/sci/ - Science & Math

>>6779516
Theoretical physics is constantly learning math on the fly, using it to capture the ideas behind theory you work on, and then forgetting about the math at the right moment, so you actually end up with some new directions.
Meanwhile, in mathematics, seven generations of PhD students will write about the exact same topic.
(In both fields, the names which people remember are the guys who work in the first generation, and the people who end the subject. Related: The initial pages of chapter 6 here are a good read:
http://arxiv.org/pdf/math.ho/9404236.pdf))

I believe the dividing of math and theoretical physics is just a self-fulfilling prophecy. It's not the subject as such that's different, it's different just from having being academically split into two broad working collectives which developed different content-producing- and results-judging habits out of their different emphasis on the sub-part of problem they work out.
This is a viewpoint I only take for some years no - at one point I was completely opposed to Arnolds (extreme) point of view
http://pauli.uni-muenster.de/~munsteg/arnold.html

>>6627562
>>6627599
The guys associated with the program
http://en.wikipedia.org/wiki/Reverse_mathematics
have worked those things out. In particular it's not possible to have a "satisfying theory" about even just finite combinatorics, if you disregard infinite sets.
It's worth reading "#0. Introduction." here:

https://groups.google.com/forum/#!original/sci.math/KQ4Weqk4TmE/LE_Wfsk00H4J

But, disturbingly, computability theory has rekt the math of even small concepts no matter how strong your tools. From an MO thread:
>...consider the expressions obtainable by addition, subtraction, multiplication, and composition from the initial functions log(2), \pi, exp(x), sin(x), and |x|. Richardson proved that there is no algorithm to (always) decide whether such an expression defines the zero function.

http://www.jstor.org/discover/10.2307/2271358?uid=3737864&uid=2&uid=4&sid=21104424034313

This implies that I can cook up two mean functions f(x) and g(x), and write down a nice one-liner:
>hey bra, is f(x)=g(x)?
You have no chance to answer it.

The idea is to modify the concept of a manifold so that <span class="math">[X^i,X^j][/spoiler] is not necessarily equal to zero. It implies an uncertainty principle for the X^i, which obscures the conventional notion of points. Write
<span class="math">[X^i,X^j]=\theta^{ij}(X)[/spoiler]
where theta denotes a general "geometric structure". From a maths pov, the study of the quantized physical phase space is a special case of this problem with theta constant (QM commutation relation)!
A motivation for fuzzy spacetimes is quantum gravity: Measurements of very small distances require high energies and then, due to gravitational singularities at very high energies, measurement of distances below a certain length scale does not make sense anymore. The idea of noncommutative geometry and its lack of points seems like a natural way out of this black hole problem.

That being said
>Has anyone ever heard of this or read related material?
The Landau problem is related to the quantum hall effect and which is roughly described as follows:
Classically, one considers an electrically charged particle moving in a homogeneous and constant magnetic field. The particle is confined to the xy-plane, while the magnetic field is pointing in the z-direction:
<span class="math">B=B_z e_z \Leftarrow A=B_z x e_y.[/spoiler]
The Lagrangian and the canonical momentum of this system are given by
<span class="math">L=m/2 \delta_{ij} \dot{x}^i\dot{x}^j-(e/c)A_i\dot{x}^i\Rightarrow\p_i=\partial L/\partial \dot{x}^i=m\delta_{ij}\dot{x}^j+(e/c)B_{ij}x^j approx (e/c)B_{ij}x^j.[/spoiler]
I have introduced the constant matrix <span class="math">B_{ij}[/spoiler] such that <span class="math">A_i=B_{ij}x^j[/spoiler] and considered the limit of small m. The Poisson brackets <span class="math">\{p_i,x^j\}=\delta_i^j[/spoiler] lead to
<span class="math">\{x^i,x^j\}=(c/e)(B^{-1})^{ij}[/spoiler]
and canonical quantization then leads to a noncommutative space.

I can drop references, but I found none of them easy going enough around the 8'th semester.
Importantly for this, as a physicist, learn ring theory!