Klein-Gordon field ($\varphi$)

A free Klein-Gordon field is a scalar obeying $(\square -m^2-\xi R)\varphi =0$ in the mostly-plus metric signature. Global hyperbolicity provides a well-defined initial-value problem. Quantization additionally needs a state or positive-norm mode splitting; the equation alone does not select particles.

For real Klein-Gordon field solutions with suitable support or boundary conditions, $\Omega ({\varphi }_1,{\varphi }_2)={\int }_{\Sigma }({\varphi }_1{n}^a{\nabla }_a{\varphi }_2-{\varphi }_2{n}^a{\nabla }_a{\varphi }_1)d\Sigma$ is a conserved symplectic form. The field equation makes its current divergence-free. Its complexification gives $(u,v{)}_{KG}=i\Omega (u^{\ast },v)$, the Klein-Gordon inner product. A compatible complex structure on the Klein-Gordon solution space is additional data needed for a particle representation.

On a scalar, the covariant wave operator is $\square \varphi =|g{|}^{-1/2}{\partial }_a(|g{|}^{1/2}{g}^{ab}{\partial }_b\varphi )$. For the metric signature $(-+++)$ it has the flat-space form $-{\partial }_t^2+{\nabla }^2$. This divergence expression is convenient for separation of variables and for integrating the Klein-Gordon equation by parts.

On the real Klein-Gordon field solution space with conserved symplectic form $\Omega$, a compatible real-linear map $J$ obeys $J^2=-I$, preserves $\Omega$, and makes $\mu (f,g)=\Omega (f,Jg)$ positive definite. Its $+i$ eigenspace in the complexified space has positive Klein-Gordon inner product with the convention $(u,v{)}_{KG}=i\Omega (\bar{u},v)$. It defines the one-particle Hilbert space and a free-field Fock vacuum.