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 ...
51
votes
6answers
854 views
What are the justifying foundations of statistical mechanics without appealing to the ergodic hypothesis?
This question was listed as one of the questions in the proposal (see here), and I didn't know the answer. I don't know the ethics on blatantly stealing such a question, so if it should be deleted or ...
37
votes
14answers
4k views
Number theory in Physics
As a Graduate Mathematics student, my interests lies in Number theory. I am curious to know if Number theory has any connections or applications to physics. I have never even heard of any applications ...
36
votes
6answers
787 views
The Role of Rigor
The purpose of this question is to ask about the role of mathematical rigor in physics. In order to formulate a question that can be answered, and not just discussed, I divided this large issue into ...
33
votes
18answers
2k views
Quantum Field Theory from a mathematical point of view
I'm a student of mathematics with not much background in physics. I'm interested in learning Quantum field theory from a mathematical point of view.
Are there any good books or other reference ...
30
votes
2answers
215 views
Physical interpretation of different selfadjoint extensions
Given a symmetric (densely defined) operator in a Hilbert space, there might be quite a lot of selfadjoint extensions to it. This might be the case for a Schrödinger operator with a "bad" potential. ...
26
votes
7answers
2k views
Is there something similar to Noether's theorem for discrete symmetries?
Noether's theorem states that, for every continuous symmetry of a system, there exists a conserved quantity, e.g. energy conservation for time invariance, charge conservation for $U(1)$. Is there any ...
26
votes
10answers
663 views
Readable books on advanced topics [closed]
I realise that there are already a few questions looking for general book recommendations, but the motivation and type of book I'm looking for here is a little different, so I hope you can indulge me.
...
25
votes
10answers
4k views
Best books for mathematical background?
What are the best textbooks to read for the mathematical background you need for modern physics, such as, string theory?
Some subjects off the top of my head that probably need covering:
...
25
votes
4answers
1k views
Trace of a commutator is zero - but what about the commutator of $x$ and $p$?
Operators can be cyclically interchanged inside a trace:
$${\rm Tr} (AB)~=~{\rm Tr} (BA).$$
This means the trace of a commutator of any two operators is zero:
$${\rm Tr} ([A,B])~=~0.$$
But what about ...
23
votes
9answers
558 views
Examples of number theory showing up in physics
My question is very simple: Are there any interesting examples of number theory showing up unexpectedly in physics?
This probably sounds like rather strange question, or rather like one of the ...
23
votes
11answers
837 views
Negative probabilities in quantum physics
Negative probabilities are naturally found in the Wigner function (both the original one and its discrete variants), the Klein paradox (where it is an artifact of using a one-particle theory) and the ...
22
votes
6answers
2k views
Formalizing Quantum Field Theory
I'm wondering about current efforts to provide mathematical foundations and more solid definition for quantum field theories. I am aware of such efforts in the context of the simpler topological or ...
22
votes
1answer
179 views
Mermin-Wagner theorem in the presence of hard-core interactions
It seems quite common in the theoretical physics literature to see applications of the "Mermin-Wagner theorem" (see wikipedia or scholarpedia for some limited background) to systems with hard-core ...
21
votes
9answers
2k views
Rigor in quantum field theory
Quantum field theory is a broad subject and has the reputation of using methods which are mathematically desiring. For example working with and subtracting infinities or the use of path integrals, ...
21
votes
13answers
2k views
Suggested reading for renormalization (not only in QFT)
What papers/books/reviews can you suggest to learn what renormalization "really" is? Standard QFT textbooks are usually computation-heavy and provide little physical insight in this respect - after my ...
20
votes
4answers
758 views
In quantum mechanics, given certain energy spectrum can one generate the corresponding potential?
A typical problem in quantum mechanics is to calculate the spectrum that corresponds to a given potential.
Is there a one to one correspondence between the potential and its spectrum?
If the ...
19
votes
2answers
46 views
Bell polytopes with nontrivial symmetries
Take $N$ parties, each of which receives an input $s_i \in {1, \dots, m_i}$ and produces an output $r_i \in {1, \dots, v_i}$, possibly in a nondeterministic manner. We are interested in joint ...
19
votes
4answers
389 views
Paradox?: What is the form of radiation experienced by a harmonically accelerated observer?
Theory predicts that uniform acceleration leads to experiencing thermal radiation (so called Fulling Davies Unruh radiation), associated with the appearance of an event horizon. For non uniform but ...
18
votes
3answers
1k views
Why are von Neumann Algebras important in quantum physics?
At the moment I am studying operator algebras from a mathematical point of view. Up to now I have read and heard of many remarks and side notes that von Neumann algebras ($W^*$ algebras) are important ...
