below some temperature), so those states are completely
inaccessible.
Now it is quite possible that in the "true" theory of quantum gravity
there really is no continuous manifold at all, that space-time is
really some sort of discrete structure. It is quite difficult (I don't
think anyone knows if it is possible) to make a theory that involves
such a discrete structure, but that also has all the symmetries that
we have come to expect of space-time. But restricting ourselves to
such theories is much too confining; it is possible that there are
many perfectly consistent, simple, theories that agree perfectly with
any experiment we have been able to do, but that have much smaller
symmetries than we expect.
If such a theory was true, then then some of our much loved
conservation laws (energy, momentum, angular momentum etc.) would also
be approximations. One would then expect to have violations of those
laws in some experiments. However these would be experiments we can't
actually do now. Planck energy collisions or the final evaporation of
black holes or something like that would be needed.
Steve Carlip wrote:
> In short, what you're suggesting is not a way of quantizing GR, but a way
> of quantizing a classical theory that is very different from GR (for one
> thing, because its space of solutions is much, much larger than that of GR).
> Before you can talk about quantization, you have to show that this classical
> theory really reduces to GR in some limit, and that all the extra degrees
> of freedom don't muck up the classical predictions.
But the correct theory might not be obtainable by quantizing GR, or by
quantizing any classical theory for that matter. It seems to me that
what we really need (given that experiments that test quantum gravity
are going to be few and far between) is some sort of taxonomy of ALL
the theories that have GR (and quantum field theory) as a large
scale/low energy limit.
Unfortunately it may be very difficult, given a fine scale theory, to
figure out its large scale consequences. I know that determining the
large scale properties of a material from its microscopic properties
can be quite hard.
Also, aren't there some results that suggest that any theory that has
anything like a spin-2 field is going to look a lot like GR at large
scales? Doesn't that suggest that the requirement that a theory reduce
to GR in a classical limit is a very WEAK constraint?
I guess getting a sufficiently small cosmological constant is a
non-trivial requirement. But still, it might be quite possible to go
from NO consistent theory of quantum gravity to millions of different,
but experimentally indistinguishable ones. Of course the first few are
always the hardest.
Ralph Hartley
Steve Carlip
> One needs to be careful in rejecting a theory just because it isn't
> invariant with respect to some symmetry. The symmetries of a fine
> scale theory can be very different from those of its large scale
> limit.
Yes, certainly. But the trick is to get from an abstract statement
like this to a specific model.
> Steve Carlip wrote:
>> In short, what you're suggesting is not a way of quantizing GR, but
>> a way of quantizing a classical theory that is very different from
>> GR (for one thing, because its space of solutions is much, much
>> larger than that of GR). Before you can talk about quantization,
>> you have to show that this classical theory really reduces to GR
>> in some limit, and that all the extra degrees of freedom don't muck
>> up the classical predictions.
> But the correct theory might not be obtainable by quantizing GR, or
> by quantizing any classical theory for that matter. It seems to me that
> what we really need (given that experiments that test quantum gravity
> are going to be few and far between) is some sort of taxonomy of ALL
> the theories that have GR (and quantum field theory) as a large
> scale/low energy limit.
This is awfully ambitious, but actually one can get part way there. There
are a few things you need to get GR as an effective field theory: a massless
spin two excitation, a way of eliminating extraneous spin zero and one
pieces, and a universal coupling to the stress-energy tensor. Getting a
spin two excitation is not too hard, though I certainly don't know of any
simple characterization of all theories with such excitations. But keeping
it massless and projecting out lower spin components is tricky; the only
way I know to do that is to impose a gauge invariance that looks at least
locally like diffeomorphism invariance. This may be familiar from QED;
the photon's masslessness is a direct consequence of gauge invariance.
Of course, as you say, the invariance may just be an approximation, but it
has to be a good enough one to agree with some pretty strong observational
limits. And it also has to extend to any matter that couples to gravitation,
because that's the only way to get a universal coupling to stress energy.
Again, this might be an approximation, but it has to be a good enough
one to explain some very stringent tests of the equivalence principle.
There is, incidentally, a startling result that a theory with a massless
graviton is qualitatively different from one with a graviton of arbitrarily
small mass: in the m->0 limit, the extra polarization state of the massive
theory doesn't decouple, at least in a Minkowski background. One result
of this ``Van Dam-Veltman-Zakharov discontinuity'' is that the bending
of light in a gravitational field with massive gravitons is only 3/4 of the
GR value, even in the m->0 limit---this is possible because the limit
isn't smooth. This might mean that you really need something like exact
diffeomorphism invariance, since a not-quite-invariant theory would
not quite project out the extra polarizations and would give a not-quite-
massless graviton. I don't think this has been fully thought out, but it at
least suggests that GR is not as universal as one might expect.
