The tachyon
A tachyon is a hypothetical particle that moves faster than light. This is often interpreted as a possibility to send superluminal signals. Investigate whether this is so.
Consider a chain of pendulums in a uniform gravitational field, suspended from a common support and connected by springs. The distance between neighboring pendulums is \(a\).
Show that the Lagrangian of this system has the form \[L = \sum_i \left(A\dot{\phi}^2_i-B(\phi_i-\phi_{i-1})^2 - C(1-\cos\phi_i)\right),\] where \(\phi_i\) is the angle by which the \(i\)th pendulum is deflected from equilibrium. Find \(A\), \(B\), \(C\).
Pass to an infinite number of pendulums in the limit \(a\to 0\). Show that the Lagrangian becomes \[\mathcal{L} = \frac{1}{2}\left(\frac{\partial\phi}{\partial t}\right)^2-\frac{1}{2}\left(\frac{\partial\phi}{\partial x}\right)^2-m^2(1-\cos\phi).\]
Obtain the equation of motion of the field \(\phi\): \[\frac{\partial^2\phi}{\partial t^2}-\frac{\partial^2\phi}{\partial x^2}+m^2\sin\phi=0.\]
For small deviations from the stable equilibrium \(\phi=0\) one obtains the ordinary Klein–Fock–Gordon equation: \[\frac{\partial^2\phi}{\partial t^2}-\frac{\partial^2\phi}{\partial x^2}+m^2\phi=0.\]
For small deviations from the unstable equilibrium \(\phi=\pi\) one obtains the Klein–Fock–Gordon equation with a negative mass squared: \[\frac{\partial^2\phi}{\partial t^2}-\frac{\partial^2\phi}{\partial x^2}-m^2\phi=0.\] Formally, a plane wave \(\phi = e^{-i\omega t+ikx}\) gives \[\omega^2 = k^2-m^2, \quad \text{and the group velocity} \quad v_g = \frac{\partial \omega}{\partial k} = \frac{k}{\omega}>1.\]
Show that the solution of the equation \[\left(\frac{\partial^2}{\partial t^2}-\frac{\partial^2}{\partial x^2}-m^2\right)\phi=\delta(t)\delta(x)\] propagates no faster than light: the retarded Green’s function vanishes outside the light cone.