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Reason for the discreteness arising in quantum mechanics?

What is the most essential reason that actually leads to the quantization. I am reading the book on quantum mechanics by Griffiths. The quanta in the infinite potential well for e.g. arise due to the ...
18
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2answers
395 views

Geometric quantization of identical particles

Background: It is well known that the quantum mechanics of $n$ identical particles living on $\mathbb{R}^3$ can be obtained from the geometric quantization of the cotangent bundle of the manifold ...
18
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1answer
661 views

Why do quasicrystals have well-defined Fourier transforms?

I was recently reading about quasicrystals, and I was really surprised to learn that even though they do not have a periodic structure, and only have long range order in a very different sense to the ...
18
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1answer
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Motivation for the use of Tsallis entropy

Every now and again I hear something about Tsallis entropy, $$ S_q(\{p_i\}) = \frac{1}{q-1}\left( 1- \sum_i p_i^q \right), \tag{1} $$ and I decided to finally get around to investigating it. I haven't ...
17
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2answers
183 views

Can symmetry generators be used for quantization?

Take the Poincaré group for example. The conservation of rest-mass $m_0$ is generated by the invariance with respect to $p^2 = -\partial_\mu\partial^\mu$. Now if one simply claims The state where ...
17
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689 views

Homotopy $\pi_4(SU(2))=\mathbb{Z}_2$

Recently I read a paper using $$\pi_4(SU(2))=\mathbb{Z}_2.$$ Do you have any visualization or explanation of this result? More generally, how do physicists understand or calculate high dimension ...
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When is Lebesgue integration useful over Riemann integration in physics?

Riemann integration is fine for physics in general because the functions dealt with tend to be differentiable and well behaved. Despite this, it's possible that Lebesque integration can be more ...
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3answers
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What is the relation between renormalization in physics and divergent series in mathematics?

The theory of Divergent Series was developed by Hardy and other mathematicians in the first half of the past century, giving rigorous methods of summation to get unique and consistent results from ...
16
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6answers
745 views

Which QFTs were rigorously constructed?

Which QFTs have mathematically rigorous constructions a la AQFT? I understand there are many such constructions in 2D, in particular 2D CFT has been extensively studied mathematically. But even in 2D ...
16
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6answers
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Applications of delay differential equations

Being interested in the mathematical theory, I was wondering if there are up-to-date, nontrivial models/theories where delay differential equations play a role (PDE-s, or more general functional ...
16
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2answers
900 views

Is the G2 Lie algebra useful for anything?

Seems like all the simpler Lie algebras have a use in one or another branch of theoretical physics. Even the exceptional E8 comes up in string theory. But G2? I've always wondered about that one. ...
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Applications of Geometric Topology to Theoretical Physics

Geometric topology is the study of manifolds, maps between manifolds, and embeddings of manifolds in one another. Included in this sub-branch of Pure Mathematics; knot theory, homotopy, manifold ...
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Representing forces as one-forms

First of all, sorry if any of those things are silly or nonsense, I'm just trying to understand better how the concepts of forms, exterior derivative and so on can be used in physics. This question ...
16
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2answers
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How do physicists use solutions to the Yang-Baxter Equation?

As a mathematician working the area of representation of Quantum groups, I am constantly thinking about solutions of the Yang-Baxter equation. In particular, trigonometric solutions. Often research ...
16
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1answer
746 views

What does it mean for a QFT to not be well-defined?

It is usually said that QED, for instance, is not a well-defined QFT. It has to be embedded or completed in order to make it consistent. Most of these arguments amount to using the renormalization ...
15
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10answers
4k views

Physics for mathematicians

How and from where does a mathematician learn physics from a mathematical stand point? I am reading the book by Spivak Elementary Mechanics from a mathematicians view point. The first couple of pages ...
15
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4answers
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Does the axiom of choice appear to be “true” in the context of physics?

