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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7 views

How are coefficients of equation of sideslip angle in the paper “A general Solution to the Aircraft Trim Problem” calculated?

I am sure many of you guys(Aerospace related) must have read the paper, "A General solution to the Aircraft Trim Problem" by Marco, Duke and Bernt. I am working with the turning of the Aircraft and I ...
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What is a NS-2 brane?

This is a question about topological string theory. The existence of a new brane called "an NS-2 brane" is predicted in (the second paragraph in the page 14 of) the paper N=2 strings and the ...
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3answers
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Line integral of a point charge

I am trying to teach myself Electrodynamics through self-study of Griffiths' Introduction to Electrodynamics, and I am having difficulty with a calculation that involves a line integral of a point ...
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Expression for sum over paths

In an introductory lecture on the path integral formalism, I came across the following. Suppose that $\gamma$'s are paths such that a particle travelling along any of them reaches the position co-...
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32 views

What are infinitesimal formally? [duplicate]

Can any problem in physics involving infinitesimal be converted to a rigorous epsilon Delta argument? Say for example finding the moment of inertia of a continuous body? Another is what is the ...
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1answer
50 views

Can a null hypersurface be foliated by spacelike sections?

Let $(M,g)$ be a $d$-dimensional Lorentzian manifold and let $\Sigma \subset M$ be a null hypersurface, which therefore has dimension $(d-1)$. We know that its normal vector $k^\mu$ is null and since ...
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1answer
47 views

Index on a compact manifold

How can the integral of a topological term (like the Nieh-Yan term) on all of a compact manifold be nonzero whereas it's a total derivative and the manifold has no boundary? I assume the manifold can ...
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1answer
46 views

Difference between two derivative operator given in Jackson's book

As I was reading Jackson (3rd edition), On page 543 I see two different types of derivatives. they are given, (11.76) $$ {\partial^\alpha} {\equiv} \frac{\partial}{\partial x_\alpha} = (\frac{\...
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2answers
59 views

Can 'distance' be mathematically described as the convolution of velocity and time, in time domain?

I have phrased the question as such, to confirm that convolution of the two functions raises the dimensionality of the convolution product. So, if I do convolution of velocity and time, then the ...
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34 views

Interpretation of random matrix eigenvectors in physics

Random matrices may be used in physics to replace Hamiltonian of complex system, for instance in nuclear physics. Eigenvalues of these matrices are simply interpreted as the energy levels (even if we'...
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Linking the de Rham bundle/complex over spacetime to the gauge bundle

In some textbooks, the Maxwell equations are stated in a very simple mathematical form (up to multiplicative constants coming from the system of units being used): $$ \begin{array} \mbox{d}F =0, \\ \...
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195 views

Magic angle graphene

In this article here, it is claimed that the following model for bilayer graphene $$\mathcal H= \begin{pmatrix} 0 & \mathcal D^*(-r) \\ \mathcal D(r) & 0 \end{pmatrix}, \mathcal D(r)=\begin{...
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46 views

Average quantity over Keplerian orbit

I have been working through some lecture notes and am quite confused on something. I am trying to understand how to average a quantity over an orbit (Keplerian) but I am struggling to get a clear idea ...
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Weight function in inner product

Until know I thought that the definition of the inner product between two functions $f(\vec{r})$ and $g(\vec{r})$ with the same domain $D:[a,b]$ was: $$\int_a^b f\cdot \overline{g} \cdot d\vec{r}$$ ...
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1answer
28 views

Angular velocity of a conic pendulum [closed]

Some year 12 circular motion questions for you. I have an experiment where an object of m mass is tied to a string of L length. Centripetal force (Fc) is known along with m and L. The object is spun ...
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1answer
63 views

What are some good references for field theory via functional analysis?

Many of the aspects of QFT are traditionally done in ways incompatible with a rigorous mathematical treatment, calling for a variety of tricks to fix essentially what was caused by unjustified ...
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1answer
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Relation between product and type of quantity?

In physics, whenever we have 3 quantities $A$, $B$ and $C$ related as $ A=BC $ where $B$ and $C$ are vector quantities and $ \theta $ is the angle between $B$ and $C$, if $A$ is proportional to $cos\...
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44 views

Taking the infinite-volume limit of a lattice fermion in different ways: does this give all unitarily inequivalent Hilbert-space representations?

