{c}
{title2=$\Box$}
{wiki}

The <laplace operator> for <Minkowski space>.

Can be nicely written with <Einstein notation> as shown at: <d'Alembert operator in Einstein notation>{full}.


d'Alembert operator

The <Klein-Gordon equation> can be written in terms of the <d'Alembert operator> as:
$$
\Box \psi + m^2 \psi = 0
$$
so we can expand the <d'Alembert operator in Einstein notation> to:
$$
\partial_i \partial^i \psi - m^2 \psi = 0
$$


Klein-Gordon equation in Einstein notation

{c}
{title2=$\partial_i \partial^i$}

Given the function $\psi$:
$$
\psi : \R^4 \to \C
$$
the operator can be written in <Planck units> as:
$$
\partial_i \partial^i \psi(x_0, x_1, x_2, x_3) - m^2 \psi(x_0, x_1, x_2, x_3) = 0
$$
often written without function arguments as:
$$
\partial_i \partial^i \psi
$$
Note how this looks just like the <Laplacian in Einstein notation>,  since the <d'Alembert operator> is just a generalization of the <laplace operator> to <Minkowski space>.


Ciro Santilli @cirosantilli 37

 Incoming links: d'Alembert operator in Einstein notation