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- 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 can't be eaten.

- 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.

- a history book.

For information on my course based on this book, see my
**PHY 610-611** page.

See also an overview of the course.

See also the
**APS News article**.

An interesting article on
open source books.

- Chapter I: cosets/projective spaces for coordinates (especially Poincaré & conformal)
- Chapter II: cosets/projective space for superconformal
- Chapter IV:
- actions in N=2 superspace
- on-shell N=4 projective superspace

- Chapter VI:
- BCFW recursion (along the lines of Vaman & Yao)
- CSW recursion
- Bloch-Nordsieck
- N=2 supergraphs

- Chapter VII: N=2 superloops
- Chapter XI:
- more on Regge limits of (non)relativistic particle theories (ladders)
- more (worldsheet) topology
- gauge-invariant vertices

- p. 16, top: enregy → energy
- p. 23, top: I might add that the caveman found fire & the wheel "counter-intuitive". We have not physically evolved significantly since then. So if you can handle learning about those things, then intuition isn't the problem.
- p. 26, bottom: arXive → arXiv
- p. 26, bottom: Another advantage -- easy to deal with when you move.
- pp. 29, 101, 104, 156: The Higgs has been found.
- 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.)The long-promised (26 years & counting!) "2nd volume" (actually 2nd edition) of the de Wit & Smith book (with a new coauthor, E. Laenen) is now said to be "forthcoming".

There are many more field theory books now, so rather than try & list them all, I'll stick to just the few I like for historical value or useful treatments of some topics. The ones I drop or don't add will be those that have the following of what I consider major shortcomings for a textbook on quantum field theory (but not necessarily a reference book):

- Treat QED 1st in complete detail using canonical quantization before coming back to path integrals for the rest of the Standard Model. As far as I'm concerned, this is simply a waste of time, as well as an obfuscation of the generality of path integrals & Yang-Mills.
- Emphasize 1 particular field theory (QED, QCD, or only scalars), treating it in full detail before coming back to the rest of the Standard Model (especially weak interactions) afterwards (even classically), if at all. Maybe the rest is considered part of another course, in which case this book can @ best be 1 volume of a set on field theory. (But try & find the others.)
- Few/no S-matrices/cross sections. Such a book is only "introductory"; again the other volume is missing.
- No 2-component spinors (& hence no spinor helicity/twistors/spacecone gauge). They're not just for supersymmetry anymore.

- p 37, bottom: Here's a useful link for TASI lectures.

- 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.

If you're not familiar with [∂/∂x,φ(x)] = φ'(x), check it by Taylor expansion (which is always OK for φ(x) about some point x₀). - p. 47, top: Spurious right quotes after {,}.
- p. 47, middle: f(ψ) = a+ψb is a more convenient ordering.
- 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. - p. 48, exercise IA2.2: This means to show ∫dψ(∂/∂ψ) = 0 ⇒ ∫dψ = ∂/∂ψ. Note that integration by parts is the same as translational invariance.
- p. 53, middle, & p. 81, exercise IB3.1a: For SU(2), f = -ε.
- p. 59, middle: There is some ambiguity in how momenta are numbered, and thus how s,t,u are defined. The default convention is to draw a planar spacetime picture of the scattering, labeling the 4 trajectories. Then (1) s is defined in terms of just the incoming (or just the outgoing) momenta, which are drawn as adjacent (as thy both come from earlier time). (2) t is defined in terms of the independent pair of adjacent particles. (3) Thus, u is defined in terms of opposite particles.
- p. 61, top, exercise IA4.7b: see exercise IA4.4 → 4.6.
- p. 66, middle: Note that p is the "total" energy-momentum, m dx/ds is the "kinetic" energy-momentum, & -qA is the "potential" energy-momentum. The kinetic part is gauge-independent, & thus directly physically measurable (in terms of the space-time path), while the total is gauge-dependent, but is what will be conserved if there are symmetries.
- p. 66, middle: effect the coordinates →
*affect*... - 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.)
- p. 68, bottom: Maybe some words on the AdS/CFT correspondence could go here.
- p. 69, bottom: The finite difference can be done the same way as the differential:
y∙y' = ee'w∙w' = -½ee'(w-w')

A similar example is nonrelativistic momenta (see exercise IA4.7): The analogous parametrization is^{2}= -½ee'(x-x')^{2}p = (m,p

^{i},E) = m(1,v,½v^{2}) → p∙p' = -½mm'(v-v')^{2} - p. 70, middle: It might be easier for some people to look at
δy
^{A}=y^{B}ε_{B}^{A}. - p. 70, bottom, exercise IA6.2: Looks like I only did this semiclassically. Let's do this quantum mechanically, like later in exercise IIB1.3. Then the 2nd constraint should be y∙r+r∙y=0 for closure of the constraint algebra. Paying careful attention to ordering, the resulting constraints in the new coordinates are then z = en+(D-2)/2 = p² = 0.
- 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 p
^{2}=0. Solve the resulting first-order differential equation by Taylor expanding λ^{a}and ζ in x. - p. 71, exercise IA6.4: Do this completely in terms of x.
- p. 72, top, new ref. 4: O. Klein,
*J. Phys. Radium*9 (1938) 1: Klein transformation. - p. 72, ref. 8: For Kastrup, 1186 → 1183.

Also, the Mack & Salam paper is available as a preprint.

And Adler's paper has an erratum,*Phys. Rev.***D7**(1973) 3821. - p. 73, bottom: For uniformity, ψ' = Mψ + V just before the exercise.
- p. 77, middle: "Irreducible representation" is first used here, but defined on p. 80, so another definition needs to be given here.
- 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.
- p. 78, middle, new exercise IB2.2:

Consider the transformation properties of these tensors under the action of the group, G_{i}' = gG_{i}g^{-1}=g_{i}^{j}G_{j}.- Prove invariance of f
_{ij}^{k}& thus η_{ij}. - Show tr
_{R}(G_{i}G_{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.

