Questions tagged [mathematical-physics]

DO NOT USE THIS TAG just because your question involves math! If your question is on simplification of a mathematical expression, please ask it at math.stackexchange.com. Mathematical physics is the mathematically rigorous study of the foundations of physics, and the application of advanced mathematical methods to problems in physics. Examples include partial differential equations (PDEs), variational calculus, functional analysis, and potential theory.

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Question about the identity operator and the bosonic ladder operators

Consider a self-adjoint operator $B$, such that for each mode $a_1,...,a_n$ [of a quantum bosonic system with Hilbert space $\cal H$ given by the corresponding Fock space] we have $B a_i B^\dagger = ...
Noobgrammer's user avatar
1 vote
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An entropy-like function on probability mass functions on finitely many points

This question is inspired by the Atiyah problem on configurations of points. If we denote by $C_n(\mathbb{R}^3)$ the (ordered) configuration space of $n$ distinct points in $\mathbb{R}^3$, then there ...
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7 votes
1 answer
302 views

Determining Bound States from Møller Operator

Hello I came across an interesting property of the Møller operator, which I summarize below: The Møller operator $\Omega^{(+)}$ maps in-states that belong to the continuum spectrum of the free ...
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2 votes
1 answer
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Equivalent definitions of Wick ordering

Let $\phi$ denote a field consisting of creation and annihilation operators. In physics, the Wick ordering of $\phi$, denoted $:\phi:$, is defined so that all creation are to the left of all ...
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Scattering matrix vs. unitary transformations

In quantum optics, the input/output bosonic modes at a beam splitter transform according to the scattering matrix $$ \begin{pmatrix} a_1 \\ a_2 \\ \end{pmatrix} = \dfrac{1}{\sqrt{...
m137's user avatar
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Multimode squeezed operator

Given CCR (bosonic) algebra, with creation / annihilation operators $a_{i}^{\dagger}, a_i$ acting on a single particle Hilbert space $\mathbb{h}$, let's introduce the multimode squeezed operator for $...
MBlrd's user avatar
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1 vote
1 answer
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Why does a singularity imply the need for a distribution?

I am following Section 11 of Prof. Etingof's MIT OpenCourseWare notes on "Geometry And Quantum Field Theory" in which he says: ...for $d = 1$, the Green's function $G(x)$ is continuous at $...
CBBAM's user avatar
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Determinant of a tensor [duplicate]

How to find the determinant of $\partial_{\lambda} \delta x^{\mu} $ where $\partial_{\lambda}$ is the four derivative and $ \delta x^{\mu}$ is the del variation in $x^{\mu}$. The answer is $\...
Wajahat's user avatar
4 votes
1 answer
393 views

When do two state functions represent the same quantum state?

According to the standard quantum mechanics, quantum states are one-dimensional subspaces of a separable Hilbert space. In practice, this Hilbert space is $L^2(M)$ where $M$ is the classical ...
mma's user avatar
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Correlation length in a 3d Ising slab with one dimension much smaller than the other two

Suppose I have a 3d Ising model on a cubic lattice, but one of its dimensions is much smaller than the other two. That is, I have an $L$ by $L$ by $L'$ slab with $L' << L$; in particular, $L'$ ...
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Why is $H = J \sum_i (S^x_i S^x_{i+1} + S^y_iS^y_{i+1})$ always gapless for any spin $S$?

In the following I have in mind antiferromagnetic spin chains in periodic boundary conditions on a chain of even length $L$. Consider the spin-$S$ spin chain $$H = J \sum_{i=1}^L (S^x_i S^x_{i+1} + S^...
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Holonomic constraints as a limit of the motion under potential

In Mathematical Methods of Classical Mechanics, Arnold states the following theorem without proof in pages 75-76: Let $\gamma$ be a smooth plane curve, and let $q_1, q_2$ be local coordinates where $...
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1 answer
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(Time-)Orientability in the Language of Fiber Bundles

I'm currently studying spin geometry through Hamilton's book Mathematical Gauge Theory. At a given point, Hamilton considers a pseudo-Riemannian manifold, which I'll take to be Lorentzian in $d=3+1$ ...
Níckolas Alves's user avatar
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1 answer
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Azimuthal coordinate operator: Hermition or not? Self-adjoint or not?

I am told that the azimuthal coordinate operator $\hat{\phi}$ is not self-adjoint. I am told this by people who I am sure know much more about this stuff than I do. To my unsophisticated mind, "...
bob.sacamento's user avatar
2 votes
0 answers
132 views

Einstein's gravity Lagrangian invariance under the change of differential structure

I came across an article claiming the appearance of singularities in the energy-momentum tensor $T_{\mu \nu}$ as a result of changing the differential structure: I wonder what symmetry or current (in ...
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Why are light rays expressed or generally shown as a sine wave? [duplicate]

Whether it be a YouTube video or book lights rays i.e(the visible specturm,uv rays) are expressed as a sine wave. Why is it so? What is the plot of its against? Why does it oscillate? Why does it ...
Shankar Shrestha's user avatar
1 vote
1 answer
31 views

Limit for big system size of Fokker-Planck eigenfunctions

I am learning how to use diagonalization methods applied to Fokker-Planck equations with Gardiner's book and these notes. The idea is to find the probability density, $ P[X_t\in[x,x+dx]]=\rho_t \, dx$,...
Javi's user avatar
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6 votes
0 answers
112 views

What does it mean for classical mechanics to be based on the category of sets?

