FIELDS

Where to get it

Sorry, we aren't mailing printed copies. If you still have computer-related problems, please contact your System Manager before e-mailing me.

What makes it different

For information on my course based on this book, see my PHY 610-611 page.
This book was the most downloaded paper from the CERN Document Server in 2000 and #4 in 2001.
See also the APS News article.
An interesting article on open source books.

Additions (v3) ☹

Chapter I

  1. p. 44, middle: Here we treat φ as both a function & an operator, which may be a little confusing, but the distinction should be clear by context. More explicitly, if x is a coordinate and X is the corresponding operator, then (for 1 coordinate, for simplicity) i[p,φ(X)] = φ'(X), where φ'(x) = ∂φ(x)/∂x. So we may sometimes write ∂φ/∂x as (∂/∂x)φ(x) to mean the ∂/∂x acts only on φ (treat φ as a wave function, e.g.), but we may also write [∂/∂x,φ(x)] when treating φ as an operator.
  2. p. 47, top: Spurious right quotes after {,}.
  3. p. 47, middle: f(ψ) = a+ψb is a more convenient ordering.
  4. p. 48, top: should be definite integration. Better yet, think of it as integration over all of a space with no boundary, like a circle; then, no boundary → no boundary terms.
  5. p. 61, top, exercise IA4.7b: see exercise IA4.4 → 4.6.
  6. p. 66, middle: effect the coordinates → affect ...
  7. p. 67, middle: The relevant symmetry is really C(P)T, because CPT is a symmetry of any local, unitary, Poincaré invariant theory in any dimension. (Here P is trivial: 1 time dimension, no space.) There is no worldline C, because everything, like x, is real. (Delete the parenthetical remark about extra variables.)
  8. p. 68, bottom: Maybe some words on the AdS/CFT correspondence could go here.
  9. p. 70, middle: It might be easier for some people to look at δyA=yBεBA.
  10. p. 70, bottom: new exercise IA6.4: Instead of using this algebraic way of finding the conformal group, one can use an analytic way: Use the definition of infinitesimal conformal transformations given above as preserving p2=0. Solve the resulting first-order differential equation by Taylor expanding λa and ζ in x.
  11. p. 71, exercise IA6.4: Do this completely in terms of x.
  12. p. 72, top, new ref. 4: O. Klein, J. Phys. Radium 9 (1938) 1: Klein transformation.
  13. p. 73, bottom: For uniformity, ψ' = Mψ + V just before the exercise.
  14. pp. 77, middle: "Irreducible representation" is first used here, but defined on p. 80, so another definition needs to be given here.
  15. p. 78, middle: The fact that any matrix that commutes with the generators in an irreducible representation is proportional to the identity is Schur's lemma. It's easy to prove by first separating out the hermitian and antihermitian parts of the matrix (in this case it's already hermitian), then diagonalizing each hermitian matrix. Separating the resultant matrix into blocks each proportional to the identity, one sees that the generators must be block diagonal with respect to them, so each block is a representation.
  16. p. 78, middle, new exercise IB2.2:
    Consider the transformation properties of these tensors under the action of the group, Gi' = gGig-1 =gijGj.
    1. Prove invariance of fijk & thus ηij.
    2. Show trR(GiGj) ≡ ηRij is invariant; as a matrix in the adjoint space, ηR' = g ηR gT = ηR. For 2 different representations, looking at the η of one times the inverse of the η of another, show using Schur's lemma that this implies that all η's are proportional.
  17. p. 80, middle: colummn → column.
  18. p. 82, bottom: strictly speaking, these integrals converge only if the matrices have positive-definite eigenvalues; the general case then follows as described below.
  19. p. 84, top: Note that this relation for the determinant provides a simpler expression for the inverse of an antisymmetric matrix directly in terms of the Pfaffian.
  20. p. 86, middle: The charge is canonically conjugate to the group coordinate θ. So if θ is compact (i.e., an "angle"), q is quantized; but if q can take continuous values, the group must be non-compact.
  21. p. 93, top: Here the Ĝ's are understood to act as differential operators, so they are not written as commutators, to emphasize that they are not matrices in the vector space.
  22. p. 109, top: There should probably be here a whole new subsection IC6 on cosets, as nonlinear realizations of symmetries, including projective representations. These will be used much later (subsection IVA3) for fields, but are often used for coordinates, e.g., for (super)conformal symmetry. A general discussion here would include the introductory part of IVA3, invariant differentials (& finite differences) and derivatives, differential form of symmetry generators, and invariant integration measure. A good example/exercise would be the explanation of exercise IA6.5, which would be moved here from p. 789. Also moved here would be the representation of the 4D conformal group as a projective space from the end of subsection IIB6 (which was applied at the end of subsection IIIB1, & to instantons in subsections IIIC5-7).
  23. p. 109, top, ref. 1: An older, longer, preprint version of that paper was recently re-released, AEC report NYO-3071.
  24. p. 109, middle, ref. 7: The 2008 edition of the Review of Particle Physics is out.

