Appendix A: Vector Analysis and Partial Differential Equations¶
Story so far: In Ch. 1 of the main text, the gradient of the gravitational potential and the Laplacian appeared, and in Ch. 2, the divergence and curl in Maxwell's equations were introduced. From Ch. 13 onward, the 2-dimensional wave equation becomes central as we deal with string vibrations.
Goals of this appendix
- Organize the tools from vector analysis and partial differential equations that appear in the main text, focusing specifically on aspects unique to string theory—the 2-dimensional wave equation, decomposition into left- and right-moving waves, and vibration modes from string boundary conditions
- General tools (partial derivatives, grad, div, curl, Laplacian, Gauss/Stokes theorems, d'Alembert solutions of the wave equation, etc.) are summarized self-consistently in General Relativity's General Relativity Appendix A, so please refer there first
🟡 Lina: If you've read General Relativity's General Relativity Appendix A, this appendix is just a quick review of the key points. The parts specific to string theory are only A.3 and A.4, so you can focus on those alone.
🔵 Kai: We already covered partial derivatives and grad/div/curl in General Relativity, right? The general solution of the wave equation is also in General Relativity's General Relativity Appendix A.
🟡 Lina: That's right. Here we avoid repetition and concentrate on what's newly needed for string theory—the structure of the 2-dimensional wave equation that describes "a 1-dimensional object called a string evolving in time."
A.1: Summary of Key Points from General Relativity Appendix A¶
🟡 Lina: Let me list the tools we'll use in this appendix. All detailed derivations, proofs, and worked examples are found in General Relativity's General Relativity Appendix A, so please refer there.
Differential Operators¶
Table A.1: Definitions and meanings of differential operators
| Operation | Definition (Cartesian coordinates) | Meaning |
|---|---|---|
| Partial derivative | \(\partial f / \partial x\): differentiate with respect to \(x\) only, holding other variables fixed | Rate of change of a multivariable function in each direction |
| Gradient | \(\nabla\varphi = (\partial_x\varphi,\, \partial_y\varphi,\, \partial_z\varphi)\) | Direction of steepest ascent and rate of change |
| Divergence | \(\nabla \cdot \boldsymbol{F} = \partial_x F_x + \partial_y F_y + \partial_z F_z\) | Strength of source/sink |
| Curl | \((\nabla \times \boldsymbol{F})_i = \epsilon_{ijk}\partial_j F_k\) | Strength and direction of vorticity |
| Laplacian | \(\nabla^2\varphi = \partial_x^2\varphi + \partial_y^2\varphi + \partial_z^2\varphi\) | Deviation from the surrounding average |
Important Identities¶
- \(\nabla \times (\nabla\varphi) = 0\) (conservative force fields are irrotational)
- \(\nabla \cdot (\nabla \times \boldsymbol{A}) = 0\) (vortices have no sources)
- \(\nabla \times (\nabla \times \boldsymbol{F}) = \nabla(\nabla \cdot \boldsymbol{F}) - \nabla^2 \boldsymbol{F}\) (used in deriving Maxwell's wave equation)
Integral Theorems¶
- Gauss's theorem: \(\displaystyle\oint_S \boldsymbol{F} \cdot d\boldsymbol{S} = \int_V \nabla \cdot \boldsymbol{F}\, dV\)
- Stokes' theorem: \(\displaystyle\oint_C \boldsymbol{F} \cdot d\boldsymbol{r} = \int_S (\nabla \times \boldsymbol{F}) \cdot d\boldsymbol{S}\)
Classification of Second-Order Partial Differential Equations¶
Table A.2: Classification of second-order partial differential equations
| Type | Standard form | Properties of solutions | Representative examples |
|---|---|---|---|
| Wave type (hyperbolic) | \(\partial_t^2 f = v^2 \nabla^2 f\) | Propagates at speed \(v\) | Electromagnetic waves, gravitational waves, string vibrations |
| Diffusion type (parabolic) | \(\partial_t f = D \nabla^2 f\) | Decays | Heat conduction, Schrödinger (imaginary coefficient) |
| Elliptic | \(\nabla^2 f = \rho\) | Static distribution | Newtonian gravity, electrostatics |
d'Alembert's General Solution for the 1-Dimensional Wave Equation¶
\(f\) is a right-moving wave, \(g\) is a left-moving wave. \(f, g\) are arbitrary twice-differentiable functions.
For the detailed derivation (using the variable substitution \(\xi = x - ct\), \(\eta = x + ct\)), see General Relativity's General Relativity Appendix A, A.7.
🔵 Kai: Everything up to here is the prerequisite, right? What's new for string theory?
🟡 Lina: Essentially, there are two key new points: bringing this 1-dimensional wave equation onto the 2-dimensional worldsheet, and the boundary conditions of the string discretizing the vibration modes. Let's look at these in A.3 and A.4.
A.2: Map of Practice Problems¶
🟡 Lina: The practice problems in this appendix are for reviewing and confirming the content of General Relativity General Relativity Appendix A in the context of string theory. If you're comfortable with the material, you can skip them; if you're unsure, solving the problems will help you review each topic.
Table A.3: Correspondence between practice problems and topics
| Practice Problem | Topic | Reference |
|---|---|---|
| A.1–A.4 | Partial derivatives (basic calculations, diffusion equation) | General Relativity General Relativity Appendix A.0 |
| A.5–A.7 | Gradient (gravitational potential, contour lines) | General Relativity General Relativity Appendix A.4.2 |
| A.8–A.11 | Divergence (concrete calculations, Coulomb electric field) | General Relativity General Relativity Appendix A.4.3 |
| A.12–A.15 | Curl (presence/absence of vortices, vector potential) | General Relativity General Relativity Appendix A.4.4 |
| A.16–A.19 | Laplacian (Laplace's equation) | General Relativity General Relativity Appendix A.4.5 |
| A.20–A.22 | Vector identities (\(\nabla\cdot\nabla\times\), \(\nabla\times\nabla\)) | General Relativity General Relativity Appendix A.5 |
| A.23–A.28 | Wave equation, standing waves, string vibration modes | This appendix A.3, A.4 |
📝 All practice problems → Appendix A Practice Problems
A.3: The 2-Dimensional Wave Equation for String Theory¶
🟡 Lina: From here on is the part specific to string theory. A string is a 1-dimensional object (a line), so we need a parameter \(\sigma\) to specify position along the string. The range of \(\sigma\) depends on the type of string (for a closed string it goes from \(0\) to \(2\pi\) completing one loop, for an open string it goes from \(0\) to \(\pi\) from one end to the other). The reason we choose these ranges is that they make the calculations clean when we expand vibration modes in trigonometric functions (which we'll cover in detail in A.4). For a closed string, choosing \(2\pi\) matches the fundamental period of \(e^{in\sigma}\) (since \(e^{in(\sigma + 2\pi)} = e^{in\sigma}\) holds automatically). For an open string, choosing \(\pi\) works well with the boundary conditions—we'll confirm the specific reasons in A.4.
