"Mathematical physics consists of the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories." (from Journal of Mathematical Physics) This tag is intended for questions on methods used ...

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Level of Rigor in Mathematical Physics

I am a physics/math undergrad and I have recently become familiar with some more rigorous formalisms of mechanics, such as Lagrangian mechanics and Noether's Theorem. However, I've noticed that the ...
21
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1answer
936 views

Why do zeta regularization and path integrals agree on functional determinants?

When looking up the functional determinant on Wikipedia, a reader is treated to two possible definitions of the functional determinant, and their agreement is trivial in finite dimensions. The first ...
20
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4answers
883 views

Mathematical and Theoretical Physics Books

Which are the good introductory books on modern mathematical physics? Which are the good advanced books? I read Whittaker's Analytical Dynamics, and I am reading Arnold's Mathematical Methods of ...
18
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3answers
548 views

Making some standard theoretical physics argument rigorous

In theoretical physics one often encounters the following rationale: if $f$ and $g$ are functions on $\mathbf{R}^n$, satisfying some technical conditions, and $\displaystyle\int_\Omega f=\int_\Omega ...
16
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3answers
1k views

A fun problem by Arnold using the Poincaré recurrence theorem

I came across this problem by V. I. Arnold while studying his classical mechanics book. Consider a sequence where the $n^{th}$ term is made up by considering the first digit of $2^n$, the first ...
15
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1answer
719 views

Stochastic interpretation of Einstein equations

Einstein's theory of gravitation, general relativity, is a purely geometric theory. In a recent question I wanted to know what the relation of Brownian motion to the Helmholtz equation is and got a ...
14
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6answers
2k views

Mathematics needed in the study of Quantum Physics

As a 12th grade student , I'm currently acquainted with single variable calculus, algebra, and geometry, obviously on a high school level. I tried taking a Quantum Physics course on coursera.com, but ...
14
votes
4answers
825 views

What is the motivation for analytic solutions in Mathematical Physics?

I am trying to understand why one cares about solving PDE's with an analytic/theoretical solution when one can use numerical methods? If you tell me, "only mathematicians try to find theoretical ...
14
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5answers
899 views

In what ways has physics spurred the invention of new mathematical tools?

I came across this comment: Mathematical rigor is not a criterion that physicists have for evaluating their theories. From a mathematical perspective, the non-rigorous theories are far more ...
14
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1answer
298 views

Legendre Transformation of a Lagrangian in Classical Mechanics

I have some questions about the Legendre Transformation of a Lagrangian in Classical Mechanics to the Hamiltonian: We start with a Lagrangian $L(q,\dot{q})=\frac{\langle \dot{q} , \dot{q}\rangle }{2} ...
13
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4answers
425 views

Correct spaces for quantum mechanics

The general formulation of quantum mechanics is done by describing quantum mechanical states by vectors $|\psi_t(x)\rangle$ in some Hilbert space $\mathcal{H}$ and describes their time evolution by ...
13
votes
2answers
1k views

What is the exact and precise definition of an ANGLE?

On wikipedea I found a definition of an Angle as such: "In order to measure an angle θ, a circular arc centered at the vertex of the angle is drawn, e.g. with a pair of compasses. The length of ...
13
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2answers
779 views

Mathematically rigorous text on classical electrodynamics.

Is there any textbook (preferably not written by a physicist) on classical electrodynamics which gives a rigorous (by the standards of pure mathematics) treatment of (a part of) the topics found in ...
12
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3answers
2k views

What's the Clifford algebra?

I'm reading a book on Clifford algebra for physicists. I don't quite understand it conceptually even if I can do most algebraic manipulations. Can some-one teach me what the Clifford algebra really ...
12
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4answers
1k views

Is this a dirac delta function?

I had this on an exam yesterday, and I'm not entirely convinced that the statement is true. We were asked to show that the function $\delta (x) = \int_{-∞}^{∞} \frac{1}{t(t-x)} dt$ is a dirac delta ...
12
votes
2answers
730 views

How to introduce stress tensor on manifolds?

