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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0answers
31 views

Chemical potentials for $D$-brane bound states

This question is about a mathematical subtlety arising in the computation of the partition function of a supersymmetric ensemble of some lower dimensional $D$-branes attached to a stack of higher ...
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3answers
2k views

Do all Noether theorems have a common mathematical structure?

I know that there are Noether theorems in classical mechanics, electrodynamics, quantum mechanics and even quantum field theory and since this are theories with different underlying formalisms, if was ...
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2answers
272 views

What are good non-paraxial gaussian-beam-like solutions of the Helmholtz equation?

I am playing around with some optics manipulations and I am looking for beams of light which are roughly gaussian in nature but which go beyond the paraxial regime and which include non-paraxial ...
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1answer
400 views

Spectral decomposition vs Taylor Expansion

This question and the comments and answers it received encouraged me to ask this question, although I know that there will be some people who think that this belongs in the math forum. But I think ...
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3answers
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Constructing Lagrangian from the Hamiltonian

Given the Lagrangian $L$ for a system, we can construct the Hamiltonian $H$ using the definition $H=\sum\limits_{i}p_i\dot{q}_i-L$ where $p_i=\frac{\partial L}{\partial \dot{q}_i}$. Therefore, to ...
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0answers
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Does microcausality plus the time-slice property imply local primitive causality?

In quantum field theory, observables are associated with regions of spacetime. One of the basic principles of relativistic quantum field theory is microcausality, which says that observables ...
3
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1answer
76 views

Axiomatic Quantum Theory

Is there any practical advantage/disadvantage between using the axioms for the Hilbert Space formulation as opposed to the C* Algebra formulation of Quantum Theory? If not, then is it possible to ...
21
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2answers
682 views

Is there a physical interpretation to invariant random matrix ensembles?

Disclaimer. I am a graduate student in pure mathematics, so my knowledge of physics more advanced than basic 1st/2nd year undergraduate physics is very limited. I welcome corrections on any ...
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1answer
47 views

Why is the end of a swinging chain diagonal when $J_0(0)$ is flat? [closed]

I'm trying to understand the Bessel function $J_0$, and how it describes the motion of a vibrating hanging chain, and I'm confused by the behavior of the free end of the chain. The derivative of $J_0(...
2
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1answer
53 views

Spacetimes with “celestial Riemann surface” other than the sphere

In the standard study of asymptotically flat spacetimes one defines null infinity demanding that topologically ${\cal I}^\pm \simeq \mathbb{R}\times S^2$ (c.f. Definition 1 of this review by Ashtekar)....
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How to know if a spinor $\Psi\left(x\right)$ is the ground state of the system?

Suppose we have time independent one-dimensional single particle Schrödinger-like equation$$-\frac{d}{dx}\left(A\left(x\right)\frac{d}{dx}\psi\left(x\right)\right)+V\left(x\right)\psi\left(x\right)=E\...
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Can we use sine and cosine graphs for varying temperatures? [closed]

I have been researching for a while looking for what to include more than just deriving the equation of Newton’s law of cooling and a website gave me inspiration by stating what if the temperature of ...
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1answer
32 views

How to handle a delta-function divergence from an infinite-dimensional trace of a quantum operator with a general continuous parameter?

Say I am working in the standard Schrödinger-style Hilbert space that corresponds to square-integrable functions on 3-space. Some normal operator $\hat{A}$ therefore has an eigenbasis indexed by 3 ...
20
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2answers
913 views

A rigorous definition of the exponential of an operator in QM?

In the Quantum Mechanics course I took, we defined the operator exponential simply as $$ \mathrm{e}^{\hat A} = \sum_{n=0}^\infty \frac{1}{n!} \hat A^n \: . $$ This is probably a good definition for ...
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45 views

What is the Weyl algebra?

I have a question about Weyl algebras. It arose reading this (page 30) and this (page 11), but I think the physical context is not very important, so I can summarize here what they say. They consider ...
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69 views

Laplace transform method to solve two coupled differential equations [migrated]

I was reading QUANTUM OPTICS for BEGINNERS by Z FICEK, in his discussion of the Jaynes-Cummings model he use Laplace transform method to solve two coupled differential equations (8.9) as shown in the ...
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2answers
171 views

Does the convergence of van Hove sequence equivalent to thermodynamic limit?