Steve Carlip
t...@rosencrantz.stcloudstate.edu
Steve Carlip <sjca...@ucdavis.edu> wrote:
>There is, incidentally, a startling result that a theory with a massless
>graviton is qualitatively different from one with a graviton of arbitrarily
>small mass: in the m->0 limit, the extra polarization state of the massive
>theory doesn't decouple, at least in a Minkowski background. One result
>of this ``Van Dam-Veltman-Zakharov discontinuity'' is that the bending
>of light in a gravitational field with massive gravitons is only 3/4 of the
>GR value, even in the m->0 limit---this is possible because the limit
>isn't smooth.
Wow! That's bizarre.
Has there been much investigation of the necessity of that "in a
Minkowski background" clause? Here's why I ask. You can think taking
m -> 0 as being like taking the length scale of the gravitational
force to infinity, right? So you can imagine that in a background
with a particle horizon things would change when that length scale got
bigger than the horizon. But of course Minkowski space doesn't have a
particle horizon.
If nothing like that does apply, then are we really forced to the
conclusion that the mass of the graviton is exactly zero?
If there's a simple explanation of why the other polarization doesn't
project out that an ignorant cosmologist like me can understand, I'd
love to hear it.
-Ted
Steve Carlip
> In article <9ca6ja$7gj$1...@woodrow.ucdavis.edu>,
> Steve Carlip <sjca...@ucdavis.edu> wrote:
>>There is, incidentally, a startling result that a theory with a massless
>>graviton is qualitatively different from one with a graviton of arbitrarily
>>small mass: in the m->0 limit, the extra polarization state of the massive
>>theory doesn't decouple, at least in a Minkowski background. One result
>>of this ``Van Dam-Veltman-Zakharov discontinuity'' is that the bending
>>of light in a gravitational field with massive gravitons is only 3/4 of the
>>GR value, even in the m->0 limit---this is possible because the limit
>>isn't smooth.
> Wow! That's bizarre.
> Has there been much investigation of the necessity of that "in a
> Minkowski background" clause? Here's why I ask. You can think taking
> m -> 0 as being like taking the length scale of the gravitational
> force to infinity, right? So you can imagine that in a background
> with a particle horizon things would change when that length scale got
> bigger than the horizon.
That's an excellent question. There's been a bunch of recent work on
exactly that issue. The answer (so far) is that classically, the m -> 0
limit is smooth if there is a cosmological constant and the limit is
taken in such a way that m^2/Lambda -> 0. The discontinuity remains
at the quantum level, though, since the extra polarizations still show
up in loops (i.e., as virtual particles). There's a new preprint by Duff
et al. that says the discontinuity first shows up at the quantum level for
M^2 -> 2 Lambda/3.
I don't tknow of any work on what happens with no cosmological constant
in a background with a particle horizon. It would be an interesting project.
> If nothing like that does apply, then are we really forced to the
> conclusion that the mass of the graviton is exactly zero?
Yes. Bizarre indeed!
Steve Carlip
Charles Torre
[Regarding the "Van Dam-Veltman-Zakharov discontinuity" in the
limit m -> 0 for gravitons.]
> The answer (so far) is that classically, the m -> 0
> limit is smooth if there is a cosmological constant and the limit is
> taken in such a way that m^2/Lambda -> 0. The discontinuity remains
> at the quantum level, though, since the extra polarizations still show
> up in loops (i.e., as virtual particles). There's a new preprint by Duff
> et al. that says the discontinuity first shows up at the quantum level for
> M^2 -> 2 Lambda/3.
>
>
>> If nothing like that does apply, then are we really forced to the
>> conclusion that the mass of the graviton is exactly zero?
>
> Yes. Bizarre indeed!
>
I'll second that. I won't swear to it, but
I dimly recall from discussions with Van Dam many years ago that
(1) This type of discontinuity is *not* present for fully
quantized photons, that is, it does not show up at tree
level or in loops. (Indeed, a common trick used in
perturbative QED is - or at least used to be - to give the
photon a small mass to define propagators and remove
infrared difficulties, then take the limit m -> 0 at the
end.)
(2) This type of discontinuity does *not* show up at tree
level for perturbatively quantized non-Abelian gauge
theory, but *does* show up in loops. So this is something
like the graviton on spacetimes with a cosmological
constant. My impression at the time of the discussion was
that, from a path integral point of view on the quantum theory,
the Faddeev-Popov determinant does not appear
correctly in the limit as m -> 0 so that the additional
degrees of freedom present when the mass is non-zero do not
properly decouple. Perhaps something similar is happening
with the graviton on spacetimes with a cosmological constant?
I never could come up with a really quick and dirty way to
explain the disontinuity for gravitons at tree level, though.
All this is very interesting, eh? From the point of view of
QED, we cannot ever really say whether the mass of the
photon is zero, or just really, really small. From the
point of view of perturbative quantum gravity the graviton
must have a zero mass, it seems.
Hmm... since the photon really comes out of electroweak
theory, I wonder if one can use the non-Abelian gauge
theory loop discontinuity to infer the masslessness of the
photon after all?
