Available from Cambridge University Press.

If you find errors/typos or wish to suggest corrections,

write to Barton Zwiebach at zwiebach@mit.edu

Corrections for the first edition (2004) are below, but have

not been updated since 2005.

Home page for Barton
Zwiebach

**Additional material of possible interest to
instructors and students**:

Here is our final test in 8.251, spring term of 2009, with solutions.

**Corrections were last updated on 06/27/2013.**

- p.24. Below eqn. (2.60) replace (2.3) by (2.58).
- p.28. Above eqn.(2.84) replace (2.3) by (2.58).
- p.39. Second sentence below (2.121), ending as

"... sines and cosines." change to

"... sines and cosines, with $ l= 0,1,2, \cdots$. " [suggestion by Y. Okawa] - p. 51. Two lines below eqn. (3.35), replace (3.15) by (3.25).
- p. 52 A change to Eqn.(3.37): change the period for a comma, leave some space, and add " i = 1, 2, \ldots , d \,, " On the line following add "where $d$ is the number of spatial dimensions." [suggestion by Y. Okawa]
- p.56. Two lines above equation (3.68), replace "subspace" by "subset". [Thanks to Richard Lebed, 07/01/09].
- p.60. Paragraph below equation (3.84), two occurrances of "$\backslash epsilon\backslash cdot\{\backslash cal\; O\}(h)$" that should be "{\cal O}(\epsilon, h)". [05/19/2009].
- p.66. Second sentence below (3.114) replace the part "Since the effective four-dimensional gravitational potential is V_g^{(5)}," by the following "Moreover, the four-dimensional gravitational potential, which is the average of V_g^{(5)} over x^4, is just V_g^{(5)}. Therefore, " [clarification suggested by Hong Liu]
- p.80. 7 lines from the bottom (4.5) should be (4.28). [05/19/2009].
- p.87. Problem 4.6, "m^2" should be just "m" in the first two displayed equations. [05/19/2009]
- p.114. One line below and two lines above equation (6.56), replace (6.5) by (6.54). [Thanks to Glenn Wouda, 05/19/2009].
- p.117. Eqn. (6.64). Replace the { } braces in the last right-hand side for parenthesis ( ). [suggestion by Y. Okawa]
- p.119. Eqn (6.69), second equation, left-hand side. replace X'^\mu by {X^\mu}' [Y. Okawa]
- p.145. Fourth line from the beginning of section 7.6. Replace "study" by "studied".[Barak Shoshani]
- p.148. Figure. 7.6, caption, 1st line, spelling mistake: change "obserber" to "observer". [Thanks to Richard Lebed, 07/01/09].
- p.150. Not a correction, but rather an improvement for part (c) of Problem 7.3. The new version would read: "(c) Consider a closed string which at t=0 is static and traces a closed curve \gamma of length \ell. How are \sigma_1 and \ell related? Find a time t_P >0 and smaller than \ell/c for which the closed string traces the curve \gamma again. Relate \vec{X} (t_P, \sigma) to the time equal to zero string \vec{X} (0,\sigma)."[05/19/2009].
- p.152. Problem 7.7, part (a), equation (1). Both the 1/4 and 1/12 should be replaced by 1/2. [Thanks to Uchupol Ruangsri, 05/19/2009].
- p.179. Above equation (9.13), sentence beginning with "Looking at (6.93)...". The reference to (6.93) is inappropriate since this expression assumes the static gauge. The sentence should be changed to "Looking at (6.49) we note that the numerator has two \sigma derivatives while the denominator has effectively one, making {\cal P}^{\tau \mu}transform like {d\over d\sigma} under reparameterizations."[05/19/2009].
- p.206. First line. Two changes. Change "fixed." for "fixed and positive." Change p^+ for p^+ > 0 .
- p.207. Eqn. (10.71) in the exponent replace "px" by "p\cdot x". [Y. Okawa]
- p.223. Second line below (11.39), replace (11.25) by (11.26).
- p.257. Two corrections.

-- Equation (12.131), sum should begin with "k=1".[Christopher Chamberland]

-- Six lines below (12.133) the L_1's in the commutator should be L_1^\perp [Y. Okawa] - p.258. After last line ending with "..show." add the following sentence. Problem 12.10 shows that the above reparameterizations preserve the constraints (9.33). [Suggestion by Hong Liu]
- p.283. Add the following sentence below (13.19).

"The reality of X^\mu (\tau, \sigma) requires \alpha_0^\mu, x_0^{L\mu}, x_0^{R\mu} be real, as well as \alpha_{-n}^\mu = (\alpha_n^\mu)* and \bar{\alpha}_{-n}^\mu = (\bar{alpha}_n^\mu)*, for n \geq 1." [suggestion by Y. Okawa] - p.285. Below eqn. (13.31) add the sentence

"Together with the hermiticity of x_0^\mu and \alpha_0^\mu, these make X^\mu (\tau, \sigma) in (13.24)

Hermitian." [suggestion by Y. Okawa] - p.313. 2nd line after (14.31), spelling, change "parenthesis" to "parentheses". [Thanks to Richard Lebed, 07/01/09].
- p.313. Eqn (14.29), for uniformity of notation replace \delta^{IJ} by \eta^{IJ} [suggestion by Y. Okawa]
- p.315. Eqn. (14.43), for uniformity of notation replace \delta^{IJ} by eta^{IJ} [suggestion by Y. Okawa]
- p.318. Line below eqn. (14.61). Replace "physical that" for "physical than" [Y. Okawa]
- p.320. Section 14.7, second paragraph, third line, replace (14.47) by (14.46).[05/19/2009].
- p.324. Second to last paragraph from the bottom, beginning as "Finally, …" The second sentence ending with "…closed unoriented strings" change the ending to "…..closed unoriented strings, also with gauge group SO(32)". [suggestion by Y. Okawa]
- p. 363. Eq. (16.37), place a loop symbol on the integral over \Gamma (as in (16.36)). [Thanks to Richard Lebed, 07/01/09].
- p.384. There are four occurrences of the cite (17.3) that are wrong:

-- three lines above eqn. (17.35) should be (17.33)

-- line above eqn. (17.36), should (17.29).

