nLab distributions are the smooth linear functionals

Idea

Historically, the theory of distributional densities (or just distributions, for short), conceived in functional analysis, was phrased in terms of continuous linear functionals on topological vector spaces of smooth functions. However, since smooth functions on smooth manifolds are the subject of differential geometry, and since spaces of smooth functions are naturally themselves generalized smooth spaces, it makes sense to ask whether distribution theory is actually a native topic to differential geometry.

In particular we may ask how distributions in the functional analytic sense relate to the smooth linear functions on smooth spaces of smooth functions. Indeed, with respect to the natural formulation of differential geometry via functorial geometry (topos theory) in terms of diffeological spaces, smooth sets etc. it turns out that distributional densities are equivalently the smooth linear functionals on smooth spaces of smooth functions.

This we discuss in the section

• Statement

below. To set the scene we briefly recall some concepts in

• Preliminaries.

Finally we discuss some

• Applications.

Preliminaries

Definition

(generalized smooth spaces)

In the following we write $\mathbf{H}$ for the category of diffeological spaces rather for the toposes of smooth sets or formal smooth set or super formal smooth sets etc. These form a sequence of full subcategory inclusions

$$SmoothManifolds \hookrightarrow DiffeologicalSpaces \hookrightarrow SmoothSets \hookrightarrow FormalSmoothSets \hookrightarrow SuperFormalSmoothSets \,.$$

The smallest of these categories of generalized smooth spaces that is needed for accomodating the spaces of smooth functions that distributions are defined on are diffeological spaces, and the reader may want to just focus on these. We mention these more general ambient categories mostly to amplify that the identificatin of distributions with smooth linear functionals remains true in these more general contexts.

Example

(smooth mapping spaces)

For $X \in \mathbf{H}$ any object, we write $[X,\mathbb{R}]$ for the mapping space (the internal hom). The underlying set is $C^\infty(X)$. If $X$ itself has $\mathbb{R}$-linear structure, we write

$$[X,\mathbb{R}]_{\mathbb{R}} \hookrightarrow [X,\mathbb{R}]$$

for the subobject of $\mathbb{R}$-linear maps.

Concretely, for $U$ a smooth manifold (or just a Cartesian space), then the sheaf $[X,\mathbb{R}]$ assigns (see at closed monoidal structure on presheaves for details)

$$[X,\mathbb{R}](U) = C^\infty(U \times X)$$

and $[X,\mathbb{R}](U) \subset C^\infty(U \times X)$ is the subset of those functions $\Phi_{(-)}(-)$ such that for all $u \in U$ the function $\Phi_u \colon X \to \mathbb{R}$ is $\mathbb{R}$-linear. The global elements $\Gamma(-)$ of the mapping space constitute the ordinary hom set

$$\Gamma [X,\mathbb{R}] \simeq \mathbf{H}(X,\mathbb{R}) \,.$$

If $X$ happens to be a compact smooth manifold then $C^\infty(X)$ carries the structure of a Fréchet manifold (see at manifold structure of mapping spaces). Under the full embdding of Fréchet manifolds into diffeological spaces, this coincides with $[X,\mathbb{R}]$ (see there).

Statement

Compactly supported distributions

Proposition

(compactly supported distributions are the smooth linear functionals)

For $n \in \mathbb{N}$, there is a natural bijection between the underlying sets of compactly supported distributions on $\mathbb{R}^n$ (this def.) and the $\mathbb{R}$-linear mapping space formed in the category $\mathbf{H}$ of generalized smooth spaces from def. :

$$\widetilde{(-)} \;\colon\; \mathcal{E}'(\mathbb{R}^n) \overset{\simeq}{\longrightarrow} \mathbf{H}([\mathbb{R}^n, \mathbb{R}], \mathbb{R})_{\mathbb{R}}$$

given by sending $\mu \in \mathcal{E}'(\mathbb{R}^n)$ to the natural transformation which on a test space $U \in$ CartSp takes a smoothly $U$-parameterized function $\Phi_{(-)}(-) \colon U \times \mathbb{R}^n \to \mathbb{R}$ to its evaluation in $\mu$ pointwise in $U$:

$$\tilde \mu(\Phi_{(-)})(u) \;\coloneqq\; \langle \mu, \Phi_{u}\rangle \,.$$

(Moerdijk-Reyes 91, chapter II, prop. 3.6)

Proof

First consider this for the case that $\mathbf{H} =$ SmoothSet (which immediately subsumes the case that $\mathbf{H} =$ DiffelogicaSpace).

