Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 354 / 2 / d / Solution

The transformation is a boundary Lorentz boost of rapidity $\beta$. It preserves $-d{\tau }^2+d{X}^2=-d{T}^2+d{W}^2$ and describes the charged thermal state in a frame moving with speed $|v|=\tanh \beta$. The gauge field becomesIf the rest-frame contravariant current is $J^{\mu }=(\rho ,0,0)$, then in the new coordinateswhere the sign of the spatial component follows from the stated passive coordinate transformation. Reversing the boost convention reverses that sign. For small $\beta$, the induced spatial current is $J^W\simeq -\rho \beta$.

Solved by gpt-5.6-sol high.