"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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22
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5answers
869 views

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 ...
18
votes
3answers
438 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 ...
14
votes
1answer
208 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} ...
12
votes
4answers
315 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 ...
12
votes
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
votes
4answers
392 views

Mathematical and Theoretical Physics Books

Which are the good introductory books on modern mathematical physics? Which are the more advanced ones? I already read Whittaker's Analytical Dynamics, and I am reading Arnold's Mathematical Methods ...
12
votes
2answers
438 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
votes
1answer
198 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 ...
11
votes
6answers
1k 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 ...
11
votes
4answers
767 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 ...
11
votes
2answers
616 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 ...
11
votes
1answer
378 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
325 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
220 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
votes
2answers
158 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
0answers
452 views

What Areas Should a Potential String Theorist Study at Graduate Level? [closed]

Next October I start a year long course in Cambridge, intended as preparation for a PhD. I chose mainly pure disciplines as an undergrad (particularly topology and analysis) but I'd really like to ...
9
votes
1answer
342 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 ...
9
votes
3answers
207 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 ...
9
votes
1answer
538 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 ...
9
votes
1answer
280 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 ...
9
votes
1answer
201 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 ...
8
votes
5answers
395 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 ...
8
votes
2answers
66 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
votes
1answer
409 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
395 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
201 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
269 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
474 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 ...
7
votes
6answers
230 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
2answers
157 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 ...
7
votes
1answer
207 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 ...
7
votes
1answer
299 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 ...
7
votes
1answer
130 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
votes
2answers
103 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
votes
1answer
343 views

Do these two sets of matrices form groups?

Stimulated by some Physics backgrounds, consider the following two sets of matrices. Notations and definitions:Let $A,B$ be two complex $n\times n$ matrices, then $\left [ A,B \right ...
6
votes
1answer
120 views

How to find interesting operators for a quantum system?

How can we find "interesting" operators for a quantum mechanical system? I can think of the following method: Given some system with an associated Hilbert space $V$ and Hamiltonian $H:V\rightarrow ...
6
votes
1answer
188 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 ...
6
votes
1answer
237 views

Set theorist as a physicist or physicist as a set theorist?

I'm majoring physics, but really interested in mathematics. I liked physics since it was really beautiful to have an analysis on a nature with mathematical tool. However, the more i study, the more ...
6
votes
1answer
50 views

Cosets for lie groups

I am looking for a general way of determining cosets for $(G\times H)/H$, where $G$ and $H$ are Lie groups. For example what are the cosets $(SU(3)\times SU(2))/SU(2)$. Is there a general method of ...
6
votes
2answers
145 views

Arnold's Trivium problem 51

Calculate $$ f(k) = \int_{-\infty}^{+\infty} e^{ikx}\frac{1 - e^x}{1+e^x}dx.$$ As far as I know, this is not a function but rather the Fourier transform in tempered distributions. 1) What is ...
6
votes
3answers
164 views

What went wrong?

Intrigued by this question, one-dimensional inverse square laws, I started to try to find an answer and came up with what follows. However, I calculated the derivatives to double check myself, and ...
6
votes
1answer
68 views

How did Le Verrier calculate Neptune's position?

In the Wikipdia article on Neptune the discovery is described as a mathematical achievement: Subsequent observations revealed substantial deviations from the tables, leading Bouvard to ...
6
votes
1answer
141 views

Mathematical background for TQFT

I am physicist. I`ve started studying Topological QFT. What would you recommend to read in mathematical field for understanding Witten’s old articles of 80s-90s? What books/articles could help form ...
6
votes
1answer
90 views

Euler Darboux PDE solution

Consider the Euler-Darboux PDE $$ u_{xy}+\frac{k}{x-y}(u_{x}-u_{y})=0 $$ What is the solution when $k>0$? All text books I have looked at give solutions for $k<0$ and I don't seem to see how I ...
6
votes
2answers
262 views

Universal Covering Group of $SO(1,3)^{\uparrow}$

I'm trying to prove that $SL(2,\mathbb{C})$ is the universal covering group for the proper orthochronous Lorentz group $SO(1,3)^{\uparrow}$. The standard way goes as follows. (1) Exhibit a real ...
6
votes
1answer
323 views

Why can algebraic geometry be applied into theoretical physics?

It is to be said at the outset that I do not have much familiarity with physics beyond what is in a semi-popular book; say, the Feynman Lectures Vol 1 and 2. As I progressed in math graduate school ...
6
votes
1answer
88 views

Formal Definition of Yang Mills Lagrangian

I have a question regarding the Lagrangian in non abelian gauge theory. Say, $G$ is the gauge group and $\mathfrak g$ the associated Lie algebra. The Lagrangian is often written as $$ \mathcal ...
6
votes
2answers
75 views

Is it possible to mathematically explain why solids go under mollification when heated?

Well, I'm sure that many people on MSE might object that this is not a math question, however, I think that there might be a well-posed mathematical answer to this question, or at least I hope so. We ...
6
votes
1answer
358 views

One of Wald's General Relativity problems

I've been working through Wald's book "General Relativity", and I'm fairly stuck on question 4. a) of chapter 6: "Let $(M,g_{ab})$ be a stationary space-time with time-like Killing field $\xi^a$. Let ...
6
votes
0answers
139 views

Renormalization for mathematicians

Can someone explain to me the processes of renormalization and regularization used in quantum field theory and similar fields in a way that a pure mathematician might make sense of it? Is there a ...