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.

Filter by
Sorted by
Tagged with
1 vote
0 answers
101 views

Mathematically Rigorous Introduction to the Standard Model [duplicate]

I am looking for textbooks, lecture notes, lecture videos on a rigorous introduction to the standard model of elementary particles. I'd prefer to not be referred to monographs for an introduction as ...
2 votes
0 answers
105 views

Mathematically Rigorous Introductory Resources for Condensed Matter Physics

I am looking for textbooks, lecture notes, lecture videos on rigorous introductions to condensed matter physics. I'd prefer to not be referred to monographs for an introduction as they tend to be ...
0 votes
0 answers
27 views

Uniqueness of form of Galilean covariant Hamiltonian

Consider the most general possible many-body quantum Hamiltonian in second quantized form (one species of spinless particle in $d\geq 1$ spatial dimensions): $$H_{\rm int} = \sum_{n,m\geq 0} \int _{\...
  • 33
1 vote
0 answers
69 views

Operator Systems vs Operator Algebras

I am someone who works in mathematics, works with operators, and tries to interpret things in terms of a "toy" QM that doesn't necessarily have anything to do with actual QM. Two things in ...
5 votes
1 answer
163 views

Operators and periodic boundary conditions

Background: In Ref. 1, a system of $N$ (identical) fermions is considered. The system is enclosed in a cubic box of volume $\Omega=L^3$ and periodic boundary conditions are employed, that is (I'll ...
5 votes
1 answer
329 views

Separable Hilbert space in quantum mechanics

When one studies quantum mechanics under a more rigorous point of view, the very first postulate states that the underlying Hilbert space $\mathscr{H}$ is separable. This means that $\mathscr{H} $ has ...
  • 815
2 votes
0 answers
22 views

How could Adrian Bejan's 'Constructal Law' be formulated mathematically? [closed]

Adrian Bejan is known for having devised a very interesting theory for the hierarchical and aesthetic structure found in nature, from the branching tree shape of river deltas to the tree shape of ...
  • 21
2 votes
1 answer
81 views

Zero frequency quantum oscillator rigorously

Problem description Consider quantum oscillator with $ \omega = 0 $. In other words, we have $$ \hat{H} = \hat{p}^2, $$ where $\hat{x}, \, \hat{p}$ are the usual coordinate and momentum operators with ...
10 votes
4 answers
934 views

Can the wavefunction be inferred from the expectation values of operators?

Preface This question is motivated by $C^*$ type treatments of quantum mechanics where operators (Basically an operator is an object that has a spectrum) are treated as fundamental and states are ...
  • 9,905
0 votes
0 answers
10 views

Branch Point and Branch cut [migrated]

The question was to find the branch point and branch cut of $\log(z-(1+z^2)^{1/2})$. My approach was first to consider the square root function. The branch point of the square root function exists ...
3 votes
0 answers
39 views

Relations between different definitions of critical temperatures

I have noticed the following definitions of critical temperature $T_c$ being used in different subject areas: MAG: The temperature below which some order parameter (e.g. magnetization or self-overlap)...
  • 673
0 votes
0 answers
36 views

References about Quantum Theory (for mathematicians) [duplicate]

I’m interested in Quantum Theory. Can anyone recommend a good book (for mathematicians) that tackles varios topics (not necessarily formally). I have in mind books that have the same idea of books ...
1 vote
1 answer
69 views

Question on ordered exponential explanation in Wikipedia

Let $\gamma:[0,1] \to \mathbb R^2$ be a path describing a rectangle with vertices $x$, $x+u$, $x+u+v$, $x+v$, where $x, u, v \in \mathbb R^2$ ($u, v$ linearly independent). Let $J:\mathbb R^2 \to \...
0 votes
1 answer
84 views

Sensor Array Position Calibration in Anisotropic Media

Problem. I have a sensor array consisting of $n \gg 4$ receivers at unknown locations $\langle x_n, y_n, z_n\rangle$ embedded in an anisotropic medium whose index of refraction varies as a known ...
  • 843
1 vote
0 answers
71 views

Construction of the Klein-Gordon field theory - what is missing?