It is quite common[1][2] in the study of physics in the context of category theory to say that one of the fundamental difference between classical mechanics and quantum mechanics is that classical ...
Slereah's user avatar
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2 votes
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The limit of time evolution operator

Through reading Nenciu's rigorous proof on the Adiabatic Theorem and Gell-Mann-Low Theorem, I found: Since the limit $t_0\to-\infty$ does not exist for $U(t,t_0,\epsilon)$, in order to make use of ...
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Is integral of energy-momentum tensor in QFT over a region $R$ self-adjoint?

Consider a quantum field theory in flat 1+1D spacetime for simplicity. Let $T_{\mu\nu}$ be the conserved symmetric stress tensor. One writes operators by integrating the tensor over the whole space, ...
physicophilic's user avatar
2 votes
0 answers
127 views
+50

Mathematical explanation of the extinction paradox

I am trying to learn properly about scattering. For this I was pointed to Wave Propagation and Scattering in Random Media by Ishimaru. I got a bit stuck in section 2-2 General properties of the Cross ...
user8469759's user avatar
10 votes
1 answer
212 views

Analytical continuation as regularization in Quantum Field Theory, the remaining questions

There is an old question posted (Regularization) which did not get an answer, about the validation of analytic continuation as regularization. It did get some discussion in the comments, referring to ...
Jos Bergervoet's user avatar
2 votes
0 answers
32 views

Mathematical references for gauge theory in condensed matter physics

I am currently trying to go through some literature on the classification of symmetry protected topological phases. Primarily, I am interested in the classical of topological phases using mathematical ...
3 votes
2 answers
178 views

Why are these unbounded operators (essentially) self-adjoint?

Can anyone provide exact mathematical reasoning as to why the following fundamental unbounded symmetric operators are essentially self-adjoint? I.e. on, their natural domains, they admit a unique ...
SiOn's user avatar
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0 answers
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Use of mathematical structure on physics [closed]

I want resources for studying in detail the connection between the mathematical structures of physical theories and said physical theories. For example, i know what a Hilbert space or a principal ...
0 votes
0 answers
23 views

Charge Density Of A Conducting Strip

I am currently doing a project which requires me to figure out the charge density of a strip. Assume that the strip is isolated in a vacuum. Assume the strip is 1 dimensional, kind of like a rod. What ...
user392135's user avatar
1 vote
2 answers
155 views

How do we know Schwinger functions exist?

Let $\mathcal{D}'(\mathbb{R}^n)$ denote the dual of $C^\infty_C(\mathbb{R}^n)$, that is distributions on the set of infinitely differentiable functions with compact support. If $d\mu$ is a probability ...
CBBAM's user avatar
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1 vote
1 answer
72 views

Conformal Transformation of Torsion

It is well known that under a conformal transformation, we have $$\tilde{g}_{\mu \nu}=\Omega^2 g_{\mu \nu}, \; ; \tilde{w}_{\mu}=w_{\mu}-\frac{1}{\alpha} \partial_{\mu} \log(\Omega^2),$$ where $\...
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4 votes
1 answer
382 views

Outer product as an operator in an infinite dimensional Hilbert space

The outer product between a bra-ket $|a\rangle\langle a|$ where if $|a\rangle\in\mathcal{H}$ and $\langle a|\in\mathcal{H}_{dual}$ is a vector in the tensor vector space formed by the Hilbert space ...
Oscarcillo's user avatar
5 votes
1 answer
218 views

How does one rigorously define two-point functions?

Let $\mathscr{H}$ be a complex Hilbert space, and $\mathcal{F}^{\pm}(\mathscr{H})$ be its associated bosonic (+) and fermionic (-) Fock spaces. Given $f \in \mathscr{H}$, we can define rigorously the ...
MathMath's user avatar
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2 votes
1 answer
105 views

Thermal ground state?

Consider a system of $N$ fermions in a periodic box $\Lambda \subset \mathbb{R}^{d}$, described by the Hamiltonian $$H_{N} = \sum_{k=1}^{N}(-\Delta_{x_{k}}-\mu) + \lambda \sum_{i< j}V(x_{i}-x_{j}) \...
MathMath's user avatar
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1 vote
2 answers
140 views

Algebraic definition of ground state

I'm currently studying arXiv: 1706.09666 [math-ph]. On page 51, the authors define what is a ground state in the algebraic approach. I quote them below. If a state $\omega$ is invariant under a one-...
Níckolas Alves's user avatar
0 votes
0 answers
59 views

Zero temperature Green function as limit of finite temperature Green function

Consider a system of $N$ fermions in a periodic box $\Lambda \subset \mathbb{R}^{d}$. The Hamiltonian of the system is: $$H_{N} = \sum_{k=1}^{N}(-\Delta_{x_{k}}-\mu) + \lambda \sum_{i< j}V(x_{i}-x_{...
MathMath's user avatar
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0 votes
0 answers
49 views

$\mathbb Z_N$ (discrete) gauge theory

I am currently trying to go through some literature on symmetry protected topological phases and gauge theories defined on lattices. I am looking for a mathematically precise reference that discusses $...
2 votes
0 answers
39 views

Has the Feynman-Kac formula been extended to the complex case?