-- line below eqn. (17.39), should be (17.33).

-- line below eqn. (17.40), should be (17.30) - p.387. Three corrections:

-- one line above eqn. (17.49), replace (17.3) by (17.30)

-- one line above eqn. (17.53), replace (17.5) by (17.47)

-- one line below eqn. (17.53), replace (17.3) by (17.30) - p.390. Paragraph above eqn. (17.64) replace the two occurrences of (17.6) by (17.63).
- p.392. Paragraph above (17.71), replace the two occurrences of (17.6) by (17.64).
- p.394. Section 17.8, three lines above and line below equation (17.76), replace (17.3) by (17.31). [05/19/2009].
- p.395. Third line above (17.80), replace "q" by "w". [Thanks to Richard Lebed, 07/01/09].
- p.396. Below eqn. (17.87) replace (17.3) for (17.31).
- p.403. Below eqn. (18.16) replace "The differential relations in (18.1) can" by "The above differential relations can".
- p.420. Line below eqn. (19.25), replace (18.1) by (18.16).
- p.424. Quick calculation 19.4, replace (19.3) by (19.37).
- p.430. Problem 19.2. Delete the last two sentences, beginning with "The back cover..." and ending with ".. of the strings." [05/19/2009].
- p.436. Eq. (20.18), replace "H_x" by "H_1" for consistent labeling. [Thanks to Richard Lebed, 07/01/09].
- p.475. Quick calculation 21.12, first line, replace (21.4) by (21.58).
- p.477. Paragraph below eqn. (21.66). Two corrections. Replace the first (21.4) by (21.56) and the second (21.4) by (21.62).
- p.478. Two corrections

-- Figure, 21.8, bottom, the second label N_2(Y= -1/2) should read N_3(Y= -1/2)

-- fifth line, replace (21.4) by (21.62). - p.483. 9th line after (21.68), change "22.4" for "23.8". [Thanks to Richard Lebed, 07/01/09].
- p.484. Fifth line from the top. Delete the period after "strengths"
- p.486. Two changes.

-- Third line from top, change 21.6 for 21.5.

-- Six lines from bottom. replace "in the range (-\Lambda_P, \Lambda_P)." for "of order \Lambda_P." [Y. Okawa]. - p.488. Eq. (21.93), last displayed equation, right-hand side, the numerator should be multiplied by 2. [Thanks to Richard Lebed, 07/01/09].
- p.494. Problem 21.5. second line, replace (n,m) for (m, n). [Thomas Klose]
- p.516. Two corrections. Two appearances of (22.6) (right above eqn.(22.132) and bottom line)
should be replaced by (22.131).

- p.517. Second line. Replace (22.13) by (22.131).
- p.527. 5th line above equation (23.1), spelling, change "parenthesis" to "parentheses".[Thanks to Richard Lebed, 07/01/09].
- p.531. Two corrections.

-- 2nd line above (23.29), replace "(14.51)" for "(15.51)". [Thanks to Richard Lebed, 07/01/09].

-- Line above (23.28) replace "such string" by "such a string" - p.538. 8 lines from the bottom, change "Section 15.5" for "Section 21.2". [Thanks to Richard Lebed, 07/01/09].
- p.545. Two lines above (23.65), replace "with higher" by "with a higher"
- p.548. 4 lines below eqn. (23.82), replace 22.6 by 25.6.[Thanks to Jeremy Green, 05/19/2009].
- p.549. 4th line after (23.84), change "(23.8)" for "(23.65)". [Thanks to Richard Lebed, 07/01/09].
- p.552. 2nd line, replace "(23.8)" for "(23.65)". [Thanks to Richard Lebed, 07/01/09].
- p.553. Two corrections.

-- Figure 23.12, bottom annotation, "spatital" should be "spatial".[05/19/2009].

-- Must change equations (23.98) and (23.99) to read like [Thanks to Thomas Klose, and Curtis Asplund]

- p.554. Two lines above beginning of section 23.10, replace "such small" by "such a small"
- p.559. Quick Calculation 23.11, second line, delete the second word, "of".[Thanks to Jeremy Green, 05/19/2009].
- p.561. Third line. Replace the whole set of sentences and equation beginning with "The AdS/CFT..."
and ending with "...principle (Problem 23.8)." Insert instead the following paragraph [Thanks to H. Liu]

- p.562. Problem 23.1, above last paragraph, definitions of X and \bar X in displayed equation, replace X^I by X^2 and X^2 by X^3. [Thanks to Jeremy Green, 05/22/2009].
- p.563. Two lines above equation (4), replace "areas" by "width"
- p.564. Problem 23.5, part (a), first line, replace (23.9) by (23.90). [Thanks to Jeremy Green, 05/22/2009].
- p.566. Problem 23.7. part (a) second line "Exhibit ... z=0." change to "Exhibit a timeline tangent at all points on the surface."
- p. 567. Delete the last line, "The estimate (3) ...temperature range."
- p.571. Equation (24.12), replace "x^\mu" by "x_0^\mu". [Thanks to Richard Lebed, 07/01/09].
- p.594. Figure 25.2(b), change a "P_2" for "p_2". [Thanks to Richard Lebed, 07/01/09].
- p.620. Two lines above (25.91), change (25.7) for (25.89).
- p.624. Four appearances of (25.8) that must be changed

-- Second line from top, change to (25.101)

-- Second line below eqn. (25.104), change to (25.103)

-- Second line above eqn. (25.105), change to (25.104)

-- Line above eqn. (25.107), change to (25.101). - p.635. Figure 26.4, bottom right, label "- ln r" on axis should be "ln r" (remove the minus sign). [05/19/2009].
- p.638. Two corrections.