To see that $\widetilde{(-)}$ is well defined, we need to check that the function

$$\array{ U &\overset{ \tilde \mu(\Phi_{(-)})}{\longrightarrow}& \mathbb{R} \\ u &\mapsto& \langle \mu, \Phi_u\rangle }$$

is smooth. But this follows immediately since $\langle \mu,-\rangle$ by definition is linear and continuous.

To see that $\widetilde{(-)}$ is indeed a bijection for each $U$ it remains that every $\mathbb{R}$-linear smooth functional (morphisms of smooth sets) of the form

$$A \;\colon\; [\mathbb{R}^n,\mathbb{R}] \longrightarrow \mathbb{R}$$

when restricted on global elements to a function of sets

$$A(\ast) \;\colon\; C^\infty(\mathbb{R}^n) \longrightarrow \mathbb{R}$$

is continuous with respect to the topological vector space structure from def. on the left.

Now by definition of the internal hom $A$ is actually “path-smooth” (this def.) and hence the statement is implied by this prop.

Finally to see that this argument generalizes to $\mathbf{H} =$ FormalSmoothSet observe that the Weil algebra of every infinitesimally thickened point is a quotient ring of an algebra of smooth functions on some Cartesian space (by the Hadamard lemma). The previous argument now applies to representatives under this quotient coprojection and one checks that it is independent of the representative chosen.

General distributions

For general distributions in the sense of continuous linear functionals on the space $\mathcal{D}(\mathbb{R}^n)$ of compactly supported smooth functions (bump functions) their equivalence to smooth linear functionals is discussed in (Kock-Reyes 04, section 2, esp. paragraph above theorem 2.3).

(…)

Applications

In quantum field theory

This has some interesting consequences. For instance a major application of distribution theory is perturbative quantum field theory, which, together with the rest of quantum physics is traditionally regarded to be a subject that makes heavy use of functional analysis. However, the development of algebraic quantum field theory with its emphasis of the abstract operator algebra of observables and de-emphasis of Hilbert spaces of quantum states (which is sometimes a convenient representation of the abstract operator algebra but more often just a distraction) combined with the development of perturbative quantum field theory, where even these operator algebras are replaced by formal star algebras of smooth functionals, has diminished the fundamental role of functional analysis in field theory. The theory of distributions survives as the tool of choice for understanding the singularity structure (or rather its absence) in the construction of the Wick algebra of quantum observables of the free fields and then in the process of renormalization via extension of distributions to the locus of coincident interaction points. But the quantum observables which are represented by these distributions are most naturally thought of as smooth functions on the space of field histories. Therefore the fact that distributions in fact are the smooth linear functionals shows that their appearance in perturbative quantum field theory is a tool for studying smooth functionals, not a concept fundamental to the theory as such.

Generalization to extensive quantities

The conception of distributions simply as elements of a double mapping space (internal hom) makes immediate that this concept exists in more generality than the traditional context. In Lawvere 86 it was suggested that distribution theory should be considered more abstractly from such a point of general “extensive quantities”. This is developed further in (Kock 11).

References

Discussion of distributions in terms morphisms out of internal homs in the smooth topos over the site of smooth loci is in

• Ieke Moerdijk, Gonzalo Reyes, around prop. II.3.6 of Models for Smooth Infinitesimal Analysis, Springer 1991

Discussion for the Cahiers topos is in

• Anders Kock, Gonzalo Reyes, Some calculus with extensive quantities: wave equation, Theory and Applications of Categories , Vol. 11, 2003, No. 14, pp 321-336 (TAC)

• Anders Kock, Gonzalo Reyes, Categorical distribution theory; heat equation (arXiv:math/0407242)

• Anders Kock, Commutative monads as a theory of distributions (arxiv/1108.5952)

using results of

• Alfred Frölicher, Andreas Kriegl, section 5 of Linear spaces and differentiation theory, Wiley 1988 (pdf)

and following the general conception of “intensive and extensive” in

• William Lawvere, Introduction to Categories in Continuum Physics, Lectures given at a Workshop held at SUNY, Buffalo 1982. Lecture Notes in Mathematics 1174. 1986

Similar sheaf theoretic discussion of distributions as morphisms of smooth spaces is in

• Frédéric Paugam, section 3.2 of Towards the mathematics of quantum field theory, 2012 (pdf)