"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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Properties of functional integration

this question comes from theoretical Physics, the issue being the so called Path Integral. The measure of this thing is something written as $[d\phi]=\prod_x d\phi(x)$ And this should be the limit ...
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1answer
62 views

How to use the Mehler kernel to get the solution of the Quantum harmonic oscillator with a given initial condition

In this wiki-article http://en.wikipedia.org/wiki/Mehler_kernel the fundamental solution of the differential equation for the Quantum harmonic oscillator is provided by the Mehler Kernel: ...
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44 views

Trigonometry, find distance of arc movement

Imagine I have the setup as follows: I want to compute the height x in State 2, depending on how much the blue axis have moved. That is, the distance ...
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1answer
58 views

The definition of scalar and vector concomitant of a metric

I'm reading Defrise-Carter's paper Conformal Groups and Conformally Equivalent Isometry Groups. One might find the paper at the following link: ...
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1answer
95 views

How to evaluate the integral $\int e^{ipx}e^{ipx} d^{3}x = 0$

I am embarrassed to ask this question. But I came across the following in a physics book: $$\int e^{ipx}e^{ipx} d^{3}x = 0$$ $d^{3}x = dydydz$, as @Semiclassical shows below. This came up in the ...
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Mathieu equation solution with non-periodic boundary conditions

I need to solve the Mathieu equation: $y''(x)+(a-2q \cos(2x)) y(x) = 0$ but with the unusal boundary condition: $y(x+\pi) = e^{i \alpha}y(x) \quad , \quad \alpha \in R$ if $\alpha = 0$ than the ...
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55 views

derivation of an equation involving the Fourier transform of the square modulus of a wave function

A textbook on electron optics states that, ignoring a factor of 2 for convenience, the result $\mathscr{F}(I(\vec{r}))=\mathscr{F}(\phi(\vec{r}))\cdot{}A(\vec{k})\cdot{}\sin[\gamma(\vec{k})]$ can be ...
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1answer
24 views

Stopping point of a sliding particle.

A particle with given mass > 0 and given coefficient of friction > 0 and given initial downward speed > 0 starts at (0,1) on the graph of y = exp(-x). The coefficient of friction applies only to those ...
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2answers
64 views

Complete ONS and pure point spectrum

In all that follows all operators are taken to be densely defined on a Hilbert space $H$. Some textbooks state that an operator $A$ on $H$ has pure point spectrum if $H$ admits a complete ONS (Hilbert ...
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3answers
72 views

integral of the sphere describing lambertian reflectance

A Lambertian surface reflects or emits radiation proportional to the cosine of the angle subtended between the exiting angle and the normal to that surface. The integral of surface of the hemisphere ...
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1answer
201 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 ...
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1answer
41 views

Function spaces for the 1-dim heat equation.

Consider the standard 1-dim heat equation: $u_t(x,t)-\alpha u_{xx}(x,t)=0$, where $u:\mathbb{R}\times\mathbb{R_+}\rightarrow \mathbb{R}$, with initial conditions $u(x,0)=g(x), x\in\mathbb{R}$ and ...
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1answer
68 views

Formulation VS Interpretation

I'm reading a book on Mathematical Physics and at some point the author says that we must distinguish between a "formulation" and an "interpretation" of a theory, although it's not easy to point what ...
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94 views

Analytical Models for Hysteresis of Complicated Systems

I’ve been working with a system that exhibits hysteresis and I’ve found that the more common models do not work for me. I am wondering if anyone is aware of other models that might be out there for ...
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104 views

Computation of the Frenet-Serret trihedron in $\Bbb L^3$ (Lorentz-Minkowski space)

Consider $\Bbb L^3 = (\Bbb R^3, \langle , \rangle)$, with the convention $$\langle (x_1,y_1,z_1), (x_2,y_2,z_2)\rangle = x_1x_2+y_1y_2 - z_1z_2$$ and $\| v \| = \sqrt{|\langle v, v \rangle|}$. Let ...
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56 views

Functional Extremum

Let a functional $H[\phi]$ of a map $\phi\in\mathbb{R}^{\mathbb{R}^4}$ be given by: $$ H(x^0) = \int_{\mathbb{R}^3} ...
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2answers
72 views

Initial value of Newton Raphson Method

I am currently studying Newton-Raphson Method. I feel that I understand the concept of it. Somehow, I am facing some question in my head about how to actually apply it. The questions that I have are ...
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56 views

Why is entropy = the Legendre transform?

