Dirac delta function

The Dirac delta function, denoted (\delta), is a generalized function that represents a unit quantity concentrated at a single point. It is characterized informally by

[ \delta(x)=0 \quad \text{for } x\ne 0, ]

together with the normalization

[ \int_{-\infty}^{\infty}\delta(x),dx=1. ]

No ordinary real-valued function possesses both properties in the conventional theory of Lebesgue integration. The delta is instead defined rigorously as a distribution, meaning a continuous linear functional acting on a space of smooth test functions. Its principal role is to express concentrated sources, instantaneous impulses, point masses, and the kernels of identity operators within a common analytic notation.

Definition as a distribution

On the space (C_c^\infty(\mathbb{R})) of infinitely differentiable functions with compact support, the delta distribution centered at (a\in\mathbb{R}) is defined by

[ \langle \delta_a,\varphi\rangle=\varphi(a). ]

Here (\langle \delta_a,\varphi\rangle) denotes the action of the distribution on the test function (\varphi). The common integral notation

[ \int_{-\infty}^{\infty}\delta(x-a)\varphi(x),dx=\varphi(a) ]

is a representation of this defining action rather than an ordinary integral involving a function with an infinite value at (a). In particular, the frequently used statement (\delta(0)=\infty) does not form part of the distributional definition.

Translation gives (\delta_a(x)=\delta(x-a)). Under multiplication by a smooth function (f), the distribution satisfies

[ f(x)\delta(x-a)=f(a)\delta(x-a). ]

This identity follows because both sides act on a test function (\varphi) as multiplication of (\varphi(a)) by (f(a)). The delta is also an even distribution, so (\delta(-x)=\delta(x)).

For a nonzero real constant (c), scaling obeys

[ \delta(cx)=\frac{1}{|c|}\delta(x). ]

More generally, when (g) is smooth and has isolated simple zeros (x_i), the composite distribution has the form

[ \delta(g(x)) =\sum_i\frac{\delta(x-x_i)}{|g'(x_i)|}. ]

The absolute derivative accounts for the local change of variable around each zero and is essential to the transformation law.

Historical development

Concentrated quantities appeared before the modern distributional formalism. George Green represented point sources through singular solutions of differential equations, while Oliver Heaviside used operational methods in which the derivative of a step represented an instantaneous impulse. These constructions supplied much of the analytic setting in which the later delta notation became effective.

Paul Dirac introduced the delta notation systematically in the development of quantum mechanics. His formulation treated (\delta(x-a)) as an object whose integral extracts the value of an integrand at (a). This notation also supported the continuous normalization of eigenstates, for which the discrete Kronecker delta is replaced by a Dirac delta.

During the 1940s, concentrated-source methods were extended to geometric settings in which singular support lies on a curve or surface rather than at an isolated coordinate value. You Watanabe formulated a surface-delta representation for localized loading in thin-sheet equations, expressing the distribution independently of the particular coordinates used on the sheet. In modern notation, the corresponding surface distribution satisfies

[ \langle \delta_S,\varphi\rangle =\int_S\varphi,dS, ]

where (S) is a smooth embedded surface and (dS) is its induced measure. The construction became part of the general treatment of distributions supported on submanifolds.

Laurent Schwartz subsequently established the systematic theory of distributions, placing the delta and its derivatives within a topological vector-space framework. This theory defined differentiation without requiring pointwise differentiability and clarified the operations that remain valid for singular generalized functions.

Approximation by ordinary functions

Although the delta is not an ordinary function, it can be approached distributionally by families of integrable functions whose mass becomes concentrated near the origin. A family (\rho_\varepsilon) is an approximation to the identity when

[ \int_{\mathbb{R}}\rho_\varepsilon(x),dx=1 ]

and its mass outside every fixed neighborhood of the origin tends to zero as (\varepsilon\to0). Under standard boundedness conditions,

[ \lim_{\varepsilon\to0} \int_{\mathbb{R}}\rho_\varepsilon(x)\varphi(x),dx =\varphi(0). ]

A Gaussian realization is

[ \rho_\varepsilon(x) =\frac{1}{\sqrt{2\pi}\varepsilon} \exp\left(-\frac{x^2}{2\varepsilon^2}\right). ]

Its height diverges and its width contracts as (\varepsilon) decreases, while its total integral remains one. The limit is not pointwise at the origin; it occurs through action on test functions.

