D'Alembert operator

There are a variety of notations for the d'Alembertian. The most common are the box symbol \Box (Unicode: ) whose four sides represent the four dimensions of space-time and the box-squared symbol \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 \Delta_M is used. Another way to write the d'Alembertian in flat standard coordinates is \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. Sometimes the box symbol is used to represent the four-dimensional Levi-Civita covariant derivative. The symbol \nabla is then used to represent the space derivatives, but this is coordinate-chart-dependent. ==Applications==

The wave equation for small vibrations is of the form : \Box_{c} u\left(x,t\right) \equiv u_{tt} - c^2u_{xx} = 0~, where is the displacement. The wave equation for the electromagnetic field in vacuum is : \Box A^{\mu} = 0 where is the electromagnetic four-potential in Lorenz gauge. The Klein–Gordon equation has the form :\left(\Box + \frac{m^2c^2}{\hbar^2}\right) \psi = 0~. ==Green's function==

The Green's function, G\left(\tilde{x} - \tilde{x}'\right), for the d'Alembertian is defined by the equation : \Box G\left(\tilde{x} - \tilde{x}'\right) = \delta\left(\tilde{x} - \tilde{x}'\right) where \delta\left(\tilde{x} - \tilde{x}'\right) is the multidimensional Dirac delta function and \tilde{x} and \tilde{x}' are two points in Minkowski space. A special solution is given by the ''retarded Green's function'' which corresponds to signal propagation only forward in time :G\left(\vec{r}, t\right) = \frac{1}{4\pi r} \Theta(t) \delta\left(t - \frac{r}{c}\right) where \Theta is the Heaviside step function. ==See also==