Many references I know on QFT start the discussion of the Klein-Gordon field theory with some discussion about harmonic oscillators. One such reference is Folland's Quantum Field Theory book. The idea ...
  • 559
1 vote
0 answers
57 views

An elementary random walk model to incorporate non-Gaussianity

I am preparing a talk for young students to introduce heterogenous dynamics in complex fluids and give them a flavour of non-Gaussianity in displacements which are defined by, $$ \alpha (t) = \frac{\...
  • 2,698
2 votes
0 answers
61 views

How do I self-study physics at the undergrad level? [closed]

I'm a new physics undergrad worried that I won't be able to learn everything I want at the university I'm going to. Basically the Institute I'm going to is applied sciences focused, and all electives ...
  • 21
3 votes
3 answers
226 views

Non-distributivity of quantum logic according to C. Piron

I'm trying to understand this highlighted sentence in Piron's "Foundations of Quantum Physics" on p. 21: I know that distributivity of a lattice means $a\land (b\lor c)=(a\land b)\lor(a\...
  • 319
2 votes
2 answers
104 views

Square root of number operator for quantum harmonic oscillator

Let $a$, $a^{\dagger}$ denote the standard annihilation and creation operators for the quantum harmonic oscillator, with $[a, a^{\dagger}] = \mathbb{I}$. The number operator is then defined as $a^{\...
  • 357
2 votes
0 answers
48 views

On the Bogoliubov-de Gennes (BdG) equation

I'm a graduate student majoring in mathematics, in particular nonlinear PDEs. So I know very little about physics, including quantum mechanics. I'm interested in the Bogoliubov-de Gennes (BdG) ...
  • 21
6 votes
3 answers
617 views

Compactification in String Theory and Compactification in Topology are they the same thing?

In topology, there is a concept of compactification which is defined as follows. A space $Z$ is a compactification of $X$ if $Z$ is compact Hausdorff and there exists an embedding $j:X \rightarrow Z $ ...
  • 109
0 votes
0 answers
31 views

Doubt on the continuity factor of Dyson mega-spheres

I) Dyson Mega-Spheres In a nice and cool recent paper, $[1]$, the authors constructed another interresting solution of general relativity; they constructed a thin-shell around a star: a dyson sphere ...
  • 2,741
6 votes
1 answer
484 views

In what sense is string theory not expected to be a QFT?

This question came to mind while reading about Weinberg's folk theorem that any quantum theory that is Poincare covariant and satisfies cluster decomposition will look like a quantum field theory at ...
  • 124
0 votes
0 answers
14 views

Intuition for the differences between two notions of quantum ergodicity: One given by weak-* convergence and one by pseudodifferential operators

Consider the two notions of quantum ergodicity of the Laplacian operator $\Delta$. (Phase space): $\Delta$ is said to be quantum ergodic (in the phase space) in a compact Riemannian manifold if there ...
1 vote
0 answers
37 views

Mathematical equivalent of Fundamental nature of charge [closed]

How to mathematically represent the fact that electric charge is a fundamental quantity? i.e. that it cannot be explained in terms of other things, for example, the normal force can be explained as ...
3 votes
0 answers
78 views

Whats is a large fields problem in RG?

I was advised on MO to link this question and reproduce it here, so here it goes. I was reading Constructive Renormalization Group by V. Rivasseau and I got some points which I would like to clarify. ...
  • 815
3 votes
1 answer
205 views

I'm getting imaginary eigenvalues of $X^2-P^2$

This is hermitian but I'm getting imaginary eigenvalues. Start with the commutator: $$[X^2-P^2,X+P]$$ $$=[X^2,P]-[P^2,X]$$ $$=X[X,P]+[X,P]X-(P[P,X]+[P,X]P)$$ $$=2i(X+P)$$ Now consider the ket $|E\...
  • 3,360
1 vote
0 answers
34 views

Quasifree states vs Gaussian states

Does anyone know the origin of the term "quasifree states" in algebraic formulation of QFT? As far as I am aware of, the definition is effectively equivalent to Gaussian states centred at ...
  • 1,518
9 votes
3 answers
1k views

Do continuous wavefunction form a Hilbert space?

In quantum mechanics we are told that the wavefunctions live in Hilbert space. the wavefunctions are continuous. It recently came to my notice that in Mathematics, there is a theorem which says ...
  • 223
1 vote
0 answers
73 views

Are mathematical physics and theoretical physics the same thing at highest levels? [duplicate]

To my understanding Mathematical physics is about how one could find a rigorous basis to understand physics/ study the mathematics used in physics. However, high level theoretical modern physics like ...
0 votes
1 answer
60 views

Motivation behind defining new measures?

Suppose we are given a measure $d\mu$. I have seen in some physics textbooks that we often define a new measure in terms of this measure, let's say $$d\rho = f\,d\mu,$$ where $f$ is some integrable ...
  • 494
2 votes
1 answer
36 views

Two questions regarding Spivak's Configuration Space

The following is from the fifth Chapter Rigid Bodies of Spivak's Physics for Mathematicians. The post consists of a statement Spivak makes -with no proof- that I do not understand. For clarity, I've ...
  • 363
2 votes
0 answers
64 views

Dynamical system model representation for the geometric motion of soap bubbles

I have been trying to describe the dynamical system model for soap bubbles proliferating, coming into existence and moving and pushing around other bubbles (i.e. water gushing into a sink, full of ...
1 vote
1 answer
64 views

Follow-up to the question of why the momentum operator is Hermitian on $\mathbb{R}$

I have a follow-up question to the following post: Imaginary Eigenvalue Of A Hermitian Operator. In the post, the question reads, The eigenfunctions of a Hermitian operator are real. But consider a ...
3 votes
1 answer
162 views

What makes coming up with a mathematically solid, non-shaky relativistic quantum field theory (RQFT) so hard?