Wikipedia writes that the Feynman-Kac formula "proves rigorously the real-valued case of Feynman's path integrals. The complex case, which occurs when a particle's spin is included, is still an ...
PretentiousPolymath's user avatar
1 vote
0 answers
31 views

Axial anomaly for odd dimension

I'm reading that many articles are using the "axial anomaly equation" (e.g. Fermion number fractionization in quantum field theory pag.142 or eq (2.27) of Spectral asymmetry on an open space)...
Davide's user avatar
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0 votes
1 answer
42 views

Time evolution of mixed state?

Suppose I have a quantum statistical mechanics system in the grand-canonical ensemble. It is given by some Hamiltonian $H = H_{0} + V$, where $H_{0}$ is the free part and $V$ an interaction. The state ...
MathMath's user avatar
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1 vote
0 answers
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Applying Kato-Rellich to the hydrogen atom model to prove stability of first kind [closed]

Trying to Understand the lower bound on the Schrodinger Operator of the Hydrogen atom. Using the kato-rellich theorem. My education has been in physics and i am slowly adding to my mathematics toolset....
Gedankenhooman's user avatar
5 votes
1 answer
313 views

Why are von Neumann algebras not suitable for dealing with Locally Covariant Quantum Field Theory in Curved Spacetime?

I recently came across this post by Valter Moretti concerning the utility of von Neumann algebras in mathematical physics. In it, he mentions The closure of von Neumann algebras with respect to the ...
Níckolas Alves's user avatar
3 votes
1 answer
133 views

Is the Godel universe Wick rotatable?

Take Wick Rotatability being as the way defined in the article by Helleland: Wick rotations and real GIT Is the Gödel universe Wick rotatable according to this definition?
Bastam Tajik's user avatar
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0 votes
1 answer
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Canonical quantization of relativistic particle using Fourier transform

Suppose I want to quantize the Hamiltonian of a relativistic particle on space-time $\mathbb{R}^{4}$. Setting $c=1$ for simplicity, the energy of the particle is given by $w(p) = \sqrt{|p|^{2}+m^{2}}$,...
MathMath's user avatar
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0 votes
1 answer
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Domain of sum of momentum operators

This is a repost from MathStackExchange (https://math.stackexchange.com/q/4840786/) where however no solution has been found so far. Given the tensor product of Hilbert spaces $\otimes_{i \in \mathcal{...
MBlrd's user avatar
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0 answers
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Why the existence of an associative OPE for a CFT$_2$ is presented as an extra axiom in this presentation?

In the book "Mathematical Introduction to Conformal Field Theory" by Schottenloher, the author introduces in Chapter 9 one axiomatic definition of what a CFT in two dimensions is. The first ...
Gold's user avatar
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7 votes
1 answer
1k views

What is the Hilbert dimension of a Fock space?

Quantum field theory in curved spacetimes is often described in the algebraic approach, which consists of describing observables as elements of a certain $*$-algebra. To recover the notion of a ...
Níckolas Alves's user avatar
1 vote
0 answers
72 views

Real algebraic formulation of quantum mechanics?

In a deterministic classical theory we describe the positions and velocities of particles using real numbers. In a non-deterministic classical theory we describe the positions and velocities of ...
Jagerber48's user avatar
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1 vote
1 answer
116 views

On the $\ast$ map in the Osterwalder-Schrader axioms

I'm studying "A Mathematical Introduction to Conformal Field Theory" by Schottenloher and there is one point on the Osterwalder-Schrader axioms that I am a bit confused about. They are ...
Gold's user avatar
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2 votes
2 answers
343 views

Quantum mechanics and derivative operators

EDITED POST Suppose we have a classical problem where the Hamiltonian is defined as: $$H = c\frac{p^2}{x}$$ This Hamiltonian emerges in the context of Hamiltonian 1D cosmology, where we define $x= a(t)...
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6 votes
0 answers
69 views

How to derive the probability distribution of reduced density matrix eigenvalues for randomly chosen pure states in Page's theorem?

Motivation I am trying to reproduce the proof in Page's theorem as conjectured in the seminal paper Average Entropy of a Subsystem by Don N. Page. It is crucial in various resolutions of black hole ...
Sanjana's user avatar
  • 414
-1 votes
1 answer
74 views

What is the physical meaning of self-adjoint operator extension?

What does it mean that there isn't any extension of a certain operator in a given domain? Does it imply that I can't apply that operator in that domain, and so that I can't measure some observables (...
hbar's user avatar
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3 votes
0 answers
122 views

Mathematical objects on crystal meltings and their relation to particle physics

I am a mathematician interested in analytic number theory, and I found the paper Dimers and Amoebae , which shows how many mathematical objects like the Mahler measure, the Ronkin function and the ...
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