SRQ seminar – December 7th, 2018

Notes from a talk in the SRQ series.

SRQ Seminar – Kotecky 20181207

Astract framework for nonperturbative renormalization

Kotecky, Preiss (1979) Ceck Jour of Phys. B

(cited by Vershik, Russian Math Surevys 72:2)

All is based on the notion of projective (inverse) limits.

Topological spaces

An example, for topologicaly spaces. \(X\) is a top space, \(\phi\) a morphism (a continuous map). Directed poset \(I\) (i.e. partially ordered set).

A projective system: a collection of spaces \((X_{\alpha})_{\alpha \in I}\) with morphisms whenever \(\alpha < \beta < \gamma\) which do:

\(\displaystyle X_{\alpha} \xleftarrow{\Phi_{\alpha, \beta}} X_{\beta} \xleftarrow{\Phi_{\beta \gamma}} X_{\gamma}\)

and \(\Phi_{\alpha, \beta} \circ \Phi_{\beta, \gamma} = \Phi_{\alpha, \gamma}\).

A projective limit is a topological space \(X\) with projections \((\pi_{\alpha})_{\alpha \in I}\) such that

\(\displaystyle \raisebox{-0.458830437351804\height}{\includegraphics[width=2.5163813459268cm,height=2.19223074904893cm]{image-1.pdf}}\)

and whenever there you have another pair \((Y, (\vartheta_{\alpha})_{\alpha})\) satisfying the same property one has a unique map \(\phi : Y \rightarrow X\) such that \(\vartheta_{\beta} = \phi \circ \pi_{\beta}\).

\(X\) is made of sequences \(x = (x_{\alpha})_{\alpha}\) such that when \(\alpha \leqslant \beta\) one has \(x_{\alpha} = \phi_{\alpha, \beta} (x_{\beta})\) and is unique up to isomorphisms.

Measurable spaces

To make sense of projective limit for measurable spaces is an old story. (Frolik, Rao, in the '70)

Consider a measurable space \((X, \mathcal{F})\) with points in \(\mathcal{F}\), i.e. for all \(x \in X\) we have \(\{ x \} \in \mathcal{F}\). Introduce the set of \(\sigma\)–additive, finite, measures \(M (X)\) on \((X, \mathcal{F})\) and \(\mathcal{B} (X)\) the set of measurable bounded functions.

Morphisms \(\phi : M (X) \rightarrow M (Y)\) with the properties: (i) linear, non-negative, contractive (\(\phi (\mu) (Y) \leqslant \mu (X)\)) (ii) continuous wrt the weak topologies.

Then the projective limit exists (by the results cited above)

\(\displaystyle (X, \pi_{\alpha}) = \lim_{\leftarrow} (X_{\alpha}, \phi_{\alpha, \beta})\)

and \(\pi_{\alpha} (\mu) \in M (X_{\alpha})\) for \(\mu \in M (X)\).

Take a non-empty \(Y\) and let \(\vartheta_{\alpha} (\rho) = \rho (Y) \mu_{\alpha}\) so we have the commutative diagram above and there exists \(\phi : \rho \mapsto \mu\).

Gibbs measures

\(S\) spin space, \(\Omega = \Pi_{x \in \mathbb{Z}^d} S\), \(\Omega_{\Lambda} = \Pi_{x \in \Lambda} S\), consider the Gibbs measure

\(\displaystyle \nu_{\lambda}^W (u) = \frac{e^{- \beta H_{\Lambda} (u|w)}}{\sum_{v \in \Omega_{\Lambda}} e^{- \beta H_{\Lambda} (v|w)}} .\)

To get all the Gibbs mesures we consider a sequence of volumes and measures on the boundary conditions \((\Lambda_n, \mu_n \in M (\Omega_{\Lambda^c}))\) and consider

\(\displaystyle \mu = \lim_{n \rightarrow \infty} \int_{\Omega_{\Lambda^c_n}} \underbrace{\left( \nu_{\Lambda_n}^{w_{\Lambda_n^c}} \otimes_{\Lambda_n} \delta_{w_{\Lambda_n^c}} \right)}_{Q_{\Lambda_n} (w, \mathrm{d} u)} \mathrm{d} \mu_n (w_{\Lambda_n^c})\)

The kernels \(Q_{\Lambda_n}\) are the specification. They are compatible: \(Q_{\Lambda} = Q_{\Lambda} Q_V\) whenever \(V \subset \Lambda\), i.e.

\(\displaystyle Q_{\Lambda} (w, f) = Q_{\Lambda} (w, Q_V (\cdot, f)) .\)

DLR says that a Gibbs state satisfies \(\mu = \mu Q_{\Lambda}\) for any \(\Lambda\).

