D'Alembert operator

The D'Alembert operator is a second order differential operator that acts on functions of dimensional spacetime (e.g. ). ${\displaystyle \Box }$ ${\displaystyle f(t,x_{1},\dots ,x_{d-1})}$ ${\displaystyle d}$ ${\displaystyle d=4}$

${\displaystyle \Box ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\sum _{i=1}^{d-1}{\frac {\partial ^{2}}{\partial x_{i}{}^{2}}}\ .}$

Its symbol (pronounced box) is similar to that of the Laplace operator ${\displaystyle \Box }$

${\displaystyle \Delta =\sum _{i=1}^{3}{\frac {\partial ^{2}}{\partial x_{i}{}^{2}}}\ ,}$

but which has clearly different properties.

The D'Alembert operator is the differential operator of the wave equation and the Klein-Gordon equation and is also called the wave operator or Quabla operator .

In physics, the convention is also used that the time coordinate is combined with the speed in the equation given above . This summary can in turn be interpreted as a route. The coordinate would be the distance that the wave traverses in time with the speed . ${\displaystyle t}$ ${\displaystyle c}$ ${\displaystyle \tau =ct}$ ${\displaystyle t}$ ${\displaystyle c}$

Lorentz invariance of the D'Alembert operator

The coefficients of the second derivatives in the wave operator are the components of the (inverse) spacetime metric

${\displaystyle \Box =\eta ^{mn}\partial _{x^{m}}\partial _{x^{n}}\ ,\ \eta ={\text{diag}}(1,-1,\dots ,-1)\ .}$

In the equally widespread convention of designating the negative of this square shape as space-time metric, stands for the negative of the D'Alembert operator defined here. ${\displaystyle {\text{diag}}(-1,+1,\dots ,+1)}$ ${\displaystyle \Box }$

Like the space-time metric , the D'Alembert operator is invariant under translations and Lorentz transformations . Applied to Lorentz chained functions it gives the same result as the Lorentz chained derivative function ${\displaystyle \eta }$ ${\displaystyle \Box }$ ${\displaystyle \Lambda }$ ${\displaystyle f\circ \Lambda ^{-1}}$

${\displaystyle (\Box f)\circ \Lambda ^{-1}=\Box \,(f\circ \Lambda ^{-1})\ .}$

Green's function

A Green function of the D'Alembert operator satisfies the definition equation as its right inverse ${\displaystyle G(t,t',x,x')}$

${\displaystyle \square (t,\mathbf {x} )G(t-t^{\prime },\mathbf {x} -\mathbf {x} ^{\prime })=\delta (t-t^{\prime })\delta (\mathbf {x} -\mathbf {x} ^{\prime })}$.

The Dirac delta distribution denotes . Since this is an operator that is not explicitly time- and location-dependent, it only depends on the differences and why we can set the deleted coordinates to zero without loss of generality . For the Fourier transform ${\displaystyle \delta }$ ${\displaystyle G}$ ${\displaystyle (t-t')}$ ${\displaystyle (x-x')}$ ${\displaystyle G(\omega ,k)}$

${\displaystyle G(t,\mathbf {x} )={\frac {1}{(2\pi )^{3}}}\iint d\omega \ d^{3}\mathbf {k} \;\mathrm {e} ^{\mathrm {i} (\omega t-\mathbf {kx} )}\ G(\omega ,\mathbf {k} )}$

the following algebraic equation then results:

${\displaystyle G(\omega ,\mathbf {k} )={\frac {1}{-(\omega /c)^{2}+k^{2}}}}$

The poles of are exactly where the dispersion relation for electromagnetic waves in a vacuum ( ) is fulfilled. The solutions of the homogeneous wave equation coincide exactly with the poles of Green's function, which is a typical resonance behavior for response functions . ${\displaystyle G(\omega ,k)}$ ${\displaystyle \omega ^{2}=c^{2}k^{2}}$

In order to be able to carry out the inverse transformation, we consider the analytical continuation of for complex frequencies. With the help of the residual calculus one can "circumnavigate" the poles , whereby different paths correspond to different boundary conditions. One differentiates: ${\displaystyle G(\omega ,k)}$ ${\displaystyle |\omega |=ck}$

Type	${\displaystyle G(\omega ,\mathbf {k} )}$	${\displaystyle G(t,x)}$
Retarded ${\displaystyle G^{+}}$	${\displaystyle {\frac {1}{-(\omega /c+\mathrm {i} \epsilon )^{2}+k^{2}}}}$	${\displaystyle {\frac {1}{4\pi x}}\delta \left(t-{\frac {x}{c}}\right)\Theta (t)}$
Advanced ${\displaystyle G^{-}}$	${\displaystyle {\frac {1}{-(\omega /c-\mathrm {i} \epsilon )^{2}+k^{2}}}}$	${\displaystyle {\frac {1}{4\pi x}}\delta \left(t+{\frac {x}{c}}\right)\Theta (-t)}$

Green's function in the frequency domain is to be understood in the limit value , which corresponds to the different paths around the poles in the integral. ${\displaystyle \epsilon \to 0^{+}}$

The factor corresponds to the law of propagation of a spherical wave. ${\displaystyle t-x/c}$

literature

• Torsten Fließbach : Electrodynamics. Textbook on theoretical physics II , 5th edition. Spectrum Academic Publishing House