- Prove invariance of f
- p. 80, middle: colummn → column.
- p. 81, exercise IB3.1a: f = -ε.
- p. 82, bottom: strictly speaking, these integrals converge only if the matrices have positive-definite eigenvalues; the general case then follows as described below.
- 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.
- 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.
- p. 89, top (twice): fundamental → defining
- 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.
- p. 102, top: The Higgs mass is 125.
- p. 104, bottom: The diagram needs more explanation regarding multiplicity. If we ignore symmetry, then (showing SU(n) indices)

⊗i

⊗j

=k

⊕i j k

⊕i j k

⊕i k j

. But there is symmetry in combined SU(n) internal & spin SU(2) indices, so we should really decomposei j k

. Then the symmetry for interchanging these pairs of indices means the separate tableaux for SU(n) & SU(2) should have the same symmetry, naively givingiα jβ kγ i j k

⊕α β γ i j k

⊕α β γ i k j

⊕α γ β i j k

. But total symmetry in the pairs of indices means the middle 2 are the same (or appear only as their sum), & shouldn't be double counted, since they are related by jβ ↔ kγ. (And the last vanishes because of 3 antisymmetrized 2-valued indices.)α β γ - 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).
- Here is a piece of that from another course.
- More can be found in this paper: probably just some pieces of sections 2-4.

- p. 109, top, ref. 1: An older, longer, preprint version of that paper was recently re-released, AEC report NYO-3071.
- p. 109, middle, ref. 7: The 2012 edition of the Review of Particle Physics is out. (It comes out every 2 years, so I should change how I cite this.)

- p. 111, bottom: In practice, it's usually more convenient to work with each component of this matrix, rather than real combinations. This is familiar from quantum mechanics courses when discussing SU(2), where "raising" and "lowering" operators are used in place of the usual vector components of angular momentum. Similar remarks apply to other groups, as seen in subsection IB5.
- 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. - p. 119, middle: In the equation for ½⊗½, there should be a ½ multiplying ψ
^{γ}χ_{γ}. - p. 119, bottom, new exercise IIA4.2:

Spherical harmonics are simple in spinor notation.- 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.
- 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.

- p. 122, above exercise IIA5.3: For SO(2,2), replace V
^{γδ'}→ V^{αβ'}. - p. 124, bottom: V should transform like ψψ̄, not ψχ̄, to make it a real 4-vector. (But that makes it an axial vector.)
- 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.)
- 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. - p. 129, middle, exercise IIA7.2: "2×2 matrix notation" means no indices, as in the previous problem. Use the results from this page (which is why this problem is here). This notation was defined @ the top of p. 124.
- p. 132, bottom: "insubsection" → "in subsection"
- p. 135, IIB2.1b: More-explicit wording -- "Find a manifestly Lorentz covariant solution to the second equation first..."
- 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^{α}p^{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). - p. 149, middle: The above signs will then appear in the complex conjugation relations, e.g., 〈pq〉*=ε(p
^{0})ε(q^{0})[qp]. - p. 154, middle: All the +'s should be ⊕'s.
- p. 157, top: I forgot to mention [p,q] = 0.
- p. 160, bottom: IJK... indices should probably be ABC... for consistency.
- p. 161, top: "M" shouldn't be used here, to avoid confusion with the M of the previous page.
- p. 161, top: The fact that a is a "bosonic index" means z
_{a}is bosonic; likewise, z_{α}is fermionic. - 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. - p. 161, bottom: A more convenient parametrization when inverses or (super)determinants are involved is the factorized form

⎛ 1 0 ⎞ ⎝ c 1 ⎠ ⎛ a 0 ⎞ ⎝ 0 d ⎠

or a similar form with the first & last factors reversed. Such forms are easy to invert and take determinants of, & the results given in the text are easily derived from them by simple redefinitions. See exercise IB3.4b, which is good for inversions as well as determinants. These forms will also have been used in new subsection IC6 (see above). They correspond to factorization of group elements into elements of the block-diagonal subgroup, & the raising and lowering operators with respect to it.⎛ 1 b ⎞ ⎝ 0 1 ⎠ - pp. 162-3: Since USp(2n) can have an indefinite "U", as USp(2n
_{+},2n_{-}), so we can also have OSp*(2m|2n_{+},2n_{-}). - 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.
- pp. 164-5: The index ordering is inconsistent in some spots; see the note above for the bottom of p. 161.
- 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.)
- 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 (spacetime) & i is fermionic (internal).

- p. 171, bottom, new exercise IIIA1.2:

Find the equations of motion from the Lagrangian L = aq^{2}+bq̇^{2}+cq̈^{2}(for constants a,b,c). - p. 181, middle: The double derivative A(∂B) - (∂A)B is 1st used here, but not defined till the top of p. 185.
- 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. - p. 183, middle: The main problem classically with "bad" high-energy behavior is the breakdown of perturbation in such couplings, which is the only known method of detailed calculation in the quantum theory: They are associated (by dimensional analysis) with higher derivatives in the action. One can only expand in dimensionless quantities (to have unit-free comparisons of different orders), which in this case would be the coupling constant times a positive power of the energy. This means the perturbation expansion would necessarily diverge at energies above the mass scale set by the dimensionful coupling.
- p. 188, middle: That should be 0 = δ∇ = δ
_{0}∇ +i[ζ^{i}G_{i},∇]. - p. 196, bottom: The τ limits of integration for the lightcone gauge should be τ
_{i}and τ_{f}. - 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.
- p. 200, top: Note that one can vary the action with respect to X(τ) and φ(x), etc., independently.
- p. 204, middle: Note that, in relating to classical mechanics by the correspondence principle, the covariant derivative corresponds to the kinetic energy-momentum, while the partial derivative corresponds to the total energy-momentum. (See the discussion for p. 66 above.) This is also reflected in the fact that it is the covariant derivative whose square gives the mass², while partial derivatives relate to translations (& thus symmetries).
- p. 209, bottom: In the last equation, that should be k = x
^{0}-t_{0}, just as explained immediately above. - p. 209, bottom, exercise IIIC2.1: The hint isn't really useful.
- 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.)
- p. 229, ref. 8: Add -- R. Capovilla, J. Dell, T. Jacobson, and L. Mason, Class. Quantum Grav. 8 (1991) 41. (They did the Yang-Mills case.)

