Questions tagged [mathematical-physics]
DO NOT USE THIS TAG for elementary physical questions. This tag is intended for questions on modern mathematical methods used in quantum theory, general relativity, string theory, integrable system etc at an advanced undergraduate or graduate level.
254
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When will two isomorphic Lie algebras have the same representation?
What feature(s) of isomorphic Lie algebras distinguish between their respective representations? When will two isomorphic Lie algebras have the same or different representations?
My particular case ...
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1
answer
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What are defining & fundamental representations?
In physics terminology, one hears of the fundamental & defining representations of lie algebras or groups - are these the same as irreducible representations?
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2
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Is $1+2+3+4+\cdots=-\frac{1}{12}$ the unique ''value'' of this series?
I'm reading about zeta-function regularization in physics and I have some mathematical doubt. I understand that, since a sum of infinite terms is not well defined in a field, a series that is ...
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votes
1
answer
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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 ...
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5
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Quantum mechanical books for mathematicians
I'm a mathematician. I have good knowledge of superior analysis, distribution theory, Hilbert spaces, Sobolev spaces, and applications to PDE theory. I also have good knowledge of differential ...
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4
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What makes the Cauchy principal value the "correct" value for a integral?
I haven't been able to find a good answer to this searching around online. There is a related old question here, but it never received much attention.
Suppose I have some physical property that I ...
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7
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Why does the "separation of variables" method for DEs work? [duplicate]
Heyho,
I am using the separation-of-variables method for quite a while now, but what was always bothering me a bit, is why is it possible to do those operations. I'll give a concrete example (source ...
user253211
17
votes
1
answer
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Can different choices of regulator assign different values to the same divergent series?
Physicists often assign a finite value to a divergent series $\sum_{n=0}^\infty a_n$ via the following regularization scheme: they find a sequence of analytic functions $f_n(z)$ such that $f_n(0) = ...
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17
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3
answers
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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 $\...
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16
votes
1
answer
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Connection between the spectra of a family of matrices and a modelization of particles' scattering?
In the excellent book "Numerical Computing with MATLAB" by Cleve B. Moler (SIAM 2004), [Moler is the "father" of Matlab], one finds, on pages 298-299, the following graphical representation (fig. 10....
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2
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What exactly are pseudovectors and pseudoscalars? And where could I read about them?
I can't find good information on the internet.
In my mathematical physics class the definition of a vector was given as:
That object with magnitude and direction which doesn't change under ...
- 365
6
votes
2
answers
999
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Conjugate Representations of Lie Algebra of Lorentz Group
I'm trying to understand the Lie algebra of the Lorentz group and am almost there, but am stuck at the final hurdle! It's easy to prove that
$$\frak so(1,3)^\uparrow_{\mathbb{C}}=sl(2,\mathbb{C})\...
- 5,105
4
votes
1
answer
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$SO(p,q)$ Fundamental Weights?
The weights in the $D^{n-1}$ and $D^{n}$ spinor representations of $SO(2n)$ are of the form
$$\frac{1}{2}(\pm e_1 \pm e_2 \pm ... \pm e_{n-1} \pm e_n)$$
such that the products of all the $\pm 1$'s are ...
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5
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Is $\nabla$ a vector?
The following passage has been extracted from the book "Mathematical methods for Physicists":
A key idea of the present chapter is that a quantity that is properly called a vector must have the ...
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1
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How to evaluate the following items in mathematical methods in physics? [closed]
I had taken a long break from math and physics for months due to stress and an illness so when I returned to it and tried answering problems given, I had difficulty figuring out how to do some of them ...
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23
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4
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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 ...
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Quantum mechanics for mathematicians
I'm looking for books about quantum mechanics (or related fields) that are written for mathematicians or are more mathematically inclined.
Of course, the field is very big so I'm in particular ...
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2
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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 ...
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3
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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}$$
Maybe you could ...
