The Nambu-Goto action for an open string of tension is S = _ _1 ^ _2 d _0^ d ,L, L =…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Particle Physics & Quantum Fields · Beyond the Standard Model

Symmetry and Flavour, Ch The relativistic string: action, light-cone quantisation, spectrum and D-branes

Problem

The Nambu-Goto action for an open string of tension T0T_0 is S=∫τ1τ2dτ∫0ℓdσ L,L=−T0(X˙⋅X′)2−X˙2X′2,S = \int_{\tau_1}^{\tau_2}\mathrm{d}\tau\int_0^{\ell}\mathrm{d}\sigma\,L, \qquad L = -T_0\sqrt{(\dot X\cdot X')^2 - \dot X^2X'^2}, where X˙μ≡∂Xμ/∂τ\dot X^\mu \equiv \partial X^\mu/\partial\tau, X′μ≡∂Xμ/∂σX'^\mu \equiv \partial X^\mu/\partial\sigma, and σ\sigma runs from one end of the string to the other. (a) Show, by differentiating LL, that Pμτ≡−∂L∂X˙μ=T0 (X˙⋅X′)Xμ′−X′2X˙μ(X˙⋅X′)2−X˙2X′2,\mathcal{P}^\tau_\mu \equiv -\frac{\partial L}{\partial\dot X^\mu} = T_0\,\frac{(\dot X\cdot X')X'_\mu - X'^2\dot X_\mu}{\sqrt{(\dot X\cdot X')^2 - \dot X^2X'^2}}, Pμσ≡−∂L∂X′μ=T0 (X˙⋅X′)X˙μ−X˙2Xμ′(X˙⋅X′)2−X˙2X′2.\mathcal{P}^\sigma_\mu \equiv -\frac{\partial L}{\partial X'^\mu} = T_0\,\frac{(\dot X\cdot X')\dot X_\mu - \dot X^2 X'_\mu}{\sqrt{(\dot X\cdot X')^2 - \dot X^2X'^2}}. (b) Vary Xμ→Xμ+δXμX^\mu\to X^\mu+\delta X^\mu in SS and integrate by parts over τ\tau and σ\sigma. With δXμ=0\delta X^\mu=0 fixed at τ1,τ2\tau_1,\tau_2, show that δS\delta S reduces to a bulk term proportional to ∂τPμτ+∂σPμσ\partial_\tau\mathcal{P}^\tau_\mu + \partial_\sigma\mathcal{P}^\sigma_\mu plus a term evaluated only at σ=0,ℓ\sigma=0,\ell. State the bulk equation of motion this requires. (c) At a free end, δXμ\delta X^\mu is unconstrained there too. State the condition this forces on Pμσ\mathcal{P}^\sigma_\mu at that end. (d) Work in the static gauge X0=τX^0=\tau, and label the string so that X˙⋅X′=0\dot X\cdot X'=0, with σ\sigma chosen so that X′μX'^\mu does not vanish at the end. Write X˙μ=(1,v)\dot X^\mu = (1,\mathbf v), where v\mathbf v is the ordinary velocity of that point of the string. Using the explicit Pμσ\mathcal{P}^\sigma_\mu of part (a), show that Pμσ\mathcal{P}^\sigma_\mu points along the string with magnitude T01−v2T_0\sqrt{1-v^2}, and hence that a free end moves at the speed of light, perpendicular to the string. (e) Suppose instead a spatial direction XiX^i is held fixed at the end. Explain in one sentence why no condition on Piσ\mathcal{P}^\sigma_i then follows, and why momentum in that direction is not conserved by the string alone.

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