\(\displaystyle \Omega_{V^c} \xleftarrow[P_{V, \Lambda}]{} \Omega_{\Lambda^c}\)

Take the measure \(\mu_{\Lambda}\) and construct

\(\displaystyle \int_{\Omega_{\Lambda^c}} \left( \nu_{\Lambda}^{w_{\Lambda_n^c}} \otimes_{\Lambda} \delta_{w_{\Lambda^c}} \right) \mathrm{d} \mu_n (w_{\Lambda^c})\)

and then forget the part inside of \(V\). So we take the marginal on \(V^c\) getting a mesure

\(\displaystyle P_{V, \Lambda} \mu_{\Lambda} = \int_{\Omega_V} \int_{\Omega_{\Lambda^c}} \left( \nu_{\Lambda}^{w_{\Lambda_n^c}} \otimes_{\Lambda} \delta_{w_{\Lambda^c}} \right) \mathrm{d} \mu_n (w_{\Lambda^c}) \in M (\Omega_{V^c}) .\)

The set of Gibbs states is the projective limit of this system.

QFT

Consider now a finite volume approximation of \(\phi^4_d\) on a lattice with spacing \(\delta\) and size \(\delta L\).

The aim is to construct a measure in \(Y =\mathcal{S}' (\mathbb{R}^d)\). Denote \(\alpha = (\delta, L)\) and consider the spaces \(Y_{\alpha}\) for the finite dimensional approximations. We have morphisms

\(\displaystyle T_{\alpha} : Y_{\alpha} \rightarrow Y\)

which allow to think all the approximation living in the same space.

\(\displaystyle \nu_{\alpha}^{(Z, m, \lambda)} = \frac{1}{Z} e^{- \left[ \sum_{x \sim y} Z \delta^{d - 2} (\varphi_x - \varphi_y)^2 + m \sum_x \delta^d \varphi_x^2 + \sum_x \delta^d \varphi_x^4 \right]} \prod_x \mathrm{d} \varphi_x .\)

The usual task is to take the limit of the sequence \(\nu_{\alpha}^{(Z_{\alpha}, m_{\alpha}, \lambda_{\alpha})}\) for any possible sequences of parameters \((Z_{\alpha}, m_{\alpha}, \lambda_{\alpha})\). I do not put restrictions on these parameters.

I say that the theory is renormalizable if the set of possible limits is a finite dimensional manifold.

We consider as spaces

\(\displaystyle Y_{\alpha} = \{ (Z, m, \lambda) \in \mathbb{R}^3 \}\)

and as morphisms (with \(\mu \in M (Y_{\alpha})\)) into probabilities in \(Y =\mathcal{S}' (\mathbb{R}^d)\) with

\(\displaystyle T_{\alpha} \mu = \int \nu_{\alpha}^{(Z, m, \lambda)} \mathrm{d} \mu (Z, m, \lambda) .\)

\(T_{\alpha}\) is the unique morphism extending \((Z, m, \lambda) \mapsto \nu_{\alpha}^{(Z, m, \lambda)}\) on \(Y =\mathcal{S}' (\mathbb{R}^d)\).

In the Gibbs situtation, the mappings \(T_{\alpha}\) are given by the integrals \(\int (\nu \otimes \delta) \mathrm{d} \mu\) as measures on \(\Omega\) and (i) \(\phi_{\alpha, \beta} = P_{V, \Lambda}\) and the projective limit \((X, \pi_{\Lambda})\), (ii) \(T_{\alpha} (\phi_{\alpha} (\mu))\) converges once \(\mu\) is a prob on \(X\) (by the DLR condition it actually do not depends on \(\alpha\)) and we can introduce the mapping \(T : M (X) \ni \mu \mapsto (T_{\alpha} (\phi_{\alpha} (\mu)))_{\alpha}\), (iii) the set of Gibbs states is simply \(\{ T \mu : \mu \in M (X) \}\).

Let's go back to field theory. Recall that \(Y_{\alpha} = \{ (Z, m, \lambda) \} \subseteq \mathbb{R}^3\) and \(\phi_{\alpha, \beta}\) is a mapping between \(M (Y_{\beta})\) into \(M (Y_{\alpha})\). We need to consider only sequences \((\alpha_n)_n\) where \(\alpha_{n + 1}\) is a refinement of \(\alpha_n\).

\(\displaystyle \phi_{\alpha_n \alpha_{n + 1}} : \delta_y \in M (Y_{\alpha_{n + 1}}) \rightarrow \mu \in (Y_{\alpha_n})\)

where the distance of \(\delta_y\) and \(\mu\) is minimal (as measures on \(\mathcal{S}' (\mathbb{R}^d)\)), i.e. \(T_{\alpha_n} \mu\) and \(T_{\alpha_{n + 1}} (\delta_y)\) have minimal distance.

It seems then that (i) and (ii) should be doable. And (iii) is totally unclear how to do.