Chapter II

  1. p. 114, bottom: Note that eiπC/2=iC, and complex conjugation just changes the sign of one component of a vector (that in the C direction), while iC is a rotation of the other 2 components into sign changes.
  2. p. 119, middle: In the equation for ½⊗½, there should be a ½ multiplying ψγχγ.
  3. p. 119, bottom, new exercise IIA4.2:
    Spherical harmonics are simple in spinor notation.
    1. Write the unit position vector (position divided by radius) as a symmetric matrix, in spherical coordinates (i.e., in tems of θ and φ). Show its components are the spherical harmonics for angular momentum 1.
    2. Explain why the totally symmetric product of L such objects gives the spherical harmonics for angular momentum L. Compare with known expressions for L = 2 and 3.
  4. p. 122, above exercise IIA5.3: For SO(2,2), replace Vγδ' → Vαβ'.
  5. p. 124, bottom: V should transform like ψψ̄, not ψχ̄, to make it a real 4-vector. (But that makes it an axial vector.)
  6. p. 125, middle: It might be better for some applications to use instead ψ→iψ, ψ̄→iψ̄. For example, this makes more sense in certain numbers of dimensions where spinors are real, so ψ̄=ψ. (But we may need to also do things like change the signs of masses.)
  7. p. 126, middle, new exercise IA5.6: Using the methods of subsection IC6 (to be added), derive 4D conformal transformations as projective transformations on 2×2 (hermitian) matrices.
  8. p. 132, bottom: "insubsection" → "in subsection"
  9. p. 135, IIB2.1b: More-explicit wording -- "Find a manifestly Lorentz covariant solution to the second equation first..."
  10. p. 148, middle: An important change of convention that will prove convenient later --- we can hide these signs to some extent by defining pαγ̇=+pαpγ̇ always, and then defining pγ̇=±(pγ)*. This allows us to do all algebra without the signs (e.g., in scattering amplitudes), until we need complex conjugation (e.g., in cross sections).
  11. p. 149, middle: The above signs will then appear in the complex conjugation relations, e.g., 〈pq〉*=ε(p0)ε(q0)[qp].
  12. p. 154, middle: All the +'s should be ⊕'s.
  13. p. 157, top: I forgot to mention [p,q] = 0.
  14. p. 160, bottom: IJK... indices should probably be ABC... for consistency.
  15. p. 161, top: "M" shouldn't be used here, to avoid confusion with the M of the previous page.
  16. p. 161, top: The fact that a is a "bosonic index" means za is bosonic; likewise, zα is fermionic.
  17. p. 161, bottom: There should be a discussion here about general conventions for index ordering. We have previously defined indices for matrices for matrix multiplication with the first index on a matrix down, second index up, so that consecutive indices can be contracted with the first of the contracted indices up, second down. If we consider matrices, or general tensors, in terms of bases of direct products of vectors (defining representation), then statistics imply that if we have different index ordering in different terms or sides of an equation, then there are signs due to relative reordering of indices. (An example of this was the rule for complex conjugation with spinor indices given in subsection IIA5.) Every time an index A has to be "pushed" past an index B to achieve the same ordering, there is the same sign that would be needed for pushing the corresponding vector in the direct-product basis. This factor is written as "(-1)AB", where each index in the exponent is assigned a value of 0 if the corresponding index on the tensor is bosonic, 1 if it's fermonic.
  18. pp. 162-3: Since USp(2n) can have an indefinite "U", as USp(2n+,2n-), so we can also have OSp*(2m|2n+,2n-).
  19. p. 164, middle: Here will go a discussion of 4D superspace(s) as superconformal cosets/projective spaces, based on new subsection IC6. This will include the definition of superspin and super-isospin. Some of this will leak into the following subsection.
  20. pp. 164-5: The index ordering is inconsistent in some spots; see the note above for the bottom of p. 161.
  21. p. 165, middle: The "reverse" statistics for (super)twistors can be explained by noting that they always carry a second spinor index, associated with the little Lorentz group. This is obvious in the 6D, but not in lower dimensions. (However, see subsection IIB6, where twistors were introduced as group elements.)
  22. p. 165, bottom: The choices of alphabets for these indices differ from the conventions used in subsection IIC3: Now A=(α,i), where α is bosonic (spactime) & i is fermionic (internal).

Chapter III

  1. p. 171, bottom, new exercise IIIA1.2:
    Find the equations of motion from the Lagrangian L = aq2 +bq̇2 +cq̈2 (for constants a,b,c).
  2. p. 183, top: It would be more general to define J as δSM/δA with respect to the rest of the action SM, as previously in this subection.
  3. p. 196, bottom: The τ limits of integration for the lightcone gauge should be τi and τf.
  4. p. 199, middle: It might be less confusing to call the matter ψ instead of φ, so as not to confuse it with the φ of the rest of this subsection.
  5. p. 200, top: Note that one can vary the action with respect to X(τ) and φ(x), etc., independently.
  6. p. 209, bottom, exercise IIIC2.1: The hint isn't really useful.
  7. p. 214, top: You can skip a step by dropping the next-to-last expression. (The same goes for the analogous equation on p. 639.)

