January 29, 2004

Pohlmeyer charges, DDF states and string-gauge duality

Posted by Urs Schreiber

It’s half past two in the morning here in central Europe and a can’t sleep because my brain is thinking about Pohlmeyer charges! Honest. I have to write this entry if only to get some sleep until my girlfriend will wake me up for breakfast in a few hours. :-)

So let’s consider the classical (super)string. Physical observables are those that are gauge invariant with respect to the worldsheet Virasoro algebra, i.e. those that Poisson-commute with all the (classical for the moment) Virasoro generators. We know that these just generate two copies of the diffeo group on the circle (for the bosonic string) so our classical physical observables are nothing but rep invariant functionals. Two very familar sorts of rep invariant functionals are

- Integrals over rep-weight 1 densities

- Wilson lines.

Pohlmeyer takes the second exit and shows that the set of all imaginable Wilson lines around the string (at fixed worldsheet time) is a full (overcomplete, actually) set of invariant classical observables. I like this somehow, because it smells of string-gauge duality, a little. Actually he considers not the Wilson lines themselves but their Taylor coefficients. Anyway, these Pohlmeyer charges are obviously classically rep invariant but have a very convoluted algebra. Thinking about it for about 15 minutes I could not figure out how to adapt the Pohlmeyer charges to the superstring (which need not mean much).

Ok. Now I take instead the first exit and construct a complete set of invariant classical observables by the first method, namely by considering integrals over fields that have classical conformal weight 1. (I’ll make all of this precise tomorrow morning, promised.) That’s very easy because this job has already been done for us by the people whose initials are DDF and whose real names I’ll look up tomorrow, if anyone insists. :-) I mean, to do so for the classical string we can straightforwardly adapt the CFT construction of DDF states. It is even easier for the classical string because we do not need to use a lightlike momentum for the classical DDF states. And even better - we get the construction for the superstring for free, because the DDF states for the superstring are just as well known. Since the real DDF states span the whole Hilbert space of the quantum string I bet that the classical DDF are complete, too.

I think that’s good, because the classical DDF states do tell us something about the string’s spectrum even in Thiemann’s approach. Furthermore, while there is no sign of critical dimension at the classical level I claim that the classical DDF states at least give us the level-matching condition! And all this for the superstring, too.

Seems to me that IF Thiemann’s approach is mathematically viable then it would profit a lot from using classical DDF states (which in his context would be the same as the quantum states, essentially), because these do tell us manifestly about the string’s spectrum and these do easily generalize to the superstring.

Or so I think. Remeber, I should really be sleeping right now ;-)

— Note added later on: —

I have taken the time to write this idea up in more detail:

Urs Schreiber, DDF-like classical invariants of (super)string.

Posted at January 29, 2004 1:47 AM UTC

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Re: Pohlmeyer charges, DDF states and string-gauge duality

This is the content of an email which I have just sent to K. Pohlmeyer and K. H. Rehren:

[begin forwarded email]

Dear K. Pohlmeyer, Dear K. H. Rehren,

last year, in a lunch-break discussion with D. Giulini at a symposium, I first heard about your approach of algebraically quantizing the Nambu-Goto action using a quantum deformation of the Poisson algebra of classical invariants.

I learned more details of your approach from the nice summary which is given in the recent paper by Thomas Thiemann.

We have been discussing this paper in some detail on a weblog called the String Coffee Table. While the central idea of this paper is problematic, Thiemann nicely highlights the independent concepts involved in the construction of Pohlmeyer charges. In the course of this discussion I began to wonder about an issue which is quite unrelated to Thiemann’s approach but pertains to your algebraic quantization program:

While the Pohlmeyer charges have certain nice properties, they, as far as I understand, satisfy a rather convoluted algebra, have no obvious relation to the usual spectrum of the quantum string and have no obvious generalization to the superstring. (Is that correct?)

Therefore I wondered why one couldn’t alternatively start the algebraic quantization procedure using a different set of invariants (which should be linear combinations of the Pohlmeyer invariants), namely the classical observables associated with the DDF-operators.

Unless I am mixed up, the classical analogues of the DDF-operators do exist, Poisson-commute with all the Virasoro constraints and are complete. They should hence, unless I am making a mistake, be an equivalent alternative starting point for algebraic quantization.

The advantage that I see in using classical DDF invariants is that they satisfy a nice Poisson algebra (isomorphic to that of worldsheet oscillators), hence manifestly know about the massive spectrum of the string (except for the ground state mass and degeneracy) and are easily generalized to the superstring.

I have tried to sketch more details of this idea in this draft. I would very much enjoy if you could find the time to have a brief look at this text and let me know your opinion on the viability of this idea. To me it seems as if it might put the algebraic quantization of the (super)string in contact with the standard quantization, but if you think that I am wrong about this please let me know.

I am posting a copy of this email to the String Coffee Table in the hope to reach a larger audience of interested people and maybe hear of further opinions.

Best regards,

Urs Schreiber

[end forwarded email]

Posted by: Urs Schreiber on February 4, 2004 12:05 PM | Permalink | Reply to this

Re: Pohlmeyer charges, DDF states and string-gauge duality

I have kindly received a reply by a collaborator of Prof. Pohlmeyer.

There are essentially two points being made:

1) I am being told that it has been studied in Pohlmeyer’s group whether a canonical quantization of the algebra of classical invariants of the Nambu-Goto string ‘with the help’ of the usual creation/annihilation operators is possible. The result found, and expected by Pohlmeyer’s group, is that this is not the case. The reason found by the group is that anomalies apear which are not reparameterization invariant.

2) Regarding my proposal to instead of the Pohlmeyer charges use the classical analogues of DDF operators as classical observables that Poisson-commute with all the Virasoro constraints it is remarked that what I like to call ‘classical DDF invariants’ only come close to being invariants, but that they require fixing conformal gauge which makes them, in contrast to the Pohlmeyer invariants, not invariant under the full gauge group.

With respect to these two points I’d have the following comments:

1) It is not yet fully clear to me what precisely has not been possible to achieve using creation/annihilation operators in the context of Pohlmeyer invariants. In Pohlmeyer’s papers the invariants are constructed in terms of the canonical fields $\pi \left(\sigma \right)±{X}^{\prime }\left(\sigma \right)$ (where $\pi \left(\sigma \right)$ is the canonical momentum to $X\left(\sigma \right)$). But these are just a linear combination of the creation annihilation operators ${a}_{n}$, ${\stackrel{˜}{a}}_{n}$. What can be written in terms of the former can be written equivalently in terms of the latter. Hence the above item 1) must refer to some other construction, but I do not yet understand which one.

2) I don’t think that it is correct to say that the construction of ‘classical DDF invariants’ requires to fix any worldsheet gauge. It is true that the usual DDF operators are usually discussed in the context of worldsheet CFT which is obtained from the Polyakov action by fixing conformal gauge. But the key idea in the construction of DDF operators, which is just to write down integrals over fields of unit weight, is quite independent of such issues and in fact only related to the form of the Virasoro algebra. This, however, follows from the Nambu-Goto action as well as from the Polyakov action (by the ADM method) without fixing conformal gauge.