I have seen two definitions of thermodynamic limit. Definition 1. This is the common definition in statistical physics textbooks, $$N \to \infty ,V \to \infty ,N/V = {\rm{constant}}$$ There are many ...
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1answer
59 views

Shadow method of solving differential equations

While reading this answer by Rishab Navneet here, it is shown how we can visualize the harmonic oscillator as the shadow of a body moving in a circle onto a line. How was it found that the plane curve ...
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1answer
144 views

Can one test an octonionic interpretation for a quantum-information conjecture, apparently valid in the real, complex and quaternionic settings?

For the values $\alpha = \frac{1}{2},1, 2$, corresponding to real, complex and quaternionic scenarios, the formulas (https://arxiv.org/abs/1301.6617, eqs. (1)-(3)) \begin{equation} \label{Hou1} P_1(\...
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1answer
150 views

Trace over configuration basis

Let us take a many-body quantum system, whose phases in the configuration basis are labeled by $\mathbf {\hat q}=(q_1,\cdots, q_N)$ and momenta $\mathbf {\hat p}=\left(-i\frac{\partial}{\partial \hat ...
3
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1answer
93 views

References on mathematical stacks for a string theory student

This question was posted on mathoverflow (here) without too much success. I'm hoping to read the famous Kapustin-Witten Paper "Electric-magnetic duality and the geometric Langlands program" ...
2
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1answer
77 views

On the number of spacetime atoms in a Calabi-Yau crystal

This question is about an application of crystallography to topological string theory. Plane partitions are discrete models of the Planck scale geometry of Calabi-Yau manifolds in the A-model ...
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62 views

Another Solution To Brachistochrone Problem

Recalling the statement of the problem : Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the ...
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4answers
1k views

Is continuity of the wavefunction “put in by hand” for the Dirac delta potentials?

In 1d, for $V(x) = g\delta(x)$, integrating the TISE yields (assuming that $\psi$ is bounded$^\dagger$, so as to suppress the term containing $E$) $$ -\frac{\hbar^2}{2m} \left( \psi'(\varepsilon) - \...
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0answers
27 views

How possible is using complex variables to model the electric field system of a dipole? [duplicate]

If one considera a dipole system, is it possible to model the system using complex variables and if ao, how can we use complex analysis to model it? I have the idea that we can model dipole moment as ...
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0answers
82 views

Is there a more theoretical approach to studying physics? [closed]

I have read some physics papers in theoretical particle physics and all of them were based on computation, they had some mile long equations and the result was always the proof of some formula. I am ...
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1answer
56 views

There can be degeneracy in 1D energy eigenfunctions even if one of them and its derivative does to zero as $x\to\infty$?

In 1D, the energy eigenvalue equation for an energy value $E$ $$ -\frac{\hbar^2}{2m}\psi''(x) + V(x)\psi(x) = E\psi(x) $$ can have at most two linearly independent solutions, $\psi_1$ and $\psi_2$, ...
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4answers
5k views

Binomial expansion of non-commutative operators

I would like to determine the general expansion of $$(\hat{A}+\hat{B})^n,$$ where $[\hat{A},\hat{B}]\neq 0$, i.e. $\hat{A}$ and $\hat{B}$ are two generally non-commutative operators. How could I ...
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0answers
20 views

Do the free particle energy eigenfunctions satisfy the closure relation?

The energy eigenfunction of the free particle ($V(x)=0$) are given by $\psi_{E, \pm}=A(E)e^{\pm ik_Ex}$ where $A(E)=\left( m/\left(8\pi^2\hbar^2 E\right) \right)$ and $k_E=\sqrt{2mE}/\hbar$ for each $...
9
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2answers
534 views

Intuitively what's the relationship between forces and connections?

In Einstein's General Relativity we relate the effects of gravity with the curvature of the Levi-Civita connection on the spacetime manifold. Also, when we get the electromagnetic tensor $F = dA$ ...
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1answer
62 views

Trying to understand how to connect the most general concept of a function to real world? [closed]

I'm a beginner wrapping my head around how general a definition a "function" really is when connected to the real world, please help. I am trying to connect the mathematical definition of a ...
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0answers
34 views

Is there a systematic way to construct a SUSY theory?