Chapter IV

  1. p. 234: I never really defined "σ model", or even said what the "σ" stood for. They are just theories of Goldstone bosons. The original theory was polynomial, as described for a vector of SO(4): The component with the vacuum value was called "σ", the 3 Goldstone bosons were identified with the π. A "nonlinear σ model" was the result of taking the mass to infinity, leaving a nonpolynomial action for the Goldstone bosons; ironically, this model contained only the π, σ having been eliminated.
  2. p. 236, middle: Of the CP(n-1) model's U(n) symmetry, the U(1) is local and the SU(N) is global.
  3. p. 236, bottom, new exercise IVA2.2:
    Consider the CP(1) model in 1 dimension:
    1. Look at new gauge invariant variables quadratic in the original fields φ. The 4 elements of this 2×2 matrix are a 3-vector "x" and its norm. (See exercise IIA3.1. φ is now an SU(2) "twistor".) Rewrite the CP(1) action in terms of this 3-vector, to find the action for a nonrelativistic particle constrained to a sphere (i.e., the case SO(3) of the previous model).
    2. Now look at the nonrelativistic action for a particle in a gravitational/Coulomb potential (~1/r), obtained from the relativistic one (section IIIB) by replacing p2 with pi2-2mE, as suggested by the discussion of subsection IA4. Instead of the usual v=1, choose v=r, so the term in the Lagrangian for the (scalar) potential becomes a constant. Then make the change of variables above in reverse, to obtain the 1D CP(1) model, but with the Lagrange multipler replaced with the original mass, and an extra term that defines the time t. In the gauge A=0 one gets a 4D harmonic oscillator (but with A still imposing a U(1) constraint at t=0).
  4. p. 241, middle: I forgot to drop 1 Lorentz index.
  5. p. 244, exercise IVA5.1: You can ignore the hint. (It doesn't seem helpful.)
  6. p. 244, middle: The action of T is linear on eiφ, so it's also useful to think of this as a U(1) nonlinear σ model, as at the beginning of IVA2.
  7. p. 248, 1st line: "for for" → "for"
  8. p. 250, middle: ∫d4x needs a 1/(2π)2.
  9. p. 251: All the δ's should have a (2π)2.
  10. p. 251, bottom: Look at δS = ∫δ(φ2gmn)δS/δ(φ2gmn).
  11. p. 253, middle: To see that the singularity is unavoidable, note that the minimum (because a>0) of the parabola is at nonpositive φ, from the energy conservation equation. (If a=0 it's a straight line, so obvious.)
  12. p. 255, middle: In defining ρ̂ (and p̂) we used units〈φ〉(essentially φ(x̊)) = 1, so to restore units we actually need to insert a G/3π (corresponding to units G = 3π). Thus Ω ≡ 2ρ̂/H2 (and σ is the simplest).
  13. p. 256, middle: ∫d4x needs a 1/(2π)2.
  14. p. 261, top: To be more explicit, the question is whether Mi* are a linear combination of Mi.
  15. p. 265, bottom: I should have given the gauge transformations explicitly. Also, the left SU(2) is the local one, while the right one is the global one.
  16. p. 276, exercise IVC1.1b: This really means to find the form of the supersymmetry transformations.
  17. p. 283, top: The 1st paragraph belongs at the end of the previous subsection.
  18. p. 283, middle: The chiral representation can also be treated as a complex gauge.
  19. pp. 278, 283 (moved to 282 with above): It should be noted that the counting of physical, auxiliary, & gauge degrees of freedom separately balances between bosons & fermions --- both these multiplets have 2 Bose + 2 Fermi physical degrees of freedom, & both have 2+2 auxiliary; the rest of the superfield also divides equally (since any superfield does). For both cases, 1/2 of the Weyl spinor is auxiliary (as seen from the lightcone analysis of subsection IIIC2). For the scalar multiplet, we have a complex auxiliary scalar; for the vector multiplet, a real scalar + 1 component of the gauge vector.
  20. p. 281, middle: "Superspin" is never defined. This will be fixed with the discussion of superspace as a coset space (see above).
  21. p. 288, top: I left out the λ's.
  22. p. 292, middle: It might be nice to give here a realistic softly broken N=4 model.