The basic idea of classical DDF invariants is quite simple, so I’ll spell it out here again:

Denote the classical worldsheet fields mentioned above by ${Y}_{±}\left(\sigma \right)$. The classical Poisson brackets are

(1)$\left[{Y}_{±}\left(\sigma \right),{Y}_{±}\left({\sigma }^{\prime }\right){\right]}_{\mathrm{PB}}=-{\delta }^{\prime }\left(\sigma ,{\sigma }^{\prime }\right)$
(2)$\left[{Y}_{±}\left(\sigma \right),{Y}_{\mp }\left({\sigma }^{\prime }\right){\right]}_{\mathrm{PB}}=0\phantom{\rule{thinmathspace}{0ex}}.$

The classical Virasoro constraints are

(3)${L}_{m}=\frac{1}{2}\int d\sigma \phantom{\rule{thinmathspace}{0ex}}{e}^{-\mathrm{im}\sigma }{Y}_{+}\left(\sigma \right){Y}_{+}\left(\sigma \right)$
(4)${\stackrel{˜}{L}}_{m}=\frac{1}{2}\int d\sigma \phantom{\rule{thinmathspace}{0ex}}{e}^{-\mathrm{im}\sigma }{Y}_{-}\left(\sigma \right){Y}_{-}\left(\sigma \right)\phantom{\rule{thinmathspace}{0ex}}.$

They generate seperately two copies of the group $\mathrm{Diff}\left({\mathrm{S}}^{1}\right)$:

(5)$\left[{L}_{m},{L}_{n}{\right]}_{\mathrm{PB}}=\left(m-n\right){L}_{m+n}\phantom{\rule{thinmathspace}{0ex}}.$

We say that any observable $A\left(\sigma \right)$ which satisfies

(6)$\left[{L}_{m},A\left(\sigma \right){\right]}_{\mathrm{PB}}={e}^{-\mathrm{im}\sigma }{A}^{\prime }\left(\sigma \right)-{w}_{+}\left(A\right)\mathrm{im}{e}^{-\mathrm{im}\sigma }A\left(\sigma \right)$
(7)$\left[{\stackrel{˜}{L}}_{m},A\left(\sigma \right){\right]}_{\mathrm{PB}}={e}^{-\mathrm{im}\sigma }{A}^{\prime }\left(\sigma \right)-{w}_{-}\left(A\right)\mathrm{im}{e}^{-\mathrm{im}\sigma }A\left(\sigma \right)$

is of (classical) weight $\left({w}_{+}\left(A\right),{w}_{-}\left(A\right)\right)$. It is obvious that integrals over fields with $\left({w}_{+},{w}_{-}\right)=\left(1,1\right)$ Poisson-commute with all Virasoro constraints:

(8)$\left({w}_{+}\left(A\right)=1,{w}_{-}\left(A\right)=1\right)⇒{\left[{L}_{m},\int d\sigma \phantom{\rule{thinmathspace}{0ex}}A\left(\sigma \right)\right]}_{\mathrm{PB}}=0={\left[{\stackrel{˜}{L}}_{m},\int d\sigma A\left(\sigma \right)\right]}_{\mathrm{PB}}\phantom{\rule{thinmathspace}{0ex}},\forall m\phantom{\rule{thinmathspace}{0ex}}.$

This is the same idea as in the construction of the quantum DDF operators, with the only difference that there one has to take care of the fact that the quantum weight of a given field may be different from its classical weight. But this makes the construction of classical DDF invariants even easier.

All that remains to be done is to find a complete basis for the space of all $\left(1,1\right)$ fields. That’s straightforward. Let

(9)${X}_{±}^{\mu }\left(\sigma \right)={\int }_{0}^{\sigma }\phantom{\rule{thinmathspace}{0ex}}{Y}_{±}^{\mu }\left({\sigma }^{\prime }\right)\phantom{\rule{thinmathspace}{0ex}}d{\sigma }^{\prime }+\sqrt{2}{X}^{\mu }\left(0\right)$

and note that ${w}_{±}\left({X}_{±}\right)=0$. This means that all fields of the form

(10)${Y}_{±}\left(\sigma \right)\mathrm{exp}\left(-\mathrm{in}k\cdot {X}_{±}\left(\sigma \right)\right)\phantom{\rule{thinmathspace}{0ex}},$

where $k$ is some target space vector (which we should choose to be lightlike in order to have no problems when quantizing) have ${w}_{±}=1$. It follows that all observables of the form

(11)$\int d\sigma \phantom{\rule{thinmathspace}{0ex}}{Y}_{+}\left(\sigma \right){e}^{-\mathrm{in}{\int }_{0}^{\sigma }d{\sigma }^{\prime }\phantom{\rule{thinmathspace}{0ex}}{Y}_{+}\left({\sigma }^{\prime }\right)}\int d\sigma \phantom{\rule{thinmathspace}{0ex}}{Y}_{-}\left(\sigma \right){e}^{-\mathrm{in}{\int }_{0}^{\sigma }d{\sigma }^{\prime }\phantom{\rule{thinmathspace}{0ex}}{Y}_{-}\left({\sigma }^{\prime }\right)}{e}^{-\mathrm{in}\sqrt{2}X\left(0\right)}$

(and similar ones with more integrals) Poisson-commute with all the Virasoro constraints (unless I am confused, that is). The field $X\left(0\right)$ is shared by both the left- and the rightmoving terms and forces us to have level matching, already classically.

(For brevity I have been slighly sloppy with notation here. For a cleaner and more detailed formulas see section 4 of my notes.)

If anyone sees any problem with the above construction I would be very grateful to learn about it!

To my mind the classical DDF invariants

1) Poisson-commute with all the constraints

2) generalize easily to the superstring

3) have a very simple and illuminating Poisson algebra, namely that of creation/annihilation operators that can easily be quantized and then has the advantage to give, by algebraic means, the same quantum theory that one usually considers for the string, except that a little more work (but the usual and well-known work) is needed to get the ground state energy and degeneracy

4) Provide a link between what has been considered as alternative quantizations of the (super)string with the usual quantization.

Posted by: Urs Schreiber on February 9, 2004 5:03 PM | Permalink | Reply to this

Re: Pohlmeyer charges, DDF states and string-gauge duality

K.-H. Rehren replies to my above letter and kindly gives permission to post it here:

[begin forwarded email]

Quantization of the string involves as the first step the determination of the quantum algebra. Pohlmeyer’s program consists in determining #only# the quantum algebra of observables. Most other approaches, including the standard approach as well as Thiemann’s, first define a larger algebra and then determine the algebra of observables as a subalgebra, invoking a symmetry principle.

The next step is the choice of a representation. Once an algebra is determined, the set of its representations is intrinsic. If the quantum algebra of observables is constructed as a subalgebra of an auxiliary larger algebra, then it may have representations which do not arise as subrep’ns of some rep’n of the larger algebra. This aspect is emphasized in Meusburger-Rehren. Thus, representing the auxiliary larger algebra to begin with, one might loose relevant representations of the observables.

Dorothea Bahns has shown in her diploma thesis, that if one quantizes classical invariant observables (Pohlmeyer charges) by embedding them into the oscillator algebra via normal odering (N.O.), then the N.O. invariants fail to commute with N.O. Virasoro constraints, and commutators of N.O. invariants among themselves yield other N.O. invariants plus “quantum corrections” which are #not# quantized classical invariants. Thus, the quantum algebra has not only relations differing from the classical ones by hbar corrections (which everybody expects), but it would have #more generators# than the classical algebra. This feature (“breakdown of the principle of correspondence”) is worse than a central extension, because the latter is a multiple of one, and as such #is# a quantized classical observable, suppressed by hbar. This feature is a property of the quantization, i.e., the very choice of the quantum algebra by replacing classical invariants by N.O. ones. One may or may not appreciate the oscillator quantization with features like this.

