What is the Connection Between Unitarity and Anomalies in Quantum Mechanics?

[marcus]

Gerard 't Hooft at Kitp25

Here is the audio etc for 't Hooft's Kavli25 talk, but it takes a few minutes to do the audio, so I am going to exerpt a few slides to give an idea of the general message.

The title of the talk was The Future of Quantum Mechanics

http://online.itp.ucsb.edu/online/kitp25/thooft/

http://online.itp.ucsb.edu/online/kitp25/thooft/oh/02.html
http://online.itp.ucsb.edu/online/kitp25/thooft/oh/03.html
http://online.itp.ucsb.edu/online/kitp25/thooft/oh/04.html

http://online.itp.ucsb.edu/online/kitp25/thooft/oh/06.html
http://online.itp.ucsb.edu/online/kitp25/thooft/oh/07.html

http://online.itp.ucsb.edu/online/kitp25/thooft/oh/13.html
http://online.itp.ucsb.edu/online/kitp25/thooft/oh/16.html
http://online.itp.ucsb.edu/online/kitp25/thooft/oh/18.html

slides 2,3,4 say:
---exerpts from 't Hooft slides---
From a pragmatic point of view there is nothing wrong with quantum mechanics...[But] the standard Hilbert space procedure for QM does not go well with gravitation, curved space-time, and cosmology.
Attempting to reconcile General Relativity with QM leads to apparently insurmountable problems...
(for anyone not happy with string theory: even with AdS/CFT, there can be
no locality in 3+1 space-time.)

Why should these problems NOT be related to the question of the foundation of quantum mechanics?

this is a minority's view point, because:
Theorem:"Hidden variables cannot be reconciled with locality and causality."
...
Perhaps Quantum Gravity can be handled by thoroughly reconsidering Quantum Mechanics itself!
---end quotes from slides 2,3,4---

Slides 6 and 7 depict two scientists who personify our expectations of Causality and Locality.

---exerpts from slides 13, 16, 18---

Quantum Mechanics is NOT a theory that describes what is really going on, not an ONTOLOGICAL theory

A key ingredient for an ontological theory: Information loss

A Quantum state is defined to be an equivalence class of states which all have the same distant future...

an apparently non-local, acausal definition!

What we call vacuum state may actually be a quantum superposition of many equivalence classes of ontological states.
---end quotes from 't Hooft---

after the talk there was a panel, including Roger Penrose and James Hartle. this has their slides and audio of their commentary
http://online.itp.ucsb.edu/online/kitp25/zee/

 

[marcus]

I wanted especially to check out what 't Hooft said yesterday at Kavli because Lubos Motl was at some pains to spin 't Hooft as a string supporter, when he posted yesterday on NEW
http://www.math.columbia.edu/~woit/blog/archives/000089.html

---quote from Lubos---
By the way, each string theorist is also using the name of 't Hooft roughly 3 times a day in average. We don't view him as a string theorist, but he would certainly joined the top ten if he described himself as a string theorist. ;-) 't Hooft is teaching a course on string theory and has extensive lecture notes, see

http://www.phys.uu.nl/~thooft/lectures/string.html

The Dutch are very good in our field, I would say, and of course support of their leaders is helpful.
---end quote---

 

[marcus]

't Hooft spoke twice at this conference
there was the 8 October 1:30 PM talk about the Future of Quantum Mechanics, slides of which are quoted here
and also he was on the 9 October 9:30 AM panel, with Ashtekar, on the topic What is Quantum Gravity?---that came right after Witten's talk

so as of now, today, Witten has given his 9:00 talk about the future of String and the panel (including Ashtekar and 't Hooft) has done its thing,
probably short remarks by each of the members.

With luck, the audio for these things should soon be appearing at
http://online.itp.ucsb.edu/online/kitp25/

But it hadnt yet, the last I looked

 

[marcus]

Witten's talk is now available for download, and the panel too I believe.

the panel session, which includes 't Hooft and Ashtekar, is at:
http://online.itp.ucsb.edu/online/kitp25/shenker/

 

[selfAdjoint]

Thanks Marcus for the links. Very interesting.