Chapter V

  1. p. 303, exercise VA2.1: S here is an unknown; only after solving the Schrödinger equation is it found to be related to the action that follows from H.
  2. p. 308, top: "δ in x" → "δ in q".
  3. p. 308, middle: The integral should be over t'', so as not to confuse with the limits of integration.
  4. p. 317, bottom: VA2.4 → VA2.1.
  5. p. 322, bottom, new exercise VB1.3: Quantize the particle with v not constrained to be positive, and show the resultant propagator is proportional to δ(p2). Also quantize the Lagrangian λαλα̇ẋαα̇, where λ & λ are twistors. Show the result is θ(p0)δ(p2), and explain the relation to the previous Lagrangian.
  6. p. 341, top: The "-" signs come from the "-i" that goes with each factor.
  7. p. 343, bottom: This argument applies only to connected graphs. The rule for disconnected graphs then follows.
  8. p. 344, bottom: More simply, giving the action a 1/ħ gives that factor to each ∫dx, & thus an ħ to each ∫dp, of which there are L-1.
  9. p. 349, bottom: That should really be pi2+mi2=0 to emphasize the fact external states can have different masses.
  10. p. 350, middle, new exercise VC4.1b: Evaluate δ/δφ(-p) on 〈φ||ψ〉. Note that it involves ψ(p), not ψ(-p).
  11. p. 355, middle: Note that for the next few subsections we stick to Minkowski space for purposes of discussing unitarity, where i's & complex conjugation are important.
  12. p. 357, bottom: IA2.3e → IA2.3d.
  13. p. 358, top: In the expression for D+, the signs for p in the δ/δφ's are wrong. (See new exercise VC4.1b on p. 350 above.)
  14. p. 362, top: Maybe a step could be added here --
    rate/particle = (probability/time)/(density × volume) = P/ρVD.
  15. p. 362, middle: That should be 〈ψ||ψ〉.
  16. p. 364, exercise VC7.1a: That 4π is only for the case of initial spin 0; otherwise the initial spin picks a direction, so the 4π should be replaced by an ∫dΩ.

Chapter VI

  1. p. 379, top: ?i → λi.
  2. p. 383, middle: We can also put matter fields into f.
  3. p. 385, top, ref. 3: Tyutin's paper has now been "published" --- arXiv:0812.0580v1 [hep-th].
  4. p. 389, middle: Note the trivial case α=∞, which corresponds to no gauge fixing, gives a propagator that blows up.
  5. p. 390, middle: See comments on p. 148 above.
  6. p. 395, middle: Actually the final result is independent of what (nonzero) constant value the scalars are given.
  7. p. 398, exercise VIB5.3: A better value might be m/2, but again the value is irrelevant.
  8. p. 404, bottom: The rule that trees go as 〈 〉2-E+[ ]2-E- can still be applied with fermions, if the subscripts on E± are applied to just the sign of the helicity, using the fact helicities ±½ always appear in equal numbers.
  9. p. 416, bottom: A point should probably be made here that the background gauge parameter is real (since the background covariant derivatives are in the real representation), while the quantum gauge parameter is chiral (since the quantum prepotential is in the chiral representation).
  10. p. 417, ref. 11: now available as hep-th/0509223
  11. p. 418, ref. 16: no space before period .
  12. p. 419, middle: After 1st paragraph of subsection 1 -- "(see subsection VC9)"
  13. p. 419, bottom, & 422, top: See comments for pp. 148-9 above.
  14. p. 423, above exercise VIC1.4: Better to use parentheses than angular brackets for the amplitude, as later.
  15. p. 424, middle: "The two cases with simple known solutions..."
  16. p. 427, above exercise VIC3.1: For consistent notation, that should be a ⊖ on the antiselfdual vertex.
  17. p. 432, middle: Left quote right above figure is backwards.
  18. p. 438, ref. 1: Phys. Rev. Lett. → Phys. Rev. D.

