Sunday, 13 November 2011

spherical harmonics and the 3-sphere

One day in summer of 2007 when I was looking for something interesting to teach those gifted kids at the school I had worked with, a picture (the one below) of the shadow of the 3-sphere grabbed my attention. I gazed upon it and was almost instantaneously reminded of pictures of the spherical harmonics, which I knew from my quantum mechanics courses. I thought, could it be that electrons are circling around the nucleus on a 4 dimensional trajectory. If this resemblance happens to be no coincidence it could make perfect sense that the contradiction between the Bohr model and a spherical cloud of probability suggested by Schrödinger's equation is none at all, since the surface of a 3-sphere is of course a 3 dimensional object. It is needless to say, that there are a lot of open questions (especially for me) yet to be answered. As an example I will give just three (at least for me) open questions:
  1. Why is it, that some picturings of those two resemble one another more than others and what are the underlying rules to that similarities respectively differences?
  2. Is this more or less resemblance still given if we depict them in other geometries i.e. hyperbolic space as necessary to get new insights to quantum-gravity?
  3. What happens to the shortcomings of the Bohr model if it is extended to 4 dimensional  space?
Definitely dispiriting is the fact, that I was not able to find anything on this topic on the internet. I suppose that the probabilistic theory of quantum physics suffices to provide answers to most questions in elementary particle physics such that most people don't bother to endeavour this topic, but nonetheless I find it to be a very attractive one to dig deeper into especially because of the possible insights into quantum-gravity it could provide.
Stereographic projection of the hypersphere's parallels (red), meridians (blue) and hypermeridians (green). Because this projection is conformal, the curves intersect each other orthogonally (in the yellow points) as in 4D. All curves are circles: the curves that intersect <0,0,0,1> have infinite radius (= straight line). 
[Source: Wikipedia]

Visual representations of the first few spherical harmonics. For each spherical harmonic, we plot its value on the surface of a sphere (front), and then in polar (back). The polar plot is simply obtained by varying the radius of the previous sphere. [Source: Mayavi]

Monday, 8 February 2010

what is geometry?

For most of my life I answered this question somewhat like: "Geometry is the study of shapes; of measurement of lengths, angles, curvatures, volume and so on." It did not occur to me that there are differences between Euclidean and e.g. spherical geometry other than the basis of the vector space. But as I had to learn there are a lot of differences between those geometries. Certainly I was not the only one caught in that assumption. For more than 2000 years the prefix Euclidean in Euclidean geometry was not needed since there were no other geometries known. Euclid was ~300 BC the first with an axiomatic attempt to geometry. His Elements had such an impact, that for the next 2000 years there were no answer to the question what is geometry other than Euclid's axioms. But then, even without asking, other geometries emerged. Spherical geometry was the most obvious but not alone. Others like hyperbolic-, projective-, Möbius- and Lie-geometry among others came into being and it became interesting again to ask: What is geometry? Finally in 1872 Felix Klein (the one with the bottle) came to a new viewpoint of geometry. In his Erlangen program he stated that there is a group of symmetries associated with every geometry. The hierarchy of the group resembles the hierarchy of the different geometries for example this means, that any theorem proven in projective geometry also holds in euclidean geometry because its associated group O(n) is a subgroup of GL(n) which is a subgroup of PGL(n+1). One major flaw of Klein's approach was that he was unable to include Riemannian geometry. Later this dilemma was fixed by Cartan and his connections which revolutionized differential geometry like Klein did with geometry.
All that I have learned about this brought me slightly to a better answer on the above question but not to fulfillment. So I conclude that I have to know more about group theory, Cartan's work
and algebraic geometry.

Tuesday, 2 February 2010

intention of this blog

Hi Folks,

since blogging is mandatory today, here is my take in this phenomenon. As usual, I've had this plan for a long time now, but (as you can see) finally it's happening. Again as usual, I start this project by saying that I'm about to start it, and tell you (my possibly/maybe reader) who I am, what the purpose of this blog is or whatever else fits into ones first posting.

Now let's begin:

Q: Who am I?
A: I don't really know. First of all human (with all the flaws attached). I'm also a guy from Berlin born back then, when there was a wall surrounding it. If you don't mind I skip the funny yet uninteresting character introducing part and go right away to the cool 21st century part. So now that you know (or would have known, if you wouldn't have pushed the skip button) how I got interested in the inner workings of nature, and thought that my purpose in life is to change society for the better, you know why I began to study physics, like the mathematics more, love music, hate all known user interfaces, like few people more than much and became a heretic of all which is.
Q: Why this blog?
A: As I said before: It's mandatory and I want to use it as an Study log for my endeavour into the geometric aspects of our world. Why a Study log? Isn't it obvious? To get this organized and to discuss my ideas with you.
Q: Why in English?
A: Oh, you noticed. Yes, as I said, I'm descendant of Berlin and thus no native speaker, but, you know, the audience is much bigger, most of the references are English and I want to master it (So please tell me if I misspelled or used false grammar).