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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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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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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Matrix derivative of a matrix with constraints

I am looking for a general method to obtain derivative rules of a constrained matrix with respect to its matrix elements. In the case of a symmetric matrix $S_{ij}$ (with $S_{ij}=S_{ji}$), one way to ...
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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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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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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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Mathematical rigorous introduction to solid state physics

I am looking for a good mathematical rigorous introduction to solid state physics. The style and level for this solid state physics book should be comparable to Abraham Marsdens Foundations of ...
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1answer
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String theory from a mathematical point of view

I have a great interest in the area of string theory, but since I am more focused on mathematics, I was wondering if there is any book out there that covers mathematical aspects of string theory. I ...
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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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Wightman quantum field - Interpretation

I have a question regarding the interpretation of the Wightman quantum field in mathematical quantum field theory. A quantum field $\phi$ is a operator-valued distribution. This means that $\phi$ is ...
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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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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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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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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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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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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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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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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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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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36 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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When does the $n$th bound state of a 1D quantum potential have $n$ maxima/minima?

In Moore's introductory physics textbook Six Ideas that Shaped Physics, he describes a set of qualitative rules that first-year physics students can use to sketch energy eigenfunctions in a 1D quantum-...
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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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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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Is the Hilbert space spanned by both bound and continuous hydrogen atom eigenfunctions?

As e.g. Griffiths says (p. 103, Introduction to Quantum Mechanics, 2nd ed.), if a spectrum of a linear operator is continuous, the eigenfunctions are not normalizable, therefore it has no ...
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Monstrous Moonshine outside of String Theory

My question concerns applications of monstrous moonshine, which is the connection between the $j$-function and the monster group. Recently, physicists have applied it to string theory and, ultimately, ...
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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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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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The role of $\dim H_n$ in the definition of asymptotically continous functions on vectors

When considering the asymptotic continuity of quantum states, one works with asymptoticallycontinuous functions. In the definition one has the following, a funtion $f$ is asymptotically cts if for a ...
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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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What are Grassmann (even/odd) numbers used in superalgebras?

Are Grassmann numbers a concept of graded Lie algebras or is something specific to superalgebras? What are they (i.e: how are they defined, important properties, etc.)? Is there a reasonable ...
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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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Normal ordering by contour integral in CFT

In chapter 6 of Di Francesco, they introduce the normal ordering $$ (AB)(w) = \oint_w \frac{ dz }{ 2\pi i (z-w) }A(z) B(w)\ .\tag{6.130}$$ So far so good. But then starting eq (6.139) $$ \oint_w \...
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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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Defining quantum effective/proper action (Legendre transformation), existence of inverse (field-source)?

Given a Quantum field theory, for a scalar field $\phi$ with generic action $S[\phi]$, we have the generating functional $$Z[J] = e^{iW[J]} = \frac{\int \mathcal{D}\phi e^{i(S[\phi]+\int d^4x J(x)\...
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Equivalent Representations of Clifford Algebra

I'm reviewing David Tong's excellent QFT lecture notes here and am a little confused by something he writes on page 94. We've considered the standard chiral representation of the Clifford Algebra, ...
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1answer
279 views

On Seiberg-Witten curves

In page 44 of Gaiotto's article "Families of $\mathcal{N}=2$ Field Theories" on Teschner's review the author writes down the pure Seiberg-Witten curve as $$ x^2 = z^3 + 2uz + \Lambda^4z $$ with the SW ...
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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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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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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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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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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
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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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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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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
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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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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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Primary constraints for constrained Hamiltonian systems

I would be most thankful if you could help me clarify the setting of primary constraints for constrained Hamiltonian systems. I am reading Classical and quantum dynamics of constrained Hamiltonian ...
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Can we deduce that particles behave as Brownian motions if the collection obeys the Einstein model?

The density dynamics of a continuum of particles with the dynamics $$dx^i_s = d w^i_s,$$ where $dw^i_s$, $0 \leq s$, $i \in \mathcal{N}$ is a standard Brownian motion, are given by the diffusion PDE $$...
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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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