Search Results

Disclaimer: This question has its motivation from physics. It is probably not entirely rigorous at the moment. I just want to clarify some steps and try to make the arguments rigorous afterwards, if p …

I started reading the paper Some Rigorous Results on the Sherrington-Kirkpatrick Spin Glass Model and I would like to clarify the notation $\langle \langle \cdot \rangle\rangle$ the authors use in th …

Let $A_{1},...,A_{n}$ be densely defined self-adjoint operators on a separable Hilbert space $\mathscr{H}$. Suppose these have a common dense domain $D\subset \mathscr{H}$ and satisfy commutation rela …

I was reading Mathematical Aspects of Quantum Field Theory by. E. de Faria and W. de Melo, and the following caught my attention. In (Hamiltonian) mechanics, the states of a system are described by p …

In his paper Constructive Renormalization Theory, V. Rivasseau describes the idea of Wilson's approach of solving path integrals step by step. In section 1.4, page 5, however, there is a statement whi …

I was reading this post from PSE and it reminded me an old question of mine, in which the use of creation and annihilation operators were discussed. Both questions got answers which agreed on the fact …

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 …

One of the most famous models in physics is the Ising model, invented by Wilhelm Lenz as a PhD problem to his student Ernst Ising. The one-dimensional version of it was solved in Ising's thesis in 192 …

Everytime I want to understand a little more about the ideas behind Renormalization Group techniques, I get troubled by a gap between the general picture one usually presents (e.g. in books or pedagog …

Let $\mathscr{H}$ be a complex Hilbert space and denote $\mathscr{H}_{n}$ the tensor product $\overbrace{\mathscr{H}\otimes\cdots\otimes\mathscr{H}}^{\text{n}}$. Denote by $\Pi_{\pm}$ the projection o …

This is somehow a more mature version of an old question of mine. I'd like to have a more clear picture of the difference between QFT and QM from a mathematical point of view. Okay, so we begin with a …

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 …

Let $m > 0$ be fixed. If $x=(x_{0},x_{1},x_{2},x_{3})$ and $y = (y_{0},y_{1},y_{2},y_{3})$ are elements of $\mathbb{R}^{4}$, we denote the Lorentz inner product by: $$ x\cdot \tilde{y} := x_{0}y_{0}-x …

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 …

In section X.7 of Reed & Simon's book there is a nice rigorous construction of the free scalar field theory which applies to the Klein-Gordon field. Question: Are there references which discuss, in an …