🔵 Kai: So \(\sigma\) is a parameter representing which position along the string we're at, and its range is determined by the shape of the string (loop or line segment).
🟡 Lina: Exactly. Furthermore, when we include time \(\tau\) as well, the motion of the string is described by two parameters \(\tau\) and \(\sigma\)—this 2-dimensional extent is called the worldsheet. For example, if you draw the motion of a point particle (a point with zero size) on a spacetime diagram, its position at each moment is a single point, so as time passes those points connect to form a single line—this is called a worldline (a concept introduced in General Relativity's General Relativity Ch. 2). Since a string is not a point but has 1-dimensional extent, its "shape" at each moment is a line segment (or loop), and as time passes these connect to trace out a 2-dimensional surface—that's the worldsheet.
🔵 Kai: A point particle traces a "line" through time, so it's a worldline; a string traces a "surface" through time, so it's a worldsheet... \(\sigma\) represents which position along the string, and \(\tau\) corresponds to time.
🟡 Lina: Exactly. The function that tells us where each point on the string is located in \(D\)-dimensional spacetime is \(X^\mu(\tau, \sigma)\) (where \(\mu = 0, 1, \ldots, D-1\) is an index running over spacetime directions). \(D\) is the number of spacetime dimensions—our everyday experience has \(D = 4\) (3 space + 1 time), but string theory requires \(D = 26\) or \(D = 10\) for consistency—we'll derive this in later chapters, so for now just remember that "we're working in general \(D\) dimensions." Classically, this \(X^\mu\) obeys a 2-dimensional wave equation:
🔵 Kai: The difference between the second derivatives in the time and space directions equals zero... In a normal wave equation like \(\partial_t^2 u = c^2 \partial_x^2 u\), the speed \(c\) appears. Why is there no \(c\) here?
🟡 Lina: Good question. In the table in A.1, we wrote the wave equation as \(\partial_t^2 f = v^2 \nabla^2 f\), right? That \(v\) is the propagation speed. In string theory, we use natural units where the speed of light \(c = 1\) (see Appendix B for details). This makes the \(c^2\) coefficient disappear, giving us the clean form where the d'Alembert operator \(\Box = \partial_\tau^2 - \partial_\sigma^2\) acts on \(X^\mu\) and gives zero. It's a wave equation in 1+1 dimensions (1 time + 1 space). The key point is that while it has formally the same structure as the ordinary wave equation, the variables have changed from \((t, x)\) to the worldsheet coordinates \((\tau, \sigma)\), and the solution \(X^\mu\) represents the spacetime coordinates themselves—that's what's new here.
⚪ Mei: So the string equation is a special case of the general wave equation \(\partial_t^2 f = v^2 \nabla^2 f\) from the A.1 table (with \(v = c = 1\), in 1 spatial dimension).
Decomposition into Left- and Right-Moving Waves¶
🟡 Lina: Applying d'Alembert's general solution (see General Relativity General Relativity Appendix A.7) directly gives:
\(X_R\) and \(X_L\) are waves that depend only on \(\tau - \sigma\) and \(\tau + \sigma\), respectively. The \(R/L\) naming varies by reference—for example, Polchinski's textbook calls the component depending on \(\tau - \sigma\) the right-mover, but some references use the opposite convention. In this book, we define \(X_R(\tau - \sigma)\) as the right-mover and \(X_L(\tau + \sigma)\) as the left-mover. It's more reliable to distinguish them by "whether the argument is \(\tau - \sigma\) or \(\tau + \sigma\)" rather than by name. In string theory, this decomposition becomes the starting point for quantization.
✅ Comprehension Check: What form does the general solution of the string's 2-dimensional wave equation \((\partial_\tau^2 - \partial_\sigma^2)X^\mu = 0\) decompose into? Also, why doesn't the speed \(c\) appear in the wave equation?
Answer
The general solution decomposes into the sum of a right-moving wave \(X_R^\mu(\tau - \sigma)\) and a left-moving wave \(X_L^\mu(\tau + \sigma)\). The speed \(c\) doesn't appear because string theory adopts natural units where the speed of light \(c = 1\).
🟡 Lina: Let me introduce a convenient change of variables here. We define new coordinates \(\sigma^+ = \tau + \sigma\), \(\sigma^- = \tau - \sigma\). These are called worldsheet light-cone coordinates. Later when we write the Lagrangian density (in the second half of A.3), we'll collectively write the worldsheet coordinates as \(\sigma^a\) (\(a = 0, 1\), i.e., \(\sigma^0 = \tau\), \(\sigma^1 = \sigma\)), but \(\sigma^\pm\) is a different coordinate system. In these coordinates, the wave equation takes the simple form \(\partial_{\sigma^+}\partial_{\sigma^-}X^\mu = 0\).
🔵 Kai: \(\sigma^+\) and \(\sigma^-\) are just \(\tau\) and \(\sigma\) added or subtracted, right? Does that alone make the equation simpler?
🟡 Lina: It does. Using the chain rule:
with a factor of \(4\). Let's see why the \(4\) appears.
First, from the chain rule:
(Since \(\sigma^\pm = \tau \pm \sigma\), we have \(\partial\sigma^\pm/\partial\tau = 1\)). Similarly, \(\partial_\sigma = \partial_{\sigma^+} - \partial_{\sigma^-}\) (since \(\partial\sigma^+/\partial\sigma = 1\), \(\partial\sigma^-/\partial\sigma = -1\)).