Chapter VII

  1. p. 444, middle: The argument P-½LD-n of the Γ function counts the overall power of p.
  2. p. 459, exercise VIIA6.1: To get the point of this problem, it's sufficient to look at the case where a scalar is exchanged only in the t channel, representing a force between 2 scalars.
  3. p. 460, ref. 7 available at this link
  4. p. 462, middle: "m → 0" needs some space.
  5. p. 474, middle & p. 475, middle: All σ's should be x's.
  6. p. 476, middle: Note that this implies φ is an angle; it's periodic with period 2π.
  7. p. 477, bottom: "The above relation" really means: Since φ no longer satisfies a free equation, consider equal-time commutators. φ is still an angle, but the canonical conjugate to φ is now π = φ̇/2β2 (not just φ̇/2), and ∂φ gets replaced with π + φ'/2. (φ̇ & φ' are the time & space derivatives.) Then integrate this "∂φ" over space to get the analog (same equal-time propagators) of the chiral φ.
  8. p. 484, exercise VIIB7.1: The analysis made for this model is actually a bit of a cheat, since each loop in φ can be written equally well with χ running in the s, t, or u direction. The generalization of this exercise makes this distinction more precise by making the diagrams used above (all χ's in the same channel) larger by powers of m or n.
  9. p. 490, bottom: After ref. 3 belong refs. 23-5 from p. 536.
  10. p. 493, bottom: Here we consider ĝ(M,ε) & g(μ,M,ε).
  11. p. 493, bottom: To see that β has no positive powers of ε, look @ the equation order-by-order in g2; it then follows inductively.
  12. p. 497, top: Unfortunately, the term "asymptotically free" used here will not be defined until subsection VIIIA3; it means β₁>0.
  13. p. 498, top: The sentences "We use... in that combination.)" would go better after exercise VIIC2.1.
  14. p. 498, bottom - 499, top: For D=0, these are ordinary integrals and derivatives, so really D → d, δ → d.
  15. p. 499, top: Actually, in this case, doing the perturbation expansion after the JWKB expansion, all the coefficients are infinite, because to get classical solutions we needed complex φ, which makes S negative.
  16. p. 499, middle: "The simplest example of a 'renormalon' problem..."
  17. p. 500, middle: What we have really done here is to insert an all-loop gluon (or similar) propagator into a diagram with other propagators of mass m. In the UV part of the calculation we neglect m.
  18. p. 502 bottom: Here we treat only the gluon coupling as ħ, ignoring the coupling to the particle of mass m. (See previous note.)
  19. p. 506, middle: Here all diagrams are closed surfaces, so tree graphs have 1 face (L=H=0). So really graphs go as NF-1 --- the equation needs an extra 1/N.
  20. p. 519, middle: Note that the coefficient of a leading divergence is the same upon dimensional reduction. However, it appears at a different order in momentum. But dimensional regularization shows only logarithmic divergences. So, e.g., to find the leading, p22 divergence in D=26 via dimensional regularization, one needs to evaluate the same graph (after dimensional reduction) in D=4, since it corresponds to an integral dDp/p4. Similarly, if one wants to analyze a quadratic divergence in D=4, one needs to look at the theory dimensionally reduced to D=2.

Chapter VIII

  1. p. 528, bottom: You may include an "i" in the rescaling of φ. Then the propagator has the same sign as the usual. The vertices will then also get an extra sign, so the net result cancels in the loop.
  2. p. 529, top: It might be clearer to consider the relevant term in the Lagrangian as ∫d4θ φ∇2φ. Then there's a d2 (on δ4(θ)) associated with each propagator, and a ∇2-d2 with each vertex (with an ∫d4θ).
  3. p. 529, bottom: The ½ at the end of the evaluation of the diagram should be there at the beginning. (The ½ produced by dd is absorbed into the definition of d2.) It comes from the symmetry between the 2 φφ propagators (or, equivalently, from treating them as φφ̄ propagators, but including a ½ to compensate for squaring the kinetic operator.)
  4. p. 532, middle: "the previous subsection" (twice) should refer to subsection VIIB5.
  5. p. 532, bottom: Here we use α = ⊕ or ⊖ to correlate with ±.
  6. p. 536, top: refs. 23-5 belong on p. 490, after ref. 3.
  7. p. 538, bottom: VA2.4 → VA2.1.
  8. p. 547, middle: Another way to understand the coupling is to interpret A = ½∂π, since they have the same gauge transformation. Thus, A is pure gauge (i.e., longitudinal).
  9. p. 550, middle: Note that the separation used here into spinless + magnetic moment contributions is exactly the one used in subsection VIIIA3 into ◻ + FS.
  10. p. 563, top: 4-2ε → 4−2ε (hyphen → minus).
  11. p. 575, middle: IXA4 → IXA3.
  12. p. 575, middle: "Explicitly, ..."
  13. pp. 579-580: The vertices should include the usual factors of the coupling g.
  14. p. 580, middle: "vanishing sources V=1" → "N=1 and V1=1 (k1=0)"
  15. p. 580, bottom - 581, top: A better explanation here is that if you try to invert a Hermitian matrix M with some vanishing eigenvalues, the best M-1 you can get can satisfy only MM-1 = projection operator killing those eigenvectors. For the tree (& 1 loop), the only such zero-mode is a constant.
  16. p. 582, top: As explained later, this is an expansion about the center of the string, not the boundary (where ∂X/∂σ=0).
  17. p. 583, bottom: Note that the divergences in the Γ function are IR (because we neglected masses to simplify the calculation), while those @ s=0 are UV (in terms of the integral over loop momentum or T).
  18. p. 584, top: Sab for spin 1/2 should have a 1/2, not 1/4.
  19. p. 584, bottom: Applying these couplings to generalize the stringy model of the previous page to maximally supersymmetric Yang-Mills, we see from subsection VIIIC4 that a physical graviton, axion, and dilaton are generated in D=10, appearing as massless, color-singlet poles.

