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Addenda

Addenda to:

 

String Theory

(also known as)

Joe's Big Book of String

I will try, for a time, to provide brief updates on recent developments and improved explanations of rough spots in the text.

Last update: April 25, 1999.

My home page, The Big Book home page, Acknowledgements, Corrections.

 

  • Volume I:
    • Section 1.4, p. 29 (4/4/99): The following point is made implicitly in several places, but it is useful to make it explicit. In unoriented theories, the orientation-reversing diffeomorphisms are gauge symmetries (configurations differing by such a diffeomorphism are physically equivalent), while in oriented theories they are not. The projection \Omega=+1, like the projection 1.4.11, is then a projection onto gauge-invariant states. (Thanks to F. Ferrari)
    • Section 2.1, p. 34 (4/25/99): For consistency with the discussion in chapter 1, let \sigma= - \sigma^1 and \tau = \sigma^0.
    • Section 2.3, p. 42 (4/4/99): Regarding the discussion of hermiticity, one can think of the Euclidean time derivative as anti-Hermitean, due to eq. A.1.36. Incidentally, the current defined in eq. 2.3.13 is actually anti-Hermitean because of the explicit "i". A more detailed discussion appears on pages 202-203, but this is perhaps too abstract for the reader at this point. (Thanks to L. Motl)
    • Section 2.4, p. 46 (12/6/98): Eq. 2.4.12 is correct but the explanation is slightly confusing: it gives eq. 2.4.12 at the point z=0, but it is equally true at any other point (just do a translation). (Thanks to M. Maggiore)
    • Section 2.4, p. 48 (4/4/99): Below eq. 2.4.24, the implicit assumptions are that T has weight (2,0) under scaling and that there are no operators of negative dimension. There are interesting exceptions to the latter (non-unitary CFTs), but eq. 2.4.24 still follows from the assumption that T generates conformal transformations. (Thanks to F. Ferrari)
    • Section 3.2, p. 84 (11/30/98): Several students have found the discussion in the final paragraph of this section confusing. It is a trivial aside, and can be omitted.
    • Section 3.3, p. 86-88 (4/4/99): My implementation of the Faddeev--Popov procedure is more roundabout than necessary. Here is a shorter version, based on a suggestion of R. Stora. First, change the definition of the Faddeev--Popov determinant:

      1 = \int [d\zeta] \Delta_{FP}(g^\zeta) \delta(g^\zeta - \hat{g})

      Since \Delta multiplies the delta-function it is defined only when its argument is on the gauge slice, but that is all that we will need. Now insert this into the functional integral 3.3.1. Use the gauge invariance of measure times e^{-S} to write

      [dX dg] e^{-S(X,g)} = [dX^\zeta dg^\zeta] e^{-S(X^\zeta,g^\zeta)}

      Now only X^\zeta and g^\zeta appear, so rename dummy variables

      (X^\zeta,g^\zeta) \to (X,g)

      and integrate over g to obtain 3.3.14. In particular, the obscure footnote on page 87 is avoided.