-- Equation (26.9), remove the loops on the first two integrals (the \gamma paths are not closed). [Thanks to Richard Lebed, 07/01/09].

-- Figure 26.6. A few radial lines do not reach the outer boundary.

**Corrections were last updated on 05/06/2005. No further updates will be done. **

Corrections by date:

**June 2004**

- p.73, below equation (4.38) delete "on the first line"
and delete "on the second line"[6/14/04].

- p.117, equation (7.2), arrow (for vector) missing on
top of v_\perp [Thanks to Siarhei Vaurynovich, 6/14/04].

- p118, equation (7.7), middle expression: the square
root in the numerator must be squared, as in (\sqrt{...})^2
[Thanks to Siarhei Vaurynovich, 6/14/04].

- p.150, lines 1 and 2, \lambda missing in the right hand side, equation should read n_\mu X^\mu = \lambda \tau [Thanks to Siarhei Vaurynovich, 6/14/04].
- p.211, immediately above the unnumbered equation (which uses two lines), replace (12.25) by (12.33).[Thanks to Mikael Djurfeldt, 6/14/04].
- p.211, second line below (12.36), delete the words "in (12.2)" [6/14/04].
- p.211, line immediately above (12.37), replace "second line in (12.2) is replaced by" by "commutator is". [6/14/04].
- p.211, line immediately below (12.38), replace "final right-hand side in (12.2)" by "result that led to (12.36)".[6/14/04]
- To facilitate the four corrections of page 211 entered
above here is a pdf showing
the corrected page.

- p.248, for clarity, replace "world-sheet" by "parameter space" immediately before equation (13.8) and immediately before equation (13.9). [6/14/04].
- p.260, section 13.4, second paragraph, replace "m_e"
by just "m" (for consistency with equations below) [6/14/04].

- p.305, problem 14.10, part (a), corrections on the
sentence beginning as: "Show that the orientifold action...".
Delete the word "simultaneous"
and add the words "either one of" between the words "about" and "the".
[6/14/04].

- p.434, equation (19.40), the "Y" should be replaced by an "X".[6/14/04].
- p.483, figure 22.1, very minor correction: the dashing
and the size of arrows should match those of figure 22.2. [6/14/04].

- p.531, Figure 23.10. The sizes of the outgoing
cylinders should match those of figures 23.11 and 23.12. So, in particular,
P_4 in figure 23.10 should be a bit smaller (consequently, both P_1
and P_2 should be smaller)[6/14/04].

- p.31, second line, replace Q^- by Q' (also in Figure 2.6, p.30). [Thanks to James Snyder, 7/04/04].
- p.45, equation (3.30), last term to the right: the \mu and \nu indices on the two eta's should be exchanged. [Thanks to James Snyder, 7/04/04].

August 2004

- p.3, second paragraph, line containing "Michael Faraday", there is a double "that", remove one! [Unbelievable that this was not caught earlier. Thanks to Jan Troost, 8/16/04].
- To facilitate the corrections of page 211 entered in June 2004 here is a pdf showing the corrected page.[8/16/04]
- p.118, equation (7.10), very minor correction: for clarity and consistency include a factor of (1/c) inside parenthesis in the middle term. [Thanks to James Snyder, 8/18/04].
- p.168, equation (10.7), the second partial derivative should
be \partial_\nu. [Thanks to James Snyder, 8/30/04].

- p.119, two lines below eqn. (7.16). "How do we recast (7.16)" should be "How do we recast (7.15)". [Thanks to Julian Vogel and Pierre Jouvelot, 9/05/04].
- p.187, eqn. (11.2), the last "\mu" should be replaced by "\nu". [Thanks to Pierre Jouvelot, 9/05/04].
- p.208, in the two lines above (12.17), for clarity, replace \xi(\tau) by \xi(\tau, \sigma), and replace \xi by \xi (\sigma). [Thanks to Pierre Jouvelot, 9/05/04].
- p.243, two lines before Equation (2), "we" is duplicated. [Thanks to Pierre Jouvelot, 9/05/04].
- p.264, in the paragraph following Equation (13.95), b^I_{-r} should be b^J_{-r}. Also, three lines before Quick Calculation 13.3, the word "states" appears duplicated. [Thanks to Pierre Jouvelot, 9/05/04].
- p.267, 4 lines before the end of the page, the word "the" appears duplicated. [Thanks to Pierre Jouvelot, 9/05/04].
- p.285, first term on the right-hand side of the equation in (14.54): the \alpha should be primed. [Thanks to Pierre Jouvelot, 9/05/04].
- p.291, line right after eqn. (14.65), the word "to" appears duplicated. [Thanks to Pierre Jouvelot, 9/05/04].
- p.299, one line before the end of the page, the word "the" appears duplicated. [Thanks to Pierre Jouvelot, 9/05/04].
- p.549, in the entry for Das and Mathur, the word "pysics" should be replaced by "physics". [Thanks to Pierre Jouvelot, 9/05/04].
- p.347, last line, replace the third word, "the", by "a closed". [see motivation below, 9/05/04].
- p.335. Four lines after (15.108), an "on" is missing in the sentence "the strings end (on) a D-brane". [Thanks to Pierre Jouvelot, 9/10/04].
- p.364. equation (16.81), an arrow is missing on top of p in d^d p. [Thanks to Pierre Jouvelot, 9/10/04].
- p.375. In (16.140), the Q_2 in the intermediate step should be Q_5. At the end of the page, last character, the "d" should be a "b", and immediately in the next page "d_f" should be "f". [Thanks to Pierre Jouvelot, 9/10/04].
- p.379, end of second line, replace 15.6 by 15.5. [Thanks to Pierre Jouvelot, 9/10/04].
- p.389 (very minor correction), Figure 17.1, the letters P and P' should be lowercase (p and p') for consistency with the explanation beginning as "A small confusion ...", in p. 390. [Thanks to Pierre Jouvelot, 9/14/04].
- p.403, line 6 of the second paragraph of Section 17.8, "radius us" should be "radius of". [Thanks to Pierre Jouvelot, 9/14/04].
- p.407, Problem 17.3, the first displayed equation should be labelled (1). [Thanks to Pierre Jouvelot, 9/14/04].
- p.383, Problem 16.6. Second line: delete "(derivatives of )"
and add "(and derivatives thereof)" right after the word "functions". Moreover,
the quoted asymptotic expansion has a missing factor and is not the most
useful. It is better to give the expansion for K_\nu without derivatives.
For this: 1) delete the prime to the left of the ~, 2) delete the
minus sign immediately to the right of the ~, 3) add a multiplicative factor
of \exp(-z), and 4) change the +3 for a -1. [9/14/04].