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10
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3
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The Inverse of a Fourth Order Tensor
Suppose that we have a fourth order tensor ${\bf{A}}$
$${\bf{A}}=A_{ijkl} {\bf{e}}_i \otimes {\bf{e}}_j \otimes {\bf{e}}_k \otimes {\bf{e}}_l$$
in the orthonormal basis $\{{\bf{e}}_1,{\bf{e}}_2,{\bf{e}...
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votes
2
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Physical interpretation of $L_1$ and $L_2$ norms
In signal analysis, students have no qualms about associating the $L_2$ norm of a square integrable function $f(t)$ as the energy associated with that signal.
A good understanding of whether a ...
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2
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Principal value of 1/x- equivalence of two definitions
As far as I know, the principal value of a non-summable function like $1/x$, denoted $\mathcal{P}(1/x)$, is a distribution that that acts on some smooth function $f$ in some test-function space and ...
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votes
2
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Regularity of an infinite series arising with the heat equation
Let $(t,y)\in(0,\infty)\times\mathbf{R}$, and $\displaystyle f(t,y) \equiv \sum_{k=-\infty}^{\infty}\frac{\exp(-(y-2\pi k)^2/2t)}{\sqrt{2\pi t}}$. This infinite series arises if one attempts to solve ...
user55315
3
votes
1
answer
306
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How to solve an ODE with $y^{-1}$ term
My major is not Mathematics, but I came across the following ODE for $y(x)$:
$$\left(y^3y^{\prime\prime\prime}\right)^\prime+\frac{5}{8}xy^\prime-\frac{1}{2}y+\frac{a}{y}=0,$$
where the prime denote ...
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3
votes
1
answer
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An elliptic integral of first kind expresses the time of motion along an elliptic phase curve in the corresponding Hamiltonian system
Arnold in his essay On teaching mathematics made the following statement:
The de-geometrisation of mathematical education and the divorce from physics sever these ties. For example, not only ...
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vote
2
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Geometry of transformed spacetimes?
The main question seeks to understand whether a conformal structure can be put on "transformed" Minkowski 2-space, which will be denoted as $\Bbb R^{1,1}:=\Bbb M^{1,1}.$ I will get more ...
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2
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Nielsen & Chuang, exercise 2.73 — Density matrix proving the minimum ensemble
I've been trying to solve exercise 2.73 (p.g 105) in Nielsen Chuang, and I'm not sure if i'v been overthinking it and the answer is as simple as i've described below or if I am missing something, or i'...
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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 ...
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votes
2
answers
8k
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Relation between $SU(4)$ and $SO(6)$
This is more of a particle physics question than maths. Since $\operatorname{SO}(6)$ and $\operatorname{SU}(4)$ are isomorphic, how are the fields (say for example scalar fields of ${\mathcal{N}}=4$ ...
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2
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What is the relation between representations of Lie Groups and Lie Algebras?
If $G$ is a Lie Group, a representation of $G$ is a pair $(\rho,V)$ where $V$ is a vector space and $\rho : G\to GL(V)$ is a group homomorphism.
Similarly, if $\mathfrak{g}$ is a Lie Algebra, a ...
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7
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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 ...
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6
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Where is the wild use of the Dirac delta function in physics justfied?
Wikipedia has a wild article about the Dirac delta function. Are the things listed correct? Or is there no proof that they are correct? For my master thesis I want to refer to rigorous proofs of these ...
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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 ...
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0
answers
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Multivariable Integral, How to compute it?
Q
How to evaluate a multivariate integral with a Gaussian weight function?
$$
\mathcal{Z_{n}}
\equiv\int_{-\infty}^{\infty}
\exp\left(-a\sum_{j = 1}^{n}x_{j}^2\right)\,
{\rm f}\left(x_{1},x_{2},\...
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1
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Why is the conjugate momentum $p(x,x')=\partial_2 L(x,x')$ an element of the cotangent space?
Imagine we have a particle described by $x \in M$, where $M$ is some manifold, then it is very intuitive I think that a velocity is an element of the tangent space at $x$, so $x' \in T_{x}M.$ Thus, by ...
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What is a good reference for rigorous Electromagnetism and Electrodynamics?
