I think we are going in circles;- this was the starting point of this strand of the debate, where I suggested that one can avoid this by abandoning the Fock rep.

It is a difference in desiderata; I observe that you can sometimes avoid abandoning Fock reps (or rather lowest-energy reps) by allowing for gauge anomalies, without giving up unitarity. This is an observation rather than a suggestion; if you look at chapter 2 of GSW, you’ll find it clearly stated that also the subcritical free string can be quantized with a ghost-free spectrum.

Of course, the free string is just a toy model – 2D gravity coupled to scalar fields. To apply analogous ideas to 4D gravity, I discovered the relevant diff anomalies in 4D (multi-dimensional Virasoro algebra), figured out how to build representations of this algebra, and found a formulation of physics to which these representations apply. The main physics lesson is that you must take the quantum nature of the observer into account – QFT is recovered in the limit that the observer is heavy and thus classical.

Among other things Hendrik suggested: “So, if one abandons the Fock representation, I think the dimension 26 restriction may be lifted.”

This looks like a nice scientific hope, (albeit seemingly far-fetched). Do you have, dear Hendrick, any evidence (rigorous or non rigorous) to support this hope? Also, wouldn’t you agree that for pursuing such an idea, it will be more efficient to still use physics’ more relaxed rigor standards rather than full mathematician’s standards of rigor?

Do you expect, Hendrick, that abandoning “Fock representation” will allow to work in any dimension and the 26 dimension restricting will be entirely lifted? Or perhaps you expect that the dimension that will burn the anomalies will depend on some parameter of your hypothetical theory, like the cc.

It is interesting that for some layperson the universe having many hidden dimensions is very appealing while for some it is a very disturbing idea. Moving from four dimensions to ten or eleven looks dramatic but does not look nearly as dramatic as QM’s dices. For me, the many dimensions is an appealing aspect of string theory (but perhaps a little arbitrary).

1. Regarding Nigel’s interesting comment (#92),

I thought that Peter’s point (#86) can be discussed as a separate issue from the issue of predictions. As for predictions, Joe explained it very nicely in his first reply. Indeed it appears to be agreed that the issue of predictions is a weak link for the string theoretic understanding/view of our universe. (And it will probably be a weak link for any alternative point of view.)

Apropos predictions it may be helpful to try to draw some lines between scientific predictions, scientific hopes, and scientific prophecies. While the borders are not always clear these three are quite different. (And all three can be of interest, value and even fun.)

Peter’s question reflects the scientific hope that the equations for SM within string theory (or any other theory of quantum gravity) will be simpler than for the SM standing alone. This may lead to some explanation for aspects of the SM which looks arbitrary, and perhaps to further understanding and some predictions within the scope of the SM as well. There is nothing wrong with this hope. It is a nice hope, and quite possibly one of the motivating hopes for string theory. (Indeed, also in Peter Shor’s amusing Amazon review of Smolin’s book that Joe mentioned this hope is described as a central motivation for ST.)

As nice as this hope is, attacking ST on the ground that it does not meet (yet or even ever) this particular hope is not convincing. Not explaining the parameters of the SM is not an obstacle for ST to explain things beyond the SM and to become a theory of quantum gravity that accommodates the standard model. Achieving this will already be a scientific landmark of the highest quality and magnitude.

String theory (but possibly also competing theories of quantum gravity that include the standard model if and when such theories will emerge,) may offer an explanation for the parameters of the SM, as representing a sort of typical or random instance of the theory, perhaps conditioned, on some basic properties of our universe. Such an explanation (and even its scientific legitimacy) is very controversial within the ST community and outside it. (Personally, such an explanation does not strike me as non-scientific.)

Another aspect of Nigel’s comment which I find problematic is that it seems that Nigel suggests a sort of definite a priori “Litmus test” for emerging scientific theories. This does not sound realistic.

Regarding Nigel’s PRL (subjective) ordeal may I say that one of the nice things about many human activities (like the academic endeavor) is the possibility to see importance and find meaning in matters that from the outside seems utterly unimportant and quite meaningless. One of the drawbacks of this nice feature is that sometimes mundane matters (like a rejection of a paper or a book) seem far too important and far too meaningful.

2. Regarding positive cc.

I do not think Joe’s and Lee’s description of the history are so different. It is agreed that positive cc was a surprise, and it is agreed that ST can accommodate positive cc. Joe views the positive cc as a place where perhaps string theory “already makes connection with observation,” and Lee regards this story perhaps as an indication how ST can accommodate anything and interprets it negatively.

(Lee’s point of view is complex. In #43,#53 Lee mentioned that the ST solution to positive cc was a success and remarked that Joe’s optimism turned out to be correct. Woit (#72) regards it as “obviously a failure and a dead end”. Clearly, the word “obviously” cannot apply to a matter where even Lee disagrees with Peter.)

In any case, both Joe’s and Lee’s interpretation are consistent with their prior approaches to the matter at hand and both do not contribute much to establish these prior approaches.

The positive cc story demonstrates a weakness of the falsifiability argument against string theory. Lee was pessimistic about the possibility that string theory can accommodates positive cc. Had it turned out that string theory cannot accommodate positive cc this would have falsified the theory.