    • Section 3.3, p. 89 (4/4/99): F. Ferrari feels that the derivation of the weights of b and c, though it gives the right answer, is rather ad hoc and suggests that a proper derivation must be within the framework of the BRST formalism; similarly for the calculation of the Weyl anomaly in section 3.4. I agree that the argument is somewhat ad hoc and that the BRST formalism is the most systematic way to proceed, but it should be possible to address the issue of gauge-independence (especially for a restricted class of gauges such as those in which the metric is fixed) without introducing the full BRST formalism (compare sections 4.1 and 4.2). I preferred this for pedagogic reasons; the logic could be tightened, but I believe that it is correct.
    • Section 3.3, pp. 97--98 (11/30/98): I state that the relation between long world-sheet cylinders and large spacetime distances will be become evident. Where exactly? Not until chapter 9. There it is shown that a long cylinder (as in figure 9.3c) gives rise to a momentum space pole (as in eq. 9.5.2); the pole is a resonance from an on-shell intermediate state, and corresponds under Fourier transform to long distance propagation. This point (and the general discussion of unitarity) got pushed back in successive drafts, though it does show up implicitly in the tree level calculations of chapter 6.
    • Section 3.6, p. 105 (4/4/99): A. Papazoglou askes for a derivation of eq. 3.6.15. Here's a hint for 3.6.15a: note that the left-hand side is the same as $\half \partial_a \delta_W \Delta(\sigma,\sigma)$ (the derivative acts on each argument, giving two terms that are equal by symmetry of \Delta).
    • Section 7.3, p. 221 (12/31/98): M. Headrick asks for more explanation of the statement that "the description of a quantum field theory as a sum over particle paths is not as familiar as the description in terms of a sum over field histories, but it is equivalent." Starting in Euclidean spacetime, consider the operator $\exp{-t(-\del_\mu \del_mu + m^2)}$. One can think of this as the evolution operator for a nonrelativistic particle moving in D space dimensions, where t is an auxiliary variable, a sort of internal time. It then has a path integral representation as in appendix A, where one sums over spacetime paths. Now integrating over t produces $(-\del_\mu \del_mu + m^2)^{-1}$, the scalar propagator. Thus we have a path sum representation for the propagator --- in fact, this is obtained in exercise 5.1, after gauge-fixing. Allowing vertices where three or more paths meet at a point, one obtains the Feynman graph expansion for scalar field theory. One can also rotate back to Minkowski spacetime, as on pp. 82--83. This can be generalized to particles with spin. However, it does not seem to be as powerful as the field representation: it does not have a natural nonperturbative extension; nonlinear symmetries are less manifest; for non-static Minkowski signature spacetimes it is hard to formulate.
    • Section 7.4, p. 223 (4/4/99): In eq. 7.4.1, the factor of 2 in the denominator has the same origin as the 1/2 in 7.3.1.
    • Section 9.8, p. 321 (4/18/99): L. Motl asks for a derivation of eq. 9.8.14. Hint: in the second line introduce an integral of {\bf k} such that "m^2" becomes "m^2 + {\bf k}^2". Then integrate over t (result comes out in terms of elementary functions). Finally, the sum on r is the series expansion of the log in the first line. (Added as exercise 9.10 in third printing).
  • Volume II:
    • Section 14.8: Under construction: I intend to give a brief explanation of the Maldacena conjecture, and its relation to Matrix Theory.
    • Section 18.3 (1/18/99): In my discussion of scales, based primarily on the perturbative heterotic string, the compactification, unification, and string scales are all near the Planck scale. So what's all the recent discussion of TeV scale string theory and millimeter dimensions? The main idea can be understood from figure 18.2. Imagine that the size of the extra dimension is increased, so the energy scale of the kink in the gravitational coupling is lowered. What happens? The gravitational coupling now meets the other three at a lower energy, before they have unified: thus we lose the simple unification of the four couplings. Let us ignore this, supposing a less standard gauge unification that produces a different unification scale (one possibility is mentioned at the end of this note). We then have two scales: the compactification scale where the gravitational coupling gets a kink, and the string scale where the gravitational coupling is of order one and meets the gauge couplings.

      So how low can these scales be? The simplest bounds are that the compactification scale be at least $10^{-4}$ eV (the inverse of 1 mm) and the string scale be at least 1 TeV. The latter comes from the non-observation of string physics at particle accelerators. The former comes from the Cavendish experiment: if the extra dimension were larger the inverse square law would be seen to turn into an inverse cube. Remarkably, it is not easy to improve on these bounds. Proton stability, cosmology, and many other constraints must be taken into account, and unless the model is very contrived these may raise the scales somewhat, but this still leaves open the possibility of a very new picture of physics above the weak scale.

      By the way, the discussion of figure 18.2 is for gauge fields in 4 dimensions and gravity in 5, but more generally the gauge fields may live in 4 and gravity in 4+k, with the rest at the string scale (that is, the gauge fields live on a 3-brane embedded in some higher space). The dimensionless gravitational coupling is $\kappa^2 E^{2+k} R^k$. Setting this to O(1) determines the approximate string scale. The reader can check that for k=2, the largest possible size (1 mm) gives also the lowest possible string scale (1 TeV). Incidentally, A. Hashimoto points out that the kink in figure 18.2 has been exaggerated, and corresponds to k = 3 rather than k = 1 as intended.

      There have been various discussions over the years of large dimensions in which both gauge fields and gravity propagate. As noted in figure 18.1, taken by itself this is inconsistent with perturbative unification of gauge interactions and gravity, because the gauge coupling becomes strong while the gravitational coupling is still very weak. Again, strong interactions allow new possibilities. One that has been recently discussed is a hybrid, a kink in the gravitational coupling at a very low scale plus a kink in the gauge couplings at a higher scale such that they still meet. It has been argued that this can produce the accelerated gauge unification that is needed when the string scale is lowered.

      One review of this subject is N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, "Phenomenology, Astrophysics and Cosmology of Theories with Sub-Millimeter Dimensions and TeV Scale Quantum Gravity," hep-ph/9807344.