18
votes
1answer
288 views
Geometric picture behind quantum expanders
A $(d,\lambda)$-quantum expander is a distribution $\nu$ over the unitary group $\mathcal{U}(d)$ with the property that: a) $|\mathrm{supp} \ \nu| =d$, b) $\Vert \mathbb{E}_{U \sim \nu} U \otimes ...
17
votes
2answers
39 views
Significance of the hyperfinite $III_1$ factor for axiomatic quantum field theory
Using a form of the Haag-Kastler axioms for quantum field theory (see AQFT on the nLab for more details), it is possible in quite general contexts to prove that all local algebras are isomorphic to ...
17
votes
5answers
813 views
Is the converse of Noether's first theorem true: Every conservation law has a symmetry?
Noether's (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law.
Is the converse true: Any conservation law of a physical ...
16
votes
6answers
305 views
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
votes
5answers
434 views
What do theoretical physicists need from computer scientists?
I recently co-authored a paper (not online yet unfortunately) with some chemists that essentially provided answers to the question, "What do chemists need from computer scientists?" This included the ...
16
votes
7answers
2k views
Classical mechanics without coordinates book
I am a math grad student who would like to learn some classical mechanics. The caveat is I am not to interested in the standard coordinate approach. I can't help but think of the fields that arise in ...
16
votes
2answers
89 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 ...
15
votes
6answers
203 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 ...
15
votes
5answers
2k views
Why does calculus of variations work?
How does it make sense to vary the position and the velocity independently?
Edit:
Velocity is the derivative of position, so how can you treat them as independent variables? Doesn't every physics ...
14
votes
4answers
59 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 ...
14
votes
1answer
54 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 ...
14
votes
2answers
124 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 ...
13
votes
5answers
131 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
votes
6answers
1k views
Quantum mechanics on a manifold
In quantum mechanics the state of a free particle in three dimensional space is $L^2(\mathbb R^3)$, more accurately the projective space of that Hilbert space. Here I am ignoring internal degrees of ...
13
votes
3answers
64 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 ...
13
votes
3answers
341 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 ...
13
votes
7answers
2k views
Applications of Algebraic Topology to physics
I have always wondered about applications of Algebraic Topology to Physics, seeing as am I studying algebraic topology and physics is cool and pretty. My initial thoughts would be that since most ...
13
votes
1answer
355 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 ...
13
votes
1answer
44 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 ...
13
votes
3answers
518 views
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 ...
13
votes
2answers
2k views
Is there a T-dual of Witten's twistor topological string theory?
In late 2003, Edward Witten released a paper that revived the interest in Roger Penrose's twistors among particle physicists. The scattering amplitudes of gluons in $N=4$ gauge theory in four ...
13
votes
1answer
429 views
How come random matrices can predict energy spectra of heavy atoms?
Some of the applications of random matrices is to find the spectra of heavy atoms in nuclear physics which are usually difficult to find otherwise.
How can starting from randomness of some kind, ...
12
votes
8answers
1k views
Why $\displaystyle i\hbar\frac{\partial}{\partial t}$ can not be considered as the Hamiltonian operator?
In the time dependent Schrodinger equation $\displaystyle, H\Psi = i\hbar\frac{\partial}{\partial t}\Psi$ , the Hamiltonian operator is given by
$\displaystyle H = -\frac{\hbar^2}{2m}\nabla^2+V$
...
12
votes
4answers
2k views
A pedestrian explanation of conformal blocks
I would be very happy if someone could take a stab at conveying what conformal blocks are and how they are used in conformal field theory (CFT). I'm finally getting the glimmerings of understanding ...
12
votes
8answers
2k views
Crash course on algebraic geometry with view to applications in physics
could you please recommend any good texts on algebraic geometry (just over the complex numbers rather than arbitrary fields) and on complex geometry including Kahler manifolds that could serve as an ...
12
votes
2answers
183 views
What is a good introduction to integrable models in physics?
I would be interested in a good mathematician-friendly introduction to integrable models in physics, either a book or expository article.
Related MathOverflow question: what-is-an-integrable-system.
12
votes
2answers
165 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 ...
12
votes
2answers
153 views
Topological twists of SUSY gauge theory
Consider $N=4$ super-symmetric gauge theory in 4 dimensions with gauge group $G$. As is explained in the beginning of the paper of Kapustin and Witten on geometric Langlands, this
theory has 3 ...
12
votes
3answers
449 views
When is Lebesque 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 ...
12
votes
3answers
1k views
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 ...
12
votes
4answers
579 views
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 ...