Chapter IX

  1. p. 587, top: It should also be pointed out that "geometry" is a classical concept. With respect to quantum theory, it's therefore more applicable to long-distance (low energy) behavior.
  2. p. 595, bottom: covariant curl should be moved to p. 597, top; see following remark
  3. p. 596, bottom: It would be useful here to put the commutator of 2 gauge transformations in covariant form. This provides an alternate definition of torsion & curvature:
    (A∙∇)Ba - (B∙∇)Aa = [A,B]a - AbBcTbca
    [A∙∇,B∙∇] = [A,B]a∇a +AaBbRabIMI
    where [A,B]m is defined with A = Am∂m. Using this, the result for the covariant curl can also be simplified, by considering AaBb∇[aCb] .
  4. p. 597, bottom: Here should go some discussion on the relation to coset spaces, as in the new subsection IC6 (see above). The treatment of flat space as the coset Poincaré/Lorentz (or similar for the other maximally symmetric spaces) is the same as the general relativity construction, but now there is a matching between covariant derivatives and Killing vectors. This coset construction explains the form of covariant derivatives: Poincaré is the symmetry group, but for any field we fix the representation of the Lorentz group (spin); this defines the coset. The covariant derivatives then take the same general form for general spaces, but not the Killing vectors (if any).
  5. p. 601, bottom, new exercise IXA3.5:
    Consider the independent components of the curvature in lower dimensions:
    1. Show the curvature vanishes in D=1.
    2. Show the curvature reduces to just the Ricci scalar in D=2. (Hint: Use ε tensors.)
    3. Show the Weyl tensor vanishes in D=3.
  6. p. 603, exercise IXA4.2: Parts should be numbered "a" & "b".
  7. p. 613, top, new exercise IXA7.3: Show this transformation of ∇ (with k=1) is consistent with (covariant) integration by parts.
  8. p. 614, IXA7.4b: The second part of this belongs in subsection IXC2, since "conformally flat" isn't defined until then.
  9. p. 616, bottom: To summarize the various manifestations of conformal invariance:
    flat space:conformal invariance
    curved space:Weyl invariance
    dilaton:dilaton decoupling
  10. p. 618, bottom: ref. 9 should include S. Deser, Ann. Phys. 59 (1970) 248.
  11. p. 621, exercise IXB1.1a: don't need to choose a gauge.
  12. p. 625, top: Here nm is understood as an arbitrary tangent vector to the geodesic; thus its direction is specified, but not its magnitude. In the massive case, we could fix the norm to be a constant; but in the massless case (null geodesic) there isn't an obvious normalization.
  13. p. 625, bottom: exercise IXA2.4 → exercise IXA2.5.
  14. p. 626, top: It would be a little simpler to do these derivations leaving choice of gauge to the last step: Using, for vanishing torsion,
    (A∙∇)Ba - (B∙∇)Aa = [A,B]a
    we have, for V for which [n,V] = 0 (where we'll eventually choose nm & Vm to be constants),
    n∙∇(n∙V) = V∙(n∙∇)n + n∙(n∙∇)V = n∙(V∙∇)n = (V∙∇)½n2 = 0
    Pulling off the constant Vm gives the desired result.
  15. p. 626, middle: The weakened assumptions include ∂mna=0.
  16. p. 628, top: This can be explained better by the above explanation for p. 626, top. The point is to prove Gaussian normal coordinates can always be chosen, so work first in a general coordinate frame, then choose n & V.
  17. p. 628, middle: This can be made simpler by the above methods:
    (n∙∇)2Va = (n∙∇)(V∙∇)na = [n∙∇,V∙∇]na = - nbVcRbcdand
  18. p. 628, below long equation: We really used the weaker conditions (n∙∇)na = ∂mnn = 0.
  19. p. 629, bottom, new exercise IXB4.1:
    Gaussian and Riemann normal coordinates are similar and generalize:
    1. Show that for Gaussian normal coordinates defined in terms of geodesics radiating from a point, the boundary conditions are again implied by the condition near that point.
    2. Show that replacing some n with n'a=fna for some function f preserves the weaker form of the geodesic condition (or relates weaker to stronger), but gives the same coordinate system.
    3. Show that for an axial coordinate system (n∙∇=n∙∂) with non-constant but still geodesic n (n∙∇n=fn) that
      n∙∂[(ema-δma)na] = (fδmn-∂mnn)[(ena-δna)na]
      and thus (ema-δma)na=0 with appropriate boundary conditions.
  20. p. 639, top: Same as for p. 214 above. Then you can drop the parenthetical remark following.
  21. p. 639, middle: The gauge choice is clearer if you look at R with 2 curved indices, where it looks exactly like the field strength for SU(2)2 Yang-Mills, and you see that the field strength for one SU(2) vanishes.
  22. p. 641, middle: The additional constraint replaces (or fixes) the projective invariance, as well as breaking conformal symmetry to the subgroup that leaves nA invariant. This construction of spaces with constant curvature is an algebraic (symmetry) one, as opposed to the previous analytic (differential) one. It's a generalization to higher dimensions (and arbitrary signature) of the construction of conic sections (intersecting a cone with a plane).
  23. p. 642, middle: The symmetry for the 3 cases (in just the spatial directions) is SO(4), ISO(3), or SO(3,1). Also, fixing the magnitude of k (by a coordinate scale transformation) is the same as fixing the unit of length. (This is classical, so there is no Planck length; we pick units with c, G, & k.)
  24. p. 644, new exercise IXC3.2b: Find the solution for a=b=0.
  25. p. 645, middle: This method will prove useful later for other cases.
  26. p. 646, exercise IXC4.2: Make it part a, and do Ω, q, and H. For part b, do the same for a cosmological constant, but a=b=0.
  27. p. 646, bottom, & p. 656, top: There are also experiments based on gravitational waves (with no significant results yet), which need only linearized gravity (not full general relativity).
  28. p. 650, middle, new exercise: Use the Weyl scale method to derive the covariant derivatives and curvature for the metric (a generalization of Schwarzschild)
    -ds2 = A2(y) dxi dxj ηij + B2(y) dyι dyκ δικ
    where the coordinates have been divided up into arbitrary numbers of x & y coordinates.
  29. p. 652, exercise IXC5.5a: The J's should be V's.
  30. p. 657, top: It might be more straightforward to do this analysis directly in the Hamiltonian formalism (cf. subsection IIIB5 for electromagnetism).
  31. p. 657, middle: v=1/m is a better gauge for the massive case; then τ=s, & the energy conservation law looks almost totally nonrelativistic, especially after the right-hand side (E2-m2)/2m is identified as (pi)2/2m in terms of the momentum at ∞.
  32. p. 658-9: There are some cheats here, in that b is treated as the minimum value of r, whereas the actual minimum is smaller by about a. For time delay, instead of expanding about r=b, one then expands about r=rmin. Then the exact square root can be expanded safely. To this order in a, the only change in the calculation is rmax→rmax+a, which affects only the √r²-1 term, adding a term a to s, but not affecting Δs. To make this clear, add a ...
  33. p. 659, new exercise IXC6.2: The variable χ is inadequate, since b/r can be greater than 1. Solve for b in terms of rmin. (The inverse is harder, solving a cubic equation.) Then define a new u = rmin/r. You will find as a convenient dimensionless expansion parameter a/(rmin-2a). Find the new result for φ(χ), which gets a new term. The final expression for the deviation is identical (to this order).
  34. p. 663, ref. 7: An English translation is available at arXiv:physics/9905030.

