Where to get it

• arXiv.org: usual formats -- pdf, ps, tex + eps figures, etc.
• archive here: tex + eps figures + pdf figures, suitable for pdftex (or TeXShop)
• pdf here: produced by pdftex (may be slightly better than arXiv.org version)

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

• It's free.
• It's fast. You can download it from arXiv.org or its mirrors, just like preprints, without a trip to the library or bookstore or waiting for an order from the publisher.
• It's electronic. You can print it, but the PDF version has many advantages, like:
  • Download it at work, home, etc. (or carry it on a USB flash drive), rather than carrying a book or printing multiple copies.
  • Get updates just as quickly, rather than printing yet again.
  • It has the usual Web links, so you can get the referenced papers just as easily.
  • It has a separate "outline" window containing a table of contents on which you can click to take the main window to that item.
  • You can electronically search (do a "find" on) the text.
  • Save trees (and ink).
  • Theft is not a problem.
  • No wear or tear.
  • No paper cuts.
• It covers many recent topics at an introductory yet nontrivial level, such as:
  • supersymmetry
  • general relativity
  • supergravity
  • strings
• It introduces many topics not appearing in other textbooks, including:
  • 1/N expansion (color ordering) in QCD, including relation to random worldsheets
  • spacecone (spinor helicity), including explicit calculations of 4- and 5-point S-matrices in Yang-Mills
  • many useful gauges, such as Gervais-Neveu, Nielsen-Kallosh, unitary lightcone, and even string gauges in gravity
  • finite N=1 supersymmetric theories
• It is NOT:
  • a history book.
    • All the other recent, comprehensive field theory texts take the "traditional" approach of covering topics in chronological (rather than logical) order, in storybook fashion. (This is strongly reminiscent of introductory classical mechanics courses that still teach Newton's laws before energy-momentum conservation.)
    • This book takes advantage of hindsight, using what we now know to be the most efficient and general approaches. (For example, these other texts still quantize QED canonically, even though they know that method is inadequate for QCD. Some even claim path integrals are less rigorous, even though constructive quantum field theory has shown the opposite to be true.)
    • Whenever I have questioned anyone who prefers the traditional approach, after eliminating all the spurious clichés, it all boils down to nostalgia. (I have even heard the excuse that it is useful to learn the less useful approaches simply because they ultimately failed --- certainly an excellent reason to relegate these topics to true history courses, for those who have the time and interest.) This really means that most professors simply teach things the way they learned them, and (ironically) will not bother to learn a new way, or even to check if it has any advantages. (Unfortunately, similar remarks often apply to research.)
  • a cookbook. Some books race to Feynman diagrams as quickly as possible, because they either consider them the only useful part of field theory, or they think such an approach is an introductory one. One consequence is that the Higgs effect must take a back seat, and thus weak interactions are underemphasized or explained more phenomenologically.
  • a concept book. All the recent texts that use a modern approach, although giving the appearance of being comprehensive except for conciseness, are curiously deficient in explicit S-matrix calculations, especially for QCD. This book both includes modern concepts and calculates with them, since the dualistic approach of concept book plus calculation book has always proved deficient for lack of two good books that work well together.
  • a survey. With few exceptions, theories are described in this book at a level that allows explicit calculations.
  • an art book. It covers topics that have proven useful, not those that have appealed to certain tastes.

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) ☹

• p. 33, top: Probably a more drastic revision of the graduate curriculum is desirable.
• p. 36: There is a new book out by M. Srednicki, Quantum Field Theory (Cambridge University, 2007) 608 pp., that doesn't really fit into any of the categories I used. (Time to re-organize again.) It's only the 2nd textbook (after mine) to have any treatment of spinor helicity. (Canonical quantization & LSZ is used for each spin 1st, but really only at the free level.)
• p 37, bottom: Here's a useful link for TASI lectures.

