nLab categorical analysis

Idea

One can base analysis on functional analysis, i.e., on the use of topological vector spaces of (generalized) functions. Completeness hypothesis give theorems like the fixed point theorem, the implicit function theorem which are fundamental tools to prove the existence and unicity of perturbatively small (small parameter) solutions of non-linear problems (e.g., partial differential equations). One also sometimes (more rarely) use related methods (compactness, etc…) to extend the local solutions to global ones. However, functional analytic methods introduce sometimes too much technical restrictions on the allowed functionals, which makes hard the presentation to mathematicians of the physicists’ methods in quantum field theory.

Another approach to analysis is given by functorial analysis, that is based on the functor of point approach to differential geometry on functional spaces. The basic idea of this approach is to use a very general class of partially defined function on functional spaces. It has the advantage of being adaptable to any non-linear (also fermionic) situation and seems particularly well suited to the study of non-perturbative quantum field theories.

The flexibility of this approach does not mean that one can avoid writing down inequalities and estimates, like in standard analysis: these are exactly the necessary steps to compute the definition domains of the objects in play.

A lot of works of physicists use a yoga of the type of categorical analysis, without computing the definition domains of the functionals in play: they just write down the formulas and compute with them. This could be a reason why mathematician often think that (perturbative) quantum field theory is not well grounded mathematically.

Example

Consider spaces modeled on smooth open subsets $U\subset \R^n$ for varying $n$, given by sheaves $U\mapsto X(U)$. Let $\pi:C=\R^3\times [0,1]\to [0,1]=M$ be a trivial bundle with coordinates $(t,x)$ corresponding to euclidean time and space coordinates. Let $L(t,x_\alpha)$ be a lagrangian density. The formula

$$S(x_u)=\int_M L(t,x_{\alpha,u}(t))dt$$

defines a partially defined functional

$$S:\Gamma(M,C)\to \R$$

whose definition domain $D\subset \Gamma(M,C)$ is given by Lebesgue’s dominated derivation theorem

$$D(U):=\{x_u\in \Gamma(M,C)(U),\;locally on U, there exists g\in L^1(M),\;|L(t,x_{\alpha,u}(t))|\leq g(t)\}$$

that is imposed to make $u\mapsto S(x_u)$ a smooth function of the parameter $u\in U$.

Related concepts

Functor of points

References

• Frédéric Paugam: see the related chapter of

  Towards the mathematics of quantum field theory.