Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 311 / 4 / i / Solution

For complex solutions of the Klein-Gordon equation, define the currentIts divergence isbecause $\square \beta =m^2\beta$ and $\square \overline{\alpha }=m^2\overline{\alpha }$ for real $m^2$. The Klein-Gordon inner product is the flux ${\int }_{\Sigma }{n}_a{j}^{a}d\Sigma$. Applying Stokes theorem to a slab between two Cauchy hypersurfaces gives equality of the fluxes, with the future normal on each surface and opposite boundary orientations.

For noncompact surfaces this requires the integrals to converge and no net flux through spatial infinity, for example spacelike compact support or suitable decay. Global hyperbolicity alone does not ensure convergence for arbitrary solutions. Under these conditions,Complex conjugation also givesThus it is a Hermitian form, antilinear in its first argument and linear in its second. It is indefinite on the full complex solution space; positivity will arise only on a chosen positive-frequency solution subspace.