Chapter I

1. p. 47, top: Spurious right quotes after {,}.
2. p. 47, middle: f(ψ) = a+ψb is a more convenient ordering.
3. p. 61, top, exercise IA4.7b: see exercise IA4.4 → 4.6.
4. 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.)
5. p. 70, middle: It might be easier for some people to look at δy^A=y^Bε_B^A.
6. 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 λ^a and ζ in x.
7. p. 71, exercise IA6.4: Do this completely in terms of x.
8. p. 73, bottom: For uniformity, ψ' = Mψ + V just before the exercise.
9. pp. 77, middle: "Irreducible representation" is first used here, but defined on p. 80, so another definition needs to be given here.
10. 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.
11. p. 78, middle, new exercise IB2.2:
    Consider the transformation properties of these tensors under the action of the group, G_i' = gG_ig^-1 =g_i^jG_j.
    1. Prove invariance of f_ij^k & thus η_ij.
    2. Show tr_R(G_iG_j) ≡ η_Rij is invariant; as a matrix in the adjoint space, η_R' = g η_R g^T = η_R. 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.
12. p. 80, middle: colummn → column.
13. p. 82, bottom: strictly speaking, these integrals converge only if the matrices have positive-definite eigenvalues; the general case then follows as described below.
14. 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.
15. 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.

Chapter II

1. p. 114, bottom: Note that e^iπC/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 ψ^γχ_γ.
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 V^γδ' → V^αβ'.
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. 132, bottom: "insubsection" → "in subsection"
8. p. 135, IIB2.1b: More-explicit wording -- "Find a manifestly Lorentz covariant solution to the second equation first..."
9. p. 148, middle: An important change of convention that will prove convenient later --- we can hide these signs to some extent by defining p^αγ̇=+p^αp^γ̇ always, and then defining p^γ̇=±(p^γ)*. 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).
10. p. 149, middle: The above signs will then appear in the complex conjugation relations, e.g., 〈pq〉*=ε(p⁰)ε(q⁰)[qp].
11. p. 157, top: I forgot to mention [p,q] = 0.
12. p. 161, top: The fact that a is a "bosonic index" means z_a is bosonic; likewise, z_α is fermionic.
13. pp. 162-3: Since USp(2n) can have an indefinite "U", as USp(2n₊,2n_-), so we can also have OSp*(2m|2n₊,2n_-).
14. pp. 164-5: The index ordering is inconsistent in some spots.
15. 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. 183, top: It would be more general to define J as δS_M/δA with respect to the rest of the action S_M, as previously in this subection.
2. p. 196, bottom: The τ limits of integration for the lightcone gauge should be τ_i and τ_f.
3. 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.
4. p. 200, top: Note that one can vary the action with respect to X(τ) and φ(x), etc., independently.
5. p. 209, bottom, exercise IIIC2.1: The hint isn't really useful.
6. 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 p_i²-2mE, 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. 244, exercise IVA5.1: You can ignore the hint. (It doesn't seem helpful.)
5. p. 248, 1st line: "for for" → "for"
6. p. 250, middle: ∫d⁴x needs a 1/(2π)².
7. p. 251: All the δ's should have a (2π)².
8. p. 251, bottom: Look at δS = ∫δ(φ²g_mn)δS/δ(φ²g_mn).
9. 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.)
10. 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).
11. p. 256, middle: ∫d⁴x needs a 1/(2π)².
12. p. 261, top: To be more explicit, the question is whether M_i* are a linear combination of M_i.
13. 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.
14. p. 276, exercise IVC1.1b: This really means to find the form of the supersymmetry transformations.
15. p. 283, top: The 1st paragraph belongs at the end of the previous subsection.
16. p. 283, middle: The chiral representation can also be treated as a complex gauge.
17. p. 288, top: I left out the λ's.
18. 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 λ_αλ_α̇ẋ^αα̇, 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. 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.
8. p. 349, bottom: That should really be p_i²+m_i²=0 to emphasize the fact external states can have different masses.
9. p. 350, middle, new exercise VC4.1b: Evaluate δ/δφ(-p) on 〈φ||ψ〉. Note that it involves ψ(p), not ψ(-p).
10. 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.
11. 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.)
12. p. 362, top: Maybe a step could be added here --
    rate/particle = (probability/time)/(density × volume) = P/ρV_D.
13. p. 362, middle: That should be 〈ψ||ψ〉.
14. 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 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. 389, middle: Note the trivial case α=∞, which corresponds to no gauge fixing, gives a propagator that blows up.
4. p. 390, middle: See comments on p. 148 above.
5. p. 395, middle: Actually the final result is independent of what (nonzero) constant value the scalars are given.
6. p. 398, exercise VIB5.3: A better value might be m/2, but again the value is irrelevant.
7. 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.
8. 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).
9. p. 417, ref. 11: now available as hep-th/0509223
10. p. 418, ref. 16: no space before period .
11. p. 419, middle: After 1st paragraph of subsection 1 -- "(see subsection VC9)"
12. p. 419, bottom, & 422, top: See comments for pp. 148-9 above.
13. p. 423, above exercise VIC1.4: Better to use parentheses than angular brackets for the amplitude, as later.
14. p. 424, middle: "The two cases with simple known solutions..."
15. p. 427, above exercise VIC3.1: For consistent notation, that should be a ⊖ on the antiselfdual vertex.
16. p. 432, middle: Left quote right above figure is backwards.
17. 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. 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.
8. p. 493, bottom: Here we consider ĝ(M,ε) & g(μ,M,ε).
9. 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.
10. p. 497, top: Unfortunately, the term "asymptotically free" used here will not be defined until subsection VIIIA3; it means β₁>0.
11. p. 498, top: The sentences "We use... in that combination.)" would go better after exercise VIIC2.1.
12. p. 498, bottom - 499, top: For D=0, these are ordinary integrals and derivatives, so really D → d, δ → d.
13. 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.
14. p. 499, middle: "The simplest example of a 'renormalon' problem..."
15. 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.
16. p. 502 bottom: Here we treat only the gluon coupling as ħ, ignoring the coupling to the particle of mass m. (See previous note.)
17. p. 506, middle: Here all diagrams are closed surfaces, so tree graphs have 1 face (L=H=0). So really graphs go as N^F-1 --- the equation needs an extra 1/N.