"Squaring \(\partial_\tau\)" means applying \(\partial_\tau\) twice in succession, i.e., \(\partial_\tau^2 = \partial_\tau \circ \partial_\tau\). Since \(\partial_\tau = \partial_{\sigma^+} + \partial_{\sigma^-}\), we expand \(\partial_\tau^2 = (\partial_{\sigma^+} + \partial_{\sigma^-})(\partial_{\sigma^+} + \partial_{\sigma^-})\). The order of partial derivatives can be exchanged (\(\partial_{\sigma^+}\partial_{\sigma^-} = \partial_{\sigma^-}\partial_{\sigma^+}\), a property called Schwarz's theorem that holds for sufficiently smooth functions), so we can treat \(\partial_{\sigma^+}\) and \(\partial_{\sigma^-}\) like variables \(A\) and \(B\) when expanding. Since commuting operators multiply by the same rules as ordinary variables, using \((A+B)^2 = A^2 + 2AB + B^2\), we get \(\partial_\tau^2 = (\partial_{\sigma^+} + \partial_{\sigma^-})^2 = \partial_{\sigma^+}^2 + 2\partial_{\sigma^+}\partial_{\sigma^-} + \partial_{\sigma^-}^2\). Similarly, squaring \(\partial_\sigma = \partial_{\sigma^+} - \partial_{\sigma^-}\) gives \((A-B)^2 = A^2 - 2AB + B^2\), yielding \(\partial_{\sigma^+}^2 - 2\partial_{\sigma^+}\partial_{\sigma^-} + \partial_{\sigma^-}^2\).
Computing \(\partial_\tau^2 - \partial_\sigma^2\), the \(\partial_{\sigma^+}^2\) and \(\partial_{\sigma^-}^2\) terms cancel, and only the cross terms remain with \(2 - (-2) = 4\), leaving \(4\partial_{\sigma^+}\partial_{\sigma^-}\)—verify the detailed calculation in the practice problem (Problem B-18. Plane Waves Satisfy the Wave Equation). In any case, since the right-hand side is zero, we can divide both sides of \(4\partial_{\sigma^+}\partial_{\sigma^-}X^\mu = 0\) by \(4\) (\(\neq 0\)) to obtain \(\partial_{\sigma^+}\partial_{\sigma^-}X^\mu = 0\). This is exactly the same variable substitution technique used in General Relativity's General Relativity Appendix A when deriving the d'Alembert solution with \(\xi = x - ct\), \(\eta = x + ct\) (since \(c = 1\), \(\sigma^- = \tau - \sigma\) corresponds to \(\xi\), and \(\sigma^+ = \tau + \sigma\) corresponds to \(\eta\)). This equation means "differentiating with respect to \(\sigma^+\) and then with respect to \(\sigma^-\) gives zero," and the functions satisfying this are limited to \(X^\mu = F(\sigma^+) + G(\sigma^-)\)—that is, the sum of a function depending only on \(\sigma^+\) and a function depending only on \(\sigma^-\). Since \(\sigma^+ = \tau + \sigma\), \(F(\sigma^+) = X_L^\mu(\tau + \sigma)\) (left-moving wave), and since \(\sigma^- = \tau - \sigma\), \(G(\sigma^-) = X_R^\mu(\tau - \sigma)\) (right-moving wave). The name "light-cone" comes from the fact that in 4-dimensional spacetime, the trajectory of light traces a cone. On the 2-dimensional worldsheet, light paths are two straight lines \(\tau = \pm\sigma\) (\(\sigma^+ = \text{const}\) or \(\sigma^- = \text{const}\)), and \(\sigma^\pm\) are precisely coordinates along these light paths. Note that the \(\sigma^\pm\) introduced here are worldsheet light-cone coordinates, which are different from the spacetime light-cone coordinates \(x^\pm = (x^0 \pm x^1)/\sqrt{2}\) introduced in Ch. 5—they share the same name, but the former are coordinates on the 2-dimensional worldsheet while the latter are coordinates in \(D\)-dimensional spacetime. Both appear in string theory, so distinguish them by context. The full application of light-cone coordinates in string theory is covered in Ch. 13 and Ch. 14.
⚪ Mei: So when we change variables to \(\sigma^\pm\), the condition "differentiating with respect to \(\sigma^+\) then \(\sigma^-\) gives zero" forces the right-moving and left-moving waves to each depend on only one variable, separating them cleanly.
🔵 Kai: I understand that changing to light-cone coordinates simplifies the equation. ...But where does this wave equation itself come from in the first place? It feels like we're just told "the string obeys a wave equation" without justification...
Derivation from the Polyakov Action (Preview)¶
🟡 Lina: Good question. Let me preview where the string's 2-dimensional wave equation comes from. We apply the principle of least action—learned in General Relativity's General Relativity Ch. 1 and Ch. 1, where the path that makes the action stationary gives the equation of motion—to the string (the specific application to strings is covered in detail in Ch. 13). For the string, after choosing a special coordinate system called the conformal gauge (discussed later), the Lagrangian density (the quantity defined at each point of the worldsheet, corresponding to the difference between kinetic and potential energy) is
There are many symbols here, so let me go through them one by one.
🔵 Kai: Please do. What is \(\partial_a X^\mu\)?
🟡 Lina: \(\partial_a X^\mu\) is the partial derivative of \(X^\mu\) with respect to the worldsheet coordinate \(\sigma^a\) (\(\sigma^0 = \tau\), \(\sigma^1 = \sigma\)), i.e., \(\partial_a X^\mu = \partial X^\mu / \partial \sigma^a\). It's potentially confusing, but \(\sigma^a\) is notation for collectively writing the two worldsheet coordinates, and when \(a = 1\), it coincides with the string position parameter \(\sigma\) mentioned earlier. And when the same index appears both up and down, we follow the convention of summing over all directions for that index (Einstein summation convention, the rule introduced in General Relativity's General Relativity Ch. 2). 🟡 Lina: For example, when we write \(\eta^{ab}\partial_a X^\mu \partial_b X^\nu\) in this expression, it means summing over all combinations where \(a\) and \(b\) each take values \(0, 1\) (\(4\) combinations in total). In fact, this expression has two levels of summation nested—first the worldsheet direction sum (\(a, b = 0, 1\)), then the spacetime direction sum (\(\mu, \nu = 0, 1, \ldots, D-1\)).
⚪ Mei: The worldsheet sum and the spacetime sum run independently.
🟡 Lina: Exactly. Let's look at them step by step.
🔵 Kai: Two levels... Should I look at the outer \(a, b\) sum first?
