Helling's post without the LM comment
---quote from Robert Helling SPS post---
Thanks for noting our paper. Unfortunately, I am about to leave
Cambridge (my next postdoc is at IU Bremen, back in Germany) and all
my papers and notes are stored away in boxes and unaccesible to me at
the moment. So I cannot answer Urs' qustions about signs and
(anti)-commutativity. He might well be right and we screwed those up
but those would be just typos and wouldn't change anything substantial
in the conclusions. Furthermore, I don't have my laptop's network
connection currently running, thus I have to use google groups rather
than my regular news reader.
> Their description of the harmonic oscillator looks particularly strange
> because if we did not agree what is the physics of the quantum harmonic
> oscillator, we could probably agree about nothing in the world.[/color]
Mybe you missed that point but our philosophy was to say "this is what
you get when you apply LQG methods to the harmonic oscillator". Be
careful with them I am general because the _physical_ consequences (esp.
the spectrum) are not what you meassure in this easy example. It was
important to us not just to say that functions that jump have no place
in physics because you could never observe them. That would be too
easy (and in fact plain wrong: If you describe a D-brane (whose
physical existence I understand Lubos does not doubt) by a skyscraper
sheef then this is done exactly by a function (of the transverse
coordinates) that is zero everywhere except at one point where it has
a finite value).
[Moderator's note:...LM]
So the point of our discussion of the harmonic oscillator is that there
are measurable consequences of doingit 'the wrong way'.
[Moderator's note:... LM]
> I am sure that many of us tried to deal with divergent sums and divergent
> integrals, and various other singular objects in various ways that can be
> proved "wrong" on physical grounds.[/color]
Actually, this is a major reason for why this formalism is more
involved than the usual one: In the algebraic language (and this is
just mathemtically more careful language, no physical difference to
the usual approach) great care is taken to avoid divergent (and
similar) sums etc so no ambiguities (or ways to do it wrong) appear
from that. This is why careful people deal with the Weyl operators
e^{ix} and e^{ip} instead of the usual x and p: The Weyl operators are
bounded and thus problems with domains of definition etc do not arise.
For example, in the position representation p is the deriviative. But
not all functions in L^2(R) are differentiable. Only a dense subset is. But all are translateble. Thus one saves some complications (if
ones intend is to be careful) if one uses the better defined Weyl
operators instead. I am not saying that it cannot be done with x and
p, it's just you either close your eyes to mathematical subtleties
(which is what we physicists do most of the time and it works fine
most of the time) or you have to deal with limits and that stuff.
[Moderator's note: ... LM]
> All Hilbert spaces obtained from these wrong assumptions are
> non-separable, unphysical, and the only way how the non-separability can
> be cured is if the resulting theory is completely topological and all
> these values of "x" are eventually unphysical, perhaps except for their
> ordering.[/color]
The separability is not the issue.
[Moderator's note:... LM]
In fact, there are (accepted) physical systems with a non-separable
Hilbert space: One (as we remark in a footnote) are Bloch electrons.
That is electrons in a periodic potential.
[Moderator's note:...LM]
Then you know that the wave function is periodic as well. Ahem no, not
quite, only the physics is periodic. So the wave function is periodic up
to a phase. And by doing the intergral over the whole infinite crystal,
you find that two wave functions with different phases are orthogonal. So
for each point in the interval [0,2pi) of phases there is an orthogonal
sector in the Hilbert space. Thus the total Hilbert space is kind of the
L^2 of the unit cell to times the number of points in that interval, clearly a non-separable space.
[Moderator's note:... LM]
You could say that this happens only in the infinite crystal size
limit.
[Moderator's note:... LM]
But this idealization people usually are happy to make. Otherwise (with an
IR cut-off) there would for example be no phase transitions. But that is a
different matter.