October 2004

- p.484, two lines before equation (22.4), missing period (.) before "In". [Thanks to Pierre Jouvelot, 10/24/04].
- p.511, 4th line of the 3rd paragraph, replace "x_2 and x_3" by "x_3 and x_4". [Thanks to Pierre Jouvelot, 10/24/04].
- p.524, 4 lines after equation (23.6), replace "clockwise" by "counterclockwise". [Thanks to Pierre Jouvelot, 10/24/04].
- p.530, Fig.23.9, top: the arrow to the left of "\delta T" extends a bit too much, it should stop right above the beginning of the lower slit.[Thanks to Pierre Jouvelot, 10/24/04].
- p.533, Fig. 23.13, caption: replace "23.12 and 23.11" by "23.11 and 23.12".[Thanks to Pierre Jouvelot, 10/24/04].
- p.536. Fig 23.16, caption, second line: the w in \eta = w/T should have a tilde.[Thanks to Pierre Jouvelot, 10/24/04].
- p.538, Fig. 23.18, a series of very minor changes: (a) cylinder shown at lower center, top face, the angle for \theta should be denoted with a small curved line rather than a little square (that incorrectly suggests right angle), (b) Im(\tau) and Re(\tau) should be I(\tau) and R(\tau), with the proper font for I and Rs. [Thanks to Pierre Jouvelot, 10/24/04].
- p.177, line immediately below eqn (10.67), replace "multiples"
by "multiplies".[Thanks to James Snyder, 10/24/04].

- p.37 Problem (2.4), (b) line before last: change "rectangular" for "square". [Thanks to Ira Rothstein, 02/16/05].
- p.98, This is not a correction but rather an additional sentence
that notes an interesting special case (alluded to in the remarks on Ch
6 at the bottom of this web page -- as noted by James Snyder). At the end
of page 98, right after "sense." add: "There is a small caveat:
at a point where the string moves with the speed of light there is no timelike
direction (see Quick Calculation 6.3).

- p.120, Equation (7.21), left-hand side: arrow missing over the X. [Thanks to Dana Mihai, 2/19/05].
- p.121, Line immediately below (7.25); add "to" after the word "assigned".[Thanks to Dana Mihai, 2/20/05].
- p.143, A very minor improvement (for clarity). Below (8.53) replace " indices" by "first index" .[Thanks to James Snyder].
- p.151 and p.154. It was called to my attention by Ira Rothstein
and Dana Mihai (02/05) that the argument for the vanishing of $n\cdot
\mathcal{P}^\sigma$ advanced in the paragraph below equation (9.18) does
not prove the claim. Indeed, the correct way to frame the issue is the following:
"We assume that $n \cdot \mathcal{P}^{\sigma}=0$

at the open string endpoints since this condition naturally guarantees the conservation of $n\cdot p$ (consider equation (8.38) dotted with $n^\mu$)." This sentence fits well in page 151, at the end of the paragraph below equation (9.4) (ending with "conservation."). Then, when we get to p.154, paragraph below (9.18), one can eliminate the sentence "Indeed, .... with $n^\mu$)." and replace it with: "As noted below equation (9.4), we assume $n \cdot \mathcal{P}^{\sigma}=0$ at the string endpoints since it guarantees the conservation of $n\cdot p$." - p. 197, Title of section 11.5: replace "generators" by "operators" (they are *translation* generators) [02/05]
- p.202, Second line above (11.85), add ", $x^-(\tau)$," right after $x^+(\tau)$. Next line: add ", (11.30), " right after (11.29) [Thanks to James Snyder].
- p. 205. This is an extra hint for a problem, which Siarhei Vaurynovich noted is otherwise fairly difficult to solve. At the end of part (b) of Problem 11.7 add the following line:[Hint: The reparameterization takes the ``hermiticized" form $\delta x^\mu (\tau) = {1\over 2} (\lambda \,\partial_\tau x^\mu + \partial_\tau x^\mu \, \lambda ) $.]
- p.212 Equation (12.46): add a prime to the sigma in
the left-hand side [Thanks to Ira Rothstein 2/21/05].