I have been wondering about the axiom of choice and how it relates to physics. In particular, I was wondering how many (if any) experimentally-verified physical theories require axiom of choice (or ...
15
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1answer
2k views

Rigged Hilbert space and QM

Are there any comprehensive texts that discuss QM using the notion of rigged Hilbert spaces? It would be nice if there were a text that went through the standard QM examples using this structure.
15
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3answers
704 views

about the Atiyah-Segal axioms on topological quantum field theory

Trying to go through the page on Topological quantum field theory - The original Atiyah-Segal axioms - "Let $\Lambda$ be a commutative ring with 1, Atiyah originally proposed the axioms of a ...
15
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2answers
586 views

Generalized Complex Geometry and Theoretical Physics

I have been wondering about some of the different uses of Generalized Complex Geometry (GCG) in Physics. Without going into mathematical detail (see Gualtieri's thesis for reference), a Generalized ...
15
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4answers
94 views

Why can't noncontextual ontological theories have stronger correlations than commutative theories?

EDIT: I found both answers to my question to be unsatisfactory. But I think this is because the question itself is unsatisfactory, so I reworded it in order to allow a good answer. One take on ...
15
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5answers
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What does a frame of reference mean in terms of manifolds?

Because of my mathematical background, I've been finding it hard to relate the physics-talk I've been reading, with mathematical objects. In (say special) relativity, we have a Lorentzian manifold, ...
15
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5answers
3k views

Reading list in topological QFT

I'm interested in learning about topological QFT including Chern Simons theory, Jones polynomial, Donaldson theory and Floer homology - basically the kind of things Witten worked on in the 80s. I'm ...
15
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3answers
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Ring theory in physics

Surely group theory is a very handy tool in the problems dealing with symmetry. But is there any application for ring theory in physics? If not, what's this that makes rings not applicable in physics ...
15
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3answers
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What is a dual / cotangent space?

Dual spaces are home to bras in quantum mechanics; cotangent spaces are home to linear maps in the tensor formalism of general relativity. After taking courses in these two subjects, I've still never ...
15
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4answers
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Introduction to string theory

I am in the last year of MSc. and would like to read string theory. I have the Zwiebach Book, but along with it what other advanced book can be followed, which can be a complimentary to Zwiebach. I ...
15
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3answers
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Why can't General Relativity be written in terms of physical variables?

I am aware that the field in General Relativity (the metric, $g_{\mu\nu}$) is not completely physical, as two metrics which are related by a diffeomorphism (~ a change in coordinates) are physically ...
15
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3answers
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Does renormalization make quantum fields into (slightly) nonlinear functionals of test functions?

Quantum fields are presented as operator-valued distributions, so that the operators in the theory are linear functionals of some test function space. This works well for free fields, giving us a ...
15
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1answer
534 views

A Theorem Due to Hodge: Hawking/Ellis

This is probably quite an obscure question but hopefully somebody has a simple answer. I'm studying the proof of the topology theorem on black holes due to Hawking and Ellis (Proposition 9.3.2, p. 335 ...
15
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1answer
2k views

Onsager's Regression Hypothesis, Explained and Demonstrated

Onsager's 1931 regression hypothesis asserts that “…the average regression of fluctuations will obey the same laws as the corresponding macroscopic irreversible process". (Here is the links to ...
15
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1answer
232 views

Miura transform for W-algebras of exceptional type

Miura transform for W-algebras of classical types can be found in e.g. Sec. 6.3.3 of Bouwknegt-Schoutens. Is there a similar explicit Miura transform for W-algebras of exceptional types, say, E6? It's ...
15
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0answers
118 views

Minimal strings and topological strings

In http://arxiv.org/abs/hep-th/0206255 Dijkgraaf and Vafa showed that the closed string partition function of the topological B-model on a Calabi-Yau of the form $uv-H(x,y)=0$ coincides with the free ...
14
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3answers
9k views

Don't understand the integral over the square of the Dirac delta function

In Griffiths' Intro to QM [1] he gives the eigenfunctions of the Hermitian operator $\hat{x}=x$ as being $$g_{\lambda}\left(x\right)~=~B_{\lambda}\delta\left(x-\lambda\right)$$ (cf. last formula on ...
14
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4answers
2k views

Where is the Atiyah-Singer index theorem used in physics?

I'm trying to get motivated in learning the Atiyah-Singer index theorem. In most places I read about it, e.g. wikipedia, it is mentioned that the theorem is important in theoretical physics. So my ...
14
votes
7answers
2k views

What are the uses of Hopf algebras in physics?

Hopf algebra is nice object full of structure (a bialgebra with an antipode). To get some idea what it looks like, group itself is a Hopf algebra, considered over a field with one element ;) usual ...
14
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5answers
1k views

What happened with Hilbert's sixth problem (the axiomatization of physics) after Gödel's work?