Meusburger and Rehren have proposed an alternative construction of the quantum algebra of the invariants by embedding into another auxiliary algebra which is the enveloppe of a Lie algebra, and showed that the quantum algebra of invariants can be consistently defined, respecting the principle of correspondence.

Because of the importance of various symmetry concepts, I include a few general mathematical reminders on states, symmetries, and anomalies (once the algebra has been fixed), which might serve to settle some disputes (mainly around diffeomorphism invariance) which I found in the round table discussion. More details can be found in Thiemann’s paper, and in textbooks on quantum mechanics and operator algebras.

A state is a positive normalized linear functional on a *-algebra. A given algebra has lots of states. Each state is realized as a vector state in #some# Hilbert space representation (reconstructed from the state itself by GNS), but one state may not be realizable as a vector state in the Hilbert space representation of another state. Thus, choosing a representation one looses most states. Eg, in case of CCR (infinite-dimensional Heisenberg), Bogolubov transformations do connect states belonging to inequivalent representations unless they are #very close# to the identity transformation.

A symmetry is an automorphism (or a group of automorphisms) of the algebra. An infinitesimal symmetry is a derivation (or a Lie algebra of derivations) of the algebra (i.e. linear maps which satisfy the Leibniz rule). A derivation may or may not exponentiate to a group of automorphisms.

A classical (infinitesimal) symmetry may or may not be realized by automorphisms (derivations) of the quantum algebra. If it is not, one has an anomaly which is a property of the quantum algebra.

If it is, then the automorphisms may or may not be implemented by unitary operators in a given representation, or the derivations may or may not be implemented by commutators with self-adjoint generators. If they are not, one has spontaneous symmetry breakdown which is a property of the representation of choice.

If they are implemented, then the unitaries may or may not satisfy the group composition law, or the generators may or may not satisfy the Lie algebra relations. If they don’t, then one has a central extension (or more generally, a cocycle). This is again a property of the representation.

If they do, then a given vector state in the representation may or may not be preserved by the unitaries. This is a property of the vector (or the state).

Once the algebra has been fixed, the set of its states and representations is intrinsically determined (though possibly difficult to compute). No-go theorems may assert which of the previous features (in increasing order of strength) can be realized in #all# or #some# or #no# states or representations of the algebra.

Examples: Poincare symmetry. Let the algebra be CCR of the time-zero scalar field, understood as Cauchy data for the solutions of Klein-Gordon. The Poincare group acts by automorphisms of this algebra. A Poincare-invariant 2-point function defines an invariant state (the vacuum) on this algebra, and by GNS one gets the Fock representation in which the Poincare symmetry is implemented, and the vacuum vector is invariant. Note: the input for these results are the algebra, the automorphisms, and the state.

Diffeomorphisms. Let the algebra be a chiral current algebra (another CCR in the abelian case). The diffeomorphism group acts by automorphisms on this algebra. In the oscillator representation, these automorphisms are even implemented by unitaries of the form exp iT(f) where T(f) is the N.O. stress tensor smeared with a function f (the infinitesimal diffeomorphism). But these unitaries fail to satisfy the group law: there is a central charge whose value is determined by the algebra (and its oscillator representation chosen). The vacuum is not invariant, and in fact #there is no# diff-invariant state at all. A no-go theorem (V. Kac) states that a unitary positive-energy (L0>0) representation of Diff without central charge must be trivial (one-dimensional). Put otherwise: c=0 is only possible if one abandons the positive-definite Hilbert space metric (ghosts), or positive energy, or unitarity.

Pohlmeyer quantization of the Nambu-Goto string attempts to avoid this no-go, by quantizing only diff-invariant classical observables and defining their quantum algebra on which the diffeomorphisms act by the trivial automorphisms (i.e. not at all). Meusburger and Rehren use an auxiliary algebra and a derivation in order to #consistently define the algebra# of observables, without referring to a representation. Thiemann seeks for the quantum algebra within an LQG type auxiliary algebra which has a unitary representation of Diff. The latter is #not# subject to the positive-energy condition. There is no reason why it should be, because the Virasoro algebra is #not# considered as a quantum field of some worldsheet CFT, but just as the generators of a symmetry. Hence implementation of the constraints with c=0 is possible.

A non-invariant state can be made invariant by hand by averaging over the group, provided the latter has a suitable measure. The resulting invariant state may belong to a #reducible# representation of the algebra (no free lunch: this happens in the case of spontaneous symmetry breakdown; the resulting unitaries connect inequivalent subrepresentation, and the symmetrized state is an incoherent mixture rather than a pure state).

In a nut-shell, “quantization” should involve 1) the choice of the quantum algebra, 2) the selection of that representation or family of representation. One would like to choose the latter such that it realizes the classical symmetry in the strongest possible form. But the choice may be limited by no-go theorems if the algebra just happens not to have representations with the desired properties.

As for the DDF-proposal: I stopped reading when seeing division (\partial k\partial X)/(k\partial X) of non-commuting operators (from the context, I understand that you #do# talk about operators here, and X is a sum of oscillator modes).

As for exponentials and their inverses: exponentiating a field which is represented as a sum of oscillators, without smearing, of course needs normal ordering. The normal ordering is multiplicative (division by the vacuum expectation value as one sees by writing the field in the exponent as a sum of its creation and annihilation part and using Baker-Campbell-Hausdorff). It is then obvious that :exp -iPhi(x): is not the inverse of :exp iPhi(x): due to N.O. String vertex operators like :exp ik.X: are objects of this kind.

On the other hand, exponentiating a #smeared field operator#, then exp -iPhi(f) is inverse of exp iPhi(f) because for operators one has functional calculus, and exp, like any other sensible function, is well-defined on self-adjoint operators without any regularization. Unitary implementers of Diff or gauge symmetries in CFT are of this kind (with Phi = stress tensor or = current).

mit besten gruessen,

karl-henning rehren

www.theorie.physik.uni-goe.de/~rehren

[end forwarded email]

Posted by: Urs Schreiber on February 17, 2004 2:30 PM | Permalink | Reply to this

Re: Pohlmeyer charges, DDF states and string-gauge duality

K.-H: Rehren wrote:

As for the DDF-proposal: I stopped reading when seeing division (\partial k\partial X)/(k\partial X) of non-commuting operators (from the context, I understand that you #do# talk about operators here, and X is a sum of oscillator modes).

There are no non-commuting operators in this expression because $k$ is lightlike, i.e. $k\cdot k=0$. The construction can for instance be found in equation (4.3.26) of Green,Schwarz & Witten.

I should emphasize that what I wrote about DDF operators is just a review of well-known facts.

Dorothea Bahns has shown in her diploma thesis, that if one quantizes classical invariant observables (Pohlmeyer charges) by embedding them into the oscillator algebra via normal odering (N.O.), then the N.O. invariants fail to commute with N.O. Virasoro constraints, […]

I think what one can do instead is the following: It can be shown that the Pohlmeyer invariants can be expressed in terms of the classical DDF invariants (section 2.3 of these notes). When we now quantize the classical DDF operators in the usual way so as to obtain the DDF operators, this induces a quantization of the classical algebra of Pohlmeyer invariants. This quantum algebra manifestly commutes with the quantum Virasoro constraints and closes on quantum-invariant terms. This quantization furthermore reproduces the usual theory.