When the quantum field theory of a free fermion field is formulated on a finite lattice, the Hilbert space is finite-dimensional. The "spectrum condition" that we normally require in QFT is ...
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Calculating exactly the divergent part of amplitudes to all loop order with DimReg

Suppose we have the $L$-loop amplitude of the form $$\mathcal{I}_L=\int \prod_{i=1}^L \frac{d^D q_i}{(2 \pi)^D} \frac{1}{q_i^2} \frac{1}{(p-\sum_{i=1}^L q_i)^2}.$$ Introducing Feynman parameters to ...
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Combinatorics identity for arbitrary value of Spin

I wanted to prove this identity for the general value of $\lambda$ $$ \sum_{n=0}^{\lambda-1} (-1)^n{\lambda-1 \choose n} {\partial^{\left(\lambda-1-n \right)}{\partial_-}^{\left(n \right)}}\left( \...
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1answer
43 views

Meaning of Kronecker Product in Partially Expanded Operators

I am studying operators in quantum mechanics and have reached confusion in the meaning of the Kronecker product of such operators. I am fairly lost so please excuse any errors in the following text. ...
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1answer
50 views

How can even there be a non-zero BMS vector field with zero asymptotic data?

Following the BMS approach, one spacetime $(M,g)$ is asymptotically flat when: We can find a Bondi gauge set of coordinates $(u,r,x^A)$ characterized by $$g_{rr}=g_{rA}=0,\quad \partial_r\det\left(\...
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Are causal sets based on set-theory?

There is a theoretical approach with the aim of formulating a quantum gravity theory which uses causal sets. Are these sets based on mathematical set theory? Are causal sets part or belong to the ...
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1answer
53 views

Horn equation (wave propagation in an object with a circular cross-section)

I have a problem with finding eigenfrequencies for wave which propagate in an object with a circular cross-section. I don't know how to start. I'll be very grateful for solution and comment or ...
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1answer
29 views

Oscillation on Angled Rails (Diff Equation)

This problem was taken from David Morin's Introduction to Classical Mechanics My attempt at solving the problem: First, I labeled all the relevant forces acting only on one of the particles of mass $...
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1answer
22 views

Are there two elements in each fiber in the fiber bundle of the tangent vectors of a circle, or infinitely many?

I'm watching this video (Frederic Schuller) and, at timestamp 9:50 have become confused about fiber bundles: https://youtu.be/UbQS40KHkH0?t=587 He says you can imagine 'turning the tangent vectors ...
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1answer
51 views

The physical meaning of maximal non-integrability of the contact structure

So, basically integrability is equivalent to the existence of an integral manifold of the distribution and I guess, the integral manifold is like a plane of motion where state moves in physical sense. ...
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Any connection between Boltzmann distribution and machine learning ``softmax'' function?

Disclaimer: I know very little physics and stumbled across this in a math text. Suppose we have a finite configuration space $\mathcal{X}$. For each configuration $x \in \mathcal{X}$, the probability ...
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32 views

Invertibility of the Legendre Transformation

The above image shows the Legendre Transformation in the context of an introduction to the Hamiltonian formalism. My question is in 4.6, wherein $u(x, y)$ has been defined; what is the guarantee ...
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22 views

Source for Learning? [duplicate]

I am an very ameteur mathematician and physicst (If I can say mathematician and physicst to myself xD). I want to learn topics in physics. Like electromagnetism, mechanic, thermodinamics etc. But I ...
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4answers
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Non-unique zero function in the function space (Hilbert space)

I have just started studying about quantum mechanics, and I was studying the definition of the inner product for functions; I am also quite new to linear algebra. While studying I think I encountered ...
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The value of Gravitational Chern Simons theory integration on some three manifolds

Consider the 3d gravitational Chern Simons theory $$S= \frac{k}{192 \pi} \int_{M_3} \mathrm{Tr}\left(\omega\; \mathrm{d} \omega + \frac{2}{3}\omega^3\right)$$ where $\omega$ is the spin-connection on ...
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170 views

Discussion: Mathematically precise physical textbooks [closed]

I am very interested in the abstract mathematical description of nature. Therefore, I have recently started to compile a list of good textbooks about physics, which have a very high level of ...
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49 views

Measure-theoretic force

I understand classical mechanics as a science of moving masses. So I decided to work out it formulation based on measure there just for fun. In this framework the classical mechanical system would be ...
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45 views

How to prove that $u(r)=k \dfrac{1}{r}$ is the only solution for the integral equation $\displaystyle\int_{V'}\rho'\ u(r)\ dV' = constant$?

