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

[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.

 

[marcus]

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.

I'm hip to what unitary means. For me the question is representation of what group?
We seem to be talking at cross purposes since Larsson makes a big point of insisting that any theory of QG have a unitary rep of the diffeo group!

Ashtekar treatment of LQG does not involve (explicitly at least) any representation of the diffeo group at all!

to me this seems the puzzle needing most urgently to be addressed, perhaps it has a simple answer

 

[selfAdjoint]

I also asked Lubos Motl on his new blog to explain the difference between time translation invariance and unitarity. He responded last Monday, scroll down to see it. He generally says what I said, but with a lot more detail and expertise.

 

[marcus]

Now I am not clear what you have been saying, selfAdjoint, because
Lubos equates "unitarity" with having "unitary time-evolution" operators.

If there are no time-evolution operators---nothing corresponding to time---then how can they expected to be unitary?

there is something we are not communicating about, so let me repeat

I may be wrong but I insist that the usual idea of "unitarity" is inseparably bound up with the idea of time, and time evolution. I got the impression you were challenging this. But Lubos bears it out, the Wiki entry on Unitarity, and anything I can remember reading by anybody. To take a radical illustration, if there is no time there can be no unitarity in the usual sense, because there can be no time-evolution.

As for the mathematical definition, I know what a unitary operator on a Hilbert space is, never any doubt about that or that normpreserving can be interpreted as probabilitypreserving.

What interests me is the appearance of a theory without time.

rovelli has a chapter on MECHANICS--- there is pre-relativistic or non-relativisitc mechanics which is about time evolution (see page 69 right at beginning of chapter 3) and there is
RELATIVISTIC MECHANICS with is NOT about time-evolution, instead (see page 84 beginning of section 3.2.4) it is about relations between observables.

chapter 3 is about the classical theory, it comes before the presentation of LQG, and it is discussing classical things

So when Rovelli says (again around page 84) that Mechanics should not be thought of as being about time-evolution but should be about relations between observables he means classical observables

and that carries over in some later chapter when he quantizes and introduces LQG.

It is pretty radical and obviously can stop conversation totally.

And then when you say "well it can't be THAT radical because Gerard 't Hooft says things like that and even David Gross was just talking about 'physics without time'. Maybe it is an idea whose time has come!"

well even when you say that some people could reply "Oh that is String theorists, it is all right if they say it, they must mean something really jazzy----it is only not all right if Rovelli says it." People are biased and sometimes will listen to something from the Bigbrains that they will not accept from the unanointed.

So I don't know what to think. I know someone who discussed this (problem about time) with Rovelli by email a year or so ago. I myself have been concerned with it for at least a year. It comes out of Gen Rel, not out of anyone approach to quantizing. It is philosophically radical.
Apparently, so I've read anyway, there was a fair amount of discussion in the 90s.
someone here mentioned Julian Barbour---but that wasnt me.

 

[selfAdjoint]

Lubos says the Hamiltonian and the Density matrix have to be unitary. He specifically says that a theory which doesn't even have time in it has to be unitary in this sense; that it has to conserve probability. Arnold Neuman has also responded to my questions over at S.P.R. and he says the same thing. Try to read these posts for what they say and not turn them into your preconceived notions. Unitarity of a theory is totally independent of any time evolution questions.

(Added)

rovelli has a chapter on MECHANICS--- there is pre-relativistic or non-relativisitc mechanics which is about time evolution (see page 69 right at beginning of chapter 3) and there is
RELATIVISTIC MECHANICS with is NOT about time-evolution, instead (see page 84 beginning of section 3.2.4) it is about relations between observables.

And if you quantize that, those relations will become operators. Do they become unitary operators? Do they conserve probability? Does Rovelli say they do?

 

[marcus]

And if you quantize that, those relations will become operators.

I do not think that correlations among observations become operators. They are relations between observables.
My understanding is that correlations between observables do not become operators, but the observables do become operators!

Observables are normally associated with hermitian operators, or self-adjoint, as you well know. If there are things in LQG that should be unitary then I have no reason to suppose that they are not unitary. Like the representation of the holonomy-flux algebra. there are plenty of unitary operators in the theory!

it is just that certain things you might expect, like the hamiltonian, are not unitary---things having to do with time evolution.
As Rovelli says, that shocks some people and they cannot assimilate it.
I have a hard enough time getting used to the idea myself--so you'd best read directly what the main authorities have to say about it.

 

[marcus]

Lubos says the Hamiltonian and the Density matrix have to be unitary.