 

[marcus]

Thanks Marcus for the links. Very interesting.

my pleasure! Peter Woit has commented on Witten's talk http://www.math.columbia.edu/~woit/blog/archives/000090.html

what I noticed about Witten's talk was that it was explicitly aimed at a non-string audience

he explicitly said this at the outset, and even warned people that since it was going to be for outsiders that the questions afterwards should be in keeping with this----he was eliciting non-technical questions (instead of those his string colleagues would want to be asking)

so Witten's message, as I take it, was "this is what we ought to be telling people who are not themselves doing research in our field but who may influence how our field is seen by the general public, the funding agencies, the academic and institutional support people, the scientific community".

As such Witten's talk doubtless served a useful purpose, indeed provided an essential piece the picture for the people there. If they were wondering what line to take with non-string academics and NSF committees etc, or with non-string colleagues at the faculty club or the dean's cocktail party, well this is it.

So it was a useful and essential talk although people may be right in saying it didnt give information from later than 1997 or whatever---original thought or big news.

the key word I heard from both Steven Weinberg and from Edward Witten at this conference on "The Future of Physics" (or as witten talk was titled, "The Future of String Theory", and there may be a tendency to equate)

the key word was "ONLY"

"the one best hope for unifying the forces"
"the only reasonable way to extend particle physics, or to go beyond the standard model"

these are not exact quotes, but give the sense of the key point they were emphasizing. IT MAY NOT (quite) WORK BUT IT IS THE ONLY HOPE

one obvious trouble with adopting that position is that it commits you to deprecate the investigation of other possible approaches-----for various reasons alternatives must be treated with condescension or simply not admitted to exist.

the silence after Ashtekar's panel talk was so remarkable, contrasted with the loud applause for the next speaker, that the panel moderator could not ignore the difference and had to make an extenuating remark.

it was at that point that I felt the wagons were in a circle and everyone outside, even Ashtekar, the token outsider, was a potential Commanche.

 

[selfAdjoint]

Well, Witten's point that superstrings was the only valid generalization of field theory was actually on the money, unless you consider supersymmetric GUTs to be generalizations. The usual argument is that you can't generalize the standard model significantly with the same tools that were used to build the standard model. This is the Coleman-Mandula theorem. Supersymmetry is the one tool beyond the standard model that field theorists are comfortable with (Frank "Tony" Smith's careful constructions with Lie Algebras are not accepted by the community). And string theory is the only generalization of what field theory talks about - from point particles to one dimensional, and now to k-dimensional objects (branes).

Plus string physics is the only accepted physics that even looks at representing both gravity and the standard model (or some GUT). LQG does none of this and follows another path. If it truly accepts non-unitarity as basic then that path is going to be long and lonely.

"Wide is the path that leads to perdition, and many are they who take it, but narrow ans steep is the path that leads to salvation, and few are they who follow it."

 

[marcus]

If it truly accepts non-unitarity as basic then that path is going to be long and lonely.

getting away from unitary time-evolution---something David Gross mentioned in his final talk summarizing questions and future directions---may be a steep and narrow path.

But it is certainly not one that loop gravitists will be traveling alone.

Just in the recent KITP25 conference I heard it mentioned by several speakers
(and you can bet they weren't talking about Loop specifically!)

Gross was especially graphic and empatic about it----talking about a physics "without time" and time being 'emergent' from some more basic timeless model.
He motivated this by remarking that in Gen Rel time is such a "tricky" concept. And he made a point of how radical a step it was because to a large extent physics is "about" this unitary time evolution and predicting the future. Time in some sense more basic to physics than space.
So he had his own point of view, but what he actually said was not really very different from what Rovelli says about physics sans time (or at least without an absolute time and unitary evolution by an ideal clock).

 

[marcus]

http://online.itp.ucsb.edu/online/kitp25/witten/oh/08.html

"It may well be that string theory is the only way to reconcile gravity
and quantum mechanics (as Weinberg suggested Thursday)..."

there is that ONLY WAY theme
it gets harped on in various forms and contexts,
and here is another important theme:

http://online.itp.ucsb.edu/online/kitp25/witten/oh/10.html

"One thing I can tell you, though, is that most string theorists suspect
that 'spacetime' is an
'emergent phenomenon'...


so again the idea recurs that time, fundamentally, does not exist
or at the very least, one would suppose, unitary evolution according
to the absolute time told by an ideal clock would not exist.

