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.
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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 ...
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What is "Bra" and "Ket" notation and how does it relate to Hilbert spaces?
This is my first semester of quantum mechanics and higher mathematics and I am completely lost. I have tried to find help at my university, browsed similar questions on this site, looked at my ...
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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 ...
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Level of Rigor in Mathematical Physics
I am a physics/math undergrad and I have recently become familiar with some more rigorous formalisms of mechanics, such as Lagrangian mechanics and Noether's Theorem. However, I've noticed that the ...
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How are topological invariants obtained from TQFTs used in practice?
Topological quantum field theories (TQFTs) are studied for different reasons, as exemplified in the following places:
Atiyah, Topological quantum field theory
Lurie, Topological Quantum Field Theory ...
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How to create new mathematics? [closed]
How do scientists and mathematicians create new mathematics for describing concepts? What is new mathematics? Is it necessarily in format of previous mathematics? Can one person make (invent or ...
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What are some math concepts which were originally inspired by physics?
There are a number of concepts which were first introduced in the physics literature (usually in an ad-hoc manner) to solve or simplify a particular problem, but later proven rigorously and adopted as ...
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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?...
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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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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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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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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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Density and dimensionality of zeros in inverse square force fields of randomly distributed sources in (at least) 1, 2 and 3 dimensions?
Background: In this answer to Are there places in the Universe without gravity? in Astronomy SE I did a quick finite 2D calculation for 20 random sources to see if there was at least one zero, and ...
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Why is this a first integral? - particle near Schwarzschild black hole
Background
I know that the Schwarzschild metric is:
$$d s^{2}=c^{2}\left(1-\frac{2 \mu}{r}\right) d t^{2}-\left(1-\frac{2 \mu}{r}\right)^{-1} d r^{2}-r^{2} d \Omega^{2}$$
I know that if I divide by $d ...
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Wave equation: predicting geometric dispersion with group theory
Context
The wave equation
$$
\partial_{tt}\psi=v^2\nabla^2 \psi
$$
describes waves that travel with frequency-independent speed $v$, ie. the waves are dispersionless. The character of solutions is ...
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Was von Neumann's 1954 ICM address "Unsolved Problems in Mathematics" outdated?
I recently tried to "explain" the generalized probability theory aspect of quantum theory (as one common part of both quantum field theory and quantum mechanics), in the sense of motivations for the ...
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Atiyah's definitions of Topological Quantum Field Theory
According to Atiyah, a TQFT is a functor from the category of cobordisms to the category of vector spaces.
How does this definition relate with the physics of quantum mechanics?
What does the ...
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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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Functional differential equation (from Quantum Field Theory).
I have a certain differential equation that includes functional derivatives. I know the solution, but I'm having a hard time to show that the equation is indeed solved by the solution. The background ...
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Integral results in difference of means $\pi(\frac{a+b}{2} - \sqrt{ab})$ [duplicate]
$$\int_a^b \left\{ \left(1-\frac{a}{r}\right)\left(\frac{b}{r}-1\right)\right\}^{1/2}dr = \pi\left(\frac{a+b}{2} - \sqrt{ab}\right)$$
What an interesting integral! What strikes me is that the result ...
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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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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 ...
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Stochastic interpretation of Einstein equations
Einstein's theory of gravitation, general relativity, is a purely geometric theory.
In a recent question I wanted to know what the relation of Brownian motion to the Helmholtz equation is and got a ...
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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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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
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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 ...
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A fun problem by Arnold using the Poincaré recurrence theorem
I came across this problem by V. I. Arnold while studying his classical mechanics book.
Consider a sequence where the $n^{th}$ term is made up by considering the first digit of $2^n$, the first terms ...
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Electrodynamics in general spacetime
Let $M\cong\mathbb{R}^4_1$ be the usual Minkowski spacetime. Then we can formulate electrodynamics in a Lorentz invariant way by giving the EM-field $2$-form $\mathcal{F}\in\Omega^2(M)$ and ...
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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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Geometric meaning of Berezin integration
Berezin integration in a Grassmann algebra is defined such that its algebraic properties are analogous to definite integration of ordinary functions: linearity (taking anticommutativity into account), ...
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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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Guidance regarding research in Mathematical Physics
I am currently a Master's student in Mathematics. The main focus of my undergraduate programme was on Mathematics. However as a part of the course, I have done some Theoretical Physics courses. In ...
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Can there be an energetically unbounded three-body orbit where escape is impossible due to conservation of angular momentum?
This question evolved from a discussion below this answer which explains (among other things) that the total energy of a system offers insight as to the possibility of one (or all) members "escaping". ...
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Learning roadmap to Topological Quantum Field Theories from a mathematics perspective
I want to learn TQFT's and am looking for review articles or books. My mathematics knowledge is limited to one year of graduate course in Algebra (Groups,Rings,Fields,Categories, Modules and ...
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A space more fundamental than Euclidean space
Summary: The mathematical physicist Paolo Budinich attributes to Élie Cartan the statement that the geometry of pure spinors is "more elementary" or more "fundamental" than Euclidean geometry, which ...
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What is the motivation for analytic solutions in Mathematical Physics?
I am trying to understand why one cares about solving PDE's with an analytic/theoretical solution when one can use numerical methods?
If you tell me, "only mathematicians try to find theoretical ...
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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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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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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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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 ...
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Arnold's theorem on action-angles.
I changed the question slightly in its form to make it more readable.
I have a question about the action-angle theorem on p. 283 in Arnold's textbook on classical mechanics.(I added the link to this ...
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Finding the radius of a neutron star that allows all points on its surface to be seen
At first glance, this fascinating question may seem better placed on Physics Stack Exchange. But, since I am only questioning the mathematics of the solution I decided it was more appropriate to place ...
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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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Legendre Transformation of a Lagrangian in Classical Mechanics
I have some questions about the Legendre Transformation of a Lagrangian in Classical Mechanics to the Hamiltonian:
We start with a Lagrangian $L(q,\dot{q})=\frac{\langle \dot{q} , \dot{q}\rangle }{2} ...
user66906
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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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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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Why are 'differential operators on manifolds' differential operators?
It is clear what is meant by a differential operator on $\mathbb{R}^n$ (or some open subset). However, it is not clear to me why differential operators on smooth manifolds are defined the way they are,...
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Products of distributions in QFT
In Quantum Field Theory quantum fields are operator valued distributions. Namely, given the Schwartz space $\mathcal{S}(M)$ defined on Minkowski spacetime $M$, fields are continuous linear maps $\phi :...
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Applications of representation theory in physics
The notes of a lecture on basic group and representation theory I attended last semester begin with a bit of motivation for the argument. They give the following examples for applications in physics:
...
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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 ...
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