I want to understand the type of stress tensor $\mathbf{P}$ in classical physics. Usually in physics it is said that the force $\text d \boldsymbol F$ (vector) acting on an infinitesimal area $\text ...
12
votes
1answer
220 views

Problem in Hamiltonian system

Not sure if this is too much physics to be here... Consider $$H:\mathbb{R}^{2N+1}\rightarrow\mathbb{R}$$ of class $C^2$, let $H(x,y,z)$ such that $x\in\mathbb{R}^N$, $y\in\mathbb{R}^N$ and ...
12
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0answers
84 views

Area enclosed by an equipotential curve for an electric dipole on the plane

I am currently teaching Physics in an Italian junior high school. Today, while talking about the electric dipole generated by two equal charges in the plane, I was wondering about the following ...
11
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3answers
250 views

Arnold Trivium Problem 39

We find in Arnold's Trivium the following problem, numbered 39. (The double integral should have a circle through it, but the command /oiint does not work here.) Calculate the Gauss integral ...
11
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3answers
334 views

Areas of contemporary Mathematical Physics

I have often heard that some developments in Physics such as Gauge Theory, String Theory, Twistor Theory, Loop Quantum Gravity etc have had a significant impact on pure Mathematics especially geometry ...
11
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1answer
258 views

Guidance regarding research in Mathematical Physics

I am currently a Master's student in Mathematics. The main focus of my undergraduate programme was on Mathematics. However as a part of the course, I have done some Theoretical Physics courses. In ...
11
votes
1answer
438 views

Quantization of angular momentum: is Dirac's proof wrong?

I'm trying to understand the physicist's proof of the theorem on the spectral structure of angular momentum operators (I'm being told that this proof is due to Dirac). I will refer to Ballentine's ...
10
votes
3answers
385 views

Meaning of $\int\mathop{}\!\mathrm{d}^4x$

What the following formula mean? $$\int\mathop{}\!\mathrm{d}^4x$$ I know that this $\int f(x)\mathop{}\!\mathrm{d}x$ is the integral of the function $f$ over the $x$ variable, but the following ...
10
votes
2answers
330 views

Existence and uniqueness of Stokes flow

What are the solution existence and uniqueness conditions for Stokes' flow? $$\begin{gathered} \nabla p = \mu \Delta \vec{u} + \vec{f} \\ \nabla \cdot \vec{u} = 0 \end{gathered}$$ Maybe you could ...
10
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2answers
227 views

Applications of Algebra in Physics

Often I have heard about the link between Algebra (in particular Representations of Groups and Algebras) and some "indefinite" field of Physics. I have a good preparation in Algebra and ...
10
votes
1answer
780 views

Hodge Star Operator

I'm trying to understand the Hodge star operation, but have come across an impasse almost immediately. I have the definition $$(\star \omega)_{a_1\dots a_{n-p}}=\frac{1}{p!}\epsilon_{a_1\dots ...
10
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1answer
417 views

Topological Quantum Field theories

I was wondering about the following on TQFTs. It is said that TQFTs have vanishing Hamiltonians $\hat{\mathcal{H}}$. Firstly, I would like to ask: Why is this so? Secondly, consider the ...
10
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1answer
215 views

Adding small correction term to ODE solution

Let $\mathbf{r}(t) = [x(t), y(t), z(t)]$ and $\mathbf{v}(t) = \frac{d}{dt}\mathbf{r}(t)$. I'm trying to solve $$ \frac{d}{dt}\mathbf{v}=\frac{q}{m}(\mathbf{v}\times\mathbf{B}) \tag{1} $$ where $q$ and ...
9
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4answers
574 views

Is there any abstract theory of electrical networks?

Designing electrical networks is among the highly mathematical engineering disciplines, which uses a vast scope of techniques from Fourier analysis and complex function theory, to logic, combinatorics ...
9
votes
1answer
354 views

Electrodynamics in general spacetime

Let $M\cong\mathbb{R}^4_1$ be the usual Minkowski spacetime. Then we can formulate electrodynamics in a Lorentz invariant way by giving the EM-field $2$-form $\mathcal{F}\in\Omega^2(M)$ and ...
9
votes
1answer
381 views

Is the Structure Group of a Fibre Bundle Well-Defined?

Am I right in thinking that the structure group of a fibre bundle is any group $G$ of homeomorphisms of the fibre $F$ such that all transition functions map into $G$? Or is $G$ somehow the minimal ...
9
votes
1answer
274 views

Riemann Zeta function, quaternions and physics

Disclaimer: This question is rather vague, and thus probably not suitable for Mathoverflow, so I prefer to ask it here. I'm sorry if it doesn't meet the standards of this site. Several years ago, I ...
8
votes
2answers
219 views

What is the simplest mathematical concept that does not map to a physical phenomenon?