- 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.
- p. 235, top: There is another way to understand this, as the parametrization of a 3D null vector (for SO(2,1)) in terms of 3D twistors. (See subsection IIC5, or exercise IIA3.1 for real ψ.) Scale the twistor to (x,1) (or (m,n) for the Pythagorean theorem).
- p. 236, middle: Of the CP(n-1) model's U(n) symmetry, the U(1) is local and the SU(N) is global.
- p. 236, bottom, new exercise IVA2.2:

Consider the CP(1) model in 1 dimension:- 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).
- 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 p
^{2}with p_{i}^{2}-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 (in addition to the constraint from v).

- p. 238, top: Note there is an ambiguity in the definition of the T's, by adding to each a bit of the H's. This corresponds to different choices of the "unitary" gauge.
- p. 238, middle: That g⁻¹dg is an element of the Lie algebra is clear from the fact g⁻¹(x)g(x+ε) differs from the identity by (i.e., is the exponential of) an infinitesimal element of the algebra. (The same is true for (dg)g⁻¹.)
- p. 241, middle: I forgot to drop 1 Lorentz index.
- p. 242, bottom: For more than 1 color, the current should have a trace for the quarks.
- p. 244, exercise IVA5.1: You can ignore the hint. (It doesn't seem helpful.)
- p. 244, middle: The action of T is linear on e
^{iφ}, so it's also useful to think of this as a U(1) nonlinear σ model, as at the beginning of IVA2. - p. 247,middle: The general case is then to start with a polynomial action for scalars, describing spontaneous symmetry breaking from G→H. As before these scalars can be decomposed into the coset G/H plus some representations of only H. Coupling Yang-Mills for group G to the G of the coset, some vectors eat the coset, leaving massless the vectors for the H subgroup. The remaining scalars are the physical Higgs. Mathematically, we take the covariant derivative ∇ for the Yang-Mills group G & the scalars g for the coset G/H, and combine them as g
^{-1}∇g. The construction is similar to that for the pure coset, but now the resulting H gauge fields include the massless vectors, while the field strengths for the coset scalars are now the massive vectors themselves. These & the physical Higgs are all singlets of the original Yang-Mills gauge group G, while the coset gauge group H has now become the new Yang-Mills gauge group. - p. 248, 1st line: "for for" → "for"
- p. 250, middle: ∫d
^{4}x needs a 1/(2π)^{2}. - p. 251: All the δ's should have a (2π)
^{2}. - p. 251, bottom: Look at δS = ∫δ(φ
^{2}g_{mn})δS/δ(φ^{2}g_{mn}). - 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.)
- 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/H
^{2}(and σ is the simplest). - p. 256, middle: ∫d
^{4}x needs a 1/(2π)^{2}. - p. 261, top: Note that φ = φ
^{T}. - p. 261, top: To be more explicit, the question is whether M
_{i}* are a linear combination of M_{i}. - p. 261, middle: This discussion could be clearer. Without masses, CP is automatic. With masses, as explained, the color group theory requires doubling, with L & R transforming as complex conjugate representations of color. We'll see on the next page that this guarantees C as well as CP (& thus P).
- 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.
- p. 273, top: This would probably be a good place to explain neutrino masses from the sterile neutrinos via the seesaw mechanism: Breaking SO(10) gives the extra right-handed neutrinos (1 per family) "Majorana" masses of the form Mν
_{R}², where M is of the order of the GUT scale, from Higgsing that symmetry. But such neutrinos can also get "Dirac" masses of the form mν_{L}ν_{R}(as for the quarks), where m is of the order of the electroweak scale, from Higgsing that. Diagonalizing the mass matrix then yields masses of the order m²/M for the observed, "mostly left-handed" neutrinos, giving a natural explanation for their smallness. The Yukawa couplings that generated the masses m are then treated as in the quark case, & include leptonic CP violation, which may be important in cosmology. - p. 273, reference 7: Also, A. Salam, Weak and electromagnetic interactions, Imperial College, 1967, unpublished, and in 1968 Nobel Symposium, Elementary Particle Theory - Relativistic Groups and Analyticity, ed. N. Svartholm (Wiley, 1968) p. 367.
- p. 276, exercise IVC1.1b: This really means to find the form of the supersymmetry transformations.
- p. 282, middle: Thus A
_{α}= e^{-Ω}(d_{α}e^{Ω}). - p. 283, top: The 1st paragraph belongs at the end of the previous subsection.
- p. 283, middle: The chiral representation can also be treated as a complex gauge.
- p. 283, IVC4.1: The parentheses mean that the d
_{α}acts only on the e^{V}. - 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.
- p. 281, middle: "Superspin" is never defined. This will be fixed with the discussion of superspace as a coset space (see above).
- p. 288, top: I left out the λ's.
- p. 290, middle: The origin in dimensional reduction is reflected in the free field equation p²+Z² = 0.
- p. 292, middle: It might be nice to give here a realistic softly broken N=4 model.

