# Classical field theory: marriage of differential geometry to functional analysis?

I have always wondered (and thoroughly - but perhaps not enough - searched for literature) how the purely geometrical formulation of classical field theory (take for example a hardcore text on this subject like https://www.amazon.com/Advanced-Classical-Theory-Giovanni-Giachetta/dp/9812838953/) can be merged into the functional analysis-based formulation of (non-interacting) quantum field theory in the Wightman axiomatization, where the "quantized fields" are Fock space operator-valued distributions. So I guess, it should suffice to "marry" functional analysis (more precisely the theory of distributions or, more generally, linear topological spaces) to the diff. geom. approach, and by this I mean define all vector spaces (sets of field configurations such as sections of vector bundles/principal/associated fiber bundles, jets and sections of jet bundles, etc.) as distribution spaces. Long story short: How exactly is the naïve relation

$$\left[\phi_A(x), \phi_B(y)\right]_{x^0 = y^0}^{PB} = f_{AB} \delta^3 (x ⃗ -y ⃗ )$$ for generic SL(2,C) indices A,B which requires the "classical fields" to be distributions merged with the functional analysis-free formulation of classical (gauge + matter + gravity) field theory which would have the same $\phi$-s as "fields"?

• The non-commuting quantum "operator-valued distributions" converge to commuting real/complex-valued distributions in the classical limit (in a suitable precise sense), and this provides the bridge between the quantum and classical theories (and we know how to do it quite satisfactorily at least for bosons). It is quite remarkable that in certain cases the classical limit theory is the naïvely expected one, despite at the quantum level non-perturbative renormalization procedures are performed, that modify the theory in a non-trivial way. – yuggib Aug 31 '17 at 7:55
• Do you have a reference (article, book) for this part: "The non-commuting quantum "operator-valued distributions" converge to commuting real/complex-valued distributions in the classical limit (in a suitable precise sense), and this provides the bridge between the quantum and classical theories (and we know how to do it quite satisfactorily at least for bosons)."? – DanielC Aug 31 '17 at 10:39
• There is some (actually not much) literature on this subject, in the mathematical physics community. A starting point reference could be the following: Ammari and Nier, 2008. – yuggib Aug 31 '17 at 12:11