nLab product of distributions

Idea

Generally

The product of two distributions is an operation that generalizes, when defined, the ordinary pointwise product of functions, if we think of distributions as generalized functions (via this prop.).

However, as opposed to ordinary functions, the product of distributions – if it is required to extend the pointwise product of functions and to satisfy associativity and the product law for derivatives of distributions – is not defined for arbitrary pairs of distributions, but only partially defined for those pairs whose Fourier transforms satisfies a certain compatibility condition. Roughly: whenever the Fourier transform around any point of one factor does not decay exponentially in one direction of wave vectors, then the Fourier transform of the other factor has to decay exponentially in the opposite direction of wave vectors.

This is known as Hörmander's criterion on the wave front set of the distributions (Hörmander90), and the study of distributions with attention to this condition is part of what is called microlocal analysis.

(If one is willing to consider changing the ordinary pointwise product of functions, then is possible to define the product operation globally on all distributions, see at Colombeau algebra.)

The product of distributions with Hörmander’s condition on their wave front set may be understood via Fourier transform, as follows.

Let $u,v \in \mathcal{D}'(\mathbb{R}^1)$ be two distributions, for simplicity of exposition taken on the real line.

Since the product $u \cdot v$, is, if it exists, supposed to generalize the pointwise product of smooth functions, it must be fixed locally: for every point $x \in \mathbb{R}$ there ought to be a compactly supported smooth function (bump function) $b \in C^\infty_{cp}(\mathbb{R})$ with $f(x) = 1$ such that

$$b^2 u \cdot v = (b u) \cdot (b v) \,.$$

But now $b v$ and $b u$ are both compactly supported distributions, and these have the special property that their Fourier transforms $\widehat{b v}$ and $\widehat{b u}$ are, in particular, smooth functions (by the Paley-Wiener-Schwartz theorem).

Moreover, the operation of Fourier transform intertwines pointwise products with convolution products. This means that if the product of distributions $u \cdot v$ exists, it must locally be given by the inverse Fourier transform of the convolution product of the Fourier transforms $\widehat {b u}$ and $\widehat b v$:

$$\widehat{ b^2 u \cdot v }(x) \;=\; \underset{\underset{k_{max} \to \infty}{\longrightarrow}}{\lim} \, \int_{- k_{max}}^{k_{max}} \widehat{(b u)}(k) \widehat{(b v)}(x - k) d k \,.$$

(Notice that the converse of this formula holds as a fact: this prop..)

This shows that the product of distributions exists once there is a bump function $b$ such that the integral on the right converges as $k_{max} \to \infty$.

Now the Paley-Wiener-Schwartz theorem says more, it says that the Fourier transforms $\widehat {b u}$ and $\widehat {b u}$ are polynomially bounded. On the other hand, the integral above is well defined if the integrand decreases at least quadratically with $k \to \infty$. This means that for the convolution product to be well defined, either $\widehat {b u}$ has to polynomially decrease faster with $k \to \pm \infty$ than $\widehat {b v}$ grows in the other direction, $k \to \mp \infty$ (due to the minus sign in the argument of the second factor in the convolution product), or the other way around.

Moreover, the degree of polynomial growth of the Fourier transform increases by one with each derivative. Therefore if the product law for derivatives of distributions is to hold generally, we need that either $\widehat{b u}$ or $\widehat{b v}$ decays faster than any polynomial in the opposite of the directions in which the respective other factor does not decay.

Here the set of directions of wave vectors in which the Fourier transform of a distribution localized around any point does not decay exponentially is called the wave front set of a distribution. Hence the condition that the product of two distributions is well defined is that for each wave vector direction in the wave front set of one of the two distributions, the opposite direction must not be an element of the wave front set of the other distribution.

In perturbative quantum field theory

The issue of mutliplying distributions has prominently been perceived in perturbative quantum field theory, where operator-valued distributions serve to give the algebra of observables such as the Wick algebra of the free fields or more generally the interacting field algebra.

A popular impression is (or has been) that the failure of distributions to have a globally defined product is a failure of the mathematical formalism to support the structures needed to model perturbative quantum field theory. But in fact the opposite is true: Handling the product of distributions correctly via proper analysis of their wave front set and handling the point-extension of distributions properly via analysis of their scaling degree leads to a mathematical rigorous construction and mathematically captures all the effects expected from the non-rigorous treatmeants, notably the renormalization freedom. This is the topic of causal perturbation theory/locally covariant perturbative quantum field theory, see there for more.

Definition

Definition

(multiplication of distributions)

Let $u,v \in \mathcal{D}'(X)$ be two distributions such that the sum of their wave front sets $WF(u) + WF(v)$ does not intersect zero (Hörmander's criterion). Then their product distribution

$$u \cdot v \in \mathcal{D}'(X)$$

is the pullback (via this prop.) of their tensor product along the diagonal map $\Delta_X \;\colon\; X \to X \times X$:

$$u \cdot v \;\coloneqq\; \Delta_X^\ast( u \otimes v ) \,.$$

Proposition

Def. is indeed well defined.

(Hörmander 90, theorem 8.2.10, Melrose 03, prop. 4.11)

Proof

We need to check that the pullback of distributions is well defined. By this prop. this means to check that the wave front set of the $u \otimes v$ does not intersect the conormal bundle of the diagonal map

Now the conormal bundle of the diagonal map consists of those pairs of covectors whose sum vanishes:

$$N^\ast_{\Delta_X} = \left\{ (x,(\xi, -\xi)) \;\vert\; \xi \in T^\ast_x X \right\} \,.$$

Moreover, by this example the wave front set of the tensor product distribution $u \otimes v$ is

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

Since any wave front set excludes the zero-section by definition, the second and the third summand in this union never intersect the above conormal bundle. The first summand intersects the above conormal bundle precisely if there is a covector in $WF(u)$ which is minus a covector contained in $WF(v)$. That this is not the case is precisely the assumption.