🟡 Lina: Yes. Let me first write out just the outer sum (\(a, b\) sum). We'll compute the original expression \(\eta^{ab}\partial_a X^\mu \partial_b X^\nu \eta_{\mu\nu}\) in the order "execute the \(a, b\) sum first, leave the \(\mu, \nu\) sum for later." Expanding the \(a, b\) sum gives \(\eta^{00}\partial_\tau X^\mu \partial_\tau X^\nu \eta_{\mu\nu} + \eta^{01}\partial_\tau X^\mu \partial_\sigma X^\nu \eta_{\mu\nu} + \eta^{10}\partial_\sigma X^\mu \partial_\tau X^\nu \eta_{\mu\nu} + \eta^{11}\partial_\sigma X^\mu \partial_\sigma X^\nu \eta_{\mu\nu}\), but since \(\eta^{ab} = \mathrm{diag}(-1,1)\) is a diagonal matrix, \(\eta^{01} = \eta^{10} = 0\) and the cross terms vanish, giving \(\eta^{ab}\partial_a X^\mu \partial_b X^\nu \eta_{\mu\nu} = -(\partial_\tau X^\mu)(\partial_\tau X^\nu)\eta_{\mu\nu} + (\partial_\sigma X^\mu)(\partial_\sigma X^\nu)\eta_{\mu\nu}\) (the \(\mu, \nu\) sum still remains—we'll execute it concretely in the next step).
⚪ Mei: I see, since all terms with \(a \neq b\) are zero, only the \(\tau\)-direction and \(\sigma\)-direction terms survive.
🟡 Lina: Exactly. Next we take the inner sum (\(\mu, \nu\) sum)—this is the operation of multiplying by \(\eta_{\mu\nu}\) and summing over \(\mu, \nu = 0, 1, \ldots, D-1\), which is a generalization of the dot product of vectors. For example, if \(D = 2\), then \(\eta_{\mu\nu}\partial_\tau X^\mu \partial_\tau X^\nu = -(\partial_\tau X^0)^2 + (\partial_\tau X^1)^2\). If \(D = 4\) (our spacetime), it becomes \(-(\partial_\tau X^0)^2 + (\partial_\tau X^1)^2 + (\partial_\tau X^2)^2 + (\partial_\tau X^3)^2\)—just 4 terms instead of 2, with the same structure.
🔵 Kai: Ah, the form is the same regardless of the dimension \(D\). Only the time component gets a minus sign, and all spatial components get plus signs.
🟡 Lina: Right. So the "\(\tau\)-direction inner product" term enters with coefficient \(+T/2\), and the "\(\sigma\)-direction inner product" term with coefficient \(-T/2\)—if you focus on just the spatial components \(X^i\), this has the same structure as the particle Lagrangian being "kinetic energy \(-\) potential energy" (handling the \(X^0\) component requires constraint conditions, which are discussed in Ch. 13). Let me explain the "index lowering" operation here. We define \(\partial_\tau X_\mu \equiv \eta_{\mu\nu}\partial_\tau X^\nu\)—by multiplying by \(\eta_{\mu\nu}\) and summing over \(\nu\), we convert an upper index to a lower one. For example, with \(D = 2\): \(\partial_\tau X_0 = \eta_{00}\partial_\tau X^0 + \eta_{01}\partial_\tau X^1 = (-1)\partial_\tau X^0 + 0 = -\partial_\tau X^0\), and \(\partial_\tau X_1 = \eta_{10}\partial_\tau X^0 + \eta_{11}\partial_\tau X^1 = 0 + (+1)\partial_\tau X^1 = \partial_\tau X^1\)—only the time component flips sign. Concretely, for \(D = 4\): \(\partial_\tau X^\mu \partial_\tau X_\mu = \eta_{\mu\nu}\partial_\tau X^\mu \partial_\tau X^\nu = -(\partial_\tau X^0)^2 + (\partial_\tau X^1)^2 + (\partial_\tau X^2)^2 + (\partial_\tau X^3)^2\), and you can see that the same sign structure (minus for the time component, plus for spatial components) obtained in the \(D = 2\) case from expanding the \(a, b\) sum holds for \(D = 4\) as well—so \(\partial_\tau X^\mu \partial_\tau X_\mu\) is shorthand for \(\eta_{\mu\nu}\partial_\tau X^\mu \partial_\tau X^\nu\). This is an application of the summation convention: "when the same letter appears as both an upper and lower index, sum over it."
⚪ Mei: With the index-lowering operation, whenever you see matching upper and lower indices, you automatically sum—it lets you write things compactly.
🟡 Lina: Using this notation to combine the two levels of summation: \(\mathcal{L} = -\frac{T}{2}\bigl[-(\partial_\tau X^\mu)(\partial_\tau X_\mu) + (\partial_\sigma X^\mu)(\partial_\sigma X_\mu)\bigr] = \frac{T}{2}(\partial_\tau X^\mu)(\partial_\tau X_\mu) - \frac{T}{2}(\partial_\sigma X^\mu)(\partial_\sigma X_\mu)\). This is how the two levels of summation are nested.