I can't believe Lubos says this! If he says the LQG Hamiltonian must be unitary then he does not know the first thing about LQG. I am far from being an expert myself but even I know that the Hamiltonian is an operator whose kernel is the physical states. A Unitary operator is injective---1-to-1--- with trivial kernel! If it were unitary then there would be no physical states at all!
The LQG Hamiltonian cannot possibly be one-to-one since it must send a huge bunch of quantum states to zero. It is a "constraint" and is called
"the Hamiltonian constraint".

It is confusing to rely on Lubos, we should use page references.
If you want i will find you a Rovelli or Ashtekar page reference about the hamiltonian constraint.
Quantization with constraints was apparently first expounded by Dirac.
I keep thinking you totally know this. How could Hamiltonian be unitary?
Am I missing something in what you say?

 

[selfAdjoint]

What's the matter with a unitary hamiltonian? It doesn't collapse stuff, it just preserves magnitude - and of course the "magnitude" of an amplitude is a probability!

Marcus when we each go "I can't believe you didn't know this" to the other, considering that we're both pretty well informed (though neither of us is a certified expert on this stuff), then I think that somehow, somewhere, we're like the two old ladies arguing across the back fence who can never agree because they are arguing from different premises.

 

[marcus]

What's the matter with a unitary hamiltonian? It doesn't collapse stuff, it just preserves magnitude - and of course the "magnitude" of an amplitude is a probability!

Marcus when we each go "I can't believe you didn't know this" to the other, considering that we're both pretty well informed (though neither of us is a certified expert on this stuff), then I think that somehow, somewhere, we're like the two old ladies arguing across the back fence who can never agree because they are arguing from different premises.

I think you are right! We definitely are bringing different premises and we are arguing as foolishly as any two back fence neighbors ever did!

I try to make sense of it in the new Ashtekar thread. didnt see your post here until just now--had to be out last night doing some volunteer work.

I believe the key split here is between the two historical approaches to QG, namely the "covariant" and the "canonical" (both names are historical accidents and potentially misleading) which have been traditionally favored by particlist and relativist respectively. You may know better but I think this is at the root of any misunderstanding about unitary time-evolution

In the one approach one can reasonably expect to get a nice unitary time-evolution operator, in the other one has the hamiltonian constraint which is zero on the physical states and one confronts the socalled "problem of time". it is worth thinking about. see if my post in the new Ashtekar thread is helpful

 

[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.

Over the past year I have always found nonunitary knows what he is talking about. so I should have paid better attention to this earlier.

There are several points here which would take a bit of deliberation.

for example "All of [Rovelli's] remarks apply equally to any version of quantum gravity."

I think the meaning here is that they apply to versions of quantum gravity which respect Gen Rel.

the principle of equivalence makes time a conundrum---it flows at different rates at different places in the gravitational field---any real physical time is extremely non-unique and contingent.

BTW in his book rovelli uses an "evolution parameter" in the Lagrangian and makes a careful distinction between that and physically measurable time. He points out that people have traditionally used the same letter "t" for both time and the (nonphysical) evolution parameter and deplores this as confusing. but the notation is conventional so there is not much one can do.
If anyone is interested I will get a page reference for that (maybe I will whether or not anyone is interested )

My suspicion is that there are really good reasons for having a "physics without time" and such a physics would have plenty of good serviceable clocks, they would just disagree all over the place. I don't mean merely in the rather tame Lorentzian way that Special Relativity clocks disagree. they would get "off" after the fashion of real physical quantum mechanisms as well as being subject to the vagaries of an evolving gravitational field. But never fear---lots and lots of clocks. Every observer would have at least one if not several! So that would be "physics without time" or as nonunitary says: with "time evolution as a not unique concept"

 

[nonunitary]

Marcus,

Thanks for the comments. I would like to add something to the discussion about diffeos as unitarily implemented in LQG. The Ashtekar-Lewandowski representaion is not only the one that implements diffeos as unitary transformations (according to the standard definition that Selfadjoint remainded us), but it has recently shown to be unique (i.e. the unique diffeo invariant representations compatiblñe with the holonomy-flux algebra), by
Lewandowski-Okolow-Sahlmann-Thiemann (the LOST result as is now known). In some previous post there was some quote about covariance of the representation. It only means that the objects used to define the holonomy-flux algebra (loops and surfaces, and internal labels) are not intrinsically defined.