We really should quote Weinberg's ONLY WAY statement, since it was the opening address and made this keynote point most radically and emphatically.

 

[marcus]

On my RealOne player the point in Weinberg's talk is called
35:00/47:46

you use the mouse to drag the button along the line until it reads
that it is in minute 35 of the 47 minute talk by Steven, then you press play.

he says:
35:15 "In other words string theory is what it is because it is the ONLY WAY of combining gravity and the standard model in 4 dimensions..."

also back at 34:25 and up ahead at 36:08 and 36:24
in this passage he uses the phrase "only way" some 5 or 6 times
probably more but I stopped counting
basically for about 2 minutes he is just repeating the main message
"only way only way only way..."


36:08 "Maybe that's all that string theory is, maybe string theory is just the only way of doing physics that includes gravity, and quantum mechanics. And if there is some other way---it that's wrong---then we all ought to try hard to find what other way there is..."

36:24 "So my guess is that string theory IS the only way, and that may be what string theory is..."

 

[selfAdjoint]

I want to pick up a point you made in a post or two back. When quantum field theorists speak of emergent time and remark that this is a problem for unitary time evolution, that's different from developing a theory that doesn't have unitarity from the gitgo. LQG doesn't fail unitarity just because it quantizes spacetime, it does so because it's development is non-unitary, at least according to Larsson, and I believe Urs is on board with this thought too.

BTW I agree with Weinberg's 36:08 comment in your post #10, just above. "..we all ought to try hard to find what other way there is...". Most of the "other way" candidates, like Smith's, are at first glance beyond the pale. Somebody should grit her/his teeth and look into these rejecta.

 

[marcus]

BTW I agree with Weinberg's 36:08 comment in your post #10, just above. "..we all ought to try hard to find what other way there is...".

there is a great quote from Feynman re how important it is to do this---where he also recognizes the serious career risk of
riding the dark horse

the way the academic ladder is set up (maybe in physics more than mathematics) can apparently offer someone a choice between boarding the bandwagon and languishing in genteel obscurity

and yet he says it is necessary, absolutely essential to progress in theoretical physics, and he urges people to do it despite the possible hardships

-----------
my hunch is that whether or not it is to young researchers advantage to check out alternatives to the bandwagon---well, that depends a lot on how the establishment is set up and who the administrators are. I have a hunch
that, say, Hermann Nicolai is making it worth some young people's while to hazard a look into alternatives. My impression is that he understands better than some others the strategy of diversifying the search and crossing lines
--------

I think it is pathetic that Harvard does not have a couple of world class young Loop Gravitists on the faculty----since they have first rate string theorists.

for one thing it would help keep Lubos in bounds
there would be someone with the authority to contradict
overthetop partisan assertions.

If someone like Nicolai were chairman of harvard physics department, there would be Loop counterparts of Lisa Randall Nima Arkani-Hamed

 

[marcus]

I guess you saw that this month's Americas conference on Quantum Gravity has a talk by Hanno Sahlmann on doing string theory using LQG methods

that is a sample of what goes on at Perimeter--where they have both stringists and loop gravitists in residence and host both string and LQG workshops (not simultaneously! not combined! but at least they do both)
and Sahlmann is talking on the loop-string.

http://www.perimeterinstitute.ca/activities/scientific/PI-WORK-2/participants.php

BTW at the same conference Smolin's talk is about deriving "physics from loop gravity"-----work was bound to get started on that, sometime or other.

I think Steven Weinberg said "only way" about 6 times in 2 minutes.

I went and got the link to the PF post about this:
https://www.physicsforums.com/showthread.php?p=336733#post336733
and the exact titles of their talks. Curious what theyll say.
Sahlmann: String Theory with LQG methods

Smolin: Physics from Loop Quantum Gravity

 

[selfAdjoint]

I will especially want to hear/read Sahlmann's talk. He has seemed to me to be the deepest and most careful theorist in that group. Smolin, well he is in the business of throwing out challenges.

Marcus, I wish you could get off the bandwagen for LQG as the alternative to String physics. It's not. LQG has no specific LQG account of particle physics, only a proposal to interface the standard model to their quantum spacetime. If you want to account for the unexplained parameters in the standard model you have to look elsewhere.