**March 2005**

- p.157. Last sentence before sect. 9.4: ${X'}^\mu$ should be replaced by ${X^\mu}'$.[Thanks to Ng Jia Tong, 03/04/05].
- p. 164. Very minor: Problem 9.3, part (b): for notational consistency, delete the parenthesis surrounding the superscripts 2 and 3 of the X's.[Thanks to Ng Jia Tong, 03/04/05].
- p. 165. Problem 9.4 part (c): replace "real" by "positive $x^1$".[Thanks to Ira Rothstein 03/06/05].
- p. 387. More than a correction, this is a clarification suggested
by Stanley Deser, who found some comments on electromagnetic duality in
p.387 somewhat misleading. If you want an improved version, do the
following changes: Second line below (17.2), add the word "familiar"
before "Lagrangian" (this is because there is a less familiar first order
Lagrangian for transverse fields that is invariant). Third line below
(17.2) up to the seventh line, delete all text from "First of all...." up
to ".... would not be invariant."(the comments on nonlocal changes
can be stated more accurately, see below). Third line below (17.3)
delete the sentence "The significant ... Lagrangian." Now, to
add some substance insert the following paragraph in between the paragraph
ending with "unchanged" and that beginning with "Consider now": ``Since
the dynamical variables in electromagnetism are the gauge potentials,
one may ask: Are there transformations of the potentials that
induce (17.2)? Not exactly, but close. One can formulate electromagnetism
(without sources) using dynamical variables $\vec{E}$ and $\vec{A}$, both
divergenceless. Duality transformations are then written as spatially
nonlocal transformations of $\vec{E}$ and $\vec{A}$. (A typical spatially
nonlocal transformation expresses the output at a point $\vec{x}$ in
terms of the values of the input at all points $\vec{x}'$.)" [Thanks to Stanley
Deser, 03/06/05]

**April 2005**

- p.53. Very minor: Equation (3.82), left hand side, superscripts on the x: reverse the order of prime and \mu. [Hyung Jun Lee, 4/20/05]
- p. 129 Line above part (c) add "with constant angular velocity" between "rotates" and "around".[Thanks to the students at MIT].
- p. 213, Very minor: Eqn (12.54), second equation, right-hand
side, superscripts on the

symbol "a": reverse the order of the star and the \mu. - p.231, Equation (12.162), add a minus sign "-" right after the equal sign (top line) [Thanks to J. Fernandez-Alcazar, who re-checked the computation, 4/05]
- p. 308, End of the second line, replace the exponent "D-4" by "4-D".
- p. 406 Table 17.2, first line (line that begins with $\tilde
H$), first term on the right-hand side: the (two) p^i's should have
no tildes. [Ira Rothstein, 04/05]

**May 2005**

- p.194. Very minor, immediately below eqn (11.47). Missing " on top of the letter "o" in Schrodinger [Thanks to Karsten Koeneke, 5/6/05]
- p. 308 Minor, second paragraph, second to last line, replace "This a" by "This is a" [Thanks to Karsten Koeneke, 5/6/05]
- p. 309 Second line, equation in parenthesis, right-hand side: replace the exponent "D-6" by "6-D" [Thanks to Karsten Koeneke, 5/6/05]

Corrections by page:

Chapter 1

- p.3, second paragraph, line containing "Michael Faraday", there is a double "that", remove one! [Unbelievable that this was not caught earlier. Thanks to Jan Troost, 8/16/04].

- p.31, second line, replace Q^- by Q' (also in Figure 2.6, p.30). [Thanks to James Snyder, 7/04/04].
- p.37, Problem (2.4), (b) line before last: change "rectangular" for "square". [Thanks to Ira Rothstein, 02/20/05].

- p.45, equation (3.30), last term to the right: the \mu and \nu indices on the two eta's should be exchanged. [Thanks to James Snyder, 7/04/04].
- p.53. Very minor: Equation (3.82), left hand side, superscripts on the x: reverse the order of prime and \mu. [Hyung Jun Lee, 4/20/05]

- p.73, below equation (4.38) delete "on the first line"
and delete "on the second line"[6/14/04].

Chapter 6

- p.98, This is not a correction but rather an additional sentence
that notes an interesting special case (alluded to in the remarks on Ch
6 at the bottom of this web page -- as noted by James Snyder). At the end
of page 98, right after "sense." add: "There is a small caveat:
at a point where the string moves with the speed of light there is no timelike
direction (see Quick Calculation 6.3).

Chapter 7

- p.117, equation (7.2), arrow (for vector) missing on top of v_\perp [Thanks to Siarhei Vaurynovich, 6/14/04].
- p118, equation (7.7), middle expression: the square root in the numerator must be squared, as in (\sqrt{...})^2 [Thanks to Siarhei Vaurynovich, 6/14/04].
- p.118, equation (7.10), very minor correction: for clarity and consistency include a factor of (1/c) inside parenthesis in the middle term. [Thanks to James Snyder, 8/18/04].
- p.119, two lines below eqn. (7.16). "How do we recast (7.16)" should be "How do we recast (7.15)". [Thanks to Julian Vogel and Pierre Jouvelot, 9/05/04].
- p.120, Equation (7.21), left-hand side: arrow missing over the X. [Thanks to Dana Mihai, 2/19/05].
- p. 129 Line above part (c) add "with constant angular velocity" between "rotates" and "around".[Thanks to the students at MIT].

- p.143, A very minor improvement (for clarity). Below (8.53)
replace " indices" by "first index" .[Thanks to James Snyder,
2//05].

