The action of a free relativistic point particle of mass can be written

PHYSICS · Multi-part · 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 action of a free relativistic point particle of mass mm can be written, before any gauge choice, as S=−m∫λ1λ2x˙μ(λ)x˙μ(λ) dλ,S = -m\int_{\lambda_1}^{\lambda_2} \sqrt{\dot x^\mu(\lambda)\dot x_\mu(\lambda)}\,\mathrm{d}\lambda, where λ\lambda is an arbitrary monotonically increasing parameter along the worldline, x˙μ≡dxμ/dλ\dot x^\mu \equiv \mathrm{d}x^\mu/\mathrm{d}\lambda, and the metric signature is (+,−,−,−)(+,-,-,-). ((a)) Define the canonical momentum pμ≡−∂L/∂x˙μp_\mu \equiv -\partial L/\partial\dot x^\mu for the Lagrangian L=−mx˙νx˙νL=-m\sqrt{\dot x^\nu\dot x_\nu}, and show that pμ=mx˙μ/x˙νx˙νp_\mu = m\dot x_\mu/\sqrt{\dot x^\nu\dot x_\nu}. ((b)) Hence show that pμpμ=m2p^\mu p_\mu = m^2 for every worldline and every choice of λ\lambda, without using the Euler-Lagrange equations that follow from SS. ((c)) State in one sentence why this result is called a constraint rather than an equation of motion.

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