Posted by: Urs Schreiber on February 17, 2004 2:59 PM | Permalink | Reply to this

Re: Pohlmeyer charges, DDF states and string-gauge duality

K-H Rehren:
>Dorothea Bahns has shown in her diploma thesis, that if one quantizes
>classical invariant observables (Pohlmeyer charges) by embedding them
>into the oscillator algebra via normal odering (N.O.), then the N.O.
>invariants fail to commute with N.O. Virasoro constraints, and
>commutators of N.O. invariants among themselves yield other N.O.
>invariants plus “quantum corrections” which are #not# quantized
>classical invariants. Thus, the quantum algebra has not only
>relations differing from the classical ones by hbar corrections
>(which everybody expects), but it would have #more generators# than
>the classical algebra. This feature (“breakdown of the principle of
>correspondence”) is worse than a central extension, because the
>latter is a multiple of one, and as such #is# a quantized classical
>observable, suppressed by hbar. This feature is a property of the
>quantization, i.e., the very choice of the quantum algebra by
>replacing classical invariants by N.O. ones. One may or may not
>appreciate the oscillator quantization with features like this.

The correspondence principle is not necessarily violated. To
construct extensions of the diffeomorphism algebra in more than 1D,
one must first expand all fields in a Taylor series around some point
q. There are no conceptual problems to express classical physics in
terms of Taylor data (q and the Taylor coefficients) rather than in
terms of fields, although there may be problems with convergence.

The reason why such a trivial reformulation is necessary is that the
higher-dimensional generalizations of the Virasoro cocycle (there are
two of them) depend on the expansion point q. The relevant extensions
are thus non-linear functions of data already present classically,
which seems consistent with the correspondence principle.

>A no-go theorem (V. Kac) states that a unitary positive-energy (L0>0)
>representation of Diff without central charge must be trivial
>(one-dimensional). Put otherwise: c=0 is only possible if one
>abandons the positive-definite Hilbert space metric (ghosts), or
>positive energy, or unitarity.

Hence we can not maintain non-triviality, anomaly freedom, positive energy, unitarity
and ghost freedom at the same time. It seems to me that giving up
anomaly freedom makes least damage, especially since we know that
anomalous conformal symmetry is important in 2D statistical models,
such as the Ising and tricritical Ising models. It is important to
realize that such models have been realized experimentally (e.g. in a
monolayer of argon atoms on a graphite substrate) and that the
non-zero conformal anomaly has been measured (perhaps only in
computer experiments). Hence anomalous conformal symmetry is not
intrinsically inconsistent.

A clarification is in order at this point. The multi-dimensional
Virasoro algebra is a kind of gravitational anomaly, but no such
anomalies exist in 4D within a field theory framework. However, it
turns out that the phrase “within a field theory framework” is a
critical assumption. As I explained above, the relevant cocycles
depend on the expansion point q, and thus they can not be expressed
in terms of the fields which are independent of q.

>Thiemann seeks for the quantum algebra within an LQG type auxiliary
>algebra which has a unitary representation of Diff. The latter is
>#not# subject to the positive-energy condition.

The passage to lowest-energy representations is the crucial
quantization step. If such representations do not appear in LQG, then
it cannot IMO be a genuine quantum theory. If LQG is essentially
classical it explains why there is no anomaly.

Re: Pohlmeyer charges, DDF states and string-gauge duality

Concerning the “breakdown of the principle of correspondence”, I was not talking about the anomalies of Diff (which is ultimately to be represented trivially in the physical Hilbert space of reparametrization invariant observables) but of the invariant charges.

Concerning the c=0 no-go theorem, I was stressing that admitting c>0 is possibly not the only way out, but Thiemann’s approach suggests another one. In this approach, THERE IS NO worldsheet CFT. Consequently, sacrificing positive energy for the worldsheet translations also means “no damage”. Crucial is only positivity of the target space energy. If this can be achieved on a Hilbert space which is not the Hilbert space of some CFT, then I can sleep peacefully.

The reasoning about 2D statistical models seems entirely besides the point to me. Nobody claims that c>0 were “intrinsically inconsistent”; on the contrary, everybody knows that it is a necessary feature of real-time QFT because (by the said no-go) otherwise the 2-point function of the stress-energy tensor were zero, and hence the stress-energy tensor itself were zero (Reeh-Schlieder).

It should be clear, to repeat it once more, that all this is about seeking for ALTERNATIVE quantizations, in which the usual CFT reasoning simply has no place a priori.

Posted by: KH Rehren on February 23, 2004 9:10 AM | Permalink | Reply to this

Re: Pohlmeyer charges, DDF states and string-gauge duality

Luboš has some ideas about Pohlmeyer’s invariants which I would like to forward, with his kind permission, to the Coffee Table:

[begin forwarded message]

Hi guys,

just a couple of points about Thiemann-related work as discussed with
Witten. It was a very small part of our discussions, but an interesting
one. Initially he was answering that it was not interesting, that it is
recommended not to read papers that obviously seem that the author does
not understand what he’s doing, and so on, and because such an answer was
not satisfactory enough, I provoked him to a more detailed answer :-) by
the question whether he thinks that Thiemann’s approach is similar to his
(+Nappi+Dolan) integrability approach to AdS5. :-)

After he was explained briefly how I understand Thiemann Pohlmeyer stuff,
we analyzed the charges - see e.g. (3.29) in Thiemann’s paper.

You integrate over sigma1 … sigmaN, which preserve a cyclical ordering,
and you put del X^{mu_1}(sigma1) … del X^{mu_N}(sigmaN) - the
holomorphic derivatives denoted Y_+ - to these points. The resulting
charges transform as N-tensors. You can do the same with Y_- variables,
the antiholomorphic derivatives of X.

I knew that the first operator N=1 is the total momentum. The second one
N=2 comes from a trivial condition on cyclical ordering of two things, so
it is still the full integral, e.g. the product P^mu P^nu of two copies of
the total momentum.

Although the cyclical ordering of 3 things sigma1,sigma2,sigma3 is a
nontrivial constraint, Witten showed that the emptiness continues. The 3rd
order Pohlmeyer charge is a tensor T^{lambda,mu,nu}. It is cyclically
symmetric, and therefore it is the sum of a totally symmetric and a
totally antisymmetric part, as we finally realized with Witten. ;-) The
totally symmetric part is equivalent to an unconstrained integral over
three sigmas, and therefore gives the cubed momentum again. The
antisymmetric part is something different.

At any rate, during the integral, you can integrate over sigma1,sigma3
freely, and sigma2 must be in between them to preserve the cyclical
ordering (123). If you integrate del X^nu(sigma2) over the interval
(sigma1,sigma3) (imagine that sigma1[end forwarded message]

[begin forwarded message]

Dear Gentlemen,

> Of course, the zero mode subtleties kill the statement that a string can…

I should clarify this comment about the zero modes. One can formally
define the total chiral Lorentz generator, as the integral of

XL^a(sigma) del X^b(sigma) - XL^b(sigma) del X^a(sigma)

but it contains the contribution of the zero modes (x0^a p^b - x0^b p^a)
which are shared between the left-movers and the right-movers. One can get
rid of this ambiguity if we e.g. antisymmetrically multiply M^{ab} and
P^c, because the antisymmetric product of P^b and P^c vanishes. This is
what happens in the antisymmetric part of the 3rd Pohlmeyer invariant.

