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 ...

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 ...

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 ...

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 ...

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, ...

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=\...

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 ...

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 “...

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.

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 ...

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} \; , \; \...

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 ...

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 ...

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
...

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, ...

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 ...

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