-charlie
Ilja Schmelzer
> There is, incidentally, a startling result that a theory with a massless
> graviton is qualitatively different from one with a graviton of arbitrarily
> small mass: in the m->0 limit, the extra polarization state of the massive
> theory doesn't decouple, at least in a Minkowski background. One result
> of this ``Van Dam-Veltman-Zakharov discontinuity'' is that the bending
> of light in a gravitational field with massive gravitons is only 3/4 of the
> GR value, even in the m->0 limit---this is possible because the limit
> isn't smooth.
Really interesting. How can this happen? I would understand other
results for gravitational waves, but for light waves?
Some reference (best on LANL)?
Ilja Schmelzer
> If I understand you correctly, you're proposing to modify the original
> Lagrangian in a way that eliminates the constraints. For example, the
> Hamiltonian constraint occurs in the Lagrangian the combination NH,
> where N is the lapse functions, essentially 1/sqrt{g^{00}}. To remove
> this as a constraint, you need to add some term that involves a time
> derivative of N, then redo the canonical analysis to find the momentum
> conjugate to N, etc.
Yep.
> Let me again stress that putting the theory on a lattice is not enough,
> even though it seems to break diffeomorphism invariance; there are
> still four sets of lattice equations that involve no time derivatives of
> the canonical variables. To eliminate the constraints, you have to add
> new time derivatives by hand.
Yep.
> Then you have a new set of questions:
>
> 1. The modified field equations are no longer those of GR. For example,
> if you add a term \lambda dN/dt to remove the Hamiltonian constraint,
> you get an equation of motion H = -d\lambda/dt rather than H=0.
> Since the initial value of \lambda is arbitrary (that's what happens
> when you remove constraints---you get new fields with arbitrary
> initial values), it certainly doesn't generally decouple from the usual
> gravitational fields. So how do you recover GR?
Let's consider here simply GET (get.ilja-schmelzer.net) as an example
of a modification of GR which removes diff invariance and introduces a
preferred frame. Indeed, the modified equations are no longer the
equations of GR. The four additional steps of freedom behave simply
like massless scalar dark matter fields.
> You might object that I chose the wrong way to eliminate the
> constraints. Maybe, but you then need to show that there's a
> ``right'' way---that there is some way to modify the classical
> Lagrangian that leads to an unconstrained system but in which the
> eight new phase space degrees of freedom, which can now have
> arbitrary initial values, still decouple well enough from the four
> standard ``graviton'' degrees of freedom to leave the classical
> theory intact.
GET is such a way IMHO.
> 2. If you eliminate the constraints, you eliminate the corresponding
> conservation laws. You said earlier that you proposed to keep some
> ``translational'' invariance to protect energy conservation. But
> then there will be a corresponding constraint.
GET has a classical global translational symmetry, nothing local.
Thus, its quantization should be no more problematic than quantization
of a classical field theory (like classical condensed matter theory).
If you tell me that even classical translational symmetry is
problematic, ok, let's break it too. Let's give the "preferred
coordinates" of GET some mass-like term. L_GET + f(sum (X^a)^2)
with some quite moderate (smooth, bounded) function f. Of course
that's not beautiful (and that's why I don't propose this as a serious
theory) and the question has to be discussed if this leads to problems
with observation. But that's only a slightly different behaviour of
the additional field steps of freedom - they become massive instead of
massless dark matter.
Of course, I'm also very interested to obtain a better understanding
of quantization problems of classical fields with classical global
translational symmetry, and the problems related with their
constraints. Because that's really the way I want to quantize
gravity.
My main "dirty quantization" scheme becomes now:
L_GR -> L_GET + f(sum (X^a)^2) -> discretization on torus, L_discrete
-> classical canonical quantization.
> In short, what you're suggesting is not a way of quantizing GR, but a way
> of quantizing a classical theory that is very different from GR (for one
> thing, because its space of solutions is much, much larger than that of GR).
> Before you can talk about quantization, you have to show that this classical
> theory really reduces to GR in some limit, and that all the extra degrees
> of freedom don't muck up the classical predictions.
GET equations reduce to GR equations for X,Y -> 0. The theory remains
different even in this limit (no nontrivial topologies allowed, a
global harmonic time coordinate exists, complete solution may be not
complete in the GR sense). But I don't know any observable fact which
allows to use these differences to falsify this theory. An observed
wormhole or causal loop would do it, but I have never seen one.
I think this challenge I'm able to meet for the "dirty theory" L_GET +
f(sum (X^a)^2) too if necessary.
Ilja Schmelzer
> That's an excellent question. There's been a bunch of recent work on
> exactly that issue. The answer (so far) is that classically, the m -> 0
> limit is smooth if there is a cosmological constant and the limit is
> taken in such a way that m^2/Lambda -> 0.
Which sign of Lambda? What about m^2/Lambda = const?
I ask because this remembers Logunovs "relativistic theory of gravity"
(hep-th/9711147). It has a mass term of type m^2(eta_ab g^ab - 2)sqrt(-g)