Dirac delta
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| Dirac delta | |
|---|---|
 Qef · CC BY-SA 3.0 · source | |
| Name | Dirac delta |
| Caption | Graphical depiction as limit of Gaussian approximations |
| Introduced | 1926 |
| Introduced by | Paul Dirac |
| Field | Mathematical analysis, Theoretical physics, Signal processing |
Dirac delta is a generalized function introduced by Paul Dirac to model an idealized point mass or point charge and to formalize impulses in classical mechanics, quantum mechanics, and electrical engineering. It acts as a linear functional that picks out the value of a test function at a point, serving as an identity for convolution and as a distributional derivative of the Heaviside step used in boundary-value problems and Green's functions. The concept unifies techniques in Fourier analysis, partial differential equations, and signal processing.
Definition and basic properties
The Dirac delta is defined informally by the sifting property ∫_{-∞}^{∞} δ(x−a) f(x) dx = f(a) for suitable test functions f, and by δ(x) = 0 for x ≠ 0 with total integral one. In one-dimensional settings this is used with kernels and Green's functions associated to linear operators such as those studied by Sofia Kovalevskaya and Joseph Fourier. Key formal properties include evenness δ(−x) = δ(x), scaling δ(kx) = |k|^{-1} δ(x) for nonzero real k, and translational covariance under translations linked to concepts used by Niels Bohr and Erwin Schrödinger in scattering theory. The delta also appears in orthogonality relations for eigenfunctions in Sturm–Liouville problems which were developed alongside work by David Hilbert and John von Neumann.
Distributional formulation
Rigour comes from the theory of distributions developed by Laurent Schwartz, where the delta is a continuous linear functional on spaces of test functions such as C_c^∞(ℝ^n). In this framework δ_a: φ ↦ φ(a) is supported at {a} and has order zero, fitting into the hierarchy of distributions that includes derivatives of δ and principal value distributions studied by Henri Poincaré and Vladimir Arnold. The topology of test function spaces used in this formulation connects to concepts advanced in functional analysis by Stefan Banach and Marshall Stone, and allows one to define convergence of approximations in the weak-* sense central to modern treatments in partial differential equations.
Representations and approximations
Numerous sequences of ordinary functions converge to δ in the distributional sense. Common approximations include Gaussian sequences exp(−x^2/(2σ^2))/(√(2π)σ) with σ→0, Lorentzian (Cauchy) kernels 1/π · (ε/(x^2+ε^2)) with ε→0, and normalized rectangular pulses used in numerical analysis and finite element methods associated to advances by Richard Courant and Kurt Friedrichs. In Fourier analysis the delta appears as the inverse transform of constant functions, linking to the work of Joseph Fourier and applications in the Fast Fourier Transform developed by James Cooley and John Tukey. Discrete analogues include Kronecker delta sequences used in combinatorics and number theory with ties to research by Paul Erdős and G. H. Hardy.
Operations and identities
As a distribution, δ obeys linearity and can be differentiated: ⟨δ', φ⟩ = −φ'(0), giving identities used in mechanics and electrodynamics explored by James Clerk Maxwell and Ludwig Boltzmann. Convolution with a test function reproduces the function: f * δ = f, an identity exploited in system theory of Norbert Wiener and control theory work influenced by Rudolf Kalman. Under coordinate transformations the Jacobian determinant appears: δ(g(x)) = Σ_i δ(x−x_i)/|g'(x_i)| for simple roots x_i, a formula used in scattering amplitude computations by Richard Feynman and in integral transforms used by Murray Gell-Mann. Multiplication of distributions is generally not defined, leading to careful use in nonlinear problems as highlighted in research by Jean Leray and Sergei Sobolev.
Applications in physics and engineering
The delta models point sources in electrostatics and point masses in Newtonian mechanics, appearing in Poisson equations and Green's functions used in techniques by George Green and Oliver Heaviside. In quantum field theory it enforces momentum conservation in Feynman diagram integrals studied by Richard Feynman and Freeman Dyson. In signal processing it represents ideal impulses and sampling using the Shannon sampling theorem attributed to Claude Shannon, and underpins filter design in electrical engineering advanced by Oliver Heaviside and Harold Black. In mechanics of continua and fracture mechanics the delta encodes concentrated forces as employed in work by Augustin-Louis Cauchy and Timoshenko. Numerical implementations for boundary integral methods and finite element modeling draw on algorithms by J. H. Wilkinson and Ivo Babuška.
Generalizations and related distributions
Higher-dimensional deltas δ(x−x_0) on manifolds interact with differential forms and volume elements in formulations by Élie Cartan and Hermann Weyl. Tempered distributions extend applicability to the Fourier transform framework used in harmonic analysis by André Weil and Lamberto Cesari. Principal value distributions, Hadamard finite part, and Colombeau algebras provide alternatives for handling products and singularities as developed by Laurent Schwartz, Jacques Hadamard, and Jean-François Colombeau. Other related objects include discrete measures in algebraic topology studied by Henri Poincaré and delta-sequences in probability theory associated with limit theorems by Andrey Kolmogorov.