Properties

General

Proposition

(wave front set of product of distributions is inside fiber-wise sum of wave front sets)

Let $u,v \in \mathcal{D}'(X)$ be a pair of distributions satisfying Hörmander's criterion, so that their product of distributions $u \cdot v$ (def. ) exists by prop. . Then the wave front set of the product distribution is contained inside the fiberwise sum of the wave front set elements of the two factors:

$$WF(u \cdot v) \;\subset\; (WF(u) \cup (X \times \{0\})) + (WF(v) \cup (X \times \{0\})) \,.$$

(Hörmander 90, theorem 8.2.10)

Proof

By def. and prop. we have $u \cdot v = \Delta_X^\ast(u \otimes v)$. By this example the wave front set of the tensor product distribution $u \otimes v$ is

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

and by this prop. we have

$$WF(\Delta_X^\ast (u \otimes v)) \subset \Delta_X^\ast WF(u \otimes v) \coloneqq \left\{ (x, k_1 + k_2 ) \;\vert\; ((x,k_1),(x,k_2)) \in WF(u \otimes v) \right\} \,.$$

More generally:

Proposition

(partial product of distributions of several variables)

Let

$$K_1 \in \mathcal{D}'(X \times Y) \phantom{AAA} K_2 \in \mathcal{D}'(Y \times Z)$$

be two distributions of two variables. For their product of distributions to be defined over $Y$, Hörmander's criterion on the pair of wave front sets $WF(K_1), WF(K_2)$ needs to hold for the wave front wave vectors along $X$ and $Y$ taken to be zero.

If this is satisfied, then composition of integral kernels (if it exists)

$$(K_1 \circ K_2)(-,-) \;\coloneqq\; \underset{Y}{\int} K_1(-,y) K_2(y,-) dvol_Y(y) \;\in\; \mathcal{D}'(X \times Z)$$

has wave front set constrained by

$$WF'(K_1 \circ K_2) \;\subset\; WF'(K_1) \circ WF'(K_2) \;\cup\; (X \times \{0\}) \times WF'(K_2)_Z \;\cup\; WF(K_1)_X \times (Z \times \{0\}) \,,$$

where on the left the composition symbol means composition of relations of wave vectors over points in $Y$. Explicitly this means that

(1)$$WF(K_1 \circ K_2) \;\subset\; \left\{ (x,z, k_x, k_z) \;\vert\; \array{ \left( (x,y,k_x,-k_y) \in WF(K_1) \,\, \text{and} \,\, (y,z,k_y, k_z) \in WF(K_2) \right) \\ \text{or} \\ \left( k_x = 0 \,\text{and}\, (y,z,0,-k_z) \in WF(K_2) \right) \\ \text{or} \\ \left( k_z = 0 \,\text{and}\, (x,y,k_x,0) \in WF(K_1) \right) } \right\}$$

(Hörmander 90, theorem 8.2.14)

$\,$

• for continuity see at Hörmander topology

Non-existence of a global product

The fact that there is no general extension of multiplication to distributions (without condition on the wave front set) is a famous no-go theorem of Laurent Schwartz.

A quick way to see the problem is the following:

Let $H(x)$ be the Heaviside function, we clearly have

$$H(x) = H^{n} (x)$$

where the product on the right side is the product of classical functions. Applying differentiation and the product rule naivly results in a contradiction immediatly:

$$\delta(x) = n H^{n-1} (x) \delta(x)$$

The inconsistency of this product is detected by the wave front sets as follows: The wave front set both of the Heaviside distribution as well as of the delta distribution on the line is $\{(0,k) \vert k \neq 0\}$ (this example and this example). Therefore for both of these distributions mutliplication is only defined, according to def. , with a distribution $u$ for which there exists a bump function $b$ with $b(0) = 1$ such that $b \cdot u$ is again a bump function. This excludes the products of these distributions with themselves and with each other.

Examples

Definition

(product of a distribution with a non-singular distributions is product of a distribution with a smooth function)

The wave front set of a non-singular distribution $u_f$ corresponding to a smooth function $f \in C^\infty(\mathbb{R}^n)$, is empty (this prop.). Therefore the product of distributions (def. ) of a non-singular distribution with any distribution $u$ is defined, and given by the product of distributions with smooth functions:

$$u_f \cdot u = f \cdot u = u(f\cdot (-)) \,.$$

Related concepts

• product of a distribution with a smooth function

• convolution product of distributions

• Hadamard distribution

• Hörmander topology

• microcausal functional, Wick algebra

• operator-valued distribution

• Fourier transform of distributions

References

• Lars Hörmander, Fourier integral operators. I. Acta Mathematica 127, 79–183 (1971) (Euclid)

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

• Richard Melrose, chapter 1 of Introduction to microlocal analysis, 2003 (pdf)

• Michael Oberguggenberger, Products of distributions, Journal für die reine und angewandte Mathematik (1986) Volume: 365, page 1-11 (EuDML)

• Michael Oberguggenberger, Multiplication of Distributions and Applications to Partial Differential Equations, Longman 1992

Slides with exposition of the role of multiplication of distributions in renormalization of Feynman diagrams via causal perturbation theory/perturbative AQFT:

• Christian Brouder, Multiplication of distributions, 2010 (pdf)