3. Regarding Clifford pingback (#89).

Let me comment that while Clifford is my world-wide favorite blogger, his description of Joe’s second reply as: “the latest installment of Joe Polchinski’s rather thorough deconstruction of the nonsense, obfuscation, selective memory, and other confusions that constitute the bulk of Lee Smolin’s attack on string theory,” does not come across as correct or fair. (What is nice about Joe’s approach is that he is trying to discuss the most interesting scientific aspects and not necessarily the weakest links in Lee’s book.)

Clifford is correct, I think, in viewing the titles of Sean’s posts as somewhat peculiar; while Sean supports the ST endeavor the titles of the two recent posts seem (unintentionally) damaging to the ST side of the “public debate”.

104 Thomas Larsson on Jun 5th, 2007 at 9:06 am

“The system here under discussion (free open bosonic string) is exactly defined by imposing constraints on an infinite set of quantum oscillators.”

You are of course free to define whatever you like,..

I don’t think so;- this definition of the string is the old one you find in Scherck and in the introduction of most string books.

…but the quantization of a classical gauge theory does not need to be a quantum gauge theory.

I cannot make much sense of this clause;- for a start, quantization is not a well-defined map from the classical to the quantum theory; there are serious existence and uniqueness problems (cf. http://arxiv.org/abs/dg-ga/9605001). I presume you mean that the quantum constraints which you define have an anomaly i.e. are second-class. But as I pointed out before, a different choice of representation may allow you to avoid this. By the way, second-class quantum constraints can also be imposed (algebraically, not as state conditions) cf. Comm. Math. Phys. 119 (1988), no. 1, 75-93, so there is another alternative as well.

It is obviously true that the constraints can only be imposed when D = 26,

I think we are going in circles;- this was the starting point of this strand of the debate, where I suggested that one can avoid this by abandoning the Fock rep.

..but the no-ghost theorem asserts that the unconstrained Hilbert space has a postive-definite inner product when D

If you don’t impose the constraints I cannot see how you can think of this as a string. It is the constraints which contain the information coupling the quantum oscillators together into the string,- without them you just have an uninteresting set of independent oscillators. In the same way that you cannot claim that a vector boson field is electromagnetism until after you have enforced the Maxwell constraint and a gauge constraint (e.g. Lorentz condition).

The system here under discussion (free open bosonic string) is exactly defined by imposing constraints on an infinite set of quantum oscillators.

You are of course free to define whatever you like, but the quantization of a classical gauge theory does not need to be a quantum gauge theory. It is obviously true that the constraints can only be imposed when D = 26, but the no-ghost theorem asserts that the unconstrained Hilbert space has a postive-definite inner product when D < 26, cf. GSW chapter 2. This may hence be viewed as the quantized subcritical string. The anomaly turns the classical gauge theory into a quantum global symmetry.

101 Thomas Larsson on Jun 4th, 2007 at 1:31 am

My understanding comes from the experimentally proven application of CFT to statphys. The relevant Hilbert spaces (minimal models) are not Fock spaces, but there is an energy bounded from below, and that’s what gives rise to anomalies. In the physical Hilbert spaces; there is no constraining to do (apart from factoring out two singular vectors), because the anomaly converts the classical conformal gauge symmetry into a quantum global symmetry.

Well, my understanding comes from quantum constraint theory. The system here under discussion (free open bosonic string) is exactly defined by imposing constraints on an infinite set of quantum oscillators. So for this system there definitely is nontrivial constraining to do, and hence one should only require something physical like a positive energy on the final system not the original one.
To give you an indication of how much freedom this gives you;- in the C*-algebra approach we deal with global (integrated) quantities, hence with the gauge transformation group (not the infinitesimal generators). So a larger set of representations of the gauge group is now available, e.g. those which are not continuous in some directions. If there are constraints, one does not need to require continuity on nonphysical objects, so this is not a problem. In fact, in our version of Gupta-Bleuler electromagnetism quoted in my previous post, we did find that the physical representations had to be discontinuous on nonphysical objects (nonregular representations). Maybe this is the resolution for the representation facts which you refer to from CFT, insofar as those facts perhaps pertain to continuous (projective) representations of the diffeomorphism group of the circle.

The situation is similar for the free subcritical string. The classical theory has a gauge symmetry, which becomes global on the quantum level because some gauge dofs (trace of the metric) become physical. Conversely, these dofs decouple in the classical limit, making the system a classical gauge theory. This is compatible with unitarity.

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97 Thomas Larsson on Jun 3rd, 2007 at 9:13 am

“So, if one abandons the Fock representation, I think the dimension 26 restriction may be lifted.”

Bad idea. It is an experimental fact that energy is bounded from below. Hence you need representations of lowest-energy type!

An “experimental” fact can only give you a requirement on your final physical system i.e. after constraining is done, not before. So the defining representation has a lot of freedom in it. In fact, in our treatment of Gupta-Bleuler in http://arxiv.org/abs/math-ph/9812022 we did obtain Fock representations on the final physical algebra, but the important fact is that these representations did NOT come from an initial Fock representation.