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For this metric, $\sqrt{-g}=a^2$, while $g^{\eta \eta }=-a^{-2}$, $g^{zz}=a^{-2}$ and $g^{ij}={\delta }^{ij}$. The massless Klein-Gordon equation is consequentlyInsert the spatial Fourier transform given in the question. Each wavenumber mode then satisfiesThus every field mode is a simple harmonic oscillator with a time-dependent angular frequency.

For the Poincaré metric on ${\mathrm{AdS}}_6$,The massless Klein-Gordon equation, evaluated with the Laplace-Beltrami operator, isorA boundary power $\varphi \sim z^{\alpha }$ obeys the indicial equationThusThe constant branch is the nonnormalizable source and the $z^5$ branch is the normalizable response. Since $m^2=\Delta (\Delta -d)=0$ with $d=5$, the roots are ${\Delta }_{-}=0$ and ${\Delta }_{+}=5$. Only standard quantization is unitary here: the putative alternative operator of dimension zero would be the identity, whereas the field is a nontrivial fluctuating operator. The permissible conformally invariant source-free boundary condition is therefore Dirichlet,More generally one may prescribe $J$ as an external source. The holographic dictionary gives

For a plane wave $\varphi =e^{i(kx-\omega t)}$, the Klein-Gordon equation givesThus the positive-frequency dispersion relation is

Let ${\Sigma }_t$ be a Cauchy hypersurface with induced metric $h_{ij}$, lapse $N$, shift $N^i$, and future unit normal $n^a$. For the normalization of the action in the question, the canonical momentum density isThe equal-time canonical commutation relations areandWith the conventional extra factor $1/2$ in the action, $\Pi$ loses the factor two.

For complex classical solutions, the Klein-Gordon inner product isThe integrand is a conserved current because both fields obey the Klein-Gordon equation. Applying the divergence theorem between two Cauchy hypersurfaces shows that the value is independent of the foliation, provided there is no boundary flux.

A free real scalar field obeys the Klein-Gordon equation and creates neutral spin-zero particles.

For the massless Klein-Gordon equation on a Kerr black hole, the covariant wave operator separates into radial and angular ordinary differential equations. With $K=(r^2+a^2)\omega -am$ and angular separation constant $\Lambda$, the radial equation is $(\Delta R^{\prime }{)}^{\prime }+[K^2/\Delta -a^2{\omega }^2+2am\omega -\Lambda ]R=0$. The angular equation is $(\sin \theta {\Theta }^{\prime }{)}^{\prime }/\sin \theta +[a^2{\omega }^2{\cos }^2\theta -m^2/{\sin }^2\theta +\Lambda ]\Theta =0$. No extra radial factor is included in the displayed ansatz.