The Pohlmeyer charges satisfy the requirement that only the internal spin
(not the orbital angular momentum) contributes, and therefore the
subtleties with the zero modes are uniquely fixed. Nevertheless it does
not change the fact that the Pohlmeyer charges are functions of the total
momentum and the total chiral spin of the string. Of course, we know that
the string states in string theory form representations of this small set
of operators (and their functions), and we also know that the condition of
“transforming under the (chiral) Poincare algebra” does not determine the
dynamics of the string completely - i.e. we know that the chosen algebra
is certainly not large enough to describe a string.

I am trying to complete the proof that all the Pohlmeyer charges can be
trivialized in this way and rewritten as a function of the total momentum
and the total internal (left-moving) spin.

Best wishes
Lubos

[end forwarded message]

Posted by: Urs Schreiber on February 13, 2004 1:58 PM | Permalink | Reply to this

Re: Pohlmeyer charges, DDF states and string-gauge duality

Luboš wrote:

I am trying to complete the proof that all the Pohlmeyer charges can be trivialized in this way and rewritten as a function of the total momentum and the total internal (left-moving) spin.

One thing at least is easy to see:

Since the Pohlmeyer invariants do not contain any zero modes of the worldsheet fields, their representation in terms of DDF invariants can only involve DDF invariants of level 0, i.e. those which Poisson-commute with the level number observable, since these are the only combinations of DDF invariants that are independent of the zero modes of worldsheet fields themselves. This should be equivalent to saying that the Pohlmeyer invariants correspond to vanishing momentum, as Lubos has already mentioned above.

Posted by: Urs Schreiber on February 13, 2004 2:19 PM | Permalink | Reply to this

Re: Pohlmeyer charges, DDF states and string-gauge duality

I am trying to complete the proof that all the Pohlmeyer charges can be trivialized in this way and rewritten as a function of the total momentum and the total internal (left-moving) spin.

I find it hard to believe this for the following reason:

The Pohlmeyer invariants give us all the Wilson loops around the string (at some fixed worldsheet time) for constant $\mathrm{GL}\left(N,R\right)$ connections with $N\to \infty$. Pohlmeyer’s claim is that by knowing all these holonomies you can reconstruct the form of the loop formed by the string (up to an arbitrary shift of the center of mass).

This claim somehow sounds quite plausible to me, though I have not studied Pohlmeyer’s proof yet. But another point in support of this claim might be the constructions used in the IKKT/IIB Matrix model. For instance when one looks at equation (2.7) of

Aoki et al., IIB Matrix Model

one finds essentially an expression for a Pohlmeyer observable. The authors show that these objects obey the equations of motion of the string.

Posted by: Urs Schreiber on February 13, 2004 2:51 PM | Permalink | Reply to this

Expressing Pohlmeyer invariants in terms of DDF observables

I have received email by H. Nicolai where he informs me that already quite a while ago, on the occasion of a talk at the Albert-Einstein Institute in Potsdam, he had asked D. Bahns if the Pohlmeyer invariants could be anything else than suitable combinations of un-quantized DDF operators. The question was answered in the negative, but this did not convince everybody.

I think I have an idea how to show that and how the Pohlmeyer invariants are related to the DDF invariants. It goes as follows:

Heuristically the DDF invariants

(1)${A}_{n}^{\mu }=\int d\sigma \phantom{\rule{thinmathspace}{0ex}}{Y}^{\mu }\left(\sigma \right){e}^{\mathrm{in}k\cdot {X}_{+}^{\mu }\left(\sigma \right)}$

(I am using the notation defined here but might have some inessential inconsistencies in some prefectors.) are nothing but ordinary oscillators ${a}_{n}^{\mu }=\int d\sigma \phantom{\rule{thinmathspace}{0ex}}{Y}^{\mu }\left(\sigma \right){e}^{\mathrm{in}\sigma }$ but reparameterized as $\sigma \to k\cdot {X}_{+}\left(\sigma \right)$.

Therefore define the invariant local fields

(2)${\stackrel{˜}{Y}}^{\mu }\left(\sigma \right):=\sum _{n=-\infty }^{\infty }{A}_{n}^{\mu }{e}^{-\mathrm{in}\sigma }=\frac{1}{2\pi }\int d{\sigma }^{\prime }\phantom{\rule{thinmathspace}{0ex}}{Y}^{\mu }\left({\sigma }^{\prime }\right)\delta \left(k\cdot {X}_{+}\left({\sigma }^{\prime }\right)-\sigma \right)\phantom{\rule{thinmathspace}{0ex}}.$

There is a subtlety when $k\cdot {X}_{+}$ is not injective. I still need to sort that out. For simplicity assume for the moment that we are evaluating these observables at points in phase space where $k\cdot {X}_{+}$ is injective. Then the above is equal to

(3)${\stackrel{˜}{Y}}^{\mu }\left(\sigma \right)=\left(\left(k\cdot {X}_{+}{\right)}^{-1}{\right)}^{\prime }\left(\sigma \right)Y\left(\left(k\cdot {X}_{+}{\right)}^{-1}\left(\sigma \right)\right)\phantom{\rule{thinmathspace}{0ex}}.$

Because this is just a linear combination of DDF invariants it still Poisson-commutes with all Virasoro constraints.

Now let $\xi$ be the function on the circle such that

(4)$\left(k\cdot {X}_{+}{\right)}^{-1}\left(\sigma \right)=\sigma +\xi \left(\sigma \right)\phantom{\rule{thinmathspace}{0ex}}.$

To first order in $\xi$ we have

(5)${\stackrel{˜}{Y}}^{\mu }\left(\sigma \right)={Y}^{\mu }\left(\sigma \right)+{\xi }^{\prime }\left(\sigma \right){Y}^{\mu }\left(\sigma \right)+\xi \left(\sigma \right){Y}^{\prime \mu }\left(\sigma \right)+𝒪\left({\xi }^{2}\right)\phantom{\rule{thinmathspace}{0ex}}.$

Hence the invariant $\stackrel{˜}{Y}\left(\sigma \right)$ is nothing but the non-invariant $Y\left(\sigma \right)$ reparameterized by a phase-space dependent function. This implies that any reparameterization invariant function $F\left(Y\right)$ of the $Y\left(\sigma \right)$ is equal to the same function of the $\stackrel{˜}{Y}\left(\sigma \right)$:

(6)$\left[{L}_{m},F\left(Y\right)\right]=0\phantom{\rule{thinmathspace}{0ex}}⇒\phantom{\rule{thinmathspace}{0ex}}F\left(Y\right)=F\left(\stackrel{˜}{Y}\right)\phantom{\rule{thinmathspace}{0ex}}.$

This applies in particular to the Pohlmeyer invariants. Therefore the prescription to express the Pohlmeyer-invariants in terms of DDF invariants should be (modulo the above subtlety):

Write down the Pohlmeyer invariants in terms of oscillators ${a}_{n}^{\mu }$ and then replace each oscillator by the respective DDF invariant ${a}_{n}^{\mu }\to {A}_{n}^{\mu }$. (The claim is that this replacement does not change the Pohlmeyer observables.)