 

[marcus]

LQG has no specific LQG account of particle physics, only a proposal to interface the standard model to their quantum spacetime. If you want to account for the unexplained parameters in the standard model you have to look elsewhere.

What proposal do you mean?

 

[nonunitary]

Hi,

Do you have the reference of who and why says that LQG is non-unitary?

 

[marcus]

I want to pick up a point you made in a post or two back. When quantum field theorists speak of emergent time and remark that this is a problem for unitary time evolution, that's different from developing a theory that doesn't have unitarity from the gitgo. LQG doesn't fail unitarity just because it quantizes spacetime, it does so because it's development is non-unitary, at least according to Larsson, and I believe Urs is on board with this thought too.
...

How is it different?

I suppose one could go back and do a historical search to find out how many years QG theorists have been saying that unitary time evolution may indicate a model is unphysical.

My perception is that Loop gravitists are not in the business of criticizing string and rarely point this out---but on one occasion last year Rovelli did speak out about it. He said that the fact that string models have unitary time evolution was grounds for suspecting something was wrong with them.
Coming from GR that's not what one expects.

As far as I know Loop has always been frankly nonunitary---but I would have to go back and read old papers to nail that down. It certainly was explicit in Rovelli's "Dialog" back in October 2003, and it was hardly news then but just something taken for granted.

With all respect to Urs, Thomas, whoever else, they seem to think they discovered something when they learned that LQG was nonunitary!
It seems they made a big fuss about discovering something that
quantum gravitists had been already frankly pointing out for some time.

Last year Rovelli's book has a big discussion of "physics without time", much of a chapter is devoted to discussing time, the different notions of time, why not to expect a fundamental theory to be one of time-evolution
(and that wasnt especially new, he has earlier papers about that, but it was part of the book because the book tries to be comprehensive) and now in October 2004 we have DAVID GROSS talking about "physics without time."

He presents it as if it is a new cutting edge recognition, looking ahead to the future of string theory. Welcome to the club
So my question is How is it different?

 

[marcus]

Hi,

Do you have the reference of who and why says that LQG is non-unitary?

I think you may know the literature better than I, so please throw in any ideas, references etc.

I will find the place in Rovelli's "Dialog" and paste it in. then there is chapter 2 of Rovelli's book.

I would ask you. Do you know any basic treatment of LQG where it says that there is unitary time-evolution? I was under the impression that a fundamental theory could not have this, but I could easily be mistaken.

(the business is different with Urs and Thomas Larsson "discovering" that LQG was nonunitary and thinking that it was somehow inadvertent and a mistake of the quantum gravitists! That is fairly recent and I have it
only second hand from selfAdjoint)

I will fetch that bit of dialog

Rovelli
http://arxiv.org/hep-th/0310077 .
October 2003

---exerpt from page 18 of "Dialog"---
Simp - ... Is loop gravity unitary?
Sal – No, as far as I understand.
Simp – This is devastating.
Sal – Why?
Simp – Because unitarity is needed for consistency.
Sal – Why?
Simp – Because without unitarity probability is not conserved.
Sal – Conserved in what?
Simp – In time.
Sal – Which time?
Simp – What do you mean “which time?”. Time.
Sal – There isn’t a unique notion of time in GR.
Simp – There is no coordinate t?
Sal – There is, but any observable is invariant under change of t, therefore everything is constant in this t just by gauge invariance.
Simp – I am confused.
Sal – I know, it is always confusing. . . Nonperturbative GR is quite different from physics on Minkowski . . .
Simp – Do we really need to get in the conceptual complications of GR?
Sal – Well, if we are discussing the theory that is supposed to merge GR and QM...
Simp – String theory merges the two without these complications.
Sal – This is why I think that string theory does not really merge GR and QM.

---end exerpt---

 

[selfAdjoint]

Last year Rovelli's book has a big discussion of "physics without time", much of a chapter is devoted to discussing time, the different notions of time, why not to expect a fundamental theory to be one of time-evolution
(and that wasnt especially new, he has earlier papers about that, but it was part of the book because the book tries to be comprehensive)

It is NOT just about time, time is a side issue. The nonunitary theory does not conserve probability, the sum of the probabilities over a complete set of alternatives doesn't add up to 100%.