Can someone give me a mathematician's explanation (and not a physicist's) as to why $$\int_{\Omega}\Psi^*(b(u(t))$$ is called the entropy where $\Psi^*$ is the Legengre transform of ...
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56 views

Integral calculation - Gravity - Free Fall

I have read this article http://physics.stackexchange.com/questions/3534/dont-heavier-objects-actually-fall-faster-because-they-exert-their-own-gravity. In the best answer by David Z - there are some ...
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37 views

Formal proof of Clebsch Gordon sum

physicist here. When looking at the irreducible representations of $so(3)$, i.e. the set of all real valued anti-symmetric matrices, one can parametrize those irreps with an index $j$ which can be ...
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1answer
48 views

An Integration Calculation

I'm just having a bit of difficulty understanding the last couple of steps made in the paper Horowitz & Hubeny - Quasinormal Modes of AdS Black Holes and the Approach to Thermal Equilibrium (p.8) ...
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48 views

Proof of Arnold-Liouville's Thm: movement in angular coordinates conditionally periodic

I'm reading Arnold's book Mathematical Methods in Classical Mechanics and got stuck on the proof of Liouville's theorem on integrable systems. The proof finishes with Problem 11: Show that the motion ...
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1answer
35 views

Homomorphism between $SL(2,\mathbb{C})$ and the Lorentz group and choice of metric

Just a quick question, I'm hoping someone can clarify how this probably small issue can be resolved. It is said that a Lorentz transformation $\Lambda$ is a linear tranformation of $\mathbb{R}^3_1$ ...
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69 views

Resources for learning Relativity

I´m looking for books to the study of Relativity. I know that this is math stack schange and not physics stack schage, but I believe that some of the users here are interesed in physical-mathematical ...
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260 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 ...
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1answer
141 views

Approximation of the Heaviside Function whose derivative has a compact support

I am looking for a smooth approximation $H_\delta$ of the Heaviside function, which has the property that $$ \lim_{\delta\rightarrow 0^+}H_\delta =H $$ in the distribution sense, and $$ ...
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2answers
61 views

What is the right domain for this Hamiltonian

I want to define a proper domain $D(H) \subset L^2$ for this Hamiltonian ( $\theta$, $\phi$ are the standard angles in spherical coordinates). Furthermore, the wave function is supposed to satisfy ...
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62 views

Where does this unit vector come from?

Although this is a physics problem, the problem is in regard to vectors. There is the relevant reference to this image here: What I need to find is the unit vector of force $F_1$. The book gives ...
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1answer
74 views

Centre of Mass and Moment of Inertia of a sphere - spherical cap

I have been given a sphere of radius a, from this sphere a cap of hight h is cut off. 1) What is the centre of mass of the rest of the sphere? 2) What is the moment of inertia regarding the axis of ...
3
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1answer
38 views

Physical reflections of prime-number distribution

Not a purely mathematical question: I have read somewhere that Atomic Orbital is closely related to the distribution of prime numbers, but I am unable to find any reference to that. Can someone ...
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0answers
52 views

How to power series expand determinants?

Say $g$ is a ($d\times d$) matrix which is given as, $g = g_0 + xg_2 + x^2 g_4 .. +x^{d/2 -1}g_{d-2}+ x^{d/2}(g_d + h_d(log (x)))$ where $d$ is an even number and each $g_i$ is a matrix (same ...
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334 views

is it necessary that curl of 2d vector is perpendicular to the plane.

I am just confused, help me guys. The question comes up, because we say that curl is either clockwise or anti-clockwise at a point.
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117 views

If physical space had cardinality larger than $\aleph_1$, would we need new math to describe dynamics? [closed]

If physical space had cardinality larger than $\aleph_1$, would we need new math to describe dynamics? For instance, would the dynamics of a sea of virtual particles of cardinality $\aleph_1$ would ...
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1answer
35 views

Question about Logistic Regression - 2

How should I tell the difference between those two formulas in the circles below. I am studying logistic regression and I have faced two different formulas from two different documents. I don't know ...
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6answers
285 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?
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2answers
81 views

Collision detection between two accelerating spheres with no initial velocity?