Posted by: Urs Schreiber on February 15, 2004 6:31 PM | Permalink | Reply to this

Re: Expressing Pohlmeyer invariants in terms of DDF observables

I have now managed to get hold of a copy of

K. Pohlmeyer and K.-H. Rehren, The Invariant Charges of the Nambu-Goto Theory: Their Geometric Origin and Their Completeness, 1988

in which the completeness of the Pohlmeyer invariants is proven. I have so far only skimmed this paper. There are some interesting comments on the shape of the surfaces (i.e. the worldsheets) described by choosing certain values for the Pohlmeyer invariants.

Meanwhile I have realized that the sketch of a proof of how to relate the Pohlmeyer invarints to the DDF invariants that I had given in my previous comment easily generalizes to finite reparameterizations.

Recall that the invariant local worldsheet fields are of the form

(1)$\stackrel{˜}{Y}\left(\sigma \right)=\left(1+{\xi }^{\prime }\left(\sigma \right)\right)Y\left(\sigma +\xi \left(\sigma \right)\right)$

for $\xi$ a phase-space dependent function. By doing a Taylor-expansion this may be rewritten as

(2)$\stackrel{˜}{Y}\left(\sigma \right)=Y\left(\sigma \right)+\sum _{n=1}^{\infty }\left(\frac{1}{n!}{\partial }_{\sigma }^{n}Y\left(\sigma \right){\xi }^{n}\left(\sigma \right)+\frac{1}{\left(n-1\right)!}{\partial }_{\sigma }^{n-1}Y\left(\sigma \right){\xi }^{n-1}\left(\sigma \right){\xi }^{\prime }\left(\sigma \right)\right)$
(3)$=Y\left(\sigma \right)+{\partial }_{\sigma }\left(\sum _{n=1}^{\infty }\frac{1}{n!}{\partial }_{\sigma }^{n-1}Y\left(\sigma \right){\xi }^{n}\left(\sigma \right)\right)\phantom{\rule{thinmathspace}{0ex}}.$

This means that the invariant local fields $\stackrel{˜}{Y}\left(\sigma \right)$ which were obtained by linearly combining DDF invariants are equal to the non-invariant local fields $Y\left(\sigma \right)$ plus a total $\sigma$-derivative. But looking at the proof for the reparameterization invariance of the Pohlmeyer invariants as in section 2.2 of my notes one sees that it says that the Pohlmeyer observables are invariant under $\delta Y={\partial }_{\sigma }\left(\mathrm{anything}\left(\sigma \right)\right)$.

Ok, so this improves the sketch of a proof that I had given last time. In conclusion, if ${Z}_{±}^{{\mu }_{1}\cdots {\mu }_{N}}\left(Y\right)$ is a Pohlmeyer invariant then ${Z}_{±}^{{\mu }_{1}\cdots {\mu }_{N}}\left(\stackrel{˜}{Y}\right)$ is an observable completely constructed from DDF observables which coincides with ${Z}_{±}^{{\mu }_{1}\cdots {\mu }_{N}}\left(Y\right)$ on the subset of phase space where the longitudinal coordinate $k\cdot {X}_{±}$ (that enters the definition of the DDF observables) defines an injective map $\sigma ↦k\cdot {X}_{±}\left(\sigma \right)$.

It is natural to conjecture that if two invariants coincide on this much of phase space that they actually coincide on all of phase space. But I need to think about that.

In any case, one can already make the following statement:

Since $\sigma ↦k\cdot {X}_{±}\left(\sigma \right)$ is injective at sufficiently large longitudinal momentum ${p}_{k}$ one can restrict attention to that part of phase space where ${p}_{k}\ge {{p}_{k}}_{\mathrm{min}}$, i.e. to points in phase space which have sufficiently large longitudinal momentum. On this subspace of phase space the fiollowing is true:

Every Pohlmeyer invariant ${Z}^{{\mu }_{1}\cdots {\mu }_{N}}\left(Y\right)$ is expressible in terms of DDF invariants and the relation is

(4)${Z}^{{\mu }_{1}\cdots {\mu }_{N}}\left(Y\right)={Z}^{{\mu }_{1}\cdots {\mu }_{N}}\left(\stackrel{˜}{Y}\right)\phantom{\rule{thinmathspace}{0ex}}.$

Since one can always boost the center-of-mass momentum without affecting any physics, this restricted phase space still captures all of the physics of the string. Restricting the phase space like this might be regarded as a weak form of gauge fixing. Sending ${p}_{k}\to \infty$ would correspond to fixing lightcone gauge.

Posted by: Urs Schreiber on February 16, 2004 12:52 PM | Permalink | Reply to this

Re: Pohlmeyer charges, DDF states and string-gauge dualityD. Bahn’s thesis appeared on the arXiv

As reported above, I have had a little conversation with Dorothea Bahns concerning the relation of Pohlmeyer invariants and DDF invariants. I was (and still am) claiming that the Pohlymeyer invariants (at least on that part of phase space where the center-of-mass momentum exceeds a certain value) can be re-expressed in terms of classical DDF invariants and that the latter have all sorts of nice properties which are not manifest or even missing for the Pohlmeyer invariants.

Now the content of D. Bahns’ thesis (dating from 1999) on the canonical quantization of the Pohlmeyer invariants has appeared in English language and in the form of a paper:

I see that part of my conversation with D. Bahns has made it into the discussion section: In the very last paragraph on p. 19 the DDF operators are mentioned.

Unfortunately, there again a wrong claim is repeated, namely that the construction of the classical DDF invariants requires to fix a worldsheet gauge, in particular conformal gauge.

This is wrong. And it is easy to see why: We can only speak of conformal gauge when dealing with the Polyakov action. But we never need to even mention the Polyakov action when deriving the Virasoro constraints and constructing the classical DDF invariants. Everybody who does not believe that this is true should work out the exercise found here. The Virasoro constraints follow of course from the Nambu-Goto action alone (and I am puzzled, because this even mentioned and used in D. Bahn’s paper).

The Pohlmeyer invariants as well as the DDF invariants are nothing but objects in the canonical quantization of the Nambu-Goto action which Poisson-commute with all the constraints, and there is absolutely no place in this construction where one would ever even have to mention any worldsheet reparameterization gauge.

(Maybe it should be remarked that one can also derive the Virasoro constraints from the Polyakov action without fixing conformal gauge by using the ADM decomposition.This can be found online on page 153 of sqm.pdf. So no matter how we look at things, the construction of observables that Poisson-commute with all the Virasoro constraints has nothing to do with fixing a reparameterization gauge on the worldsheet.)

What Dorothea Bahns does in her paper is to show that when quantizing the Pohlmeyer invariants by applying the usual frequency normal ordering with respect to the worldsheet oscillators breaks their algebra and in particular introduces quantum corrections to the algebra which are not invariant. She concludes from this observation that

The results presented here show that the canonical approach and the algebraic quantization are inequivalent.

(Here by ‘canonical approach’ the usual quantization of the string is meant.)

My claim is that this is not the right conclusion to draw from the above mentioned observation.

I claim that we can regard the Pohlmeyer invariants as Polynomials in the DDF invariants and that these have the well known quantization and ordering which fully agrees with all the usual results. This way the Pohlmeyer invariants can be quantized nicely and consistently and reproduce all the standard results. They are also nicely generalized to the superstring and everything is find.

I was still hoping to make a relation of the Pohlmeyer invariants to the IIB Matrix model manifest, which might make them more interesting but which has kept me so far from putting the notes

U. Schreiber, DDF-like classical invariants of (super)-string

on the preprint server.