And Rovelli may talk about the absence of time in his book, but it does not, at least the draft I saved does not, contain any of the words unitary, unitarity, nonunitary or nonunitarity. I repeat, these are two different issues.

 

[marcus]

The Dialog here was originally part of Rovelli's book---in October/November drafts---but in the December 31 draft, which I saved the Dialog was removed. I understand that a publisher had asked for it and Rovelli decided to let it be published separately.

...Rovelli
http://arxiv.org/hep-th/0310077 .
October 2003

---exerpt from page 18 of "Dialog"---
Simp - ... Is loop gravity unitary?
Sal – No, as far as I understand.
Simp – This is devastating.
Sal – Why?
Simp – Because unitarity is needed for consistency.
Sal – Why?
Simp – Because without unitarity probability is not conserved.
Sal – Conserved in what?
Simp – In time.
Sal – Which time?
Simp – What do you mean “which time?”. Time.
Sal – There isn’t a unique notion of time in GR.
Simp – There is no coordinate t?
Sal – There is, but any observable is invariant under change of t, therefore everything is constant in this t just by gauge invariance.
Simp – I am confused.
Sal – I know, it is always confusing. . . Nonperturbative GR is quite different from physics on Minkowski . . .
Simp – Do we really need to get in the conceptual complications of GR?
Sal – Well, if we are discussing the theory that is supposed to merge GR and QM...
Simp – String theory merges the two without these complications.
Sal – This is why I think that string theory does not really merge GR and QM.

---end exerpt---

I will look for some other things that are in the 31 December draft for you.

 

[marcus]

It is NOT just about time, time is a side issue. The nonunitary theory does not conserve probability, the sum of the probabilities over a complete set of alternatives doesn't add up to 100%.

And Rovelli may talk about the absence of time in his book, but it does not, at least the draft I saved does not, contain any of the words unitary, unitarity, nonunitary or nonunitarity. I repeat, these are two different issues.

It took me less than a minute to find the discussion of unitarity in my draft of Rovelli.

I just looked in the Table of Contents to where it said "section 10.1.3
Space, time, and unitarity"

I hope you have the 31 December 2003 draft. I think it is still available at Rovelli's site and it was the most recent draft made available. I would recommend printing out this version. it is a good book and having it in paper, with the table of contents, makes stuff real quick to find.

 

[marcus]

And Rovelli may talk about the absence of time in his book, but it does not, at least the draft I saved does not, contain any of the words unitary, unitarity, nonunitary or nonunitarity. I repeat, these are two different issues.

rovelli's book does contain the word unitary and the word unitarity. it is easy to find. you just go to section 10.1.4
I wish you had the most recent available addition so we could all be
"be on the same page"
Anyway, you are mistaken about it not containing any of those words.

I also believe you are mistaken to disconnect the two issues with such emphasis. the two issues, as rovelli discusses them in that section, are very intimately related.

here are quotes from the section of the book I mentioned:

"In conventional QM and QFT, unitarity is a consequence of the time translation symmetry of the dynamics. In GR there isn't, in general, an analogous notion of time translation symmetry. Therefore there is no sense in which conventional unitarity is necessary in the theory.

One often hears that without unitarity a theory is inconsistent. This is a misunderstanding..."

"... the idea of the necessity of unitary time evolution...is...inappropriate to describe general relativistic quantum physics."

selfAdjoint, notice that he is not even talking about LQG here, he is a relativist (expert in GR) and he is talking about GR----he says there is no time translation symmetry notion in GR analogous to that in QM and QFT.
Unitary time evolution would be appropriate in Quantum Gravity only if the classical theory had time translation symmetry which it does not. The two issues are inseparably linked, your assertion to the contrary, with all respect, notwithstanding.

Indeed it seems self-evident that the two issues----time and unitary-time-evolution---are inseparably bound up with each other regardless of what Rovelli says about it. We don't have to take his word for it---we can see at once for ourselves that they are linked.

 

[selfAdjoint]

Unitarity is not important in GR because it is a classical theory. But conservation of probability is important in quantized theories because probabilities is what those theories produce.