- 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. Also, in solving for the normalization, the useful identities appear @ the top of p. 306.
- p. 304, bottom: Note that it's odd powers in Δφ of the exponential (the integrand) that vanish, not the exponent (action).
- p. 308, top: "δ in x" → "δ in q".
- p. 308, middle: The integral should be over t'', so as not to confuse with the limits of integration.
- p. 312, middle: Note that δχ is proportional to a δ function in time (for functional differentiation), which is what makes V ≠ I.
- p. 313, top: Note the relation of this expansion for time development to the one on pp. 188-9.
- p. 317, bottom: VA2.4 → VA2.1.
- 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(p
^{2}). Also quantize the Lagrangian λ_{α}λ_{α̇}ẋ^{αα̇}, where λ & λ are twistors. Show the result is θ(p^{0})δ(p^{2}), and explain the relation to the previous Lagrangian. - p. 323, exercise VB2.1: A quote is backwards.
- p. 324, bottom: The overall signs of the last two expressions for Δ
_{A}are inconsistent with the 1st expression. Also, the sign of the inhomogeneous term varies for the various propagators, but that term is obvious from hitting the 1st expression of each with p²+m². - p. 326, middle: It would have made more sense to define M as a contravariant vector, since the hypersurface is naturally defined as a covariant vector (as Hodge dual to a (D-1)-form; see subsection IC2).
- p. 336, top: That should be 〈φ||ψ〉 & 〈φ*||ψ〉.
- p. 336, middle: Probably "M" is a poor choice of letter here, since it has a different use on the preceding & following pages.
- p. 338, bottom: Not every φ in Z gets replaced by a wave function; but the only terms in Z that survive are the ones whose order in φ is the same as the number of external particles.
- p. 341, top: The "-" signs come from the "-i" that goes with each factor.
- p. 343, bottom: This argument applies only to connected graphs. The rule for disconnected graphs then follows.
- 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.
- p. 349, bottom: That should really be p
_{i}^{2}+m_{i}^{2}=0 to emphasize the fact external states can have different masses. - p. 350, middle, new exercise VC4.1b: Evaluate δ/δφ(-p) on 〈φ||ψ〉. Note that it involves ψ(p), not ψ(-p).
- 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.
- p. 357, bottom: IA2.3e → IA2.3d.
- 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.) - p. 358, bottom: We have gone from the S-matrix
*S*to*S*[φ] to Z[φ]. For purposes of defining unitarity & causality,*S*[φ] can be defined in various ways, but the most convenient way is, as for Z, expanding the quantum fields about a background in the interactions. Then it's easy to go to the Z form of these identities, since Z[φ] = 〈0|*S*[φ]|0〉 for that definition. Inserting the identity between*S*^{†}&*S*^{†}as a sum over states as performed by D₊, the translation then follows. - p. 359, middle: Corresponding to the sign error on p. 324, if the signs of the latter 2 expressions there are changed, then the sign for Δ
_{A}here is. It doesn't affect anything, because that's the 1 propagator that isn't used. - p. 359, bottom: Note that we're looking @
*S*^{†}*S*(not*SS*^{†}), so the sum is over final states (not initial), hence always positive energy flows from*S*to*S*^{†}. - p. 361, middle: Note that the minus sign for each cut fermion loop is exactly the extra sign for converting the usual numerator factor for negative-energy fermions into a sum over states. (See subsection VB3.)
- p. 362, top: Maybe a step could be added here --

rate/particle = (probability/time)/(density × volume) = P/ρV_{D}. - p. 362, middle: That should be 〈ψ||ψ〉.
- p. 362, bottom: The final factors of 1/n! are only for the
*total*cross section. Then one sums over identical particles in permuted positions, which is multiple counting, since the amplitude is already (anti)symmetric under the permutation. - 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.
- p. 371, reference 9: to Goldstone, Salam, & Weinberg also appears in IVA, ref. 2.

- p. 374. top: As usual, the constraints define a Lie algebra. The situation here is one where the original Hilbert space contains no singlet representations, or we want a non-singlet. The values of G₀ define the representation. In terms of the future subsection IC6, this happens in the coset construction, where the constraints are covariant derivatives.
- p. 375, bottom: Note that for any A, [Q,{Q,A}] = [Q²,A] = 0 as a consequence of the Jacobi identity & Q² = 0.
- p. 375, bottom: Products of physical operators are also physical: [Q,AB] = [Q,A]B + A[Q,B] = 0, &
δ(AB) = [Q,λ
_{A}]B + A[Q,λ_{B}] = [Q,λ_{A}B+Aλ_{B}]. - p. 376, middle: Note that if we write Ĝ
_{i}= G_{i}+ G̃_{i}, then Q = c^{i}(G_{i}+½G̃_{i}). - p. 379, top: ?
_{i}→ λ^{i}. - p. 379, middle: The disappearance of B & c̃ refers to the classical action.
- p. 380, top: This would be a good place for a table relating Λ's in the H & L formalisms:
**L**λc̃ (λ-f)c̃ λ̇c̃ **H**0 fb λb - p. 383, middle: We can also put matter fields into f.
- p. 385, top, ref. 3: Tyutin's paper has now been "published" --- arXiv:0812.0580v1 [hep-th].
- p. 389, middle: Note the trivial case α=∞, which corresponds to no gauge fixing, gives a propagator that blows up.
- p. 390, exercise VIB2.2: There are 2 simple ways to do this exercise -- use Lorentz covariance, then fix the 2 coefficients by multiplication; or use the explicit formula for inverses from subsection IB3.
- p. 390, middle: See comments on p. 148 above.
- p. 391, top: Should be "... more derivatives in the gauge
*condition*...". - p. 393, middle, new exercise VIB3.2:

Treat propagators in path-integral language as the insertion of 2 fields into the path integral. Show that BRST invariance relates the propagators of the ghosts to those of the bosons. Check that these relations hold for the gauges of the previous problem. - p. 395, middle: Actually the final result is independent of what (nonzero) constant value the scalars are given.
- p. 397, top: The last 2 terms in the equation should come with "-" signs.
- p. 398, exercise VIB5.3: A better value might be m/2, but again the value is irrelevant.
- p. 402, top: To be perfectly general, we should include scalars. Then the statement is that we use the momentum of an external line with non-positive helicity to define external line factors for lines with positive helicity, and non-negative for negative.
- p.402, middle: The derivation of the 〈 〉[ ] rule needs improvement: 1st, the external line factors aren't derived till the next page. 2nd, counting momenta can best be explained by dimensional analysis, e.g., looking @ the original gauge-invariant action.
- 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. - p. 409, exercise VIB8.2: "Arbitrary gauge-invariant Yang-Mills action" means any functional of the Yang-Mills field invariant under the full nonabelian gauge transformations. "An Abelian quantum gauge transformation" means just the Abelian part of the full nonabelian quantum gauge transformation.
- 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).
- p. 416, bottom: The NK ghost needs some comments -- the ◻ introduced in gauge fixing needs to be modified as discussed on p. 438.
- p. 417, ref. 11: now available as hep-th/0509223
- p. 418, ref. 16: no space before period .
- p. 419, middle: After 1st paragraph of subsection 1 -- "(see subsection VC9)"
- p. 419, bottom, & 422, top: See comments for pp. 148-9 above.
- p. 423, above exercise VIC1.4: Better to use parentheses than angular brackets for the amplitude, as later.
- p. 423, bottom: Note that these "Maximal Helicity Violating" diagrams all have exactly 1 non-self-dual vertex. Since the ⊖ reference line must be attached to an antiselfdual vertex, no 4-point vertices contribute to any of these amplitudes (& only 1 antiselfdual vertex).
- p. 424, middle: "The two cases with
*simple*known solutions..." - p. 425, bottom: It might be simpler to replace "N-1" with "j-i".
- p. 427, above exercise VIC3.1: For consistent notation, that should be a ⊖ on the antiselfdual vertex.
- p. 428, bottom: If we interpret the amplitude itself as expressed in terms of field strengths (as above for the nonsupersymmetric case), highest order in p
^{i}is lowest helicity. But if we convolute with external-line wave functions and then integrate over p^{i}, then highest order in p^{i}in those wave functions is highest helicity. - p. 432, middle: Left quote right above figure is backwards.
- p. 432, middle: The argument for loop signs (as opposed to just switching lines) isn't correct: Unlike the cutting rules (subsection VC6), there is no minus sign for each cut fermion loop, since the sum is now over initial & final states, and there are no loops with sums over fermions with both signs of the energy.
- p. 435, bottom: Overall sign of propagator exponent is wrong.
- p. 436, middle: All propagators have overall signs wrong.
- p. 438, top: This modification of ◻ needs to be used for the NK ghost derivation on p. 416.
- p. 438, ref. 1: Phys. Rev. Lett. → Phys. Rev. D.

- p. 444, middle: The argument P-½LD-n of the Γ function counts (-½) the overall power of p.
- p. 451, bottom: The optical theorem was originally derived in optics. This is due to the relationship of quantum mechanics to classical electromagnetism (& thus scattering amplitudes in both), as described in exercise IIIA1.3:

As a result, the more modern, quantum mechanical derivation of the optical theorem is simpler than the old optics one, even when applied to electromagnetism. (Similar remarks apply to many other topics common to wave equations, such as expansions in spherical coordinates.)wave function ↔ electromagnetic field (as complex 3-vector) probability (density) ↔ energy (density) Schrödinger equation ↔ (time-derivative ½ of) Maxwell's equations - p. 451, bottom: Because the initial & final states are the same, this is "forward scattering" -- the "incoming" particle is undeflected, so its scattering angle vanishes.
- p. 458. top: There are 6 diagrams.
- 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.
- p. 460, ref. 7 available at this link
- p. 462, middle: "m → 0" needs some space.
- p. 462, bottom: The neglect of massless tadpoles is not exactly correct when IR divergences are involved. For example, the usual massless 1-loop tadpole in D=2 is both IR & UV divergent. The whole graph can still be taken to vanish, but only because the nontrivial IR & UV divergences cancel each other. But the UV divergence needs to be renormalized, while the IR one must be canceled by other means. This can be seen by the method described: Taking the lim m²→0 is singular near D=2, as there are both a 1/ε term & a ln(m²) term, representing the 2 types of divergences. (See the calculation on the next page, but evaluate for D=2.)
- p. 470, middle: A
_{2}is really a function of x & p, or x^{2}, p^{2}, & x∙p. - p. 474, middle & p. 475, middle: All σ's should be x's.
- p. 476, middle: Note that this implies φ is an angle; it's periodic with period 2π.
- 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/2β
^{2}(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. - p. 480, bottom: UV divergence is for ε≤0, IR is for ε≥0.
- 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*. - p. 490, reference 1: See note for p. 371.
- p. 490, bottom: After ref. 3 belong refs. 23-5 from p. 536.
- p. 490, ref. 4: There was also a paper by A.G. Izergin & V. E. Korepin (1979); see arXiv:1310.1575.
- p. 493, bottom: Here we consider ĝ(M,ε) & g(μ,M,ε).
- p. 493, bottom: To see that β has no positive powers of ε, look @ the equation order-by-order in g
^{2}; it then follows inductively. - p. 497, top: Unfortunately, the term "asymptotically free" used here will not be defined until subsection VIIIA3; it means β₁>0.
- p. 498, top: The sentences "We use... in that combination.)" would go better after (old) exercise VIIC2.1.
- p. 498, bottom - 499, top: For D=0, these are ordinary integrals and derivatives, so really D → d, δ → d.
- 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.
- p. 499, middle: "The simplest example of a
*'renormalon'*problem..." - 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.
- p. 502 bottom: Here we treat only the gluon coupling as ħ, ignoring the coupling to the particle of mass m. (See previous note.)
- p. 506, top: There are too many Euler's theorems, so sometimes this is called "Euler's formula". The Euler number is sometimes called the "Euler characteristic".
- 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.

