Our discussion sessions have been going well. I hope we can dedicate more and more time and effort to what's been going on in these sessions. This has really provided us with a good platform where we have the freedom to think and question without having to take anything for granted. So far its been very unconventional. We are in fact unlearning a lot, and these sessions are the only place where this kind of unlearning, I feel, has been welcome. So far, so good.
I will now be putting down here what I have been doing in a little more detail.
Session 1 : Space, time and Structure
Our high school physics begins with Newton's laws of Motion. By the end of MSc. we would have investigated motion a little further and spoken about it in terms of Lagrangian and Hamiltionian. When Einstein came up with his theory of relativity, he showed that space and time actually had a structure. A structure that evolves dynamically. This should perhaps be enough motivation to question about the very structure of the space and time on which motion actually takes place. Doing this won't be an easy task. We can't observe physical effects of time to model its structure, and yet everything that happens is parametrised using time. One big leaps in our understanding of time came when Einstein said that the faster you move in space, the slower you move in time. And this stitches space and time forever. You can't talk about space or time, you can only talk about space-time. Also, he showed that mass has something to do with space-time. The closer you are to a heavy mass, the slower the time flows for you.
We express time as the amount distance travelled with a certain velocity. And velocity as the amount distance travelled in certain time. If you try to talk about the structure of space and time in terms of distance and velocity, this is as far as you'll get. The only thing that can be further broken down here is distance. A space where we can talk about distance between two points is called a metric space. We seem to live on such a space. To be even able to talk about why our space the way it is, we need to talk about a more general space which does not have the notion of distance on it. This general space is known as a topological space. We go on later to define a metric on it.
I hope in the first session I was successful in motivating the reason behind going into some amount of topology and differential geometry. Though these are some purely mathematical structures, I feel that when used by physicists, they should be powerful tools to gain new insights into any physical concept and to look at physics in a unified sense.
Towards the end of the first session we tried entering into a little bit of set theory. And how sets are looked at in a topology. I ended with a few definitions like open and closed sets.
The assignment (which was successfully completed by all) was to look up a few theorems and lemmas, more importantly 'The Axiom of Choice' , Zorn's lemma. Also, to look up how a vector space and an algebra is technically defined. There was another long term assignment given which was to think if it was possible to arrive at the fact that there had to be a observer independent constant velocity given that space and time both are relative in our universe.
References: (1) Naive Set theory by P. Halmos (2) Differential Geometry and Lie Groups For Physicists by Marian Fecko (3) Wikipedia:)
I have spoken for three other sessions, the summary's for which will soon be put up. Our new aim is to be able to make lecture notes as well and turn them in to soft copies.
Karthik has been doing a wonderful job with classical mechanics. He has motivated us to think of a lot of things that haven't been thought of before. Shruthi has been showing failures of classical mechanics and has been trying to bring in the notion of wave-particle duality and making us think as to why a new theory was needed . And Harshini has been dealing with the mathematics of QM and the how's and why's of it. She did a nice derivation today which she will probably summarise later.
I urge everyone to post the summaries of their talks. It really has been a wonderful time discussing and opening our eyes to the fact that we really don't know anything.
ciao
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