Chapter X

  1. p. 664, middle: As for gravity (see above), we can again define the form of the covariant derivatives by relating to cosets. So for G/H we choose G as Poincaré + supersymmetry + R-symmetry, while H is Lorentz + R-symmetry to define the superspin & super-isospin of the multiplet. Again, the form of these coset covariant derivatives for flat space generalizes to arbitrary spaces.
  2. pp. 679-80, ref. 6: The links have changed. ("kiss_prepri" should be replaced with "kiss_prepri.v8".)
  3. p. 691, top: For consistency with earlier notation, eq]β̊α should be eq]αβ̊

Chapter XI

  1. p. 725, middle: Notice that we are discussing high-energy behavior in the s channel corresponding to bound states in the t channel.
  2. p. 725, middle: An alternative is to replace s with u. Then there can be poles in just t & u, and none in s, so the limit s→+∞ can be taken safely. In that limit, with t fixed, u→-∞.
  3. p. 725, bottom: This is actually a Mellin transform, as applied for other purposes in subsection VIIC3.
  4. p. 731, top: h=w=c=0 is the sphere. (The torus, h=1, has no curvature, so χ=0.)
  5. p. 731, middle: I may add more details here about how the different kinds of tadpoles contribute to the curvature and combine to form the different topologies mentioned.
  6. p. 736, middle: VIC4 → VIIC4
  7. p. 737, top: σε[0,π] → σ∈[0,π] (epsilon → element).
  8. p. 737, bottom: The worldsheet symmetries should be CT & P: P doesn't involve complex conjugation and, as for the particle, there is no C.
  9. p. 752, ref. 5: Nambu's work has been published: "Duality and Hadrodynamics", notes prepared for the Copenhagen High Energy Symposium, 1970, in Broken Symmetry, Selected Papers of Y. Nambu, eds. T. Eguchi and K. Nishijima (World Scientific, 1995) p. 280.
    Hara probably doesn't belong here.
  10. p. 756, bottom: A couple of δ's are missing on the bottom of the δ/δ's.
  11. p. 757, top; There is a more geometrical way to think about this σ gauge fixing. (The τ fixing is pretty obvious.) Say you have a differential (1-)form J defined on the worldsheet. Then you might want to define σ, up to a constant, by dσ ≡ J. This requires dJ = 0. If we write J in terms of its (Hodge) dual as J = Jmεmndσn, then dJ = 0 is the conservation equation ∂mJm = 0, and dσ = J is Jm = δ0m. In our case, this has the interesting interpretation that while τ is identified with X+, σ is identified with the T-dual of X+.
  12. p. 763, middle: The closed-string tachyon has M2=-4α'-1, as easily seen from the formulas that follow.
  13. p. 766, top: "roation" → "rotation"
  14. p. 766, top: This reality condition holds only for conformal weight 0; there is an obvious extra power of z otherwise.
  15. p. 767, bottom: now z = τ+iσ.
  16. p. 773, middle: The 〈TT〉 calculation is easy to generalize to Q2=0 by inspection. For the bosonic string, we can look at Q = cT(X) + ½cT(cb). The 1-propagator terms give the classical calculation, while the 2-propagator terms give 〈TT〉, with the ½'s performing the same function of canceling the 2's from contracting either of the 2 c's with a b. For the general case, with Q given as in subsection VIA1, we see Q2=0 reduces to 〈ĜĜ〉.
  17. p. 775, top: The calculation is a little obscure here because of the δ's: As explained earlier, δ's should be eliminated by contour integration; then examine [∮λ1T̂,∮λ2T̂]. Note that, since we are really looking at just equal-time commutation relations, any corresponding interaction terms added to the action are irrelevant for evaluating propagators.
  18. p. 775, middle: For the vector vertex there is also a "cross term" where 1 propagator from T contracts with A(X) & 1 with ∂X, giving a ∂∙A term. This implies the gauge condition ∂∙A=0. However, this condition can be relaxed (& the ◻A=0 equation made gauge invariant) when ghosts are taken into account. (The Nakanishi-Lautrup field responsible for closing the BRST algebra of the gauge field resides in the ghost sector.)