- 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, p
^{22}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 d^{D}p/p^{4}. Similarly, if one wants to analyze a quadratic divergence in D=4, one needs to look at the theory dimensionally reduced to D=2. - 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.
- p. 529, top: It might be clearer to consider the relevant term in the Lagrangian as
∫d
^{4}θ φ∇^{2}φ. Then there's a d^{2}(on δ^{4}(θ)) associated with each propagator, and a ∇^{2}-d^{2}with each vertex (with an ∫d^{4}θ). - 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 d
^{2}.) 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.) - p. 532, middle: "the previous subsection" (twice) should refer to subsection VIIB5.
- p. 532, bottom: Here we use α = ⊕ or ⊖ to correlate with ±.
- p. 536, top: refs. 23-5 belong on p. 490, after ref. 3.
- p. 538, bottom: VA2.4 → VA2.1.
- 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).
- p. 550, middle: Note that the separation used here into spinless + magnetic moment contributions is exactly the one used in subsection VIIIA3 into ◻ + FS.
- p. 559, top: Note that perturbation with respect to this potential is reliable only for
*ln g*small. This suggests confinement can be treated only for*g ≈ 1*. Various other approaches (such as random lattice quantization of strings and supersymmetric nonlinear σ models) suggest that*g ≈ ∞*, the theory related to*g ≈ 0*by electric-magnetic duality, is likewise a normal field theory, not exhibiting confinement perturbatively in*1/g*. - p. 563, top: 4-2ε → 4−2ε (hyphen → minus).
- p. 575, middle: IXA4 → IXA3.
- p. 575, middle: "Explicitly, ..."
- pp. 579-580: The vertices should include the usual factors of the coupling g.
- p. 580, middle: "vanishing sources V=1" → "N=1 and V
_{1}=1 (k_{1}=0)" - 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. - p. 582, top: As explained later, this is an expansion about the center of the string, not the boundary (where ∂X/∂σ=0).
- pp. 582-3: At the bottom of p. 582 the example is for a tree, while the example on p. 583 is for a loop.
- p. 583, bottom: The "external vector" is really a scalar.
- 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).
- p. 584, top: S
_{ab}for spin 1/2 should have a 1/2, not 1/4. - 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.

- 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.
- p. 595, middle: We can similarly write
T
_{ab}^{m}= +∇_{[a}e_{b]}^{m}. - p. 595, bottom: covariant curl should be moved to p. 597, top; see following remark
- 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

where [A,B]^{a}- (B∙∇)A^{a}= [A,B]^{a}- A^{b}B^{c}T_{bc}^{a}

[A∙∇,B∙∇] = [A,B]^{a}∇_{a}+A^{a}B^{b}R_{ab}^{I}M_{I}^{m}is defined with A = A^{m}∂_{m}. Using this, the result for the covariant curl can also be simplified, by considering A^{a}B^{b}∇_{[a}C_{b]}. - 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).
- p. 601, bottom, new exercise IXA3.5:

Consider the independent components of the curvature in lower dimensions:- Show the curvature vanishes in D=1.
- Show the curvature reduces to just the Ricci scalar in D=2. (Hint: Use ε tensors.)
- Show the Weyl tensor vanishes in D=3.

- p. 603, exercise IXA4.2: Parts should be numbered "a" & "b".
- p. 613, top, new exercise before IXA7.3: Show this transformation of ∇ (with k=1) is consistent with (covariant) integration by parts.
- p. 613, bottom: new exercise after IXA7.3: Consider "conformal Killing vectors", defined as preserving covvariant derivatives up to Weyl scale. Write this definition as a commutator (modifying that in subsection IXA2 by the definition of Weyl scale above). Find the "conformal Killing equations".
- p. 614, IXA7.4b: The second part of this belongs in subsection IXC2, since "conformally flat" isn't defined until then.
- p. 616, bottom: To summarize the various manifestations of conformal invariance:
flat space: conformal invariance curved space: Weyl invariance dilaton: dilaton decoupling - p. 618, bottom: ref. 9 should include S. Deser,
*Ann. Phys.*59 (1970) 248. - p. 621, exercise IXB1.1a: don't need to choose a gauge.
- p. 624, IXB2.1b: (dx/ds)
^{2}= -1 (in Minkowski space). - 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. - p. 625, bottom: exercise IXA2.4 → exercise IXA2.5.
- 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

we have, for V for which [n,V] = 0 (where we'll eventually choose n^{a}- (B∙∇)A^{a}= [A,B]^{a}^{m}& V^{m}to be constants),n∙∇(n∙V) = V∙(n∙∇)n + n∙(n∙∇)V = n∙(V∙∇)n = (V∙∇)½n

Pulling off the constant V^{2}= 0^{m}gives the desired result. - p. 626, middle: The weakened assumptions include ∂
_{m}n^{a}=0. - 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.
- p. 628, middle: This can be made simpler by the above methods:
(n∙∇)

^{2}V^{a}= (n∙∇)(V∙∇)n^{a}= [n∙∇,V∙∇]n^{a}= - n^{b}V^{c}R_{bcd}^{a}n^{d} - p. 628, below long equation: We really used the weaker conditions (n∙∇)n
^{a}= ∂_{m}n^{n}= 0. - p. 629, bottom, new exercise IXB4.1:

Gaussian and Riemann normal coordinates are similar and generalize:- 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.
- 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. - Show that for an axial coordinate system (n∙∇=n∙∂) with non-constant but still geodesic n (n∙∇n=fn) that
n∙∂[(e

and thus (e_{m}^{a}-δ_{m}^{a})n_{a}] = (fδ_{m}^{n}-∂_{m}n^{n})[(e_{n}^{a}-δ_{n}^{a})n_{a}]_{m}^{a}-δ_{m}^{a})n_{a}=0 with appropriate boundary conditions.

