Recently I have been reading Kevin Costello's book (draft) Renormalization of Quantum Field Theories, which claims to work out some foundations of perturbative quantum field theory following the "Wilsonian philosophy". I don't understand this stuff well enough to really say much, and hopefully someone else can say more; but I think the basic idea is to, instead of doing integrals over infinite dimensional spaces such as $C^\infty(M)$, do integrals over finite dimensional "approximations" of these infinite dimensional spaces, for example, the space of functions $C^\infty(M)_{\leq \Lambda}$ of energy $\leq \Lambda$, where $\Lambda$ is some constant. I think energy $\leq \Lambda$ means you take the Laplacian of $M$ (corresponding to a Riemannian metric, which is probably fixed from the beginning), and you take eigenfunctions of the Laplacian corresponding to eigenvalues $\leq \Lambda$. (Someone should correct me if I'm wrong.) Then, a low energy theory should be related in an appropriate way to (indeed it should be determined by) the higher energy theories.

I might be wrong, but my impression is that it is "impossible" to make a rigorous definition of the path integral: There are various problems with defining the appropriate measures on infinite dimensional spaces. Therefore, if we wish to make path integrals "rigorous", we must find some other means to define it, or find some alternative "roundabout" solution, such as the Wilsonian idea. But again I am not an expert on this; these are just my (very) naive impressions.

There is also the Atiyah-Segal axiomatization of (topological) quantum field theory. Perhaps this can also be viewed as a "roundabout" solution to "defining" the path integral: It avoids having to define path integrals, and instead axiomatizes the properties that should hold if the path integral could be rigorously defined. Check out Atiyah's original paper and Segal's notes. One way that higher categorical stuff arises is via the "locality" property/assumption of (T)QFTs. For more on this, see for example Jacob Lurie's paper on TFTs (available on his webpage), and the references therein.

Recently I have been reading Kevin Costello's book (draft) Renormalization of Quantum Field Theories, which claims to work out some foundations of perturbative quantum field theory following the "Wilsonian philosophy". I don't understand this stuff well enough to really say much, and hopefully someone else can say more; but I think the basic idea is to, instead of doing integrals over infinite dimensional spaces such as $C^\infty(M)$, do integrals over finite dimensional "approximations" of these infinite dimensional spaces, for example, the space of functions $C^\infty(M)_{\leq \Lambda}$ of energy $\leq \Lambda$, where $\Lambda$ is some constant. I think energy $\leq \Lambda$ means you take the Laplacian of $M$ (corresponding to a Riemannian metric, which is probably fixed from the beginning), and you take eigenfunctions of the Laplacian corresponding to eigenvalues $\leq \Lambda$. (Someone should correct me if I'm wrong.) Then, a low energy theory should be related in an appropriate way to (indeed it should be determined by) the higher energy theories.

I might be wrong, but my impression is that it is "impossible" to make a rigorous definition of the path integral: There are various problems with defining the appropriate measures on infinite dimensional spaces. Therefore, if we wish to make path integrals "rigorous", we must find some other means to define it, or find some alternative "roundabout" solution, such as the Wilsonian idea. But again I am not an expert on this; these are just my (very) naive impressions.

There is also the Atiyah-Segal axiomatization of (topological) quantum field theory. Perhaps this can also be viewed as a "roundabout" solution to "defining" the path integral: It avoids having to define path integrals, and instead axiomatizes the properties that should hold if the path integral could be rigorously defined. Check out Atiyah's original paper and Segal's notes. One way that higher categorical stuff arises is via the "locality" property/assumption of (T)QFTs. For more on this, see for example Jacob Lurie's paper on TFTs (available on his webpage), and the references therein.

Recently I have been reading Kevin Costello's book (draft) Renormalization of Quantum Field Theories, which claims to work out some foundations of perturbative quantum field theory following the "Wilsonian philosophy". I don't understand this stuff well enough to really say much, and hopefully someone else can say more; but I think the basic idea is to, instead of doing integrals over infinite dimensional spaces such as $C^\infty(M)$, do integrals over finite dimensional "approximations" of these infinite dimensional spaces, for example, the space of functions $C^\infty(M)_{\leq \Lambda}$ of energy $\leq \Lambda$, where $\Lambda$ is some constant. I think energy $\leq \Lambda$ means you take the Laplacian of $M$ (corresponding to a Riemannian metric, which is probably fixed from the beginning), and you take eigenfunctions of the Laplacian corresponding to eigenvalues $\leq \Lambda$. (Someone should correct me if I'm wrong.) Then, a low energy theory should be related in an appropriate way to (indeed it should be determined by) the higher energy theories.

Recently I have been reading Kevin Costello's book (draft) Renormalization of Quantum Field Theories, which claims to work out some foundations of perturbative quantum field theory following the "Wilsonian philosophy". I don't understand this stuff well enough to really say much, and hopefully someone else can say more; but I think the basic idea is to, instead of doing integrals over infinite dimensional spaces such as $C^\infty(M)$, do integrals over finite dimensional "approximations" of these infinite dimensional spaces, for example, the space of functions $C^\infty(M)_{\leq \Lambda}$ of energy $\leq \Lambda$, where $\Lambda$ is some constant. I think energy $\leq \Lambda$ means you take the Laplacian of $M$ (corresponding to a Riemannian metric, which is probably fixed from the beginning), and you take eigenfunctions of the Laplacian corresponding to eigenvalues $\leq \Lambda$. (Someone should correct me if I'm wrong.) Then, a low energy theory should be related in an appropriate way to (indeed it should be determined by) the higher energy theories.

I might be wrong, but my impression is that it is "impossible" to make a rigorous definition of the path integral: There are various problems with defining the appropriate measures on infinite dimensional spaces. Therefore, if we wish to make path integrals "rigorous", we must find some other means to define it, or find some alternative "roundabout" solution, such as the Wilsonian idea. But again I am not an expert on this; these are just my (very) naive impressions.

There is also the Atiyah-Segal axiomatization of (topological) quantum field theory. Perhaps this can also be viewed as a "roundabout" solution to "defining" the path integral: It avoids having to define path integrals, and instead axiomatizes the properties that should hold if the path integral could be rigorously defined. Check out Atiyah's original paper and Segal's notes. One way that higher categorical stuff arises is via the "locality" property/assumption of (T)QFTs. For more on this, see for example Jacob Lurie's paper on TFTs (available on his webpage), and the references therein.