 

[selfAdjoint]

Marcus,

Thanks for the comments. I would like to add something to the discussion about diffeos as unitarily implemented in LQG. The Ashtekar-Lewandowski representaion is not only the one that implements diffeos as unitary transformations (according to the standard definition that Selfadjoint remainded us), but it has recently shown to be unique (i.e. the unique diffeo invariant representations compatiblñe with the holonomy-flux algebra), by
Lewandowski-Okolow-Sahlmann-Thiemann (the LOST result as is now known). In some previous post there was some quote about covariance of the representation. It only means that the objects used to define the holonomy-flux algebra (loops and surfaces, and internal labels) are not intrinsically defined.

nonunitary, we know about the uniqueness theorem. Where in the works of Ashtekar and Lewandowski do you find that their representation is "the one that implements diffeos as unitary transformations ". I can't find a reference to this at the arxiv.

 

[nonunitary]

Hi,
Eqs. (4.54) and (4.55) of the latest Ashtekar-Lewandowski review.

 

[marcus]

in case anyone is reading along
and wants to look up the reference. A-L review article is
http://arxiv.org/gr-qc/0404018
I believe the equations nonU mentioned are on page 40

 

[selfAdjoint]

Thomas Larsson has now posted on s.p.r. a long and beautifuly clear essay in answer to my question. In it we see why the Hamiltonian in GR is just a constraint, H = 0, why 't Hooft was so despreate at Kitpi, and why the missing factor in our discussions, Marcus, was locality. Along the way we learn is that since (as nonunitary showed me) the Ashtekar-Lewandowski representation IS unitary, therefore it is the trivial representation so the Hamiltonian reduces to that constraint and there is no well defined time or energy, and as a result Larsson's definition of locality is violated. His definition of locality depends on a notion of closeness, not on a metric, so it's preserved by diffeos, since while they move points around, they don't, being "diffeo", violate closeness. So A-L representation based physics, can't support universal closeness.

Is this the underlying reason that Rovelli introduced his generalized diffeomorphisms ("chunkymorphisms"), which would violate closeness at a finite number of points but preserve it elsewhere? In other words, since our approach is doomed to have this flaw, let's reduce it at least to a set of measure zero so our integrals can breeze by it.

Well, as they say, Read the Whole Thing - closely!

 

[marcus]

...
Is this the underlying reason that Rovelli introduced his generalized diffeomorphisms ("chunkymorphisms"), which would violate closeness at a finite number of points but preserve it elsewhere? In other words, since our approach is doomed to have this flaw, let's reduce it at least to a set of measure zero so our integrals can breeze by it.

That is an interesting idea!
The classical 1915 GR theory is invariant under generalized diffeomorphisms ("chunkymorphisms"), and so Rovelli is within his rights to use the full symmetry of the classical theory when quantizing.

the customary restriction to the more limited class was in part an historical accident I suppose----they are nice mathematically.

Locality is a problem in classical GR because the manifold you start with has no metric. Not easy to say what locality is when there is no distance function defined.

BTW does String theory in its current Avatars have locality? the string world sheet doesn't seem very pointlike, and Gary Horowitz new paper seems to be saying there isn't any spacetime anyway, only this hologram 2D surface at infinity.

Anyway it seems like the ship of physics is adrift from the idea of locality
and maybe it goes back to 1915.
Gen Rel is invariant under chunkymorphisms and you seem to be saying here that this already violates "closeness" (by which I think you mean locality). I do not see your reasoning, but it is very possible.

GR is becoming more influential in people's thinking and this may be forcing people to change their ideas about locality.

... the Ashtekar-Lewandowski representation IS unitary,..

I am glad you have come to this conclusion

LQG has lots of unitary operators--you and I have gone over the unitary representation connected with work of L-O and S-T months ago, for example.
But it does not have unitary hamiltonian/ unitary time evolution (my only point). so I guess we agree!

BTW the combined L-O-S-T paper is not yet posted on arxiv. this is what is supposed to tie up the whole package, which currently is still spread out among several papers by Sahlmann Thiemann Lewandowski Okolow in various permutations and combinations of authorship

 

[selfAdjoint]

LQG has lots of unitary operators--you and I have gone over the unitary representation connected with work of L-O and S-T months ago, for example.
But it does not have unitary hamiltonian/ unitary time evolution (my only point). so I guess we agree!

But note Larsson's caviat: The representation that hath unitarity but hath not an anomaly is the trivial representation! In view of this the proof that the A-L representation is unique does not now seem so striking; of course the trivial representation is unique! And the symptom of this is the lack of a hamiltonian!

Locality without a metric is not difficult; topologists do it all the time. Think of Hausdorf spaces. Larsson uses a closeness criterion that he gets from the connection.

