Laplacian in Einstein notation (${\partial }_i{\partial }^i$)

Consider a real valued function of three variables:

Its Laplacian can be written as:

It is common to just omit the variables of the function, so we tend to just say: or equivalently when referring just to the operator:

Given the function $\psi$: the operator can be written in Planck units as: often written without function arguments as: 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.

The Klein-Gordon equation can be written in terms of the D'alembert operator as: so we can expand the D'alembert operator in Einstein notation to: