D'Alembert operator

In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box: ${\displaystyle \Box }$), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator (cf. nabla symbol) is the Laplace operator of Minkowski space. The operator is named after French mathematician and physicist Jean le Rond d'Alembert.

In Minkowski space, in standard coordinates (t, x, y, z), it has the form

Here ${\displaystyle \nabla ^{2}:=\Delta }$ is the 3-dimensional Laplacian and η^μν is the inverse Minkowski metric with

Note that the μ and ν summation indices range from 0 to 3: see Einstein notation.

(Some authors alternatively use the negative metric signature of (− + + +), with ${\displaystyle \eta _{00}=-1,\;\eta _{11}=\eta _{22}=\eta _{33}=1}$.)

Lorentz transformations leave the Minkowski metric invariant, so the d'Alembertian yields a Lorentz scalar. The above coordinate expressions remain valid for the standard coordinates in every inertial frame.

There are a variety of notations for the d'Alembertian. The most common are the box symbol ${\displaystyle \Box }$ (Unicode: U+2610 ☐ BALLOT BOX) whose four sides represent the four dimensions of space-time and the box-squared symbol ${\displaystyle \Box ^{2}}$ which emphasizes the scalar property through the squared term (much like the Laplacian). In keeping with the triangular notation for the Laplacian, sometimes ${\displaystyle \Delta _{M}}$ is used.

Another way to write the d'Alembertian in flat standard coordinates is ${\displaystyle \partial ^{2}}$. This notation is used extensively in quantum field theory, where partial derivatives are usually indexed, so the lack of an index with the squared partial derivative signals the presence of the d'Alembertian.