- p.150, lines 1 and 2, \lambda missing in the right hand side, equation should read n_\mu X^\mu = \lambda \tau [Thanks to Siarhei Vaurynovich, 6/14/04]
- p.151 and p.154. It was called to my attention by Ira Rothstein
and Dana Mihai (02/05) that the argument for the vanishing of $n\cdot
\mathcal{P}^\sigma$ advanced in the paragraph below equation (9.18) does
not prove the claim. Indeed, the correct way to frame the issue is the following:
"We assume that $n \cdot \mathcal{P}^{\sigma}=0$

at the open string endpoints since this condition naturally guarantees the conservation of $n\cdot p$ (consider equation (8.38) dotted with $n^\mu$)." This sentence fits well in page 151, at the end of the paragraph below equation (9.4) (ending with "conservation."). Then, when we get to p.154, paragraph below (9.18), one can eliminate the sentence "Indeed, .... with $n^\mu$)." and replace it with: "As noted below equation (9.4), we assume $n \cdot \mathcal{P}^{\sigma}=0$ at the string endpoints since it guarantees the conservation of $n\cdot p$." - p.157 Last sentence before sect. 9.4: ${X'}^\mu$ should be replaced by ${X^\mu}'$.[Thanks to Ng Jia Tong, 03/04/05].
- p. 164 Very minor: Problem 9.3, part (b): for notational consistency, delete the parenthesis surrounding the superscripts 2 and 3 of the X's.[Thanks to Ng Jia Tong, 03/04/05].
- p. 165, Problem 9.4 part (c): replace "real"
by "positive $x^1$".[Thanks to Ira Rothstein 03/06/05].

- p.168, equation (10.7), the second partial derivative should be \partial_\nu. [Thanks to James Snyder, 8/30/04].
- p.177, line immediately below eqn (10.67), replace "multiples"
by "multiplies".[Thanks to James Snyder, 10/24/04].

- p.187, eqn. (11.2), the last "\mu" should be replaced by "\nu". [Thanks to Pierre Jouvelot, 9/05/2004].
- p.194. Very minor, immediately below eqn (11.47). Missing " on top of the letter "o" in Schrodinger [Thanks to Karsten Koeneke, 5/6/05]
- p. 197, Title of section 11.5: replace "generators" by "operators" (they are *translation* generators) [02/05].
- p.202, Second line above (11.85), add ", $x^-(\tau)$," right after $x^+(\tau)$. Next line: add ", (11.30), " right after (11.29) [Thanks to James Snyder].
- p. 205. This is an extra hint for a problem, which Siarhei Vaurynovich
noted is otherwise fairly difficult to solve.
At the end of part (b) of Problem 11.7 add the following
line:[Hint: The reparameterization takes the ``hermiticized"
form $\delta x^\mu (\tau) = {1\over 2} (\lambda \,\partial_\tau x^\mu + \partial_\tau
x^\mu \, \lambda ) $.]

Chapter 12

- p.208, in the two lines above (12.17), for clarity,
replace \xi(\tau) by \xi(\tau, \sigma), and replace \xi by \xi (\sigma).
[Thanks to Pierre Jouvelot, 9/05/2004].

- p.211, immediately above the unnumbered equation (which uses two lines), replace (12.25) by (12.33).[Thanks to Mikael Djurfeldt, 6/14/04].
- p.211, second line below (12.36), delete the words "in (12.2)" [6/14/04].
- p.211, line immediately above (12.37), replace "second line in (12.2) is replaced by" by "commutator is". [6/14/04].
- p.211, line immediately below (12.38), replace "final right-hand side in (12.2)" by "result that led to (12.36)".[6/14/04].
- p.212 Equation (12.46): add a prime to the sigma in the left-hand side [Thanks to Ira Rothstein 2/21/05].
- p. 213, Very minor: Eqn (12.54), second equation, right-hand
side, superscripts on the

symbol "a": reverse the order of the star and the \mu. - p.231, Equation (12.162), add a minus sign "-" right after the equal sign (top line) [Thanks to J. Fernandez-Alcazar, who re-checked the computation, 4/05]
- p.243, two lines before Equation (2), "we" is duplicated. [Thanks to Pierre Jouvelot, 9/05/2004].

- p.248, for clarity, replace "world-sheet" by "parameter space" right before equation (13.8) and right before equation (13.9). [6/14/04].
- p.260, section 13.4, second paragraph, replace "m_e" by "m" (for consistency with equations below) [6/14/04].
- p.264, in the paragraph following Equation (13.95), b^I_{-r} should be b^J_{-r}. Also, three lines before Quick Calculation 13.3, the word "states" appears duplicated. [Thanks to Pierre Jouvelot, 9/05/2004].
- p.267, 4 lines before the end of the page, the word "the"
appears duplicated. [Thanks to Pierre Jouvelot, 9/05/2004].

- p.285, first term on the right-hand side of the equation in (14.54): the \alpha should be primed. [Thanks to Pierre Jouvelot, 9/05/2004].
- p.291, line right after eqn. (14.65), the word "to" appears duplicated. [Thanks to Pierre Jouvelot, 9/05/2004].
- p.299, one line before the end of the page, the word "the"
appears duplicated. [Thanks to Pierre Jouvelot, 9/05/2004].

- p.305, problem 14.10, part (a), corrections on the sentence
beginning as: "Show that the orientifold action...".
Delete the word "simultaneous" and
add the words "either one of" between the words "about" and "the". [6/14/04].

- p. 308, End of the second line, replace the exponent "D-4" by "4-D".
- p. 308 Minor, second paragraph, second to last line, replace "This a" by "This is a" [Thanks to Karsten Koeneke, 5/6/05]
- p. 309 Second line, equation in parenthesis, right-hand side: replace the exponent "D-6" by "6-D" [Thanks to Karsten Koeneke, 5/6/05]
- p.335. Four lines after (15.108), an "on" is missing in the sentence
"the strings end (on) a D-brane". [Thanks to Pierre Jouvelot, 9/10/04].

- p.347, last line, replace the third word, "the", by "a closed". [see motivation below, 9/05/04].

Chapter 16

- p.364. equation (16.81), an arrow is missing on top of p in d^d p. [Thanks to Pierre Jouvelot, 9/10/04].
- p.375. In (16.140), the Q_2 in the intermediate step should be Q_5. At the end of the page, last character, the "d" should be a "b", and immediately in the next page "d_f" should be "f". [Thanks to Pierre Jouvelot, 9/10/04].
- p.379, end of second line, replace 15.6 by 15.5. [Thanks to Pierre Jouvelot, 9/10/04].
- p.383, Problem 16.6. Second line: delete "(derivatives of )"
and add "(and derivatives thereof)" right after the word "functions". Moreover,
the quoted asymptotic expansion has a missing factor and is not the most
useful. It is better to give the expansion for K_\nu without derivatives.
For this: 1) delete the prime to the left of the ~, 2) delete the
minus sign immediately to the right of the ~, 3) add a multiplicative factor
of \exp(-z), and 4) change the +3 for a -1. [9/14/04].

Chapter 17

- p. 387 More than a correction, this is a clarification
suggested by Stanley Deser, who found some comments on electromagnetic duality
in p.387 somewhat misleading. If you want an improved version, do the
following changes: Second line below (17.2), add the word "familiar"
before "Lagrangian" (this is because there is a less familiar first order
Lagrangian for transverse fields that is invariant). Third line below
(17.2) up to the seventh line, delete all text from "First of all...." up
to ".... would not be invariant."(the comments on nonlocal changes can
be stated more accurately, see below). Third line below (17.3) delete
the sentence "The significant ... Lagrangian." Now, to add some
substance insert the following paragraph in between the paragraph ending
with "unchanged" and that beginning with "Consider now": ``Since the
dynamical variables in electromagnetism are the gauge potentials,
one may ask: Are there transformations of the potentials that
induce (17.2)? Not exactly, but close. One can formulate electromagnetism
(without sources) using dynamical variables $\vec{E}$ and $\vec{A}$, both
divergenceless. Duality transformations are then written as spatially
nonlocal transformations of $\vec{E}$ and $\vec{A}$. (A typical spatially
nonlocal transformation expresses the output at a point $\vec{x}$
in terms of the values of the input at all points $\vec{x}'$.)" [Thanks
to Stanley Deser, 03/06/05]

- p.389 (very minor correction), Figure 17.1, the letters P and P' should be lowercase (p and p') for consistency with the explanation beginning as "A small confusion ...", in p. 390. [Thanks to Pierre Jouvelot, 9/14/04].
- p.403, line 6 of the second paragraph of Section 17.8, "radius us" should be "radius of". [Thanks to Pierre Jouvelot, 9/14/04].
- p. 406 Table 17.2, first line (line that begins with $\tilde
H$), first term on the right-hand side: the (two) p^i's should have
no tildes. [Ira Rothstein, 04/05]

- p.407, Problem 17.3, the first displayed equation should
be labelled (1). [Thanks to Pierre Jouvelot, 9/14/04].

Chapter 18

Chapter 19

- p.434, equation (19.40), the "Y" should be replaced
by an "X". [6/14/04].

Chapter 21

Chapter 22

- p.483, figure 22.1, very minor correction: the dashing and the size of arrows should match those of figure 22.2. [6/14/04].
- p.484, two lines before equation (22.4), missing period (.) before "In". [Thanks to Pierre Jouvelot, 10/24/04].
- p.511, 4th line of the 3rd paragraph, replace "x_2 and x_3" by
"x_3 and x_4". [Thanks to Pierre Jouvelot, 10/24/04].

- p.524, 4 lines after equation (23.6), replace "clockwise" by "counterclockwise". [Thanks to Pierre Jouvelot, 10/24/04].
- p.530, Fig.23.9, top: the arrow to the left of "\delta T" extends a bit too much, it should stop right above the beginning of the lower slit.[Thanks to Pierre Jouvelot, 10/24/04].
- p.531, Figure 23.10. The sizes of the outgoing cylinders should match those of figures 23.11 and 23.12. So, in particular, P_4 in figure 23.10 should be a bit smaller (consequently, both P_1 and P_2 should be smaller)[6/14/04].
- p.533, Fig. 23.13, caption : replace "23.12 and 23.11" by "23.11 and 23.12".[Thanks to Pierre Jouvelot, 10/24/04].
- p.536. Fig 23.16, caption, second line: the w in \eta = w/T should have a tilde.[Thanks to Pierre Jouvelot, 10/24/04].
- p.538, Fig. 23.18, a series of very minor changes: (a)
cylinder shown at lower center, top face, the angle for \theta should
be denoted with a small curved line rather than a little square (that
incorrectly suggests right angle), (b) Im(\tau) and Re(\tau) should be
I(\tau) and R(\tau), with the proper font for I and Rs. [Thanks to Pierre
Jouvelot, 10/24/04].

References

- p.549, in the entry for Das and Mathur, the word "pysics"
should be replaced by "physics". [Thanks to Pierre Jouvelot, 9/05/2004].

Here is the quiz that was given to students of 8.251 in the spring term of 2004.

