Questions tagged [mathematical-physics]

DO NOT USE THIS TAG for elemetary 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.

26
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1answer
1k views

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 ...
20
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10answers
15k views

Do infinity and zero really exist? [closed]

From the first day that I entered college, I was wondering about the relationship between some basic mathematical abstract concepts and nature. I'm going to explain them here and you may find them a ...
2
votes
2answers
2k views

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 ...
14
votes
4answers
2k views

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 ...
14
votes
3answers
717 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 $\...
15
votes
1answer
428 views

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) = ...
2
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5answers
1k views

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 ...
-2
votes
1answer
96 views

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 ...
21
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4answers
1k 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 ...
13
votes
2answers
593 views

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 ...
4
votes
2answers
4k views

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 ...
4
votes
2answers
3k views

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 ...
5
votes
2answers
328 views

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 ...
2
votes
0answers
162 views

Is this equal ? (I found it on this website)

I found this equation on this website! I would like to know it its true or not? And how can proof or disprove it?! Euler-Mascheroni constant expression, further simplification $$\ln(n)=\lim_{M\...
25
votes
4answers
3k views

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 ...
21
votes
7answers
10k 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 ...
17
votes
3answers
2k views

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 ...
13
votes
2answers
3k 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 ...
16
votes
2answers
3k views

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 ${\mathcal{N}}=4$ ...
14
votes
7answers
3k views

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 ...
7
votes
0answers
294 views

Multivariable Integral, How to compute it?

Can anybody please tell me, 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 ...
5
votes
2answers
699 views

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 ...
11
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1answer
3k views

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?
1
vote
4answers
2k views

Proportional to 2 Separate Variables vs. Proportional to Product of 2 Variables

I've always seen the following in physics and math textbooks but never understood the process by which it was mathematically deducted: $A \propto B$ $\space$ and $\space$ $A \propto C \space\space\...
9
votes
1answer
124 views

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 (I have ...
3
votes
0answers
158 views

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, ...
1
vote
1answer
118 views

$k$-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 ...
2
votes
1answer
233 views

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 ...
1
vote
0answers
453 views

Calculate the components of a unit vector that lies in the xy-plane and makes equal angles with the positive directions of the x- and y-axes

It's just a doubt. $\vec { A } = A \hat{ x } + A\hat { y} $ $ A\hat { y} =A\hat{ x} $ $\vec { A } = A \hat { x } + A\hat { x} $ Now finding the magnitud of the vector, $A=\frac { 1 }{ \sqrt { 2 ...
0
votes
1answer
631 views

Center of mass in a straight rod

I got an assignment to prove that in a straight homogeneous rod, you can always choose a coordinate system in such a way that $$\int_S x_1 \, dx_1 \, dx_2=0 $$ $$\int_S x_2 \, dx_1 \, dx_2=0 $$ $$\...
140
votes
2answers
13k views

Open problems in General Relativity

I would like to know if there are some open mathematical problems in General Relativity, that are important from the point of view of Physics. Is there something that still needs to be justified ...
68
votes
4answers
14k views

String Theory: What to do?

This is going to be a relatively broad/open-ended question, so I apologize before hand if it is the wrong place to ask this. Anyways, I'm currently a 3rd year undergraduate starting to more seriously ...
18
votes
4answers
7k 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 is?...
14
votes
2answers
2k views

Prerequisite for Takhtajan's “Quantum Mechanics for Mathematicians”

I want to know the math that is required to read Quantum Mechanics for Mathematicians by Takhtajan. From the book preview on Google, I gather that algebra, topology, (differential) geometry and ...
34
votes
11answers
7k views

Find the height of a bar, given the lengths of shadows cast by it and another bar [duplicate]

What is the height of the red bar? My try: with respect to the picture, it seems for the green bar $\frac{h}{H}=\frac{2}{3}$. So, I think that ratio is the same for the red bar, and the height of the ...
9
votes
1answer
1k views

Guide to mathematical physics?

I am currently a math phd student specializing in algebraic geometry aspiring to work at the boundaries of the the fields of mathematics and physics and so, was looking into the field of mathematical ...
14
votes
5answers
1k views

In what ways has physics spurred the invention of new mathematical tools?

I came across this comment: Mathematical rigor is not a criterion that physicists have for evaluating their theories. From a mathematical perspective, the non-rigorous theories are far more ...
7
votes
2answers
1k views

Intuition behind definition of spinor

Some time ago I searched for the definition of spinors and found the wikipedia page on the subject. Although highly detailed the page tries to talk about many different constructions and IMHO doesn't ...
11
votes
4answers
2k 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 ...
7
votes
4answers
4k views

Gentle introduction to fibre bundles and gauge connections

To better understand papers like this for example, which makes heavy use of fibre bundles and gauge connections to represent gauge fields, I am looking for a nice introduction to this topic. The only ...
11
votes
4answers
952 views

Gathering books on Lorentzian Geometry

I find it very hard to find books on Lorentzian Geometry, more focused on the geometry behind it, instead of books that go for the physics and General Relativity approach. More specifically, I'm ...
11
votes
2answers
830 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 ...
4
votes
4answers
2k views

What is Octave Equivalence?

This is an updated copy of a question I asked on Physics Stack Exchange not too long ago. Since I work primarily in mathematics, I thought it would be a good idea to ask it here as well (especially ...
8
votes
1answer
636 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 ...
12
votes
2answers
943 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 ...
8
votes
2answers
3k views

What are elements in $SU(1, 1)$?

I am reading some papers in physics. I don't know some notations in those papers. For example, $SU(1, 1)$, $U(1)$. I think these are Lie groups which consist of matrices. But I don't know what kind of ...
7
votes
2answers
793 views

Reference request: toric geometry

What is a good book on algebraic geometry, with focus on toric varieties, similar both in the philosophy and in the prestige of the authors to Modern Geometric Structures and Fields by Novikov and ...
3
votes
2answers
886 views

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 ...
4
votes
1answer
110 views

Is there a 3D shape with a flat face throughout which one would experience constant “downward” acceleration?

A spaceman restricted to the center of his platform A person standing on a thin disk in space will experience gravitational acceleration exactly normal to the surface only when he is situated exactly ...
4
votes
3answers
849 views

What do physicists mean with this bra-ket notation?

In Quantum mechanics we said that $\langle x'|\psi \rangle = \psi(x)$, where $\langle \phi|\psi \rangle $ is the dot product in $L^2(\mathbb{C})$. I found out, that this is true, if you set x' to ...