> The states in this model represent unphysical mixtures of a
> hugely infinite number of superselection sectors - it is another
> description of Helling et al. comments about "discontinuity" of
> Thiemann's representation, I think. Each of these sectors is made of a
> single state. In physics, it is legitimate to study each single
> superselection sector separately - and if these superselection sectors are
> made of a single state, the theory is physically vacuous.[/color]
At least in the mathematical sense (and that is supposed to coincide
with the physical sense), a super-selection sector is a representation
of the quantum algebra. To states are in different sectors if they
are in inequivalent representations.
> > Everybody knows that first quantization is a mystery.[/color]
>
> "Why the world is quantum?" may be a mystery and the most counterintuitive
> insight about the world, but the mathematical operation behind the first
> quantization does not seem mysterious in any way, and it also does not
> seem ambiguous. Moreover, I don't know why you chose the "first
> quantization" because even it is a mystery, it is a smaller mystery than
> the "second quantization". ;-)[/color]
Was this just a joke? If not, here is why people say this (and usualy
this continues with "second quantization is a functor"): Of course if
your classical system has R^n as configuration space and its cotangent
bundle as the symplectic space then every child knows how to quntize:
Take L^2(R^n) as your Hilbert space and replace all x's by
multiplication operators and all p's by derivatives. Oh, and when
there's an ordering ambiguity, follow one or the other prescription
(but do that consistently).
However, what do you do, if I just give you some symplectic space and
don't tell you which are the simple preferred position and momentum
coordinates (and that's what x and p are, just coordinates). And as
the real world does not come with coordinates written on everything
one should have some recepy how to deal with this more general
situation. And then check that this reduces to the usual story (or an
equivalent one) in the simple situation. And this is what we have done
in the paper.
It is known, that there is no unique way to do this map from a
symplectic space to a Hilbert space with operators in general. There
are further choices involved.
[Moderator's note:... LM]
> I did not quite like their comments describing the algebras with different
> central charges as different "representations" (with an exclamation mark).[/color]
When we say algebra, we mean the C*-algebra of the observables. And
those are indeed the same (and only the represenations differ).
[Moderator's note:... LM]
These algebras are the algebras of the X's (or the W(f) after some
massaging). Then this Weyl algebra has representations. And on those
representations there is a symmetry (Lie-)algebra acting by unitary
operators. And this symmetry algebra is some Virasoro alegbra in both
cases. But these symmetry algebras have different central charges in
the two representations of the Weyl algebra. So: There are two kinds
of algebras, don't confuse them.
Furthermore, even if we didn't talk about it, in the Virasoro algebra
the central charge is just an abstract element usally called c. It
commutes with everything, so in an irreducible representation it is
represented by a number. And again, this number depends on the
representation. This number, together with h, the eigenvalue of L_0 in that representation, label a highest weight representation of the
algebra.
[Moderator's note:... LM]
> The only way how can one understand this sentence is that they claim that
> the Virasoro algebras with different central charges are isomorphic to
> each other.[/color]
Nobody claimed that. As I just said: In the algbra, c is an abstract
element, it becomes a number only in a representation. And nobody
claims that representations with different c are equivalent.
[Moderator's note:... LM]
> There is a common theme in Thiemann's papers which, I'm afraid, may
> unfortunately be shared by Helling and Policastro - which is that they
> often look at the "classical limit" of an algebra, and treat all the
> modifications implied by quantum mechanics as unimportant - and perhaps
> annoying? - details whose relevance for any of their conclusions goes to
> zero.[/color]
Could you be more specific with this claim?
[Moderator's note:... LM]
> By the way, this purely quantum viewpoint will be even more important if
> we want to get more insight into the (2,0) theory or even M-theory at the
> generic point of the moduli space - because these theories (at least in
> some backgrounds) clearly indicate that they cannot be fully obtained from
> a classical theory by quantization - and they almost certainly cannot be
> obtained from a *unique* classical theory.[/color]
Quantization is a game that always involves a classical system.
However nobody claimed that every qunatum system arises from the
quantization of a classical theory.
[Moderator's note:... LM]
-----end quote---