- p. 639, top: Same as for p. 214 above. Then you can drop the parenthetical remark following.
- 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. - 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). - 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.)
- p. 644, new exercise IXC3.2b: Find the solution for a=b=0.
- p. 645, middle: This method will prove useful later for other cases.
- 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.
- 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).
- p. 650, middle, new exercise: Use the Weyl scale method to derive the covariant derivatives and curvature for the metric (a generalization of Schwarzschild)
-ds where the coordinates have been divided up into arbitrary numbers of x & y coordinates.^{2}= A^{2}(y) dx^{i}dx^{j}η_{ij}+ B^{2}(y) dy^{ι}dy^{κ}δ_{ικ} - p. 652, exercise IXC5.5a: The J's should be V's.
- p. 657, top: It might be more straightforward to do this analysis directly in the Hamiltonian formalism (cf. subsection IIIB5 for electromagnetism).
- 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 (E
^{2}-m^{2})/2m is identified as (p^{i})^{2}/2m in terms of the momentum at ∞. - pp. 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 ... - 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). - p. 662, bottom: A related statement is that none of these alleged black holes have been determined not to be naked singularities: Some solutions of classical general relativity have no event horizons, but only singularities.
- p. 663, ref. 7: An English translation is available at arXiv:physics/9905030.

- 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.
- pp. 679-80, ref. 6: The links have changed. ("kiss_prepri" should be replaced with "kiss_prepri.v8".)
- p. 691, top: For consistency with earlier notation, e
_{q]β̊α}should be e_{q]αβ̊}. - p. 706, middle: its → their
- p. 717, exercise XC6.4: An easier way to do this, instead of using the parametrization @ the top of the page, is to instead
- Start from the form of the gravity action with a Weyl compensator. Then choose the scale gauge where the above ψ=1. (The compensator takes its place.)
- The only non-gravitational c (with respect to 1 less dimension) is then the usual F (for A). A nice way to think of this is as a torsion. Either way, the reduction is R → R - ¼F² (together with the Weyl compensator).