I'll write the question but I'm not fully confident of the premises I'm making here. I'm sorry if my proposal is too silly. Hilbert's sixth problem consisted roughly about finding axioms for physics ...
14
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1answer
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Self-adjoint and unbounded operators in QM

An operator $A$ is said to be self-adjoint if $(\chi,A\psi)=(A\chi,\psi)$ for $\psi, \chi \in D_A$ and $D_A=D_{A^\dagger}$. But for the free particle momentum operator $\hat{p}$ these inner products ...
14
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1answer
137 views

Instanton Moduli Space with a Surface Operator

I would like to understand the mathematical language which is relevant to instanton moduli space with a surface operator. Alday and Tachikawa stated in 1005.4469 that the following moduli spaces are ...
14
votes
1answer
474 views

Self-dual Maxwell equations, the second homology group, and topological invariants of a four manifold

In Witten's paper Quantum Field Theory and the Jones Polynomial, he mentioned that: Geometers have long known that (via de Rham theory) the self-dual and anti-self-dual Maxwell equations are ...
14
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1answer
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Discreteness of set of energy eigenvalues

Given some potential $V$, we have the eigenvalue problem $$ -\frac{\hbar^2}{2m}\Delta \psi + V\psi = E\psi $$ with the boundary condition $$ \lim_{|x|\rightarrow \infty} \psi(x) = 0 $$ If we ...
14
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1answer
972 views

Fundamental question about the Buckingham $\pi$ theorem (dimensional analysis)

I have a rather fundamental question about the Buckingham $\pi$ theorem. They introduce it in my book about fluid mechanics as follows (I state the description of the theorem here, because I noticed ...
14
votes
1answer
470 views

What is Motivic mathematics and how is it used in physics?

In a few videos I've seen where he discusses the new approach to calculating the super Yang Mills scattering amplitudes, Nima Arkani-Hamed sometimes alludes to the use of Motivic methods as being ...
14
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0answers
423 views

Hypersingular Boundary Operator in Physics

This has been a question I've been asking myself for quite some time now. Is there a physical Interpretation of the Hypersingular Boundary Operator? First, let me give some motivation why I think ...
13
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5answers
455 views

Other processes than formal power series expansions in quantum field theory calculations

I am not sure if this question is too naive for this site, but here it goes. In QFT calculations, it seems that everything is rooted in formal power series expansions, i.e. , what dynamical systems ...
13
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4answers
847 views

Can momentum have a complex expectation value?

I'm making examples of wave functions to incorporate in a QM exam. I came up with the following wave function, which gives me some troubles: $$\psi(x,0) = \begin{cases} A(a-x), & -a \leq x \leq ...
13
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6answers
2k views

Does the Banach-Tarski paradox contradict our understanding of nature?

Since the Banach-Tarski paradox makes a statement about domains defined in terms of real numbers, it would appear to invalidate statements about nature that we derived by applying real analysis. My ...
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3answers
254 views

Status of local gauge invariance in axiomatic quantum field theory

In his recent review... Sergio Doplicher, The principle of locality: Effectiveness, fate, and challenges, J. Math. Phys. 51, 015218 (2010), doi ...Sergio Doplicher mentions an important open ...
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2answers
1k views

Literature on fractal properties of quasicrystals

At the seminar where the talk was about quasicrystals, I mentioned that some results on their properties remind the fractals. The person who gave the talk was not too fluent in a rigor mathematics ...
13
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5answers
6k views

Fourier transform of the Coulomb potential

When trying to find the Fourier transform of the Coulomb potential $$V(\mathbf{r})=-\frac{e^2}{r}$$ one is faced with the problem that the resulting integral is divergent. Usually, it is then argued ...
13
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3answers
980 views

$\mathrm{SU(3)}$ decomposition of $\mathbf{3} \otimes \mathbf{\bar{3}} = \mathbf{8} \oplus \mathbf{1}$?

I have a question about the tensor decomposition of $\mathrm{SU(3)}$. According to Georgi (page 142 and 143), a tensor $T^i{}_j$ decomposes as: \begin{equation} \mathbf{3} \otimes \mathbf{\bar{3}} = ...