The meaning of each symbol is:
- \(T\): string tension
- \(\eta^{ab}\): the 2-dimensional Minkowski metric of the worldsheet, with components \(\eta^{ab} = \mathrm{diag}(-1, 1)\) (a \(2 \times 2\) matrix with diagonal entries \(-1, 1\)). The indices \(a, b\) run over worldsheet directions \((\tau, \sigma)\). \(-1\) corresponds to the time direction (\(\tau\)) and \(+1\) to the space direction (\(\sigma\))—this is the 2-dimensional version of the special relativity metric \(\mathrm{diag}(-1,1,1,1)\). The upper indices (\(\eta^{ab}\)) are notation representing components of the "inverse matrix," and for the Minkowski metric, the component values are the same whether indices are up or down (i.e., \(\eta_{ab} = \eta^{ab} = \mathrm{diag}(-1, 1)\)). In general curved spacetime, upper and lower differ, but for now it's OK to think "upper and lower have the same numerical values"
- \(\eta_{\mu\nu}\): the \(D\)-dimensional spacetime Minkowski metric, with components \(\eta_{\mu\nu} = \mathrm{diag}(-1, +1, +1, \ldots, +1)\). Only the time direction has \(-1\)—this is the convention in special relativity for "distinguishing time from space," and it serves to take the spacetime inner product of \(\partial_a X^\mu\) and \(\partial_b X^\nu\)
From this \(\mathcal{L}\), applying the Euler-Lagrange equation (the equation of motion derived from the condition of extremizing the action) gives the following result. Intuitively, writing down the condition that at each point of the string, "the acceleration in the time direction" balances "the restoring force from tension in the space direction" gives the wave equation—the same principle as when each point of a guitar string vibrates with tension and inertia in balance. The step-by-step derivation from the Lagrangian density through the Euler-Lagrange equation to the equation of motion is done carefully in Ch. 13, so here please just accept the form of the equation of motion. However, one caveat—this equation of motion only holds after fixing the worldsheet metric to the conformal gauge. "Gauge fixing" is the operation of choosing a particular computationally convenient form from among physically equivalent descriptions; here we're choosing the worldsheet metric to be flat Minkowski form (\(\eta^{ab} = \mathrm{diag}(-1,1)\)). Why this is allowed and what it physically means are covered in detail in Ch. 13. In conformal gauge, \(\eta^{ab}\) is a constant independent of position, so the equation of motion obtained from the Euler-Lagrange equation is:
Since \(\eta^{ab}\) is a constant diagonal matrix, summing over \(a, b\) using the summation convention, the \(a \neq b\) terms vanish, giving \(\eta^{00}\partial_0\partial_0 X^\mu + \eta^{11}\partial_1\partial_1 X^\mu = (-1)\partial_\tau^2 X^\mu + (+1)\partial_\sigma^2 X^\mu = 0\). Multiplying both sides by \(-1\) gives \(\partial_\tau^2 X^\mu - \partial_\sigma^2 X^\mu = 0\), so:
🔵 Kai: Ah, so that's why "string vibration = wave equation on the worldsheet" is connected... But the \(\eta^{ab}\) and \(\eta_{\mu\nu}\) appearing in the Lagrangian density have the same symbol but different index positions. Are they different things?
🟡 Lina: Good observation. \(\eta^{ab}\) and \(\eta_{\mu\nu}\) are different metrics. \(\eta^{ab}\) is the worldsheet metric (2-dimensional, \(\tau, \sigma\)) and is a \(2 \times 2\) matrix; \(\eta_{\mu\nu}\) is the spacetime metric (\(D\)-dimensional) and is a \(D \times D\) matrix. The symbol \(\eta\) is the same because both are "flat Minkowski metrics." Whether indices are up or down is the distinction between contravariant (up) and covariant (down), which was covered in General Relativity's General Relativity Ch. 5. Here, just remember "\(\eta^{ab}\) is for the worldsheet, \(\eta_{\mu\nu}\) is for spacetime—they're different objects with different index ranges."
🔵 Kai: The same symbol \(\eta\) but different objects with different dimensions... So if the worldsheet weren't flat but curved, the \(\eta^{ab}\) part would change to a more complicated metric?
🟡 Lina: Exactly. In general, we write the worldsheet metric as \(h_{ab}(\tau, \sigma)\) to describe a curved surface. However, in string theory, "gauge fixing" allows us to choose \(h_{ab} = \eta_{ab}\)—that's why we can get by with just the flat \(\eta^{ab}\) for now. This mechanism is covered in detail in Ch. 13.
A.4: Boundary Conditions and Vibration Modes of Strings¶
🟡 Lina: The general solution of the 1-dimensional wave equation involved arbitrary functions \(f, g\), but real strings have boundary conditions. Whether it's a closed string (a loop with ends connected) or an open string (with free ends) changes the conditions, and that determines the discretization of vibration modes.
Periodic Boundary Condition for Closed Strings¶
For a closed string, \(\sigma\) goes around with period \(2\pi\):
From this periodicity, \(X^\mu\) can be expanded in a Fourier series in \(\sigma\). A Fourier series expansion represents a periodic function as a sum of trigonometric functions (\(\sin\) and \(\cos\)). Just as the sound of a musical instrument can be expressed as a superposition of the fundamental and overtones, any periodic function can be expressed as an infinite sum of \(\sin(n\sigma)\) and \(\cos(n\sigma)\) (\(n = 1, 2, 3, \ldots\)) with appropriately chosen amplitudes—this is Fourier's theorem. The proof is covered in university mathematics, so here we'll use the fact that "a function with period \(2\pi\) can be decomposed into oscillation components with integer \(n\)" as a tool.
🔵 Kai: It's a bit surprising that any wave shape can be represented as a sum of \(\sin\) and \(\cos\). But if I think of musical sounds as superpositions of overtones, it makes sense.
🟡 Lina: Good image. Let me introduce one more tool here. Using Euler's formula \(e^{i\theta} = \cos\theta + i\sin\theta\), we can combine \(\cos\) and \(\sin\) into a single complex exponential \(e^{i\theta}\). The proof is covered in university mathematics (you can derive it by substituting \(x = i\theta\) into the infinite series expansion of \(e^x\)), but here think of it as "a formula that packages \(\cos\) and \(\sin\) into one exponential function." The reason for combining them is that exponential functions don't change form when differentiated (\(d(e^{i n\sigma})/d\sigma = in\, e^{in\sigma}\)), making differentiation and summation calculations much easier. Using this complex number notation, let's combine the Fourier series expansion with the wave equation solution conditions step by step. First, from the periodic boundary condition, the Fourier components in the \(\sigma\) direction are limited to the form \(e^{in\sigma}\) (\(n\) is an integer). Next, for each component to satisfy the wave equation \((\partial_\tau^2 - \partial_\sigma^2)X = 0\), we need to multiply \(e^{in\sigma}\) by appropriate time dependence. Trying \(e^{-in\tau}e^{in\sigma} = e^{-in(\tau - \sigma)}\), \(\partial_\tau^2\) gives \((-in)^2 = -n^2\) and \(\partial_\sigma^2\) gives \((in)^2 = -n^2\), so the difference is zero—it's indeed a solution. Similarly, \(e^{-in(\tau + \sigma)}\) is also a solution—thus we obtain right-moving waves (depending on \(\tau - \sigma\)) and left-moving waves (depending on \(\tau + \sigma\)).
⚪ Mei: The periodic boundary condition restricts the \(\sigma\) direction to \(e^{in\sigma}\), and the wave equation determines the form of the time part—combining two conditions narrows down the mode shapes.
🟡 Lina: Exactly. However, this alone doesn't contain information about where the string as a whole is in space, or in which direction it's moving overall. The oscillation components only represent "changes in the string's shape," and we need to separately specify where the center of mass is (\(x^\mu\)) and in which direction and how fast it's moving (uniform motion proportional to momentum \(p^\mu\)). Adding these:
Here the sum \(\sum_{n \neq 0}\) means summing over all nonzero integers \(n = \pm 1, \pm 2, \pm 3, \ldots\). From Euler's formula, \(e^{-in(\tau-\sigma)} = \cos[n(\tau-\sigma)] - i\sin[n(\tau-\sigma)]\), so \(e^{-in(\tau-\sigma)}\) represents a wave oscillating at frequency \(|n|\), expressed in complex number form.
🔵 Kai: Wait, there's an imaginary unit \(i\) in the expression. Won't the position \(X^\mu\) become imaginary?
🟡 Lina: Good question. Since \(X^\mu\) represents position in spacetime, it must be real. The \(\alpha_n^\mu\) themselves are generally complex numbers whose values are determined by the string's initial conditions (what shape it had and how it was moving initially). The condition guaranteeing that "\(X^\mu\) as a whole is real" is \(\alpha_{-n}^\mu = (\alpha_n^\mu)^*\) and \(\tilde{\alpha}_{-n}^\mu = (\tilde{\alpha}_n^\mu)^*\) (complex conjugate—for a complex number \(z = a + bi\), flipping the sign of the imaginary part gives \(z^* = a - bi\)). The same condition is needed for both right-moving and left-moving waves. Let's see this concretely—for example, adding the \(n = 1\) and \(n = -1\) terms of the right-moving wave: \(\frac{i}{1}\alpha_1^\mu e^{-i(\tau-\sigma)} + \frac{i}{-1}\alpha_{-1}^\mu e^{i(\tau-\sigma)} = i\alpha_1^\mu e^{-i(\tau-\sigma)} - i(\alpha_1^\mu)^* e^{i(\tau-\sigma)}\), which has the form \(z - z^* = 2i\,\mathrm{Im}(z)\), so the result is real. The same mechanism works for the left-moving \(\tilde{\alpha}_n\) terms. For general \(n\), the same mechanism cancels the imaginary parts. The details of this condition are confirmed in Ch. 13.
Here's a summary of each symbol: - \(x^\mu\): initial position of the string's center of mass - \(\alpha'\): Regge slope parameter (a constant related to the string tension \(T\) by \(\alpha' = 1/(2\pi T)\). It represents the "softness" of the string. The reason for using \(\alpha'\) rather than \(T\) itself is that mode expansions and mass formulas are more concise when written in terms of \(\alpha'\)—historically it was introduced as the slope of Regge trajectories) - \(p^\mu\): momentum of the string's center of mass (\(2\alpha' p^\mu \tau\) represents uniform motion of the center of mass. Why the coefficient is \(2\alpha'\) naturally emerges when defining momentum from the Polyakov action—the derivation is in Ch. 13, so for now accept it as "a conventional normalization involving \(\alpha'\)." Note that in this book we adopt the convention of writing the center-of-mass term as \(2\alpha' p^\mu \tau\) for both closed and open strings) - \(\alpha_n^\mu\): the \(n\)-th mode expansion coefficient of the right-moving wave (the part depending on \(\tau - \sigma\)) - \(\tilde{\alpha}_n^\mu\): the \(n\)-th mode expansion coefficient of the left-moving wave (the part depending on \(\tau + \sigma\)) - The coefficients \(i\) (imaginary unit), \(1/n\), \(\sqrt{2\alpha'}\), and the \(2\alpha'\) in the center-of-mass term are all conventional normalizations. The \(1/n\) weights modes so that "higher frequency (larger \(n\)) modes have smaller amplitude"—this is chosen so that when we quantize in later chapters (Ch. 13), the commutation relations take the clean form \([\alpha_m, \alpha_n] \propto m\delta_{m+n}\). The \(i\), \(\sqrt{2\alpha'}\), and \(2\alpha'\) are chosen for the same reason. For now, just think "the coefficients are chosen for convenience when quantizing later"
The fact that left and right are independent (decomposable) is a characteristic feature of closed strings.
⚪ Mei: To summarize, the mode expansion of a closed string consists of four parts: "center-of-mass position \(x^\mu\) + uniform center-of-mass motion \(2\alpha' p^\mu \tau\) + right-moving oscillations (sum of \(\alpha_n\)) + left-moving oscillations (sum of \(\tilde{\alpha}_n\))." The periodic boundary condition enables Fourier series expansion, and oscillation frequencies are discretized to integers \(n\).
🔵 Kai: Just to confirm—\(\alpha_n^\mu\) and \(\tilde{\alpha}_n^\mu\) can take completely different values even for the same \(n\), right? Since you said left and right are independent for closed strings... But why can left and right remain independent for closed strings?
🟡 Lina: A closed string is a loop, so it has no "ends." Without ends, there's nowhere for waves to reflect—so right-moving and left-moving waves don't interfere with each other, and \(\alpha_n^\mu\) and \(\tilde{\alpha}_n^\mu\) are completely independent coefficients. In contrast, open strings have ends where reflection occurs, coupling left and right—that difference becomes important in the second half of A.4.
🔵 Kai: No ends means no reflection—so left and right don't mix and remain independent.
✅ Comprehension Check: What is the periodic boundary condition for closed strings? And what structure does it create in the mode expansion?
Answer
For closed strings, \(\sigma\) goes around with period \(2\pi\), so the periodic boundary condition \(X^\mu(\tau, \sigma + 2\pi) = X^\mu(\tau, \sigma)\) is imposed. This condition allows \(X^\mu\) to be Fourier-expanded, resulting in a structure where right-moving mode coefficients \(\alpha_n^\mu\) and left-moving mode coefficients \(\tilde{\alpha}_n^\mu\) exist independently.