 

[marcus]

Unitarity is not important in GR because it is a classical theory. But conservation of probability is important in quantized theories because probabilities is what those theories produce.

Please read what Rovelli says more carefully

You should have the 31 December draft. It has been available free online for over 9 months----by agreement with the Cambridge U. P.

Rovelli has answered your objection, but instead of responding to what he says and proceeding to the next round you are repeating your same objection.

We are in section 10.1 "The physical picture of Loop Gravity"
He is giving an argument that "there is no sense in which conventional unitarity is necessary" in LQG.

selfAdjoint, please continue! Maybe Rovelli is wrong! Maybe you will win the argument! But please get the recent draft of his book. And please listen to what he says.

 

[nonunitary]

Hi,

I think what Rovelli means is precisely that it really does not make sense to ask for unitary evolution for a theory that has time evolution, classically, as a not unique concept. This is the celebrated problem of time. Rovelli is not carefull in making the distinction between gauge time evolution and 'real time evolution' in GR. This has to do witgh the qustion of the topology of the space (closed, asymptotically flat, etc). Any way, all of his remarks apply equaly to any version of quantum gravity. As fas as I know, nobody has ever written a time evolution operator for LQG, so the question of whether opne can make it a unitaty operator has not shown up at the practical level. Saying that LQG is not unitary
is, IMHO, completely out of place.

 

[marcus]

Hi,

I think what Rovelli means is precisely that it really does not make sense to ask for unitary evolution for a theory that has time evolution, classically, as a not unique concept. This is the celebrated problem of time. Rovelli is not carefull in making the distinction between gauge time evolution and 'real time evolution' in GR. This has to do witgh the qustion of the topology of the space (closed, asymptotically flat, etc). Any way, all of his remarks apply equaly to any version of quantum gravity. As fas as I know, nobody has ever written a time evolution operator for LQG, so the question of whether opne can make it a unitaty operator has not shown up at the practical level. Saying that LQG is not unitary
is, IMHO, completely out of place.

Thanks nonunitary,
and I think you were running to grab coffee and maybe teach a class because your orthography here is very hurried, so with permission I will correct the spelling.

I think what Rovelli means is precisely that it really does not make sense to ask for unitary evolution for a theory that has time evolution, classically, as a not unique concept. This is the celebrated problem of time. Rovelli is not carefull in making the distinction between gauge time evolution and 'real time evolution' in GR. This has to do with the qustion of the topology of the space (closed, asymptotically flat, etc). Any way, all of his remarks apply equaly to any version of quantum gravity. As fas as I know, nobody has ever written a time evolution operator for LQG, so the question of whether one can make it a unitary operator has not shown up at the practical level. Saying that LQG is not unitary is, IMHO, completely out of place.

 

[selfAdjoint]

Please read what Rovelli says more carefully

You should have the 31 December draft. It has been available free online for over 9 months----by agreement with the Cambridge U. P.

Rovelli has answered your objection, but instead of responding to what he says and proceeding to the next round you are repeating your same objection.

We are in section 10.1 "The physical picture of Loop Gravity"
He is giving an argument that "there is no sense in which conventional unitarity is necessary" in LQG.

selfAdjoint, please continue! Maybe Rovelli is wrong! Maybe you will win the argument! But please get the recent draft of his book. And please listen to what he says.

Okay, I downloaded the Dec 30 draft, and found what he says.

In conventional QM and QFT unitarity is the consequence of the time translation symmentry. In GR there isn't, in general, an analogous notion of time translation symmetry. Therefore there is no sense in which unitarity is necessary to the theory.

In my view the conclusion doesn't follow that because time translation symmetry cannot be defined in a theory, therefore that theory doesn't need unitarity. GR hasn't got TTS and therefore it hasn't got U; but that's OK for GR because as I said before (sorry if you're tired of hearing it) GR is classical. You have to show in a separate argument that this happy result works for a QUANTUM theory, of gravity or anything else.

But not to rely on my feeble understanding of the question, I posted the same quote to Thomas Larsson's thread over on S.P.R. and asked him to comment. We'll see if he will give us the benefit of his understanding.

