nLab
wavefront set

Contents

Idea

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).

Definition

Motivation

The definition of wavefront sets is motivated by a version of a Paley-Wiener theorem that characterizes smooth compactly supported functions ( n\mathbb{R}^n \to \mathbb{R}) by a growth condition on their Fourier transform \mathcal{F}:

Theorem

(Paley-Wiener-Schwartz theorem)

The vector space C 0 ( n)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 FF which satisfy the following estimate: there is a positive constant BB such that for every integer m>0m \gt 0 there is a constant C mC_m such that:

F(z)C m(1+|z|) mexp(B|Im(z)|) F(z) \le C_m (1 + |z|)^{-m} \exp{ (B \; |\operatorname{Im}(z)|)}

Smoothness

We call a smooth compactly supported function that is identically 11 in a neighbourhood of a point x 0x_0 a cutoff function at x 0x_0. Let U nU \subset \mathbb{R}^n be open, we identify the cotangent bundle of UU with U× nU \times \mathbb{R}^n. A subset of U× nU \times \mathbb{R}^n is said to be conic if it is stable under the transformation

(x,ζ)(x,ρζ)withρ>0 (x, \zeta) \mapsto (x, \rho \zeta) \quad \text{with} \; \rho \gt 0

Note that a conic subset is uniquely determined by its intersection with the unit sphere bundle U×S n1U\times S^{n-1}.

Definition

Let ff be a distribution and (x 0,ζ 0)(x_0, \zeta_0) with ζ 00\zeta_0 \neq 0 be a point of the cotangent bundle of UU. ff is smooth in (x 0,ζ 0)(x_0, \zeta_0) if there is a cutoff function χ\chi in x 0x_0 and an open cone Γ 0\Gamma_0 in n\mathbb{R}^n containing ζ 0\zeta_0 such that for every m>0m \gt 0 there is a nonnegative constant C mC_m such that for all ζΓ 0\zeta \in \Gamma_0:

(χf)(ζ)C m(1+ζ) m \| \mathcal{F}(\chi f) (\zeta) \| \le C_m (1 + \| \zeta \|)^{-m}

where (χf)\mathcal{F}(\chi f) is the Fourier transform (of the variable ζ\zeta) of the function χf\chi f (of the variable xx).

Definition

A distribution ff is smooth in a conic subset Γ\Gamma of the cotangent bundle of UU if ff is smooth in a neighbourhood of every point in Γ\Gamma.

Wavefront set

Let U nU \subseteq \mathbb{R}^n be an open subset, T *UT^* U its cotangent bundle and ff be a distribution on UU. The complement of the union of all conic subsets of T *UT^* U where ff is smooth is the wavefront set WF(f)WF(f). Since the wavefront set is therefore itself conic, it is equivalently determined by a subset of the unit sphere bundle of T *UT^* U.

(Hörmander 70 (2.4.1), Hörmander 90, section 8.1)

This definition turns out to make invariant sense (Hörmander 90, p. 256).

Examples

Example

(wave front set of delta distribution)

For nn \in \mathbb{N}, consider the delta distribution

δ(0)𝒟( n) \delta(0) \in \mathcal{D}'(\mathbb{R}^n)

on nn-dimensional Cartesian space, given by evaluation at the origin. Its wave front set is

WF(δ(0))={(0,k)|k n{0}} n× nT * n. WF(\delta(0)) = \left\{ (0,k) \;\vert\; k \in \mathbb{R}^n \setminus \{0\} \right\} \subset \mathbb{R}^n \times \mathbb{R}^n \simeq T^\ast \mathbb{R}^n \,.
Proof

First of all the singular support of δ(0)\delta(0) is clearly singsupp(δ(0))={0}singsupp(\delta(0)) = \{0\}, hence the wave front set vanishes over n{0}\mathbb{R}^n \setminus \{0\}.

At the origin, any bump function bb supported around the origin with b(0)=1b(0) = 1 satisfies bδ(0)=δ(0)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 δ^(0)\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.

Example

(wave front set of Heaviside distribution)

Let H𝒟( 1)H \in \mathcal{D}'(\mathbb{R}^1) be the Heaviside distribution given by

H,b 0 b(x)dx. \langle H, b\rangle \coloneqq \int_0^\infty b(x)\, d x \,.

Its wave front set is

WF(H)={(0,k)|k0}. WF(H) = \{(0,k) \vert k \neq 0\} \,.
Example

For (X,e)(X,e) a globally hyperbolic spacetime and PP 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 ff to Pf=0P 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.

Example

(wave front set of tensor product distribution)

Let u𝒟(X)u \in \mathcal{D}'(X) and v𝒟(Y)v \in \mathcal{D}'(Y) be two distributions. then the wave front set of their tensor product distribution uv𝒟(X×Y)u \otimes v \in \mathcal{D}'(X \times Y) satisfies

WF(uv)(WF(u)×WF(v))((supp(u)×{0})×WF(v))(WF(u)×(supp(v)×{0})), WF(u \otimes v) \;\subset\; \left( WF(u) \times WF(v) \right) \cup \left( \left( supp(u) \times \{0\} \right) \times WF(v) \right) \cup \left( WF(u) \times \left( supp(v) \times \{0\} \right) \right) \,,

where supp()supp(-) denotes the support of a distribution.

(Hörmander 90, theorem 8.2.9)

Properties

Proposition

(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))

Proposition

(derivative of distributions retains or shrinks wave front set)

Taking derivatives of distributions retains or shrinks the wave front set:

For u𝒟( n)u \in \mathcal{D}'(\mathbb{R}^n) a distribution and α n\alpha \in \mathbb{N}^n a multi-index with D αD^\alpha denoting the corresponding partial derivative, then

WF(D αu)WF(u). WF(D^\alpha u) \subset WF(u) \,.

(Hörmander 90, (8.1.10), p. 256)

References

General

The concept of wave front set is due to

  • Lars Hörmander, Linear differential operators, Actes Congr. Int. Math. Nice 1970, 1, 121-133 (pdf)

A textbook account for distributions on open subsets of Euclidean space is in

  • Lars Hörmander, section 8.1 of The analysis of linear partial differential operators, vol. I, Springer 1983, 1990

and for distributions more generally on smooth manifolds is in

  • Lars Hörmander, The analysis of linear partial differential operators, vol. III, Springer 1994

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

In quantum field theory

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

  • Marek Radzikowski, Micro-local approach to the Hadamard condition in quantum field theory on curved space-time, Commun. Math. Phys. 179 (1996), 529–553 (Euclid)

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

For more see the references at locally covariant perturbative quantum field theory.

Revised on November 12, 2017 10:31:27 by Urs Schreiber (94.220.55.151)