In microlocal analysis, the wave front set (Hörmander 70) of a generalized function such as a distribution or a hyperfunction is a characterization of the singularity structure of the generalized function, hence of how it deviates from being an ordinary smooth function.
The wave front set is the sub-bundle of the cotangent bundle that consists of all those directions (non-zero covectors) such that the local Fourier transform of the distribution is not rapidly decaying in this direction (Hörmander 90, section 8.1). Such covectors are stable under multiplication by positive scalars, so the wave front set can also be considered as a sub-bundle of the unit sphere bundle of the cotangent bundle.
The projection of the wave front set down to the base space is the singular support of the distribution. The additional information in the “wave front” covectors over this singular support may be understood as providing the directions of propagation of these singularities. This is made precise by the propagation of singularities theorem
A notorious issue with distributions is that, when thought of as generalized functions, generally neither their composition of distributions? nor their pointwise product of distributions is defined. However, closer inspection shows that the obstruction to these operations being defined for any given pair of distributions is exactly characterized by the wave front set:
For instance the product of distributions is well defined precisely if the sum of their wave front sets does not intersect the zero-section (Hörmander's criterion, Hörmander 90, theorem 8.2.10).
The definition of wavefront sets is motivated by a version of a Paley-Wiener theorem that characterizes smooth compactly supported functions ($\mathbb{R}^n \to \mathbb{R}$) by a growth condition on their Fourier transform $\mathcal{F}$:
(Paley-Wiener-Schwartz theorem)
The vector space $C_0^{\infty}(\mathbb{R}^n)$ of smooth compactly supported functions (bump functions) is (algebraically and topologically) isomorphic, via the Fourier transform, to the space of entire functions $F$ which satisfy the following estimate: there is a positive constant $B$ such that for every integer $m \gt 0$ there is a constant $C_m$ such that:
We call a smooth compactly supported function that is identically $1$ in a neighbourhood of a point $x_0$ a cutoff function at $x_0$. Let $U \subset \mathbb{R}^n$ be open, we identify the cotangent bundle of $U$ with $U \times \mathbb{R}^n$. A subset of $U \times \mathbb{R}^n$ is said to be conic if it is stable under the transformation
Note that a conic subset is uniquely determined by its intersection with the unit sphere bundle $U\times S^{n-1}$.
Let $f$ be a distribution and $(x_0, \zeta_0)$ with $\zeta_0 \neq 0$ be a point of the cotangent bundle of $U$. $f$ is smooth in $(x_0, \zeta_0)$ if there is a cutoff function $\chi$ in $x_0$ and an open cone $\Gamma_0$ in $\mathbb{R}^n$ containing $\zeta_0$ such that for every $m \gt 0$ there is a nonnegative constant $C_m$ such that for all $\zeta \in \Gamma_0$:
where $\mathcal{F}(\chi f)$ is the Fourier transform (of the variable $\zeta$) of the function $\chi f$ (of the variable $x$).
A distribution $f$ is smooth in a conic subset $\Gamma$ of the cotangent bundle of $U$ if $f$ is smooth in a neighbourhood of every point in $\Gamma$.
Let $U \subseteq \mathbb{R}^n$ be an open subset, $T^* U$ its cotangent bundle and $f$ be a distribution on $U$. The complement of the union of all conic subsets of $T^* U$ where $f$ is smooth is the wavefront set $WF(f)$. Since the wavefront set is therefore itself conic, it is equivalently determined by a subset of the unit sphere bundle of $T^* U$.
(Hörmander 70 (2.4.1), Hörmander 90, section 8.1)
This definition turns out to make invariant sense (Hörmander 90, p. 256).
(wave front set of delta distribution)
For $n \in \mathbb{N}$, consider the delta distribution
on $n$-dimensional Cartesian space, given by evaluation at the origin. Its wave front set is
First of all the singular support of $\delta(0)$ is clearly $singsupp(\delta(0)) = \{0\}$, hence the wave front set vanishes over $\mathbb{R}^n \setminus \{0\}$.
At the origin, any bump function $b$ supported around the origin with $b(0) = 1$ satisfies $b \cdot \delta(0) = \delta(0)$ and hence the wave front set over the origin is the set of covectors along which the Fourier transform $\hat \delta(0)$ does not suitably decay. But this Fourier transform is in fact a constant function and hence does not decay in any direction.
(wave front set of Heaviside distribution)
Let $H \in \mathcal{D}'(\mathbb{R}^1)$ be the Heaviside distribution given by
Its wave front set is
For $(X,e)$ a globally hyperbolic spacetime and $P$ a hyperbolic differential operator such as the wave operator/Klein-Gordon operator, then the propagation of singularities theorem says that the wave front set of any solution $f$ to $P f = 0$ is a union of lightlike geodesics and their cotangent vectors.
Specifically for the Klein-Gordon operator such ditributional solutions include the causal propagator and the Feynman propagator.
(wave front set of tensor product distribution)
Let $u \in \mathcal{D}'(X)$ and $v \in \mathcal{D}'(Y)$ be two distributions. then the wave front set of their tensor product distribution $u \otimes v \in \mathcal{D}'(X \times Y)$ satisfies
where $supp(-)$ denotes the support of a distribution.
(empty wave front set corresponds to ordinary functions)
The wave front set of a compactly supported distribution is empty precisely if the distribution comes from an ordinary smooth function (hence a bump function).
e.g. (Hörmander 90, below (8.1.1))
(derivative of distributions retains or shrinks wave front set)
Taking derivatives of distributions retains or shrinks the wave front set:
For $u \in \mathcal{D}'(\mathbb{R}^n)$ a distribution and $\alpha \in \mathbb{N}^n$ a multi-index with $D^\alpha$ denoting the corresponding partial derivative of distributions, then
Hence if $P$ is any differential operator with smooth function coefficients, then
(Hörmander 90, (8.1.10) (8.1.11), p. 256)
(wave front set of convolution of compactly supported distributions)
Let $u,v \in \mathcal{E}'(\mathbb{R}^n)$ be two compactly supported distributions. Then the wave front set of their convolution of distributions is
The concept of wave front set is due to
A textbook account for distributions on open subsets of Euclidean space is in
and for distributions more generally on smooth manifolds is in
A history of the concept of wave front sets with extensive pointers to the literature is given in Hörmander 90, p. 322-324.
See also
The application of microlocal analysis via wave front sets to the discussion of n-point functions in quantum field theory and especially quantum field theory on curved spacetimes originates with the results of
which were first picked up in
C. Moreno, Spaces of positive and negative frequency solutions of field equations in curved space- times. I. The Klein-Gordon equation in stationary space-times, II. The massive vector field equations in static space-times, J. Math. Phys. 18, 2153-61 (1977), J. Math. Phys. 19, 92-99 (1978)
Jonathan Dimock, Scalar quantum field in an external gravitational background, J. Math. Phys. 20, 2549-2555 (1979)
and brought into context with the Hadamard distributions needed for the construction of Wick algebras in
A textbook account amplifying this usage (on Minkowski spacetime) in the mathematically rigorous construction of perturbative quantum field theory via causal perturbation theory is in
Günter Scharf, Finite Quantum Electrodynamics -- The Causal Approach, Berlin: Springer-Verlag, 1995, 2nd edition
Günter Scharf, Quantum Gauge Theories -- A True Ghost Story, Wiley 2001
For more see the references at locally covariant perturbative quantum field theory.