 

[marcus]

But note Larsson's caviat: The representation that hath unitarity but hath not an anomaly is the trivial representation! In view of this the proof that the A-L representation is unique does not now seem so striking; of course the trivial representation is unique! And the symptom of this is the lack of a hamiltonian!

strange he would say that
Smolin discussing the same thing --- page 11 of Invitation, 3 lines past equation (12) --- says words to the effect that:

"It is less trivial to prove, but still true, that there is no anomaly."

Forgive me if I am skeptical that Larsson or anyone can show in this case that the unitary representation of the diffeomorphisms is the trivial representation!

Both Ashtekar and Smolin are defining their unitary rep of the diffeos in analogous fashion, nearly the same notation. It seems to be important to prove that there is no anomaly and Smolin takes the trouble to mention this.

Thomas Larsson seems to be flying in the face of a lot of expert academic folk (who do not think their unitary rep of the diffeos is equal to the trivial representation). It would be interesting if he would give online references (links) and spell out his definitions.

 

[marcus]

here is the sort of thing Thomas says that makes me suspicious. this is from a post of his on SPR, I have bolded part for emphasis.

"The secret reason why canonical quantization of diff-invariant
theories in more than 2D fails is that the relevant diffeomorphism
group anomaly is little known. The diffeomorphism generators should
be represented by unitary operators on a conventional Hilbert space,
and all non-trivial such representations are anomalous. Since neither
the string theory nor LQG camps care about these anomalies in 4D,
they cannot do canonical quantization."

why does he say "conventional"----why not simply say Hilbert space.
does he have anything special in mind?

and the ordinary meaning of a non-trivial unitary rep is one that is not constantly equal to the identity.

it sounds strange that only Thomas should be right and all the String theorists and Loop gravitists should be wrong, because they don't understand the secret reason.

Also he is much more pessimistic than, for example, I am, bout string!
here is thomas' 12 October post at NEW. I have bolded some key sentences for emphasis.

His post was in response to: If you were Witten, what would you do to "fix up" string theory as it's known today (besides fixing up diffeomorphism anomalies)?
What would convince you to change your mind and be in support of string theory?

---quote from Larsson---
In the unlikely event that string theory acquired massive experimental support, I guess that I would have to believe in it. But the present situation is rather the opposite.


The construction of a quantum theory with some prescribed symmetries is, from my perspective, the same thing a constructing the representation theory of the group of symmetries. There is really a 1-1 correspondence:
1. Given a quantum theory, its symmetry group acts by a unitary representation on the Hilbert space.
2. Given a unitary representation of some group, the Hilbert space on which it acts is the Hilbert space of some quantum theory.


In particular, the Hilbert spaces of the fully interacting gauge-invariant or diff-invariant theories carry unitary representation of the groups of gauge transformations and diffeomorphisms. Perhaps one should factor out gauge symmetries, although I don't see why - it is definitely not necessary for consistency (unitarity). But this is really irrelevant for the argument. The anomalies must be there at least before factoring them out, so if you cannot write down the anomalies in the first place, you lose.


I am pretty sure that there is no way to fix string theory. The representations look the way they do, and their Hilbert spaces look rather like fixed versions of field theory. I don't see any way to "fix" SU(2) to allow for unitary spin-1/4 representations either.


I don't have a clue what I would do if I were Witten, and I don't really care. It's not my problem.

Posted by: Thomas Larsson at October 12, 2004 12:20 PM
---end quote---

I am sorry but this is just too over-the-top for me. Perhaps can you, selfAdjoint, provide some corroborations in the form of peer-reviewed articles that back up Thomas claims?

 

[selfAdjoint]

The connection between unitarity of the representation on Hilbert space and the existence of an anomaly is I believe known, if not well-known. Urs has cited it too, and provided me (I think it was he) with an old paper from Communications in Mathematical Physics (99, 103-114 (1985)), Hamiltonian Interpretation of Anomalies by Philip Nelson and Luis Alvarez-Gaume, which derives the result (it is not available online, at least not without a subscription to the journal). Larsson's derivation in the post I linked to is clearer: if the four dimensional diffeomorphism group had a unitary representation without an anomaly, so would its 1-dimensional subgroups by simple restriction. But we know the only unitary representation of the`1 dimensional diffeo group without an anomaly is the trivial representation.

Notice there is today (10/16/04) a reply to Larsson by Arnold Neumayer which clearly distinguishes the two senses of unitarity that were confusing us, that of the S-matrix or Hamiltonian and that of the representation in Hilbert space. BTW, I think you are overinterpreting Larsson's reference to a conventional Hilbert space, IMHO he just means not one of the various "rigged" Hilbert spaces that have been introduced recently.