 

[marcus]

..., I posted the same quote to Thomas Larsson's thread over on S.P.R. and asked him to comment. We'll see if he will give us the benefit of his understanding.

now we are cooking!

these are interesting questions and it is good to be able to consider them together.

just for being specific, I presume that what we mean by a theory having UNITARITY means the same as its having a UNITARY TIME-EVOLUTION operator on its hilbertspace

that is, LQG can have plenty of unitary operators that we just are not talking about---but when we say that the theory is not unitary or that it does not have unitarity we mean that it lacks a time-evolution operator of the specified sort.

now we see a really simpleminded connection----maybe in this form too simple to apply even, but one can ask: suppose there is a theory which does not have any idea of time---how then can it have time-evolution operator?---how then can it be expected to have unitarity?!

i do not want to say that this form of the statement applies precisely to LQG, I say it to point up the connection between a theory having unitarity and a theory having a well-delineated notion of time, a kind of standard clock.

I am looking forward to seeing some response from SPR---good idea to put the question there!

 

[nonunitary]

Marcus,

Thanks a lot! please do correct all my mistakes.
I think the discussion is sharpening. And that is good.

 

[selfAdjoint]

that is, LQG can have plenty of unitary operators that we just are not talking about---but when we say that the theory is not unitary or that it does not have unitarity we mean that it lacks a time-evolution operator of the specified sort.

Yes, your statements so far have indicated that you take that meaning. I on the other hand am not so concerned about time evolution as I am about non-unitarity in basic quantum events. I sure hope Larsson gives us some direction on this, because I think we're both arguing a bit over our heads.

 

[marcus]

... I on the other hand am not so concerned about time evolution as I am about non-unitarity in basic quantum events.

?

what kind of operator is a "basic quantum event" operator?

understandably one wants certain operators to be unitary on the state space because, for instance, norm-preserving means probability preserving.
but what are the "basic quantum event" operators you are concerned might not be unitary?

it would help make what you are saying NOT be over my head, for one, if you would say or give an example.

 

[marcus]

I found the Thomas Larsson thread I think you were talking about!
SPR is hard to use because it may take 4 or 5 days for something to post, or so I find, so i don't watch it too carefully. I hope Thomas Larsson can come here, so we can actually talk (more in "real time")

Now I understand that you (and Larsson) are more worried about DIFFEOMORPHISM INVARIANCE. When you said "unitarity" I was misled to think you meant unitary time-evolution. IIRC in LQG one actually does not want to find unitary representation of the diffeomorphism group----AFAIK this is just something that Larsson believes, which may not be true.

I think Larsson gets off on the wrong foot already in the 3rd or 4th sentence of the first post of that thread. I will show you where. The thread (I am guessing this is the one you refer to altho you didnt name it) is, I think,
"Diffeomorphism, LQG, and positive energy"
and already right at the start he says this:

---quote SPR Larsson thread---
"This paper gives a good background for a general discussion of
canonical quantization of general-covariant theories like general
relativity. At some stage in this process, we want to find a
unitary representation of the diffeomorphism generators on some
Hilbert space. Ideally, we want our representations of the
diffeomorphism group to be non-trivial, irreducible, unitary,
anomaly-free, and of lowest-energy type. Unfortunately, a theorem
states that no such representation exists, which is major
complication..."

---end quote---

I have bolded to show what I think is his mistaken assumption.

With all respect to Larsson, whose views on string theory are knowledgeable and penetrating, I do not believe it is true that
"At some stage in this process, we want to find a
unitary representation of the diffeomorphism generators on some
Hilbert space."

In other words, I reckon the Larsson thread is based on a misconception at the very start, besides which it seems more concerned with the work of string theorists (like Helling, Policastro) and string-related writing than with the main LQG canon. However it would be very nice if one could talk to Larson and help him to make contact with mainstream LQG!

I would like to ask that anyone interested in diffeomorphisms and LQG look at two short passages on page 41 and page 56 of a standard Ashtekar pedagogical reference. this will give a good idea of the role of diffeos in LQG. This is dated April 2004 and is perhaps more up to date than Rovelli's December 2003 draft.

