Everything below is to be viewed in the Euclidean setting with $d$ dimensions and all measures are understood to be Borel measures.

The usual problem of Quantum Field Theory is to make sense of Feynman Path Integrals like

\begin{equation} \int \exp \left[- \frac{m^2}{2} \int \phi^2 + \frac{Z}{2} \int \phi \Delta \phi - \lambda \int \phi^4 \right] \mathrm{d} \phi \end{equation}

where the outer integral is supposed to run over some space of test functions. It is well-known that this does not really work. What one can do, however, is for $\lambda = 0$ to produce a "cylinder set probability measure" on e.g the space $\mathcal{S}$ of Schwartz functions. More generally this works for any continuous positive definite bilinear form on $\mathcal{S}$ and the resulting cylinder set measure may be pushed forward to $\mathcal{S}'$ where it miraculously becomes a true Gaussian probability measure. Hence, for such theories we know what we are doing!

With $\lambda \neq 0$ this is no longer the case. In order to ensure the finiteness of expressions of the form $\int \phi^4$ we would like $\phi$ to live in $\mathcal{S}$. To this end, we pick two non-negative mollifiers $\chi, \xi \in \mathcal{D}$ with $\xi \left( 0 \right) = 1$. Furthermore, for any $\epsilon > 0$ and $\Lambda > 0$, define

\begin{aligned} \chi_\epsilon \left( x \right) &= \epsilon^{-d} \chi \left( \frac{x}{\epsilon} \right) \\ \xi_\Lambda \left( x \right) &= \xi \left( \frac{x}{\Lambda} \right) \\ \end{aligned}

and consider the continuous (? - it should certainly be Borel measurable) mapping

\begin{aligned} \mathcal{M} \left( \xi_\Lambda \right) \mathcal{C} \left( \chi_\epsilon \right) : \mathcal{S}' &\to \mathcal{S} \\ T &\mapsto \xi_\Lambda \cdot \left( \chi_\epsilon \ast T \right) \, . \end{aligned}

We may now take a Gaussian measure $\mu$ on $\mathcal{S}'$ corresponding to some bilinear theory and regularize it as $\left[ \mathcal{M} \left( \xi_\Lambda \right) \mathcal{C} \left( \chi_\epsilon \right) \right]_\ast \mu$ to a measure on $\mathcal{S}$. One can then define measures

\begin{equation} \nu_{\epsilon, \Lambda} = \exp \left[ -\lambda \int \phi^4 \right] \cdot \left[ \mathcal{M} \left( \xi_\Lambda \right) \mathcal{C} \left( \chi_\epsilon \right) \right]_\ast \mu \end{equation}

normalize them, push them to $\mathcal{S}'$ again and study their behaviour as $ \left( \epsilon, \Lambda \right) \to \left( 0, \infty \right)$. But this also does not work as it just produces the expected divergences coming from attempting to multiply distributions. Instead, one has to give up on keeping the model parameters $m, Z, \lambda$ constant and let them depend on $\epsilon$ and $\Lambda$ (i.e on cutoffs) instead. One might then hope that for some nice $\left( \epsilon, \Lambda \right)$-dependence a limit (or at least a cluster point) $\nu_{0, \infty}$ indeed exists.

From a Wilsonian point of view, we would however like to split this $\nu_{0, \infty}$ into "fast" and "slow" modes depending on $\epsilon$ and $\Lambda$. That is, we want to fix a collection $\left( \mu_{\epsilon, \Lambda} \right)_{\epsilon \ge 0, \Lambda \le \infty}$ of probability measures on $\mathcal{S}'$ such that for all $\epsilon, \delta > 0$ and all $\Lambda, \kappa > 0$

• $\nu_{0, \infty}$ factorizes over $\mu_{\epsilon, \Lambda}$, i.e $\nu_{0, \infty} = \tilde{\nu}_{\epsilon, \Lambda} \ast \mu_{\epsilon, \Lambda}$ for some probability measure $\tilde{\nu}_{\epsilon, \Lambda}$ on $\mathcal{S}'$
• $\mu_{\epsilon, \Lambda} \ast \mu_{\delta, \kappa} = \mu_{\epsilon+\delta, \frac{\lambda \kappa}{\Lambda + \kappa}}$ (or someway similar)
• $\lim_{\epsilon \to 0, \Lambda \to \infty} \mu_{\epsilon, \Lambda} = \mu_{0, \infty}$

where somehow $\nu_\epsilon$ and $\tilde{\nu}_\epsilon$ should be related. Then we obtain a renormalization group equation

\begin{equation} \tilde{\nu}_{\epsilon, \Lambda} = \tilde{\nu}_{\epsilon + \delta, \frac{\lambda \kappa}{\Lambda + \kappa}} \ast \mu_{\delta, \kappa} \end{equation}

as well as

\begin{equation} \tilde{\nu}_{0, \infty} = \mu_{0, \infty} \ast \lim_{\epsilon \to 0, \Lambda \to \infty} \tilde{\nu}_{\epsilon, \Lambda} \end{equation}

But, presuming the existence of some nice QFT corresponding to $\nu_0$

• Why would such a decomposition with respect to a such a family $\mu_{\epsilon, \Lambda}$ exist? i.e why can we separate fast and slow modes in an arbitrary way?
• How do we relate $\nu_{\epsilon, \Lambda}$ and $\tilde{\nu}_{\epsilon, \Lambda}$ i.e how do we understand the map from bare to renormalized quantities and/or vice versa?

PS: The above is pretty much as far as my level of mathematics goes. Please try to not go too much further.