This is something I know of but I'm not quite sure I understand the details. Particularly, when it comes to interacting RQFTs, such as even QED, where some posts here have pointed out that it cannot ...
0 votes
1 answer
88 views

As written, how is the spectral theorem useful?

I am self-teaching myself QM using Brian Hall's Quantum Theory for Mathematicians. He devotes a good chunk of the book to motivating, explaining, and proving the spectral theorem in multiple ways. The ...
  • 494
0 votes
0 answers
37 views

$q$-dilogartihm function power series

I was reading the $q$-dilogarithm function (Faddev-Kashaev) https://arxiv.org/abs/hep-th/9310070 How can I derive $$\frac{1}{\Psi(x)}=\sum_{n=0}^{\infty} \theta^{n} x^{n} /(\theta)_{n}$$ by using the ...
2 votes
0 answers
33 views

How fast can the spectral gap of a translation-invariant Hamiltonian close?

Consider an arbitrary local Hamiltonian defined on some lattice of size L where the local Hilbert space dimension associated with each site on the lattice is finite. If there is no constraint on the ...
  • 2,797
1 vote
0 answers
18 views

Some questions related to the energy of a viscoelastic bar

Let's consider a problem of free longitudinal vibrations of an one dimensional infinitely long elastic relaxing bar (Maxwell material) with constant cross section. The bar's displacement $u$ and ...
  • 111
6 votes
1 answer
267 views

Gauge Theory determined by Gauge Group and Representation: What about specifying the bundle?

I have the following question. In physics, when one talks about (Yang-Mills) gauge theories, one often states that it is enough to specify the following data: The gauge group $G$, which is usually a ...
  • 626
1 vote
2 answers
125 views

Doubt on the geometry of "quantum phase space"

In Jose & Saletan's "Classical Dynamics", they show the global structure of Hamiltonian mechanics: you then have a $Q$ manifold (configuration space), and the phase space structure is ...
  • 2,741
1 vote
1 answer
65 views

The abstract state of a particle

I recently started learning about quantum physics. In the book, Quantum physics by H.C. Verma, the author explains that there are many ways to represent the state of a particle. The wave function $\...
4 votes
0 answers
58 views

Quantum Particle in a Fractal Box

I was thinking about particle in a 2D box the other day, and I realize that it shapes actually affect its energy and wavefunction. Therefore I thought to myself, what if a particle is inside a fractal ...
3 votes
1 answer
53 views

Resources on Post-Einsteinian Results in GR

What are some good books, lecture notes, articles, etc. that can be used as introduction to the landscape of major results in general relativity since Einstein? In terms of the timeline, I'm thinking ...
0 votes
0 answers
22 views

How was the minimal model with a boundary related to the D brane?

Quote my advisor: The D brane was the boundary of the CFT However, in the development of the rational CFT, such as the minimal model, the D brane was not realized. Thus, when the boundary CFT was ...
4 votes
0 answers
43 views

Green's function existence vs explicit description

Are there examples in physics where the mere existence of a Green's function on some domain (for some PDE) has useful applications? Or is it true that in literally all applications of Green's ...
1 vote
0 answers
58 views

Contractivity of trace distance in infinite dimensions

The trace distance is heavily used in physical applications and is defined being half of the trace norm $T(\rho, \sigma) := \frac{1}{2} Tr\left[\sqrt{(\rho-\sigma)^{\dagger}(\rho-\sigma)}\right]$. It ...
  • 397
6 votes
1 answer
151 views

Interacting QFT construction on curved spacetime

As far as I can tell, most of the concrete models considered in (rigorous) QFT on curved spacetime are either free or perturbative. In fact the only construction of an interacting QFT on curved ...
  • 223
0 votes
0 answers
33 views

Euler equation of motion for fluids

I was seeing some proofs of Euler equation of motion for fluids online and most of the videos drew this figure in which they consider infinitesimal cylindrical element. My question:- Now it mentions ...
  • 420
1 vote
2 answers
397 views

It is possible to have a drag force which is non-Lipschitz?

When working with the Drag Force is typical used on classical mechanics systems the following: For high speeds it is used the Drag equation which says the drag is proportional to the squared of the ...
  • 77

1
2 3 4 5
44