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Let $\mathcal{H}$ be a finite-dimensional inner product space over $\mathbb{C}$. Suppose $A_{1},...,A_{N}$ are linear operators on $\mathcal{H}$ such that: $$\{A_{i},A_{j}\} = 0 \quad \mbox{and} \quad …

Let $\Omega_{0}:=\{-1,1\}$ be a single spin space. If $\Lambda \subset \mathbb{Z}^{d}$ is a fixed finite set, take $\mathcal{F}_{0}$ to be the $\sigma$-algebra $2^{\Omega_{0}}$ on $\Omega_{0}$. We def …

I'm looking for rigorous discussions on the derivation of the Euler-Lagrange equation for field as it is usually discussed in classical field theory books. More precisely, if the action is given by: $ …

This is actually a more elaborated version of a previous question of mine, which is now deleted. First, some quick notations: (1) $\Omega_{0} := \{-1,1\}$ and $\mathcal{F}_{0} := 2^{\Omega_{0}}$ are, …

Let $\mathscr{H}$ be a complex Hilbert space and $A$ be an Hermitian operator $A: \mathscr{H}\to \mathscr{H}$. Suppose, for a moment, that $A$ has a set of discrete eigenvalues $\{\lambda_{n}\}_{n\in …

This is a second part of my previous question. I'm trying to figure it out by myself how to deduce Hamilton's equations in classical field theory as it is usually obtained in physics books. Notation: …

I have been taking a look at some papers in constructive quantum field theory and I got the impression that there is a systematic of estimating things like e.g the effective action or the free energy …

In QFT, one usually talks about operator-valued distributions. But let's take, for instance, $L^{2}(\mathbb{R}^{3})$ and its associated Fock space $\mathcal{F} = \bigoplus_{n=0}^{\infty}L^{2}(\mathbb{ …

Let $\mathscr{H}_{1}$ and $\mathscr{H}_{2}$ be Hilbert spaces. If $\psi_{1}\in \mathscr{H}_{1}$ and $\psi_{2}\in \mathscr{H}_{2}$, define $\psi_{1}\otimes \psi_{2}$ to be a function on $\mathscr{H}_{1 …

I was reading Leon Takhtajan's book on quantum mechanics and, at some point, he states the J. von Neumann Theorem. The first part of this theorem is as follows. For every self-adjoint operator $A$ on …

Let $\varepsilon > 0$, $L \gg 1$ and define the torus $\mathbb{T} = \varepsilon \mathbb{Z}^{d}/L\mathbb{Z}^{d}$. Let $K$ be a smooth, strictly decreasing function. To make things easier, consider an e …

I was reading Constructive Renormalization Group by V. Rivasseau and I got some points which I would like to clarify. On page 2, Rivasseau talks about the large field problem and, if I understood it c …

I was reading these short notes on the Borel functional calculus where the author discusses the uniqueness property of this calculus for both bounded and unbounded self-adjoint operators. When it come …

This is basically the same question I made on physics stack exchange The spectrum of the Hamiltonian in quantum mechanics, but I got no answers so far and decided to move it to mathoverflow with some …

In his book The Principles of Quantum Mechanics, Dirac states: "We call a real dynamical variable whose eigenstates form a complete set an observable." To Dirac, any observable has a complete set of …