Ashtekar and Lewandowski
Background Independent Quantum Gravity
http://arxiv.org/gr-qc/0404018

---quote from page 41---
Let us summarize our discussion of quantum kinematics for background independent theories of connections. In section IVC1, we introduced a Lie algebra of holonomy and flux functions on the classical phase space [48]. In the subsequent sub-sections, we constructed a natural, diffeomorphism covariant representation of the quantum analog of this holonomy flux algebra. For pedagogical reasons, we chose a constructive approach and developed the theory step by step starting from quantum mechanics...


...The non-trivial fact is that the structure of Cyl is such that the spectrum is easy to exhibit: it is precisely our space A [40]. Thus, the representation of the algebra of elementary variables we constructed step by step is in fact rooted in the general Gel’fand representation theory. Even though this procedure is quite general and well-motivated, one can nonetheless ask why we did not adopt the more general algebraic approach but focused instead on a specific representation. Interestingly, several partial uniqueness theorems have been established indicating that the requirement of general covariance suffices to select a unique cyclic representation of the kinematic quantum algebra [55–58]. This is the quantum geometry analog to the seminal results by Segal and others that characterized the Fock vacuum in Minkowskian field theories...
---end quote---

now there is a second relevant passage on page 56

---quote from page 56---
"Let us summarize. The basic idea of the procedure used to solve the diffeomorphism constraint is rather simple: One averages the kinematical states with the action of the diffeomorphism group to obtain physical states. But the fact that this procedure can be implemented in detail is quite non-trivial. For example, a mathematically precise implementation still eludes the geometrodynamics program. Furthermore, even the final answer contains certain subtleties. We will conclude by pointing them out..."
---end quote---

Now Larsson presumes that at some point LQG needs to exhibit a
"representation of the diffeomorphism group [which is] non-trivial, irreducible, unitary,...etc etc..."
and AFAIK that is simply wrong.

Am I missing something that Ashtekar is doing and that he didnt tell us about? All i see is that the representation is INVARIANT under diffeomorphisms.
Now maybe you are going to tell me that when Ashtekar says
"...we constructed a natural, diffeomorphism covariant representation of the quantum analog of this holonomy flux algebra."
that this implies also a representation of the diffeo group! even tho he didnt say it. that would be a clever mathematical subtlety.
and maybe Larsson can give arguments why this representation of the diffeo group is bad or fatal or whatever.

but first let's see if it isn't true, as I am claiming, that nowhere is Ashtekar talking about a rep of the diffeo generators, or the diffeo group, and that it does not seem to be a part of the program. Is this not kosher for some reason? SHOULD it be part of the program? Love to hear your and Larsson opinion.

BTW let's not wait for larsson, if it is SPR it may take days

 

[selfAdjoint]

A Unitary transformation of a group or algebra is one where all the transformations in the representation are unitary. A transformation A is unitary if there is a transformation U sunch that $$A = UAU^{\dagger}$$, where $$A^{\dagger}$$ denotes the adjoint. Ashtekar has not claimed this property for his representation.

 

[selfAdjoint]

A Unitary representation of a group or algebra is one where all the transformations in the representation are unitary. A transformation A is unitary if there is a transformation U sunch that $$A = UAU^{\dagger}$$, where $$A^{\dagger}$$ denotes the adjoint. Ashtekar has not claimed this property for his representation.

 

[marcus]

I see that SPR thread was the one you meant and that your post appeared today (13 October). Bravo. Hopefully he will reply before long.

Larsson seem less concerned with unitary time-evolution and more concerned with what I take to be his own narrow definition of
"canonical quantization" which he insists must involve a unitary representation of the group of diffeomorphisms on the manifold.

With other people, when I hear them say "is the theory unitary? does it have unitarity?" what it mostly means AFAIK is "does it have unitary time evolution?" But here we are getting into a different issue.

Rovelli, altho he talks about unitary time-evolution, does not talk about a unitary rep of the diffeo group on the hilbert space. So if we want to connect with Larsson we may need to use some other source----the best i can offer so far is the Ashtekar April 2004 paper (reasonably authoritative, recent, tries to be as rigorous and selfcontained as possible, intended to be introductory)

Since you proposed consulting Larsson as, I guess, a trustworthy LQG critic, I have taken a look at what he is saying lately. he seems very concerned with his particular idea of "canonical" and pointing out how string and loop both fail----could we have a semantic problem here?

I will get some Larsson quotes to illustrate.