Consider a hollow spherical charge with density $\rho'$ continuously varying only with respect to distance from the center $O$. $V'=$ yellow volume $k \in \mathbb {R}$ $\forall$ point $P$ inside ...
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87 views

Why is integral of product of a test function and derivative of Dirac-delta function seems to diverge? [closed]

Suppose,we have to evaluate the integral $\int_{-\infty}^{\infty}f(x)\delta'(x)dx$ Traditionally to solve this,we integrate by parts so that the integral is equal to$-f'(0)$,which is finite if $0$ is ...
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1answer
19 views

Measuring phase constants from the sine function

In simple harmonic motion, is the phase, by definition, always measured using the sine function? I'm asking because a question came up that provided $\omega$ and the amplitude, and also specified the ...
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30 views

How to calculate the total gravitational potential energy of a vertical object (do we use integration?)

Hello I was reading another question asked by zach466920, and when he was trying to calculate the total GPE of a water 'tower', he used this explanation: He basically used integration to calculate ...
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18 views

Relation between Scattering Matrix and Correlation matrix

Scattering matrix is the matrix which transform an input vector to an output vector. On the other hand Correlation matrix is the matrix of auto-correlation and cross correlation functions. Where we ...
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1answer
31 views

Does the concept of infinity have any relevancy or application in Physics and applied Physics? [duplicate]

Does the concept of infinity have any relevancy or application in Physics and applied Physics? I must admit that I am not particularly knowledgeable in the area of Physics, but I have never seen the ...
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1answer
130 views

Past boundary of $\mathcal{I}^+$ and future boundary of the hyperboloid resolving $i^0$

Let us consider Minkowski spacetime. Let $(u,r,x^A)$ be retarded coordinates with $x^A$ coordinates on the sphere. Future null infinity is described here as the $r\to \infty$ limit with $(u,x^A)$ ...
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2answers
172 views

Physical meaning of the operator $\exp(-a {\hat{p}}^2)$

I am curious about the physical meaning of the operator $\exp(-a {\hat{p}}^2)$ with $a$ being a positive constant. With respect to the coordinate basis, I find that $\langle x |\exp(-a {\hat{p}}^2)|x' ...
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43 views

Introductory texts on the mathematical formalism of quantum mechanics [duplicate]

I would like to eventually understand the formal mathematical construction of quantum mechanics. I have almost worked my way through Griffith's book and want a more mathematical approach now that I'm ...
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44 views

Time reversal in second quantization

Let $\mathscr{H} = L^2(\mathbb{R}^d)$ denote the Hilbert space of single-particle states and let $\mathscr{F}$ denote the corresponding Fock space (let's say fermion). Then the time-reversal operator $...
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Calculation of one-point functions in causal perturbation theory

How are one-point functions evaluated in causal perturbation theory? I'm not sure where my mistake is in following the standard procedure. Take the first-order coupling $T_1=\lambda \phi^3$. Within ...
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2answers
67 views

Can the real-time Green's function be written in the form of path integral on the real axis? [closed]

In every textbook, the path integral of the Green's function is written in imaginary-time. I wonder whether we could write real-time green function in the path integral form.
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1answer
84 views

Mathematical understanding of band energy

Let me first give a sketch of how I understand band energy mathematically. It is not exactly rigorous, but probably could be made rigorous under suitable conditions. Let $H$ denote the Hamiltonian on ...
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59 views

What is the advantage of using Kruskal coordinates?

(as opposed to Eddington-Finkelstein coordinates) The EF coordinates already take care of the coordinate singularity so I dont see a point for using Kruskal coordinates.
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134 views

Why boundary terms make the variational principle ill-defined?

Let me start with the definitions I'm used to. Let $I[\Phi^i]$ be the action for some collection of fields. A variation of the fields about the field configuration $\Phi^i_0(x)$ is a one-parameter ...
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141 views

What is density matrix in QFT?

In quantum mechanics exist fundamental object Density matrix. (See for introduction last chapter in Principles of Quantum Mechanics by David Skinner). Density matrix nesesary to describe systems even ...

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