2. The Bosonic String
The bosonic string is defined by the area of its worldsheet swept out in spacetime. We develop the Nambu-Goto and Polyakov actions, derive the equations of motion, and quantize the theory to discover the critical dimension $D = 26$.
The Nambu-Goto Action
A relativistic point particle minimizes its worldline proper length. By analogy, a string minimizes the area of its worldsheet. Parametrize the worldsheet by coordinates$(\tau, \sigma) = (\sigma^0, \sigma^1)$. The induced metric on the worldsheet is:
$$h_{\alpha\beta} = \partial_\alpha X^\mu \partial_\beta X_\mu = \eta_{\mu\nu}\frac{\partial X^\mu}{\partial\sigma^\alpha}\frac{\partial X^\nu}{\partial\sigma^\beta}$$
The Nambu-Goto action is the worldsheet area weighted by the string tension:
$$S_{\text{NG}} = -T\int d^2\sigma\,\sqrt{-\det(h_{\alpha\beta})} = -\frac{1}{2\pi\alpha'}\int d^2\sigma\,\sqrt{(\dot{X}\cdot X')^2 - \dot{X}^2 X'^2}$$
where $\dot{X}^\mu = \partial_\tau X^\mu$ and$X'^\mu = \partial_\sigma X^\mu$. The square root makes this action difficult to quantize directly.
The Polyakov Action
Introduce an independent worldsheet metric $h_{\alpha\beta}$ as a dynamical field. The classically equivalent Polyakov action is:
$$S_P = -\frac{T}{2}\int d^2\sigma\,\sqrt{-h}\,h^{\alpha\beta}\partial_\alpha X^\mu\partial_\beta X_\mu$$
This action has three local symmetries: (1) worldsheet diffeomorphisms, (2) Weyl invariance$h_{\alpha\beta} \to e^{2\omega}h_{\alpha\beta}$, and (3) spacetime Poincare invariance. The equation of motion for $h_{\alpha\beta}$ gives the constraint:
$$T_{\alpha\beta} = \partial_\alpha X^\mu\partial_\beta X_\mu - \frac{1}{2}h_{\alpha\beta}h^{\gamma\delta}\partial_\gamma X^\mu\partial_\delta X_\mu = 0$$
The equation of motion for the embedding fields $X^\mu$ is the 2D wave equation:
$$\Box X^\mu = \frac{1}{\sqrt{-h}}\partial_\alpha\left(\sqrt{-h}\,h^{\alpha\beta}\partial_\beta X^\mu\right) = 0$$
Boundary Conditions
For open strings ($0 \le \sigma \le \pi$), the boundary term in the variation vanishes if we impose either:
$$\text{Neumann:}\quad \partial_\sigma X^\mu\big|_{\sigma=0,\pi} = 0 \qquad\text{(free endpoints)}$$
$$\text{Dirichlet:}\quad \delta X^\mu\big|_{\sigma=0,\pi} = 0 \qquad\text{(fixed endpoints — D-branes!)}$$
For closed strings ($0 \le \sigma \le 2\pi$), the fields are periodic:$X^\mu(\sigma + 2\pi, \tau) = X^\mu(\sigma, \tau)$.
Mode Expansion and Quantization
In conformal gauge ($h_{\alpha\beta} = \eta_{\alpha\beta}$), the solution for an open string with Neumann boundary conditions is:
$$X^\mu(\sigma,\tau) = x^\mu + 2\alpha' p^\mu\tau + i\sqrt{2\alpha'}\sum_{n\neq 0}\frac{\alpha^\mu_n}{n}\,e^{-in\tau}\cos(n\sigma)$$
Canonical quantization promotes the modes to operators with:
$$[\alpha^\mu_m, \alpha^\nu_n] = m\,\delta_{m+n,0}\,\eta^{\mu\nu} \qquad [x^\mu, p^\nu] = i\eta^{\mu\nu}$$
The Virasoro generators enforce the residual gauge constraints:
$$L_n = \frac{1}{2}\sum_{m=-\infty}^{\infty}\alpha_{n-m}\cdot\alpha_m \qquad [L_m, L_n] = (m-n)L_{m+n} + \frac{c}{12}m(m^2-1)\delta_{m+n,0}$$
Critical Dimension: D = 26
The physical state conditions require $L_n|\text{phys}\rangle = 0$ for$n > 0$ and $(L_0 - a)|\text{phys}\rangle = 0$. The normal ordering constant $a$ is:
$$a = \frac{D-2}{2}\sum_{n=1}^{\infty}n = \frac{D-2}{2}\cdot\left(-\frac{1}{12}\right) = -\frac{D-2}{24}$$
using zeta-function regularization $\zeta(-1) = -1/12$. The no-ghost theorem (absence of negative-norm states) requires:
$$a = 1 \quad\Longrightarrow\quad \frac{D-2}{24} = 1 \quad\Longrightarrow\quad D = 26$$
In $D = 26$, the bosonic string has a tachyon at level 0 with$M^2 = -1/\alpha'$, signaling an instability of the vacuum. This is resolved by the superstring.
Closed String and Level Matching
For a closed string with periodic boundary conditions$X^\mu(\sigma + 2\pi, \tau) = X^\mu(\sigma, \tau)$, the mode expansion splits into independent left-movers and right-movers:
$$X^\mu = X_L^\mu(\tau + \sigma) + X_R^\mu(\tau - \sigma)$$
with separate oscillator algebras $\alpha^\mu_n$ (right) and$\tilde{\alpha}^\mu_n$ (left). The mass-shell condition and level-matching constraint are:
$$\alpha' M^2 = 2(N + \tilde{N} - 2) \qquad N = \tilde{N}$$
The level-matching condition $N = \tilde{N}$ is a consequence of the residual gauge invariance under rigid $\sigma$-translations. For strings on a compact dimension of radius $R$, the mass formula generalizes to include winding:
$$\alpha' M^2 = \frac{\alpha' n^2}{R^2} + \frac{w^2 R^2}{\alpha'} + 2(N + \tilde{N} - 2) \qquad N - \tilde{N} = nw$$
The winding number $w$ counts how many times the string wraps the compact dimension — this is a topological quantum number with no point-particle analogue.
Simulation: String Vibration Modes
Visualizing the first 5 harmonics for both open strings (Neumann boundary conditions:$\partial_\sigma X = 0$ at endpoints) and closed strings (periodic: left-movers + right-movers).
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