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20 votes
Accepted

Formal mathematical definition of renormalization group flow

The renormalization group (RG) as a geometric flow (like the Ricci flow) is a very special case of the RG, namely, the one corresponding to the nonlinear sigma-model (NLSM) in two dimensions with ...
Abdelmalek Abdesselam's user avatar
11 votes

Formal mathematical definition of renormalization group flow

Classical field theories (Lagrangian variational principles) sometimes come in families. The families may be finite dimensional, or also infinite dimensional. One could even take the family to consist ...
Igor Khavkine's user avatar
10 votes

Simple example of renormalization

The simplest and earliest example I know regarding the renormalization group idea is the following. Suppose we want to study some feature $\mathcal{Z}(\vec{V})$ of some object $\vec{V}$ which is in a ...
Abdelmalek Abdesselam's user avatar
9 votes

Renormalization in physics vs. dynamical systems

The renormalization approach to dynamical systems pioneered by Chen, Goldenfeld and Oono [1] applies the Gell-Mann and Low renormalization group from quantum physics [2] to extract the global behavior ...
Carlo Beenakker's user avatar
7 votes

Non-perturbative Renormalization in the sense of Polchinski's equation. Do we have a mathematical formulation?

Good question! Before going further in your investigations on rigorous nonperturbative implementations of the renormalization group (RG) philosophy used for the construction of QFTs in the continuum, ...
Abdelmalek Abdesselam's user avatar
5 votes
Accepted

Wick product of free fields and wave front sets in the sense of Lars Hörmander

The answer to both questions is no. This is due to two facts: The Klein-Gordon two-point distribution $\omega_2(x,y)=\langle\Omega,\phi(x)\phi(y)\Omega\rangle$ in $\mathbb{R}^4$, where $\Omega_1=\...
Pedro Lauridsen Ribeiro's user avatar
5 votes

From the conceptual idea of the RG to its actual implementation

Q: "What is one looking for in a typical RG research problem?" One typically hopes to find that the combination of coarse-graining (e.g. by removing high wave number components) and ...
Carlo Beenakker's user avatar
4 votes

Renormalization in physics vs. dynamical systems

There’s a bit of a terminological collision going on here. Physicists often use the term “renormalization” to refer the process of removing infinities from QFT calculations and to renormalization or “...
Aaron Bergman's user avatar
4 votes
Accepted

Is the underlying set of every renormalization group countable and finite?

No, the renormalization group of a continuum field theory contains continuously parameterized scale-changing transformations—hence an uncountable number of them.
Buzz's user avatar
  • 1,372
3 votes

Is there any case of remormalization in which we have to solve it by ways in two different systems?

A charitable reading suggests that you are referring to the $\zeta$ function regularization $$ \sum_{i=1}^\infty i = \lim_{s\to -1} \sum_{i=1}^\infty i^{-s}=\zeta(-1)=-\frac{1}{12} $$ which occurs in ...
gmvh's user avatar
  • 2,798
3 votes

Some identities with the Riemann-Hurwitz zeta function

The definition of the Hurwitz-Riemann $\zeta$ function is: \begin{eqnarray} \zeta(s, x) = \sum_{n=0}^{\infty} \frac{1}{(n+x)^s} \quad , \quad x>0 \end{eqnarray} with $s \in \mathbb{C} \; , \; \...
Herman Jaramillo's user avatar
2 votes
Accepted

How can one recover/obtain information from the renormalization group procedure?

The limiting function $V^\ast$ is such that any further convolutions of $e^{-V^\ast}$ with $\mu$ return $e^{-V^\ast}$, so $Z^\ast(\phi)=e^{-V^\ast(\phi)}$. To obtain critical properties, you need the ...
Carlo Beenakker's user avatar
2 votes
Accepted

Cluster expansion, Mayer expansion and perturbative renormalization group

I think there is a misunderstanding here on what is the expansion parameter. Perturbative renormalization expands in a power series of the interaction strength (the coupling parameter $\lambda$); this ...
Carlo Beenakker's user avatar
1 vote
Accepted

The ultraviolet limit as a limiting case of the renormalization group flow?

The telescopic argument you mentioned is incorrect. If you carefully look at Rivasseau's notations, you will see that the first term $C^{0}(p)$ in the sum $$ \sum_{j=0}^{\rho}C^{j}(p) $$ is defined as ...
Abdelmalek Abdesselam's user avatar
1 vote
Accepted

Singular Radon probabilities on $[0,1]^d$. Is conditioning on $x_i = \alpha$ well-defined?

$\newcommand{\B}{\mathcal B}$ First here, $[0,1]^d$ is a Polish space (i.e., a separable complete metric space). So, $[0,1]^d$ is a Radon space, and hence any (Borel) probability measure is Radon. So, ...
Iosif Pinelis's user avatar
1 vote

Why does the type-A subdivision algebra look like the Rota-Baxter algebra axiom?

This is too long for a comment. The reason for my question in comments was as follows. In some cases (Gerstenhaber algebras, Poisson algebras, etc.) an operad describing a certain algebraic structure ...
Vladimir Dotsenko's user avatar

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