Chapter 1

Concerning my citation of Biot-Savart (1820) in page 3, Blake Stacey [MIT senior] wrote:

"For history's sake, it might be worth making clear that Jean Baptiste Biot (1774-1862) and Felix Savart (1791-1841) were two distinct people--say, by changing the hyphen to an "and". Biot sure lived well: the mica-based mineral biotite and a lunar crater are both named after him. It doesn't look like Savart picked up as many honors."[8/18/04]

Chapter 2

After noting that the light-cone energy is (-p_+) in (2.91) it may be useful to emphasize to theChapter 3

students that in a Lorentz frame (-p_0) is the energy. [6/14/04]

Some students find the discussion of orbifolds in the final paragraph of section 2.7 a little insufficient

to deal with problems 2.4 and 2.7. A nice additional example that can be discussed in class is that of

a "cone" with angle at the apex equal to "2 pi/N". This orbifold is obtained by the identification

z \sim z \exp(i 2 pi/N) on the complex coordinate z. [6/14/04]

Question(by Andrew Mamo, MIT undergrad, on 12 Feb 2004): When we saw how the gravitational constant G requires different units in different dimensions, I wondered why no similar constant is necessary for the electric field. If the electric potential is analogous to the gravitational potential, and if the charge density has different units in different dimensions, why is there no need to have a dimension-specific constant to keep units matching, the way we had for gravity?

My reply:It is just a matter of convenience and the fact that charge is less familiar a concept than mass. The first hint for the answer is that (Quick Calculation 3.6) the units of charge *are* different in different dimensions. So, it turns out that both sides of the equation \laplacian Phi = - \rho have the same units and no additional constant is needed. While the gravitational potential is defined to have the same units in all dimensions, the electric potential, which is energy per unit charge, has different units in different dimensions. Since charges have different units in different dimensions, if you are given a 5-dimensional charged particle, you have to do a nontrivial calculation to find the 4-dimensional effective charge after compactification. In gravity, we want to keep the units of mass the same in all dimensions (because of F= ma, which exists independent of gravity). As a result, we had no choice but to have a dimensionality dependent Newton constant. [6/14/04]

A note on pedagogy. In section 3.6 there is a very brief introduction to the language of General Relativity. I made no attempt to prove that Newtonian gravity arises as a limit of General Relativity. This is because I did not want to delay the beginning of our study of strings. A point relevant to the Newtonian limit comes up in problem 5.7. where the student derives the geodesic equation. In Chapter 10 we discuss gravity fields and graviton states in some detail. An instructor that wishes to discuss the Newtonian limit to convince students that the h_{\mu\nu} field is the gravitational field, may find it useful to introduce this material as a digression, perhaps through problem sets. A recent pedagogical discussion can be found in Hartle's "Gravity" (see section 22.4).[8/16/04]

Chapter 4

Chapter 5

Chapter 6

It is stated at the bottom of page 98 that at any point on the world-sheet there are spacelike and timelike directions. There is, however, an exception. For open strings any point on the boundaries of the world-sheet represents an open string endpoint. At any (free) open string endpoint we have a spacelike direction and a null direction (the tangent to the boundary), since the endpoint moves with the speed of light (section 6.9). There is no timelike direction. This is actually the situation considered in Quick calculation 6.3. [9/05/04, Thanks to James Snyder for the comment.]

Chapter 7

Problem 7.1, part (a) The hyperplane is not supposed to be a D-brane. As stated in the problem, the endpoints are completely free, as if we had a spacefilling D-brane. It is not true that an open string must lie at all times fully on a D-brane that does not fill space.[8/18/04]

An application of problem 7.3.[03/07/05]

Consider a closed string that at time t=0 has zero velocity and is stretched along the

perimeter of a square of side L/4. Choose a convenient set of coordinate axes and describe

$\vec{F}(u)$ and $\vec{F}'(u)$. Explain why, at any time, the string is composed by a set of

piecewise linear parts. Draw a few sketches showing the shape of the string as it contracts

down to zero size.

Chapter 8

Chapter 9

Chapter 10

Chapter 11

In the above list of "corrections" an extra hint has been added for part (b) of Problem 11.7 .

Chapter 12

Chapter 13

Chapter 14

Chapter 15

Problem 15.5.The conservation of the string current in (15.11) holds only for closed strings. For open strings there are additional contributions to the string current that are needed to obtain conservation. The statement of the problems is better modified from "..to the case of the string" to read "... to the case of a closed string".

Question[P. Jouvelot, 9/05/04]: Why is there no "Kalb-Ramond coupling constant" in the term (15.3) as there is one for electromagnetism in (15.1), via "q"?

Answer:In string theory there is no coupling that is independent of the string coupling g, so introducing a coupling with some different label in (15.3) could give a wrong impression. Any coupling that you may introduce in (15.3) is compatible with (15.4), where the second term has no coupling. This is because a rescaling of the Kalb-Ramond field B can be used to move the coupling from the second to the last term in (15.4), where it would simply affect what we call \kappa. As stated below (15.6), \kappa is a calculable function of g. Note, incidentally, that in a similar spirit the charges at the end of the string were conventionally chosen to be equal to plus or minus one in equation (15.54).

Chapter 16A little abuse of notation[9/10/04, P. Jouvelot] In page 323 there are two occurances where a repeated index is not supposed to be summed over. In (15.68), there are three "m" indices -- the last one is not part of the sum. On the line before (15.69), the two "m" indices are not summed over.

Problem 16.6has an important correction (see above). The partition function can be expressed in terms of the modified Bessel functions and its derivatives. The required asymptotic expansion is that of K_\nu without derivatives [9/14/04].

Chapter 17

Some modifications of p.387 were suggested above (see corrections) to clarify the discussion of electromagnetic duality. A useful (and quite brief) reference is Deser, S. (1982). ``Off-shell electromagnetic duality invariance," J. Phys.A:Math. Gen. {\bf 15} 1053. [03/07/05]

Chapter 18

Comment[suggested by P. Jouvelot, 9/14/04]. In section 18.2, to obtain equation (18.28) we must use the fact that the group elements U_1 and U_2 (and their inverses) commute. This is the case because they are complex numbers. The group U(1) is an abelian group: the multiplication is commutative.

Chapter 19

Chapter 20

Chapter 21

Chapter 22

Chapter 23