For the sake of simplicity, I am considering a 0+0d scalar field theory with multiple bosonic and fermionic fields/variables. The fields are coupled together up to a certain order (say 4) with ...
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0answers
28 views

Why is $f\in C_{\infty}=\{f\in C(\mathbb{R})\mid \lim_{R\to\infty} \sup_{∣∣x∣∣>R} ∣f(x)∣=0\}$ [duplicate]

Today in my mathematical quantum theory lecture our professor told us without any explanation, that if $f\in L^1(\mathbb{R}) \cap C^1(\mathbb{R})$ and $f'\in L^1(\mathbb{R})$ it follows that $f\in C_{\...
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0answers
42 views

$f\in L^1(\mathbb{R}) \cap C^1(\mathbb{R})$ and $f'\in L^1(\mathbb{R})$ [closed]

Today in my mathematical quantum theory lecture our professor told us without any explanation, that if $f\in L^1(\mathbb{R}) \cap C^1(\mathbb{R})$ and $f'\in L^1(\mathbb{R})$ it follows that $f\in C_{\...
4
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1answer
96 views

Number of bras and kets

My quantum mechanics teacher told us during the class that there were many more "bra" than "kets", but I confess that I don't quite understand this. Indeed, in quantum mechanics, ...
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0answers
39 views

What does the Hausdorff-Young inequality tell us?

The Hausdorff-Young Inequality relates the size of a function and its Fourier coefficients. What is meant the by "the size of a function"?
8
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1answer
222 views

Allowed anyons for Chern-Simons at level $k.$

Ref.1. proves that the allowed representations of Chern-Simons $\mathrm{SU}(2)_k$ are those with dimension $$ \dim(R)\le k+1\tag{7.53} $$ Question: Is the generalisation of $(7.53)$ to arbitrary $N$ ...
3
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1answer
482 views

Functional derivative commutes with total derivative

I have a question about a rule from the calculus of variations. Assume we consider the space of differentiable functions on $C^1(\mathbb{R})$ (or for the sake of simplicity the smooth functions $C^{\...
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0answers
25 views

On Goldstone fermions / goldstino: SUSY breaking

There are statements about Goldstone fermions, or goldstino, seem confusing to me. (1) Goldstone boson requires a continuous symmetry spontaneously broken. Does Goldstone fermion imply continuous SUSY ...
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0answers
46 views

Polarization procedure in geometric quantization

The geometric quantization can be considered as an approach the formalize the way of associating a quantum theory corresponding to a given classical theory. Suppose we start with a sympetic manifold $(...
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2answers
32 views

Orthogonality of two randomly chosen velocity vectors in the kinetic theory of gas and the relative velocity

I was looking for the average value of the relative velocity of ideal gas molecules in the kinetic theory. The following is from this article. The vector notation is not mathematically rigorous, just ...
11
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4answers
2k views

Does Hestenes Zitterbewegung Explain why complex numbers appear in QM?

This question may fit better in the discussion of "Why Complex variables are required by QM?", but it also relates to the extent to which arguments by Hestenes are accepted in mainstream physics and ...
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3answers
4k views

Electromagnetism for mathematicians

I am trying to find a book on electromagnetism for mathematicians (so it has to be rigorous). Preferably a book that extensively uses Stoke's theorem for Maxwell's equations (unlike other books that ...
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0answers
49 views

Convergence of $c^* Ac$ in second quantization

Let $A$ denote a bounded operator on a complex separable Hilbert space $\mathscr{H}$. Let $\mathscr{F} = \bigoplus \mathscr{F}_N$ be the Fock space generated by $\mathscr{H}$ where $\mathscr{F}_N$ is ...
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1answer
54 views

Dirac delta equalities in physics

Earlier I asked this question on the Math Exchange but I'm looking for a physics point of view. How do you interpret an equation like $$x^n \delta(x) = 0, \qquad n\in \mathbb{N},$$ around $x=0$? Why ...
2
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2answers
84 views

Confusion about the dimension of a Hilbert Space in Quantum Mechanics [duplicate]

In Quantum Mechanics, the quantum state of the physical system lives in an infinite-dimensional Hilbert space and can be written in terms of two different bases, the position basis (uncountably ...
2
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1answer
228 views

Can one force the octupole moments of a charge distribution (neutral and with vanishing dipole moment) to vanish using a suitable translation?

In a previous question, I noted that if you have a charge distribution with nonzero charge, then it is possible to choose an origin (at the centre of charge) which makes its dipole moment vanish, and ...
0
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1answer
45 views

What equation should I use if I am estimating the distance of a rocket's landing point to the point it was launched from?

I am trying to figure out how to estimate how far (in meters) that a solid fuel model rocket will land from its launching point. To do this, I have figured out I will need to estimate the maximum ...
3
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1answer
111 views

Does one ever need infinitely many cohomologies?

In a theory containing gauge fields or higher-form gauge fields, if the background spacetime is a complicated manifold, a nice way to represent the configuration of the gauge field mathematically is ...
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1answer
46 views

Superalgebra and BRST supersymmetry 1: odd vs even

It is said that most of theories, the even elements of the superalgebra correspond to bosons and odd elements to fermions (but this is not always true; for example, the BRST supersymmetry is the other ...

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