  19. p. 777, top: XIIC1.1 → XIC1.1.
  20. p. 781, middle: How the closed-string rules are obtained from the open-string ones is explained at the beginning of subsection XIC5; that explanation really belongs here.
  21. p. 787, middle: This might be a good place to put a conformal field theory discussion of BRST, and its use to derive gauge-invariant vertex operators from p. 851 of subsection XIIB8, including the example of the massless vector vertex.
  22. p. 787, bottom: That V for the closed string should be a W.
  23. p. 788, middle: There is a sign error here. The vertices are ordered from right to left in the amplitude, so we should evaluate 〈C(zN)C(zN-1)C(z1)〉. We then get a +zN2 that agrees with the previous calculation.
  24. p. 789, top: This coset will have been explained in new subsection IC6.
  25. p. 792, middle: too many N's (α'M2 is in terms of the number operator, not the number of external lines.)
  26. p. 793, exercise XIIC1.1: should, of course, be XIC1.1.
  27. p. 794, top: really |w|<1 & Re T > 0 to avoid singular behavior @ w=1 (T=0)
  28. p. 797, bottom: The relation to temperature shows up if you multiply by e-βm, & integrate over m to define the partition function for temperature 1/β, which diverges for temperature above m0.
  29. p. 798, middle: For the twisted case ρ̄', the reflection about the boundary, is not just the complex conjugate; but for open string calculations with external open-string states, it will still be the same point on the boundary as ρ'.
  30. p. 801, top, and p. 805, middle: When I say "f" here, I really mean "wf24".
  31. p. 801, middle: A small cheat -- in the cases considered later, c̃τ̃+d̃ is pure imaginary for the open string, so we can use the absolute value for the 12th power; for the closed string, everything gets an absolute value anyway.
  32. p. 801, middle: 3) Actually, you might as well consider all of the usual UV field theory divergences.
  33. p. 802: The analysis here is the same as that at the end of subsection XIC2 for the Hagedorn temperature -- The transformation on w switches UV divergences to IR poles.
  34. p. 802, middle: We begin with the planar loop.
  35. p. 804, middle: What I meant to say here was that the usual Schwinger parametrization converges in Euclidean space except for tachyons, so one can blame the tachyon divergence on that. But a dilaton tadpole will always diverge, since both the momentum and mass vanish. That's why in field theory it's necessary to avoid massless particles with vacuum values by expanding around the correct vacuum, which bosonic string theory doesn't.
  36. p. 816, bottom: The dilaton contribution for the nonplanar case vanishes in the limit m → ∞.

Chapter XII

  1. p. 829, exercise XIIA4.1a: IA2.3e → IA2.3d.
  2. p. 839, top: Note that Φ=|i>Φi & ΦT=Φi<i|, where Φi (not Φi) is Hermitian. Also, keeping track of i's & signs is a bit easier if one works in terms of φ & ψ to start with, then plugs in appropriate i's for reality at the end.
  3. p. 839, middle: Really we should have QΦ=-Q̂Φ (Q-hat), & similarly for J & Ĵ, as explained in subsection IC1.
  4. p. 851: Most of this page would fit better on p. 787 of subsection XIB7.
  5. p. 851, top: That should be "[Q,W} = - ∂V", as in the middle of the page.
  6. p. 851, middle: The integral should be ∮dσ; in the z-plane it would then be ∮dz z; but more general gauges can be considered.
  7. p. 863, middle; p. 864, top: Strictly speaking, there should be some factors of 1/2 here for the quadratic terms.

Index

  1. p. 883: "superspin", after we define it (see above); also "super-isospin"