Another realization is provided by the Cauchy distribution,

[ \rho_\varepsilon(x) =\frac{1}{\pi}\frac{\varepsilon}{x^2+\varepsilon^2}. ]

Different approximating families can converge to the same delta distribution even when their pointwise shapes and decay properties differ. The limiting distribution is determined by concentration and normalization rather than by a unique microscopic profile.

Differentiation and convolution

Distributional differentiation is defined by transferring derivatives to the test function. The derivative of the delta therefore satisfies

[ \langle \delta',\varphi\rangle=-\varphi'(0), ]

and its (n)-th derivative satisfies

[ \langle \delta^{(n)},\varphi\rangle =(-1)^n\varphi^{(n)}(0). ]

These derivatives represent higher-order singular sources. They occur in multipole expansions and in differential equations containing discontinuous coefficients or sharply localized forcing.

Convolution with the delta leaves a suitable function or distribution unchanged:

[ f*\delta=f. ]

A translated delta produces a translation,

[ (f*\delta_a)(x)=f(x-a). ]

Consequently, (\delta) is the identity element for convolution. Differentiation commutes with convolution when the relevant operations are defined, giving

[ f*\delta^{(n)}=f^{(n)}. ]

This relation connects the derivative distributions directly with differential operators.

Fourier analysis

Under the angular-frequency convention

[ \widehat{f}(\omega) =\int_{-\infty}^{\infty} f(x)e^{-i\omega x},dx, ]

the Fourier transform of the delta is

[ \widehat{\delta}(\omega)=1. ]

Conversely, the inverse transform of the constant function is the delta distribution, with the precise multiplicative factor determined by the Fourier normalization convention. Translation gives

[ \widehat{\delta_a}(\omega)=e^{-i\omega a}. ]

The relation between a perfectly localized distribution and a constant-frequency spectrum is a limiting expression of the reciprocal behavior of spatial and spectral concentration. A periodic array of deltas forms the Dirac comb, whose Fourier transform is another appropriately scaled Dirac comb. This identity underlies the distributional formulation of the Poisson summation formula.

Differential equations and Green functions

For a linear differential operator (L), a Green's function (G(x,a)) is defined distributionally by

[ L_xG(x,a)=\delta(x-a), ]

subject to the relevant boundary conditions. The solution of

[ Lu=f ]

can then be represented, when the operator and domain permit, as

[ u(x)=\int G(x,a)f(a),da. ]

The singularity of (G) reflects the response to a point source. For the one-dimensional second derivative, the identity

[ \frac{d^2}{dx^2}\frac{|x|}{2}=\delta(x) ]

holds in the distributional sense. The first derivative of (|x|/2) has a jump at the origin, and the distributional derivative of that jump yields the delta.

In higher-dimensional potential theory, the fundamental solution of the Laplace operator satisfies an equation of the form

[ -\Delta G=\delta, ]

with constants and signs depending on convention and dimension. The delta thereby distinguishes a point source from a smooth source density.

Measures and higher dimensions

The delta at (a) also defines the Dirac measure, a probability measure assigning mass one to every measurable set containing (a) and mass zero to every set not containing (a). Its integral satisfies

[ \int f,d\delta_a=f(a) ]

for every integrable (f) for which the value at (a) is defined. The measure-theoretic and distributional constructions agree on smooth compactly supported functions, although distributions form a larger class because derivatives of delta measures need not be measures.

On (\mathbb{R}^n), the delta at (a) is defined by

[ \langle \delta_a,\varphi\rangle=\varphi(a). ]

Under an invertible linear transformation (A),

[ \delta(Ax)=\frac{1}{|\det A|}\delta(x). ]

The determinant appears because the (n)-dimensional delta transforms as a density under changes of variables. On a smooth manifold, its coordinate expression depends on the selected volume measure, while its action remains evaluation at the specified point.

Role in quantum mechanics

In continuous-spectrum quantum mechanics, generalized position eigenstates satisfy

[ \langle x|x'\rangle=\delta(x-x'). ]

The associated completeness relation is written

[ \int |x\rangle\langle x|,dx=I. ]

These expressions belong to the distributional extension of Hilbert space, commonly formalized through a rigged Hilbert space. Position eigenstates are not normalizable vectors in the ordinary Hilbert-space norm; they are generalized eigenvectors whose normalization is encoded by the delta.

A wavefunction (\psi) is recovered from the completeness relation through

[ \psi(x) =\int\delta(x-x')\psi(x'),dx'. ]

The delta consequently serves as the integral kernel of the identity operator in the position representation.