Boundary Conditions for Open Strings¶
For open strings, conditions are imposed at both ends \(\sigma = 0, \pi\). There are two representative types:
- Neumann boundary condition: \(\partial_\sigma X^\mu|_{\sigma=0, \pi} = 0\) (the ends can move freely)
- Dirichlet boundary condition: $X^\mu|_{\sigma=0, \pi} = $ constant (the ends are pinned to specific positions in spacetime and cannot move as time passes. The object to which they are pinned is called a D-brane—details in Ch. 14)
Under Neumann boundary conditions, the left- and right-moving waves "reflect" at the ends and couple, giving the mode expansion:
As with closed strings, \(\sum_{n \neq 0}\) sums over all \(n = \pm 1, \pm 2, \ldots\). You might think "since \(\cos(-n\sigma) = \cos(n\sigma)\), can't we write it with only positive \(n\)?" but the time part \(e^{-in\tau}\) changes with the sign of \(n\) (\(e^{-i\tau}\) and \(e^{+i\tau}\) are different), so positive and negative \(n\) are independent terms. In fact, you could rewrite using only positive \(n\) with \(\cos(n\tau)\) and \(\sin(n\tau)\), but keeping the complex exponential form \(e^{-in\tau}\) makes the commutation relations cleaner when quantizing in later chapters (Ch. 13), so we deliberately use both positive and negative \(n\).
🔵 Kai: I see—in the \(\sigma\) direction it's \(\cos\), so positive and negative give the same thing, but the \(\tau\)-direction \(e^{-in\tau}\) gives different waves depending on the sign of \(n\), so both are needed.
🟡 Lina: Right. However, for \(X^\mu\) to be real, the positive \(n\) and negative \(n\) terms must pair up to cancel the imaginary parts—that condition is \(\alpha_{-n}^\mu = (\alpha_n^\mu)^*\). Let's look at this concretely for \(n = 1\) and \(n = -1\). The \(n = 1\) term is \(\frac{i}{1}\alpha_1^\mu e^{-i\tau}\cos\sigma\), and the \(n = -1\) term is \(\frac{i}{-1}\alpha_{-1}^\mu e^{i\tau}\cos\sigma = -i(\alpha_1^\mu)^* e^{i\tau}\cos\sigma\) (using \(\alpha_{-1} = \alpha_1^*\)). Adding these gives \(i\alpha_1^\mu e^{-i\tau}\cos\sigma - i(\alpha_1^\mu)^* e^{i\tau}\cos\sigma\), which has the form \(z - z^* = 2i\,\mathrm{Im}(z)\), so the result is real. The same mechanism cancels imaginary parts for general \(n\) (exactly the same argument as for the closed string mode expansion).
⚪ Mei: Positive and negative \(n\) pair up to cancel the imaginary parts, keeping \(X^\mu\) as a whole real. The same mechanism as for closed strings.
🟡 Lina: The reason \(\cos(n\sigma)\) appears is that differentiating \(\cos(n\sigma)\) with respect to \(\sigma\) gives \(-n\sin(n\sigma)\), and at \(\sigma = 0\) we have \(\sin(0) = 0\), while at \(\sigma = \pi\) we have \(\sin(n\pi) = 0\) (since \(n\) is an integer), so the Neumann condition \(\partial_\sigma X^\mu = 0\) is automatically satisfied. Conversely, satisfying \(\sin(n\pi) = 0\) requires \(n\) to be an integer—this is one reason for the discretization of frequencies. Note that the \(n = 0\) component corresponds not to oscillation but to center-of-mass motion (the \(x^\mu + 2\alpha' p^\mu \tau\) part), so only \(n \neq 0\) terms appear in the oscillation mode sum. \(\alpha'\) is the Regge slope parameter explained in the closed string section (\(\alpha' = 1/(2\pi T)\)). It's the same constant for both closed and open strings, appearing in the same way.
🔵 Kai: In the closed string case there were separate \(\alpha_n\) and \(\tilde{\alpha}_n\), but for open strings there's only \(\alpha_n\). Why does it reduce to one set? Also, I notice the time part is \(e^{-in\tau}\) instead of \(e^{-in(\tau - \sigma)}\).
🟡 Lina: For open strings, waves reflect at the ends, so right-moving and left-moving waves can't remain independent. The reflection couples them together, so the independent mode coefficients are consolidated into one set. Regarding the time part, in fact \(e^{-in(\tau-\sigma)} + e^{-in(\tau+\sigma)} = e^{-in\tau}(e^{in\sigma} + e^{-in\sigma}) = 2e^{-in\tau}\cos(n\sigma)\), so the result of adding right-moving and left-moving waves takes the form \(e^{-in\tau}\cos(n\sigma)\). The factor of \(2\) is absorbed into the normalization.
⚪ Mei: So for closed strings, left and right can oscillate independently, but for open strings, reflection at the ends couples left and right, resulting in mode coefficients consolidating into just one set \(\alpha_n\).
✅ Comprehension Check: What physical constraints do Neumann and Dirichlet boundary conditions respectively impose on the ends of open strings?
Answer
The Neumann boundary condition \(\partial_\sigma X^\mu|_{\sigma=0,\pi} = 0\) means the string ends can move freely. The Dirichlet boundary condition $X^\mu|_{\sigma=0,\pi} = $ constant means the string ends are fixed at specific positions (on a D-brane). Under Neumann conditions, left- and right-moving waves reflect at the ends and couple, consolidating mode coefficients into one set (\(\alpha_n\) only). Under Dirichlet conditions, reflection at the ends also occurs, similarly consolidating mode coefficients into one set (the specific form of the mode expansion is covered in Ch. 14).
Discretization of String Vibration Modes¶
🟡 Lina: For both closed and open strings, imposing boundary conditions (periodicity or end conditions) discretizes the frequencies \(\omega_n = n\) (in natural units) to integer values \(n = 1, 2, 3, \ldots\). This is what creates the "quantum states" of the string.
Incidentally, for an open string with both ends fixed (Dirichlet-Dirichlet boundary condition), the situation is the same as a musical instrument's string. To make the discussion clearer, let's take a specific direction out of the \(D\)-dimensional \(X^\mu\) (say the \(\mu = 1\) component) and write the \(n\)-th vibration mode as \(f_n(\sigma, \tau)\). Since the ends can't move, we need \(f_n = 0\) at \(\sigma = 0\) and \(\sigma = \pi\). Since \(\sin(n\sigma)\) is zero at \(\sigma = 0\) and also at \(\sigma = \pi\) (when \(n\) is an integer), each vibration mode takes the form \(f_n(\sigma, \tau) = A_n\sin(n\sigma)\cos(n\tau) + B_n\sin(n\sigma)\sin(n\tau)\) (\(A_n\), \(B_n\) are constants determined by initial conditions). For example, if the string is plucked with zero initial velocity, \(B_n = 0\) and only \(f_n = A_n\sin(n\sigma)\cos(n\tau)\) remains. You'll verify that this actually satisfies the wave equation in practice problem A.28. Under Neumann conditions it's \(\cos(n\sigma)\), under Dirichlet conditions it's \(\sin(n\sigma)\)—the spatial function changes with boundary conditions, but in both cases \(n\) is discretized to integers.