Posted by: Urs Schreiber on March 13, 2004 3:44 PM | Permalink | PGP Sig | Reply to this

Re: Pohlmeyer charges, DDF states and string-gauge duality

On sci.physics.strings Luboš has started an intersting discussion on Pohlmeyer invariants. Because google groups hasn’t yet begun archiving sci.physics.strings, I want to prevent this discussion from being lost in the near future by copying it here to Coffee Table:

Luboš writes: Dear Ladies and Gentlemen, I noticed that Urs Schreiber has an article today

http://www.arxiv.org/abs/hep-th/0403260

Congratulations. (By the way, try to please this format to refer about the articles and any websites: a nearly empty line, in between two empty lines, starting with a tabulator, and containing the URL, for example with the “abs” home page of the article.)

I want to ask (Urs Schreiber, and perhaps others) several questions about the recent status of these issues, namely:

* Is there now a general consensus that the Pohlmeyer charges are enough to construct all Virasoro-invariant operators?

* Can DDF be used as the basic blocks to construct all Virasoro-invariant operators?

* Can you construct the DDF operators from the Pohlmeyer charges, and vice versa? (This is just to check the consistency of the previous answers.)

* Are you free to impose the standard stringy commutation relations between the Pohlmeyer generators, and use them to analyze the spectrum in standard string theory?

* Are the spaces that you obtain by applying the theorems (about the existence of a representation) separable? Does this answer depend on whether you use the standard commutators, or the simplified Poisson brackets?

* If Pohlmeyer charges can be used to organize the standard stringy spectrum, how should I imagine their eigenvalues and eigenvectors?

* Which Pohlmeyer charges can be expressed using the (chiral) Poincare (zero mode) generators?

* Do the Pohlmeyer charges imply a separate argument on integrability? In fact, I must ask a more dumb question: What “integrability” means exactly? Is it the ability to write all correlators in a compact form? What does it mean “compact form”; which functions are allowed?

It could be enough to start this thread. Thanks, Lubos

Urs writes:

‘Lubos Motl’ schrieb:

* Is there now a general consensus that the Pohlmeyer charges are enough to construct all Virasoro-invariant operators?

Because of the fact that the Pohlmeyer invariants never involve the 0-mode x of the worldsheet fields X they all have vanishing level number. So you can in principle never express any combination of DDF invariants of nonvanishing level using Pohlmeyer invariants. A little reflection shows that you cannot even express all of the DDF invariants of vanishing level by means of the Pohlmeyer invariants.

What I demonstrate in the above paper are that the Pohlmeyer invariants are a subset of the set of all DDF invariants. The converse is not true, i.e. the set of Pohlmeyer invariants is strictly smaller than the set of DDF invariants (“much” smaller, indeed).

But still, Pohlmeyer et al. claim that the set of Pohlmeyer invariants is complete (overcomplete even, if I recall correctly) in the sense that classically the knowledge of the values of all the Pohlmeyer observables allows you to reconstruct the shape of the string’s worldsheet.

* Can DDF be used as the basic blocks to construct all Virasoro-invariant operators?

The key argument in the above paper is that when you have any functional F of the left- or right-moving worldsheet fields dX, bar dX (\mathcal{P}_\pm in the above paper) which is Virasoro invariant, then you can re-express this in terms of DDF invariants. So all Virasoro-invariant operators that are such functionals can be built from DDF building blocks.

I would suspect that this includes all invariant operators, but I have no proof for that.

* Can you construct the DDF operators from the Pohlmeyer charges, and vice versa? (This is just to check the consistency of the previous answers.)

You cannot. The Pohlmeyer invariants are a proper subset of all possible invariants that can be built by using DDF building blocks.

* Are you free to impose the standard stringy commutation relations between the Pohlmeyer generators, and use them to analyze the spectrum in standard string theory?

My argument is that since it is possible to build the Pohlmeyer invariants from DDF invariants, we can quantize the latter in the completely standard stringy way and thereby automatically get a quantization of the Pohlmeyer invariants which conforms this standard quantization. Indeed, I believe that this ‘solves’ Pohlmeyer’s program, which had the goal to find a consistent quantum deformation of the algebra of Pohlmeyer invariants.

But I don’t know how much the Pohlmeyer invariants themselves are useful to analyze the spectrum of the string. If I didn’t know their relation to the DDF invariants I wouldn’t see how they would be related to the string’s spectrum at all.

Maybe another important question would be: What is the quantum analog of the claim that classically the Pohlmeyer invariants are complete?

I haven’t checked it yet, but I could imagine that it might be true that knowing the expectation values

(1)$〈\psi \mid P\mid \psi 〉$

for _all_ Pohlmeyer operators P (quantized using the quantization of the DDF invariants) for a given state $\mid \psi 〉$ of the string’s Hilbert space allows you to reconstruct the state $\mid \psi 〉$. I.e. if I hand you all the complex numbers \langle\psi| P |\psi\rangle for all P you know which state $\mid \psi 〉$ I used to compute these values.

Of course if one is interested in just reconstructing states from their expectation values one could simply choose the expectation values of suitably projectors on states of fixed oscillator content. But these projectors are not invariant, as opposed to the Pohlmeyer invariants.

Note that even if the above conjecture is true, I wouldn’t know if it is particularly useful for anything. It might be nothing more than a nice toy example for how one can find complete sets of gauge invariant observables is a covariant quantum theory.

This is a general statement that I make: I don’t know if the Pohlmeyer invariants are really useful for anything. All that I know is how they can be re-expressed in terms of DDF invariants. But in the discussion section of the above paper I have included some speculations about what the Pohlmeyer invariants might be good for.

* Are the spaces that you obtain by applying the theorems (about the existence of a representation) separable? Does this answer depend on whether you use the standard commutators, or the simplified Poisson brackets?

As you know, Thomas Thiemann has tried to quantize the Pohlmeyer invariants by representing them as operators on a non-seperable Hilbert space. This is based on results by Rehren. But one has to be aware that the work by Rehren contains an unproven assumption, the so-called “quadratic generation hypothesis”. As long as this hypothesis is not proven it is not clear that these representations even exist.

I think the bottom line is that by just staring at the - extremely convoluted - classical algebra of Pohlmeyer invariants the task of finding a consistent quantum deformation is hopelessly difficult. I have no idea if this program, which Pohlmeyer et al. have followed for many years now without full success so far, can be fulfilled in any other way except for using the DDF operators and representing the Pohlmeyer invariants on the ordinary Hilbert space of the string this way. This of course makes the consistent quantization of their algebra a triviality, because the algebra of the DDF invariants is so simple!

* If Pohlmeyer charges can be used to organize the standard stringy spectrum, how should I imagine their eigenvalues and eigenvectors?

The only thing I can say here is to point again to the curious fact that all the Pohlmeyer invariants have level 0. This implies in particular that you can never get any excited string state by acting with Pohlmeyer invariants on the vacuum state. This is of course totally different for general DDF operators. Therefore you can never “walk through the spectrum” of the string by applying Pohlmeyer operators to a given state.

* Which Pohlmeyer charges can be expressed using the (chiral) Poincare (zero mode) generators?

It is obvious that this can be done for Pohlmeyer invariants of order 0, 1, and 2. You also told me that together with Edward Witten you showed that the same holds at order 3. You said you don’t think that it continues to hold at order 4.