- p. 721: Somewhere there should be a discussion of what "strong" & "weak" coupling mean in the theory of "strong interactions". As we saw in subsection VIIC4, there are 2 couplings relevant to confinement: Ng² & 1/N (where N is the number of colors & g is the Yang-Mills coupling). But by dimensional transmutation (subsection VIIB3) Ng² gets replaced by the QCD scale M, which is a mass, not a coupling. (It is a property of free hadrons.) On the other hand, 1/N (where N is usually 3) is effectively small. So from the point of view of confinement, QCD describes a theory of somewhat weak interactions between hadrons. But if we want to study chiral symmetry breaking (subsection IVA4), then we need to consider the masses m of the quarks, & we then have the dimensionless coupling M/m, which is large for the light quarks. (This is equivalent to the statement that the running coupling g becomes large at small energies.) So the strong interactions are actually strong only when considering the ground-state hadrons, whose properties are best described in terms of chiral symmetry breaking. But "confinement" is a property of the excited states, where this coupling becomes weak. A related statement is that any attempt to describe confinement will fail unless it takes dimensional transmutation into account.
- p. 725, middle: Notice that we are discussing high-energy behavior in the s channel corresponding to bound states in the t channel. So it might be better to switch s & t for comparing to nonrelativistic potential scattering (including the figure). In that case, there is no large-t limit involved (unless one analytically continues to unphysical t); instead one writes the amplitude as a sum over trajectories (leading & daughters). But in the relativistic case, using crossing symmetry, one can switch s & t, and then it makes sense to take s → ∞, where the leading trajectory dominates.
- 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→-∞.
- p. 725, bottom: This is actually a Mellin transform, as applied for other purposes in subsection VIIC3.
- p. 730, bottom: IXA7.3 → IXA7.6.
- p. 731, top: h=w=c=0 is the sphere. (The torus, h=1, has no curvature, so χ=0.)
- 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. See these notes.
- p. 736, middle: VIC4 → VIIC4
- p. 737, top: σε[0,π] → σ∈[0,π] (epsilon → element).
- 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.
- p. 738: The analysis in this subsection of closed = open ⊗ open is based on a flat space analysis. There can be modifications from compactification on nontrivial backgrounds when considering lower dimensions, especially since for closed superstring theories the open string states exist only formally, & can't be coupled to nontrivial closed string backgrounds.
- p. 743, top: That should be "∫dx √-G because G", where G is the determinant of the space-time metric, not to be confused with the superfield "G" appearing later.
- p. 743, middle: The dilaton coupling as defined here (i.e., with this dependence on G) depends on the measure for the worldsheet path integral, and the regularization of worldsheet loops. This ambiguity can be avoided by defining the coupling directly to the ghosts as mentioned previously; integrating out the ghosts at 1 (α') loop gives the ambiguous curvature coupling. The loop is defined here to preserve T-duality, and is the definition consistent with the lightcone gauge (where this dilaton doesn't appear; neither do the ghosts).
- 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. - p. 756, bottom: A couple of δ's are missing on the bottom of the δ/δ's.
- 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 = J
^{m}ε_{mn}dσ^{n}, then dJ = 0 is the conservation equation ∂_{m}J^{m}= 0, and dσ = J is J^{m}= δ_{0}^{m}. In our case, this has the interesting interpretation that while τ is identified with X^{+}, σ is identified with the*T-dual*of X^{+}. - p. 763, middle: The closed-string tachyon has M
^{2}=-4α'^{-1}, as easily seen from the formulas that follow. - p. 766, top: "roation" → "rotation"
- p. 766, top: This reality condition holds only for conformal weight 0; there is an obvious extra power of z otherwise.
- p. 767, bottom: now z = τ+iσ.
- p. 770, middle: From here through the rest of this section, it really would have been better if I had used the opposite sign convention for T (& BRST Q).
- p. 773, middle: The 〈TT〉 calculation is easy to generalize to Q
^{2}=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 Q^{2}=0 reduces to 〈ĜĜ〉. - p. 774, top: A better explanation of why the μ term doesn't contribute to the propagator is that the μRφ term vanishes in flat (2D) space: Its only affect is an "improvement term" in the energy-momentum tensor, one that is separately conserved.
- 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 [∮λ
_{1}T̂,∮λ_{2}T̂]. 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. - 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.)
- p. 777, top: XIIC1.1 → XIC1.1.
- p. 778, middle: Arbitrary loops for particles
*can*be treated by 1st-quantization, by methods very similar to those used for strings. See, e.g., P. Dai & W. Siegel, hep-th/0608062 and Dai, Y.-t. Huang & Siegel, hep-th/0807.0391. - 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.
- p. 782, bottom: This isn't generally true -- The expressions found for α(t) from nonrelativistic potential scattering (e.g., Coulomb, as described in exercise XIA1.3) or summing ladder diagrams (e.g., Higgs) typically go to a negative constant (usually -1) as t→-∞, unlike the string result. If such trajectories are substituted into the Beta function, then the limit t→-∞ is effectively the same as t fixed (because α(t) is fixed), so the result is the same as Regge behavior. Thus, if still α(s)~s as s→∞, then the fixed angle limit is a negative power of s, as desired.
- p. 786, bottom: This is a lousy discussion of the vector vertex. See instead these notes.
- 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.
- p. 787, bottom: That V for the closed string should be a W.
- 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(z
_{N})C(z_{N-1})C(z_{1})〉. We then get a +z_{N}^{2}that agrees with the previous calculation. - p. 788, bottom: The condition that ad-bc = 1 can be weakened to the condition that it's positive, since rescaling a,b,c,d by the same real number changes the right-hand side by a positive factor; the constraint thus eliminates redundancies. However, ad-bc = -1 (e.g.) includes a reflection of the real axis, & can't be obtained continuously from the identity; it would reverse the cyclic ordering of the vertices. Thus, the Sp(2) transformations can change any 3 z's to fixed ones in the same cyclic order.
- p. 789, top: This coset will have been explained in new subsection IC6.
- p. 789, bottom, new exercise XIB7.3: Instead of mapping the open-string worldsheet to the upper-half plane, we could use instead the disk with unit radius. Show that the form of SL(2) that maps the unit circle to itself is SU(1,1) with diagonal group metric.
- p. 792, middle: too many N's (α'M
^{2}is in terms of the number operator, not the number of external lines.) - p. 793, middle: Note that the contribution from the oscillators of 1 boson is the same as the inverse of that from the oscillators of 1 fermion, as expected from statistics. However, the contribution from 1 boson compactified on a circle, including 0-modes, is the same (not the inverse) as that from 2 fermions (including 0-modes in some way), as expected from bosonization, as explained in the following subsection.
- p. 793, exercise XIIC1.1: should, of course, be XIC1.1.
- p. 794, top: For you statistical mechanics,
*V*(T) was the "canonical partition function" (with 1/T the "temperature"), while Z(T,ρ) is the "grand canonical partition function" (with ρ/T the "chemical potential"). - p. 794, top: really |w|<1 & Re T > 0 to avoid singular behavior @ w=1 (T=0)
- 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 m_{0}. - 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'.
- p. 801, top, and p. 805, middle: When I say "f" here, I really mean "wf
^{24}". - 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.
- p. 801, middle: 3) Actually, you might as well consider all of the usual UV field theory divergences.
- 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.
- p. 802, middle: We begin with the planar loop.
- 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.
- p. 805, bottom: fundamental → defining
- p. 807, middle: To agree with previous notation, "z" should be called "ρ" here.
- p. 808, top: Really we have gauged z
_{2}=2πi; then z_{1}=-T. - p. 812, bottom: "dilaton or
*graviton*coupling" - p. 816, bottom: The dilaton contribution for the nonplanar case vanishes in the limit m → ∞.

- p. 825, middle: ∂
_{+}→ ∂^{+}. - p. 829, exercise XIIA4.1a: IA2.3e → IA2.3d.
- p. 839, top: Note that Φ=|
^{i}>Φ_{i}& Φ^{T}=Φ^{i}<_{i}|, where Φ^{i}(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. - p. 839-40,3,5,7,53-4,8,62,4-5: The sign conventions for the antifields of the Yang-Mills ghost & antighost used throughout section XIIB-C are inconsistent with those for the more general case in section XIIC, and should be reversed.
- p. 839, middle: Really we should have QΦ=-Q̂Φ (Q-hat), & similarly for J & Ĵ, as explained in subsection IC1.
- p. 848, middle: Note that the solution for the cohomology of Q is trivial in this case, following the method of subsection XIIB4, since S
^{⊕-}is now of the same form as in exercise VIA1.2. - p. 851: Most of this page would fit better on p. 787 of subsection XIB7.
- p. 851, top: That should be "[Q,W} = - ∂V", as in the middle of the page. But really instead we should change our sign convention for T & Q for the string.
- 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.
- p. 858, middle: As described above, the signs are off for the antifields of the Yang-Mills ghost & antighost in the expansion of Φ, but OK in the few lines following.
- p. 863, middle; p. 864, top: Strictly speaking, there should be some factors of 1/2 here for the quadratic terms.

- p. 881: Include VIIB5 for "Normal ordering".
- p. 883: "superspin", after we define it (see above); also "super-isospin"