nLab wave equation

Idea

For $(X,g)$ a pseudo-Riemannian manifold and $\Box : C^\infty(X) \to C^\infty(X)$ its Laplace operator, the wave equation on $X$ is the linear differential equation

$$\Box_g f = 0 \,.$$

where $\Box_g$ denotes the wave operator /Laplace operator, a hyperbolic differential operator.

For $m \in \mathbb{R}$ the inhomogenous equation

$$\Box_g f = m^2 f$$

is called the Klein-Gordon equation.

Properties

Fundamental solution

On a globally hyperbolic spacetime the wave equation/Klein-Gordon equation has unique advanced and retarded Green functions.

Their difference is the Peierls bracket which gives the Poisson bracket on the covariant phase space of the free scalar field. This in turn defines the Wick algebra of the free scalar field, which yields the quantization of the free scalar field to a quantum field theory.

Bicharacteristic flow and propagation of singularities

The bicharacteristic strips of the Klein-Gordon operator are cotangent vectors along lightlike geodesics (this example).

Related concepts

• wave

• plane wave,

  wave vector, wavelength, frequency

• superposition

• linear differential equation

• scalar field

• Green's function

• Feynman propagator

• spectral geometry, hearing the shape of a drum

• heat equation

• Klein-Gordon equation

• sine-Gordon equation

• Schrödinger equation

• Dirac equation

References

• F. Friedlander, The Wave Equation on a Curved Space-Time, Cambridge: Cambridge University Press, 1975

• Howard Georgi, The Physics of Waves, Prentice Hall (1993) $[$web, pdf$]$

• Christian Bär, Nicolas Ginoux, Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, ESI Lectures in Mathematics and Physics, European Mathematical Society Publishing House, ISBN 978-3-03719-037-1, March 2007, Softcover (arXiv:0806.1036)

• Nicolas Ginoux, Linear wave equations, Ch. 3 in Christian Bär, Klaus Fredenhagen, Quantum Field Theory on Curved Spacetimes: Concepts and Methods, Lecture Notes in Physics, Vol. 786, Springer, 2009

• Sergiu Klainerman, chapter 4, section 3 of Lecture notes in analysis, 2011 (pdf)