All I can say is that the claim by Pohlmeyer et al. to have proveen that the Pohlmeyer invariants are classically complete would be in contradiction to the claim that they are all trivial (only consist of 0-modes). But I haven’t checked this proof or even looked at it in detail.

But, as I said before, I find this claim very plausible, because of the relation of the Pohlmeyer invariants to Wilson loops along the string. But that’s just my intuition, nothing more.

* Do the Pohlmeyer charges imply a separate argument on integrability? In fact, I must ask a more dumb question: What “integrability” means exactly? Is it the ability to write all correlators in a compact form? What does it mean ‘compact form’; which functions are allowed?

Integrability of any system means that you have as many (Poisson-)commuting invariants of motion as there are degrees of freedom of the system. In the case of constrained systems, such as the string, these invariants must also (Poisson-)commute with all the constraints.

Many thanks for these interesting questions!

Creighton Hogg writes:

Alright, I’m going to exercise my license to ask stupid questions and ask for what you think the DDF invariants are good for? I’ll be honest, I don’t really understand what they are, but it looks like they’re objects that can be used to quantize a string theory in a different way. If that’s true, then what advantage does this have over the classic way of quantizing the string as done in GSW or Polchinski? Is this related to Thiemann’s claims about being able to quantize the string with LQG methods and get a different physical system?

I just want to be clear now and in the future that none of the questions I ever ask are meant as attacks. When I ask what something is good for, it’s because I honestly just don’t get it.

Urs writes:

I don’t quite see the distinction that you make here. If you quantize the string covariantly (OCQ or BRST quantization as opposed to lightcone gauge quantization) then the DDF operators are the operators that generate the physical spectrum of the string. And they are used a lot in GSW, for instance in the proof of the no-ghost theorem (see p.113 of GSW and the discussion leading there).

Also note that the DDF operators are essentially just integrated vertex operators. They show up all over the place in string theory. For instance in hep-th/0303222 DDF operators are used to understand the spectrum of the string on pp-wave backgrounds.

I don’t see how one can study the spectrum of string in a covariant way without using the DDF operators. They generate the entire physical spectrum and allow you to derive the critical dimension in OCQ.Of course very often people use lightcone gauge instead to study the string’s spectrum. In these cases no DDF operators will be mentioned.

Note that from the worldsheet point of view the DDF operators (or rather their Fourier transformations) are the “quasi-local gauge invariant observables” of the 1+1d quantum gravity theory. It is not physically meaningful to ask for the string’s momentum density at sigma=2.1 (at a fixced worldsheet time) where sigma is the spatial parameter along the string. But it does make sense to as for the string’s momentum density at a point where certain oscillations of the string have a certain value. This question is invariantly defined. And that’s, heuristically, the question answered by the DDF operators. See pp. 13 of hep-th/0403260 for more details on this aspect.

Is this related to Thiemann’s claims about being able to quantize the string with LQG methods and get a different physical system?

Maybe I should emphasize, that nothing of what I write in hep-th/0403260 has anything to do with Thomas Thiemann’s ideas on how to quantize the string, which I think are physically wrong.

The reason I cite Thomas Thiemann’s paper is that it gives the most comprehensive review of the work that has been done on Pohlmeyer invariants. He and other people working on Pohlmeyer invariants are hoping that using these invariants one can find ‘alternative quantizations’ of the string. The message of my hep-th/0403260 is that this is an ill-founded hope, since the Pohlmeyer invariants are nothing but special cases of DDF invariants! Correctly quantizing the Pohlmeyer invariants, I claim, yields nothing but the standard string theory in an unorthodox way.

Creighton Hogg writes:

After reading what you said here, and

http://golem.ph.utexas.edu/string/archives/000301.html

I realize that I was confused and thought that the DDF operators were different from the operators used in OCQ. Sorry about that, but thank you for clearing this up for me.

So if I’m understanding correctly now, your claim is that the application of LQG quantization techniques using Pohlmeyer invariants to string theory should not be able to produce any new physics because it only amounts to the normal covariant quantization, but using a different “basis” of DDF operators?

Urs writes:

Yes, more or less.

The Pohlmeyer program is a priori something totally unrelated to LQG approaches. Pohlmeyer’s idea was to quantize a system by first deriving its algebra of classical invariants and then looking for a consistent promotion of this Poisson algebra to a commutator algebra of operators.

While Pohlmeyer et al. didn’t succeed in finding this operator algebra they believe(d?) that it does exist and does allow to, for instance, get rid of the critical dimension (for some reason that escapes me).

Thomas Thiemann tried to make use of these conjectures in his ‘LQG-string’ paper (http://golem.ph.utexas.edu/string/archives/000299.html) by trying to represent the Pohlmeyer invariants on his Hilbert space. Since LQG uses a ‘relaxed’ version of canonical quantization of constraints (http://groups.google.de/groups?selm=c3goq8%2418le%241%40f04n12.cac.psu.edu) which is not physics as we know it, I think that Thiemann’s quantization of the string is not viable irrespective of how it is related to Pohlmeyer’s program.

But what I claim is that there is at least one solution to Pohlmeyer’s program, and that this solution does reproduce the usual results by expressing the Pohlmeyer invariants in terms of DDF invariants. As long as no other solutions to the program can be found (and they have not be found) it is futile to speculate if they would allow “new physics”, as you say.

So, yes, my claim is that the most obvious solution of Pohlmeyer’s ‘alternative quantization’ of the string is just the ordinary well-known theory written in an unusual way.

Of course this is a result interesting only for those who were interested in Pohlmeyer invariants in the first place.

That’s why one might think about if the Pohlmeyer invariants, even when understood as just a curious subset of the DDF invariants, are of any further interest in themselves. As I discuss in the conluding section of hep-th/0403260 I have two maybe interesting observations concerning the Pohlmeyer charges, which might indicate a genuine relevance of these objects:

1) They remind me of the IIB Matrix Model (http://golem.ph.utexas.edu/string/archives/000314.html). In that model strings appear as Wilson loops of a Yang-Mills theory of large-N _constant_ gauge fields. A similar statement is true for the Pohlmeyer invariants: They, too, are essentially Wilson loops of large-N constant gauge connections along the string.

2) They remind me of spectral geometry in the sense of Connes’ NCG. (http://golem.ph.utexas.edu/string/archives/000322.html). That’s because if you generalize the Pohlmeyer invariants to the superstring (which is easily possible when expressing them in terms of DDF invariants as discussed in section 2.3.3 of hep-th/0403260) they look like linear combinations of terms of the form

[G,c1][G,c2]…[G,cp] ,

where G is the supercharge on the string’s worldsheet, i.e. the Dirac-Ramond operator. But expressions as this are known to be generalized differential forms in spectral noncommutative geometry. You can read the super-Virasoro constraints as saying that these forms are _closed_. Note that the “ci” above are integrals of weight h=1/2 fields and that hence G is _nilpotent_ on these.

I believe that the fact that hence the Pohlmeyer invariants can be regarded as generalized closed differential forms on loop space (the configuration space of the string) fits into general observations that you can regard the super-Virasoro constraints as a system of Dirac-Kaehler constraints on loop space. I have a very detailed discussion of this perspective and its applications in hep-th/0401175 and hep-th/0311064.

Posted by: Urs Schreiber on March 26, 2004 5:21 PM | Permalink | PGP Sig | Reply to this