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. 538, bottom: VA2.4 → VA2.1.
7. 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).
8. p. 563, top: 4-2ε → 4−2ε (hyphen → minus).
9. p. 575, middle: IXA4 → IXA3.
10. p. 575, middle: "Explicitly, ..."
11. pp. 579-580: The vertices should include the usual factors of the coupling g.
12. p. 580, middle: "vanishing sources V=1" → "N=1 and V₁=1 (k₁=0)"
13. 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.
14. p. 582, top: As explained later, this is an expansion about the center of the string, not the boundary (where ∂X/∂σ=0).
15. 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).
16. p. 584, top: S_ab for spin 1/2 should have a 1/2, not 1/4.
17. 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∙∇)B^a - (B∙∇)A^a = [A,B]^a - A^bB^cT_bc^a
   [A∙∇,B∙∇] = [A,B]^a∇_a +A^aB^bR_ab^IM_I

   where [A,B]^m is defined with A = A^m∂_m. Using this, the result for the covariant curl can also be simplified, by considering A^aB^b∇_[aC_b] .
4. 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.
5. p. 603, exercise IXA4.2: Parts should be numbered "a" & "b".
6. p. 613, top, new exercise IXA7.3: Show this transformation of ∇ (with k=1) is consistent with (covariant) integration by parts.
7. p. 614, IXA7.4b: The second part of this belongs in subsection IXC2, since "conformally flat" isn't defined until then.
8. p. 616, bottom: To summarize the various manifestations of conformal invariance:

   flat space:	conformal invariance
   curved space:	Weyl invariance
   dilaton:	dilaton decoupling

9. p. 621, exercise IXB1.1a: don't need to choose a gauge.
10. p. 625, top: Here n^m 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.
11. p. 625, bottom: exercise IXA2.4 → exercise IXA2.5.
12. 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∙∇)B^a - (B∙∇)A^a = [A,B]^a

    we have, for V for which [n,V] = 0 (where we'll eventually choose n^m & V^m to be constants),

    n∙∇(n∙V) = V∙(n∙∇)n + n∙(n∙∇)V = n∙(V∙∇)n = (V∙∇)½n² = 0

    Pulling off the constant V^m gives the desired result.
13. p. 626, middle: The weakened assumptions include ∂_mn^a=0.
14. 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.
15. p. 628, middle: This can be made simpler by the above methods:

    (n∙∇)²V^a = (n∙∇)(V∙∇)n^a = [n∙∇,V∙∇]n^a = - n^bV^cR_bcd^an^d

16. p. 628, below long equation: We really used the weaker conditions (n∙∇)n^a = ∂_mnⁿ = 0.
17. 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=fn^a 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∙∂[(e_m^a-δ_m^a)n_a] = (fδ_mⁿ-∂_mnⁿ)[(e_n^a-δ_n^a)n_a]

       and thus (e_m^a-δ_m^a)n_a=0 with appropriate boundary conditions.
18. p. 639, top: Same as for p. 214 above. Then you can drop the parenthetical remark following.
19. 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.
20. p. 641, middle: The additional constraint replaces (or fixes) the projective invariance, as well as breaking conformal symmetry to the subgroup that leaves n^A 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).
21. 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.)
22. p. 644, new exercise IXC3.2b: Find the solution for a=b=0.
23. p. 645, middle: This method will prove useful later for other cases.
24. 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.
25. 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).
26. 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 dx^j η_ij + B²(y) dy^ι dy^κ δ_ικ where the coordinates have been divided up into arbitrary numbers of x & y coordinates.
27. p. 652, exercise IXC5.5a: The J's should be V's.
28. p. 657, top: It might be more straightforward to do this analysis directly in the Hamiltonian formalism (cf. subsection IIIB5 for electromagnetism).
29. 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 ∞.
30. 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=r_min. Then the exact square root can be expanded safely. To this order in a, the only change in the calculation is r_max→r_max+a, which affects only the √r²-1 term, adding a term a to s, but not affecting Δs. To make this clear, add a ...
31. 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 r_min. (The inverse is harder, solving a cubic equation.) Then define a new u = r_min/r. You will find as a convenient dimensionless expansion parameter a/(r_min-2a). Find the new result for φ(χ), which gets a new term. The final expression for the deviation is identical (to this order).
32. p. 663, ref. 7: An English translation is available at arXiv:physics/9905030.

Chapter X

1. pp. 679-80, ref. 6: The links have changed. ("kiss_prepri" should be replaced with "kiss_prepri.v8".)
2. p. 691, top: For consistency with earlier notation, e_q]β̊α should be e_q]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, bottom: This is actually a Mellin transform, as applied for other purposes in subsection VIIC3.
3. p. 731, top: h=w=c=0 is the sphere. (The torus, h=1, has no curvature, so χ=0.)
4. 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.
5. p. 736, middle: VIC4 → VIIC4
6. p. 737, top: σε[0,π] → σ∈[0,π] (epsilon → element).
7. 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.
8. 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.
9. p. 763, middle: The closed-string tachyon has M²=-4α'^-1, as easily seen from the formulas that follow.
10. p. 766, top: This reality condition holds only for conformal weight 0; there is an obvious extra power of z otherwise.
11. p. 767, bottom: now z = τ+iσ.
12. 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]. Solve the locality condition before the conformal weight condition; then it's really W (not W̃) that has weight 1. 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.
13. 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.)
14. p. 777, top: XIIC1.1 → XIC1.1.
15. 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.
16. 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.
17. p. 787, bottom: That V for the closed string should be a W.
18. p. 792, middle: too many N's (α'M² is in terms of the number operator, not the number of external lines.)
19. p. 793, exercise XIIC1.1: should, of course, be XIC1.1.
20. p. 794, top: really |w|<1 & Re T > 0 to avoid singular behavior @ w=1 (T=0)
21. 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.
22. p. 801, top, and p. 805, middle: When I say "f" here, I really mean "wf²⁴".
23. 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.
24. p. 801, middle: 3) Actually, you might as well consider all of the usual UV field theory divergences.
25. 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.
26. p. 802, middle: We begin with the planar loop.
27. 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.
28. p. 816, bottom: The dilaton contribution for the nonplanar case vanishes in the limit m → ∞.

Chapter XII

1. p. 839, top: Note that Φ=|ⁱ>Φ_i & Φ^T=Φⁱ<_i|, where Φⁱ (not Φ_i) 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.
2. p. 839, middle: Really we should have QΦ=-Q̂Φ (Q-hat), & similarly for J & Ĵ, as explained in subsection IC1.
3. p. 851: Most of this page would fit better on p. 787 of subsection XIB7.
4. p. 851, top: That should be "[Q,W} = - ∂V", as in the middle of the page.
5. p. 863, middle; p. 864, top: Strictly speaking, there should be some factors of 1/2 here for the quadratic terms.