Klein–Gordon operator

Definition

On a Lorentzian manifold $(M,g)$ , the minimally coupled Klein–Gordon operator of mass $m\geq0$ is

P_m=\Box_g+m^2,

where $\Box_g=-\operatorname{tr}_g\nabla d$ is the d’Alembert operator. More generally, a curvature coupling gives

P_{m,\xi}=\Box_g+m^2+\xi\,\operatorname{Scal}_g

for a real coupling constant $\xi$ .

Symbol and hyperbolicity

The mass and curvature terms have order zero, so

\sigma_2(P_{m,\xi})(x,\zeta) =-g_x^{-1}(\zeta,\zeta).

Consequently every $P_{m,\xi}$ is a normally hyperbolic operator, with the same characteristic null cone and finite propagation speed as the massless wave operator.

Minkowski dispersion relation

On Minkowski spacetime,

P_m=\partial_t^2-\sum_j\partial_{x_j}^2+m^2.

For a plane wave $e^{-i\omega t+i k\cdot x}$ , the equation $P_m\phi=0$ becomes

\omega^2=|k|^2+m^2.

Convention warning

If a source uses $\widetilde\Box_g=\operatorname{tr}_g\nabla d=-\Box_g$ with the same $(-+\cdots+)$ signature, it usually writes the equivalent equation as $(\widetilde\Box_g-m^2)\phi=0$ . Neither the sign of the mass term nor the symbol $\Box$ is meaningful without the wave-operator convention.

References

Christian Bär, Nicolas Ginoux, and Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, European Mathematical Society, 2007. Publisher record. Relevant: §§3.2 and 4.3.
Robert M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, University of Chicago Press, 1994. Publisher record. Relevant: Chapters 3–4.