✅ Comprehension Check: Why are string vibration modes discretized (frequencies limited to integer values \(n = 1, 2, 3, \ldots\))?
Answer
Only solutions satisfying the boundary conditions (periodicity for closed strings, or end conditions for open strings) are physically allowed, so out of continuous frequencies, only integer values survive. This is the same principle as why only the fundamental and overtones are allowed for a musical instrument's string.
🔵 Kai: I understand that vibration modes are discretized. But why does "different vibration pattern = different particle"? What's fundamentally different from musical string overtones sounding like different notes?
🟡 Lina: Good question. When you quantize the classical string vibration modes, the excitation number of each mode determines the particle's mass and spin—particles like gravitons and gauge bosons emerge as different vibrational states of the string. The difference from musical overtones is that in string theory, vibrational energy is directly connected to mass through \(E = mc^2\). The detailed quantization is developed in Ch. 13 and Ch. 14.
🔵 Kai: Since vibrational energy becomes mass, different vibration patterns appear as particles with different masses... But if the fundamental vibration \(n = 1\) and the overtone \(n = 2\) determine mass, then since there are infinitely many string vibration modes, wouldn't there be infinitely many particle types? But we only observe finitely many particles.
🟡 Lina: Sharp observation. Actually, larger \(n\) modes have larger masses, becoming super-heavy particles around the Planck mass. Only the lightest few modes are observable with current accelerators—so it appears finite. The details will be confirmed in Ch. 14 when we calculate the mass spectrum.
🔵 Kai: So in theory there are infinitely many, but they're too heavy to be detected, so they're just "invisible"... But conversely, if a higher-energy accelerator is built in the future, could heavy mode particles be discovered? Could that be experimental verification of string theory?
⚪ Mei: To summarize, larger \(n\) modes are heavier and enter a regime unobservable with current technology—so "there are infinitely many particle types, but only the light few are visible."
🟡 Lina: Exactly. And the Planck energy is more than \(10^{15}\) times greater than current accelerators, so direct verification is currently very difficult. That's precisely why indirect evidence—such as consistency with predictions at low energies—becomes important. We'll discuss this in later chapters too.
🔵 Kai: \(10^{15}\) times... So even if string theory is correct, the straightforward verification of "directly finding heavy string modes" isn't feasible for a long time. Is that OK as science? I've heard that theories that can't be falsified aren't science.
🟡 Lina: That's a very essential question. The problem of experimental verification of string theory—including the relationship with falsifiability—is discussed head-on in Ch. 22, so look forward to it.
🔵 Kai: ...From the discussion so far, the mode structure differs between closed and open strings (left-right independent vs. coupled into one set)—does that mean the types of particles produced are different too?
🟡 Lina: Yes, exactly. From closed strings, gravitons (spin 2) emerge; from open strings, gauge bosons (spin 1) emerge—the difference in mode structure directly corresponds to the difference in particle spin. We'll confirm this in Ch. 14 after quantization.
🔵 Kai: The difference in mode structure directly corresponds to spin differences... Is it something like: closed strings have independent left and right, so there are more mode combinations allowing spin 2, but open strings have only one set so they're limited to spin 1?
🟡 Lina: Intuitively, that image is correct. For closed strings, you can "multiply" left and right modes to construct spin 2 states—the precise discussion is deferred to Ch. 14, but the direction is right.
🔵 Kai: I'll look forward to seeing the mechanism of combining left and right to make spin 2 in Ch. 14. ...One more confirmation—for open strings with Dirichlet conditions, how does the mode expansion change? I understand that \(\cos\) becomes \(\sin\), but what happens to the center-of-mass motion part \(2\alpha' p^\mu \tau\)?
🟡 Lina: Good question. In directions where Dirichlet conditions are imposed, the end positions are fixed, so the string as a whole can't move in that direction—meaning momentum in that direction is zero. The center of mass doesn't move in that direction, and only the oscillation part is expanded in \(\sin(n\sigma)\). On the other hand, in directions where Neumann conditions are imposed, the center of mass can move normally. In actual D-branes, conditions differ by direction, so the detailed form will be written out in Ch. 14 when we discuss D-branes.
🔵 Kai: I see—since some directions have Neumann and others have Dirichlet conditions, the string can move in some directions but not others—that's what determines the "dimension" of the D-brane.
⚪ Mei: Summarizing the key points of this appendix—the string's 2-dimensional wave equation can be decomposed into left- and right-moving waves via the d'Alembert solution; for closed strings, the periodic boundary condition gives independently existing mode coefficients \(\alpha_n\) and \(\tilde{\alpha}_n\). For open strings, reflection at the ends couples left and right, consolidating mode coefficients into one set, with expansion in \(\cos(n\sigma)\) for Neumann conditions and \(\sin(n\sigma)\) for Dirichlet conditions. In both cases, boundary conditions discretize frequencies to integers, which becomes the origin of particle masses and spins.
📝 Exercises:
- Verification of the wave equation, decomposition into standing waves, boundary conditions for string vibration modes → Problem B-18. Plane Waves Satisfy the Wave Equation, Problem B-19. Complex Exponential Wave Satisfies the Wave Equation, Problem M-5. d'Alembert Solution \(g(x - vt)\), Problem B-20. Classification of Partial Differential Equations, Problem M-6. Decomposition of Standing Waves, Problem M-7. Boundary Conditions for String Vibration Modes
Preview of Next Chapter¶
In Appendix B, we list the values of physical constants appearing in the main text—the speed of light \(c\), Planck's constant \(\hbar\), Newton's gravitational constant \(G\), etc.—and organize the correspondence between SI units and natural units. What does it mean to "set \(c = 1\)"? Let's see how the choice of unit system changes the way we see physics.
References¶
- General Relativity's Appendix A "Vector Analysis" — Self-contained exposition of partial derivatives, grad, div, curl, Laplacian, vector identities, integral theorems, and d'Alembert solutions of the wave equation (common hub for all 4 volumes)
- David Tong, Lectures on String Theory, Ch.2–3 — String wave equation, mode expansion, light-cone quantization
- Barton Zwiebach, A First Course in String Theory, Ch.6–8 — Relativistic strings, Neumann/Dirichlet boundary conditions, quantization
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