Is there any good book on Electromagnetism from a more mathematical point of view? By this I mean a book which makes use of differential forms and maybe De Rham cohomology. I was also searching for ...
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5
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If instantaneous rates of change aren't that rigorous, how correct is the usage of instantaneous rates of change (like velocity) by physicists?
According to this answer, instantaneous rates of change are more intuitive than they are rigorous.
I tend to agree with that answer because, in the Wikipedia article on differential calculus, they ...
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1
answer
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Showing Jacobi identity for Poisson Bracket
We were given the following problem: show that $[A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0$ where $[A,[B,C]]$ et cetera are Poisson brackets. As I understand it this is a poisson bracket (where $\mathcal{H}...
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Cylinder-ray intersections equation
I found an article involving infinite cylinder-ray intersections, and I don't know how they develop this equation:
$$(q - p_a - (v_a, q - p_a)v_a)^2 - r^2 = 0$$
In the end of the first page I quote:
...
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1
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Is a "spinor" an element of the Spin group, or an object that transforms under the Spin group—or both?
I've been researching spinors, and I'm a bit confused by some of the terminology. In some cases, spinors seem to be presented as elements of the Spin group, whereas in others they seem to be presented ...
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1
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What really is a path-ordered exponential?
In some texts about gauge theories in Physics I've found one object called a path-ordered exponential which I'm not sure what it means. As I understood, the idea is as follows: let $G$ be a Lie group ...
- 26k
4
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1
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Which of the following is gradient/Hamiltonian( Conservative) system
The question that I have to solve is found below. However, I do not know how to start the solution since I am unsure about the defintion of a Gradient/Hamiltonian System. What must I check first to ...
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1
answer
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Are the physics and math definitions of a complex representation equivalent?
I was astonished to read at Wikipedia that
The term complex representation has slightly different meanings in mathematics and physics. In mathematics, a complex representation is a group ...
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2
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Using the commutator to show unit vectors in polar coordinates are a noncoordinate basis?
I'm having trouble getting the algebra right here and don't know where I'm going wrong:
Show that the unit basis vector fields for polar coordinates in the Euclidean plane,
$$\hat{\mathbf{r}} = \cos\...
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votes
3
answers
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Evaluate the Integral using Contour Integration (Theorem of Residues)
$$
J(a,b)=\int_{0}^{\infty }\frac{\sin(b x)}{\sinh(a x)} dx
$$
This integral is difficult because contour integrals normally cannot be solved with a sin(x) term in the numerator because of ...
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3
votes
0
answers
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Approximating definite integral over infinitesimal interval (reformulated)
Pursuant to helpful comments by user254433, I have decided to take another swing at this problem while reformulating it with a simplified example.
(Reformulated) General Problem: Generally speaking, ...
- 427
2
votes
1
answer
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Choice of the First Term in Legendre Polynomials
The two solutions of the Legendre's Differential Equation obtained by series solution method are :
and
Now according to my textbook, for the useful polynomial for n equal to a positive integer, the ...
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2
votes
1
answer
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Prerequisites for ‘Quantum field theory and representation theory: a sketch’ [arXiv:hep-th/0206135]
I'm interested in reading Dr. Peter Woit's article, Quantum field theory and representation theory: a sketch [hep-th/0206135].
What math and physics background would be needed?
(A list of topics ...
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1
vote
1
answer
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Limit of probability density function as random variable approaches +/- infinity
Consider a complex-valued function $\Psi(x,t)$ such that $|\Psi|^2$ is a probability density function for $x$ (for any time $t$).
In his introductory Quantum Mechanics book, David J Griffiths writes ...
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1
vote
1
answer
254
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$k$-space tensor integral in statistical physics
$$Q=\int_{\text{all space}} \frac{\hbar \nu_g \mathbf{k}\mathbf{k}}{\exp[(\hbar \nu_g |\mathbf{k}|-\mathbf{k}\cdot\mathbf{u})/k_B T]-1}d\mathbf{k} $$
Please help me to integrate the above tensor ...
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