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Let us consider the Minkowski space $(\mathbb{R}^{4},\eta)$ and the mass shell $H_{m}$, $m\ge 0$, given by: \begin{eqnarray} H_{m}:=\{x=(x_{0},x_{1},x_{2},x_{3}) \in \mathbb{R}^{4}: \hspace{0.1cm} x\c …

Statistical mechanics is all about taking thermodynamic limits and, as far as I know, there are more than one way to define such limits. Consider the following theorem: Theorem: In the thermodynamic l …

In what follows, I'm following Folland's book and Reed & Simon. Notation: Points in $\mathbb{R}^{4}$ are denoted by $p =(p_{0},p_{1},p_{2},p_{3})$. Also, I'm using Reed & Simon's notation for the Lore …

This is probably a very basic question but I tried physics stack exchange already and I got no answers, so I'm asking the same question here. I was reading this article and the author considers the fo …

I'm sorry if this is not the right place to ask this question but I've been struggling with this for days now (and I think this is too technical/specific for math stack). Notation: A conjugation $C$ o …

I think I figured it out. Let $f \in \mathscr{H}$ be arbitrary, where $\mathscr{H}$ here is a complex Hilbert space. Then $f$ can be written uniquely as: $$f = f_{1} + if_{2} $$ where $f_{1},f_{2} \in …

One way to approach QFT in mathematical terms is provided by the so-called Gårding-Wightman axioms, which defines in rigorous mathematical terms what a quantum field theory is supposed to be. If I'm n …

Let $\Lambda \subset \mathbb{Z}^{d}$ be a finite set and $\varphi = (\varphi_{x})_{x\in \Lambda} \in \mathbb{R}^{|\Lambda|}$. Let $F^{\Lambda}=F^{\Lambda}(\varphi)$ be a real-valued global function, b …

This is related to my previous question. I'm having some problems understanding the local to global program discussed in Brydge's lecture notes. We are assuming $C=C_{1}+\cdots+C_{N}$ is a covariance …

I have already addressed this problem on my previous question but I still have trouble understanding Brydges' RG maps on his lecture notes, so I'll try to elaborate my question a little better. Let $ …

In the past months, I've been trying to understand the so-called Sine-Gordon transformation, so I've posted some questions here about this topic. I also did an extensive research about this subject, s …

Before introducing block spin transformations in chapter four of Random Walks, Critical Phenomena and Triviality in Quantum Field Theory, the authors state the following: "In this chapter we sketc …

Mayer expansions and the Hamilton–Jacobi equation by D. Brydges and T. Kennedy begins mentioning that many problems in statistical mechanics and QFT center on the analysis of integrals of the form: \b …

Let me discuss two possible constructions of Gaussian measures on infinite dimensional spaces. Consider the Hilbert space $l^{2}(\mathbb{Z}^{d}) := \{\psi: \mathbb{Z}^{d}\to \mathbb{R}: \hspace{0.1cm} …

I've came across an identity once (I don't remember where) concerning convolutions of Gaussian measures. If I'm not mistaken, this identity was \begin{eqnarray} (\mu_{C}*f)(y) = \exp\bigg{[}\frac{1}{ …