We have two non-touching spheres of radii r1 & r2 are lying in space at rest. Both of them are then given accelerations a1 & a2 respectively at time t=0. Find whether they will ever come in ...
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1answer
26 views

Understand the English paragraph on association rule.

I am currently studying Association Rule Pattern Mining. I am reading the explanation on wikipedia about it. Somehow, I feel like I have a problem in understanding the paragraph below. Can somebody ...
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1answer
51 views

Orthogonality of associated Legendre polynomials

Let $P_n(x)$ be the $n$-th degree Legendre polynomial. Let $k$ be a nonnegative integer less than or equal to both $n,m$. How to prove that $$ \int_{-1}^1 (1-x^2)^k D^kP_n(x) D^kP_m(x)\,dx = ...
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1answer
68 views

About reparametrization of timelike curves in $\mathbb{L}^3$ (Lorentz-Minkowski space)

I think there is something wrong with the proof this text gives of Lemma $2.1.5$, in pages $19$ and $20$, for timelike curves. I used another function, and it seems to work. Either I'm wrong, or he ...
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26 views

An identity relating sum of number of partitions to sum of number of parts

I encountered this identity while studying about the Kac determinants in CFT. $$\sum_{\{n_1 + \cdot \cdot \cdot + n_k \}} k = \sum_{pq \le N} P(N-pq)$$ Here $P(N-pq)$ is the number of partitions of ...
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1answer
52 views

Are these derivatives correct?

Given the map $E: \mathbb{R}^3 \times \mathbb{R} \rightarrow \mathbb{R}^3$, such that (notice that this excercise is taken from Physics(Electrodynamics)). $$E(r,t) = -\frac{1}{4 \pi \varepsilon_0} ...
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1answer
35 views

Uniform convergence on functions

For a $f(x)=x^n, x\in[0,0.5]$ I know that $d_\infty (x^n,0)=sup|x^n-0|=\frac{1}{x^n} \rightarrow 0 $, when $n\rightarrow \infty$ and $x\in[0,0.5]$. For $\{x^n-x^{n+1}\}, x\in[0,1]$ this isn't the ...
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1answer
60 views

Mathematics/Mechanics Problem

I would like to ask you if anybody could help me with this problem. So far i know that the positions where B and A have to meet are at distances L and L+2r
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1answer
38 views

Reference request: what is the relation between classical r-matrices and quantum R-matrices?

I learned from a professor that $$ R=Id+(q-1)r+ o(q-1), $$ where $R$ is a quantum $R$-matrix and $r$ is the corresponding classical $r$-matrix. Here $o(q-1)$ denotes a term of the form $A(q-1)^2$, ...
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1answer
54 views

What is the explicit formula for classical r-matrices?

It is said that classical r-matrices are those satisfy the classical Yang-Baxter equation $[r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] = 0$, where $r \in \mathfrak{g} \otimes \mathfrak{g}$. ...
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59 views

Using the Hodge theorem to decompose the metric tensor

There has been a previous discussion about concrete constructions using the Hodge theorem , Construction of Hodge decomposition Let me try to ask here about a specific case where one is trying to ...
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1answer
88 views

Calculating a double pendulum

consider the following situation of a double pendulum. We found the moving equations as $$ \ddot{\theta_1}=-L_1\sin\theta_1 + \frac{m_2}{m_1}\cos\theta_2\sin(\theta_2-\theta_1),\\ ...
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1answer
30 views

Find $\psi\in L^2$ so that the overlap $\langle \psi(x+\delta),\psi(x-\delta)\rangle$ is as small as possible

Is there a way to find / characterise functions $\psi\in L^2$ which make the value of this integral small? $$ \text{Re}\int_{\mathbb R} \overline{ \psi (x+\delta)}\,\psi(x-\delta)\;\text d x $$ ...
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41 views

Why is the space of trajectories for a particle a cotangent bundle?

In Paugams book, Towards the Mathematics of Quantum Field Theory, he writes the following in the introduction: Let us first illustrate this very general notion by the simple example of Newtonian ...
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97 views

Are there any legitimate examples of applications of set theory to Physics?

Sets are of course ubiquitous, so in this sense set theory is applicable to physics. But I'm thinking more like techniques like forcing, or constructs like cardinals or ordinals. Intuitively it ...