One of my colleagues argues that everything in math proves something in the physical world. For instance, he claims that the existence of math to describe fractals proves the infinite divisibility of ...
8
votes
1answer
435 views

Hamiltonian for Geodesic Flow

I'm trying to prove that geodesic flow on the cotangent bundle $T^* M$ is generated by the Hamiltonian vector field $X_H$ where $$H = \frac{1}{2}g^{ij}p_i p_j$$ but I am stuck. Could somebody show ...
8
votes
2answers
81 views

Proof of a theorem about oscillation [duplicate]

There is a theorem in page 100 of Arnold's Mathematical Methods of Classical Mechanics, which says that: If $\cfrac{dx}{dt} = f(x) = Ax + R_2(x)$, where $A = \cfrac{\partial f}{\partial x}|_{x = ...
8
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1answer
623 views

Applications of representation theory in physics

The notes of a lecture on basic group and representation theory I attended last semester begin with a bit of motivation for the argument. They give the following examples for applications in physics: ...
8
votes
2answers
429 views

What is a particle mathematically?

In quantum field theory, what is a particle mathematically? How would you explain to someone who kows alot of math but no physics what a particle is? A simple example model would suffice.
8
votes
1answer
344 views

What is quantum field in terms of mathematics?

I am reading a book on quantum field theory, while I have never been trained as a physicist. I found a big gap in language and have trouble understanding what physicists mean by "quantum field". If I ...
8
votes
1answer
282 views

Maurer-Cartan 1-form

Can anyone help me with the following? Let $\rho$ be the right-invariant Maurer-Cartan 1-form $$\rho = dg\ g^{-1}$$ I want to show that the MC equation $$d\rho - \rho \wedge\rho = 0$$ holds. So ...
8
votes
1answer
141 views

Derivation of Schrödinger's equation

I recall a famous quote of the late physicist Richard Feynman: Where did we get that from? It's not possible to derive it from anything you know. It came out of the mind of Schrödinger. This ...
8
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1answer
613 views

Does apparent retrograde motion of planets begin and end at quadrature?

I've read it several places that the apparent retrograde motion of planets (during which they seem, as viewed from Earth, to move in the opposite sense of their normal "direct" orbital motion against ...
8
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0answers
315 views

Overview of nonlinear analysis, differential equations (ODE and PDE), dynamical systems, and mathematical physics, and their relationships [closed]

The fields of (i) nonlinear analysis, (ii) ODE and PDE, (iii) dynamical systems, and (iv) mathematical physics (e.g., electromagnetism, general relativity, gravitation, etc,...) are very huge, ...
7
votes
6answers
322 views

Proving that $E=mc^2$

What are the axioms of special relativity? Is there a book or paper that introduces the theory of special relativity in a rigorous manner, and proves that $E=mc^2$ after appropriate definitions?
7
votes
3answers
181 views

Gathering books on Lorentzian Geometry

I find it very hard to find books on Lorentzian Geometry, more focused on the geometry behind it, instead of books that go for the physics and General Relativity approach. More specifically, I'm ...
7
votes
2answers
118 views

In what sense does $\sum_{k=0}^{\infty} 2^{2k} = - {1 \over 3}$?

In The Road to Reality Penrose remarks on an identity written down by Euler which is "obviously wrong" and yet correct "on some deeper level". He makes reference to the series again when discussing ...
7
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1answer
288 views

What are spinors mathematically?

In the wikipedia article on spinors a number of mathematical definitions are given of spinors which I find slightly confusing. There are essentially two frameworks for viewing the notion of a ...
7
votes
1answer
785 views

Some questions about representations of $SO(6)$

I would like to know the proof/explanation for the following three properties of the representation of $SO(6)$, What is the importance of symmetric traceless tensors of arbitrary rank w.r.t $SO(6)$ ...
7
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1answer
160 views

Positivity of the Coulomb energy in 2d

Let $$D(f,g):=\int_{\mathbb{R}^3\times\mathbb{R}^3}\frac{1}{|x-y|}\overline{f(x)}g(y)~dxdy$$ with $f,g$ real valued and sufficiently integrable be the usual Coulomb energy. Under the assumption ...
7
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2answers
143 views

When does a PDE solve a variational problem?

I understand that for a functional $J[f]$ on the space of differentiable functions $f$ on some domain, setting $\delta J[f]|_{f=f_0} = 0$ yields a (possibly nonlinear) partial differential equation in ...
7
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1answer
172 views

Stefan-Boltzmann Constant and Stefan's Law

The following argument is from my textbook, An Introduction to Thermal Physics by Daniel Schroeder. If you are familiar with the derivation of Stefan's Law from the energy density of a photon gas, you ...