Wednesday, 5 January 2011

Happy new Year

Well another new year starts doesn't feel like it in gloomy Edinburgh going to work in the dark and coming home back in the dark is guaranteed to make me grumpy and also lethargic. I managed to stick to my time table for watching the Ring cycle but my other ambitions took a back burner and I basically vegetated for most of the Christmas break. The materials for M208 have arrived but as I already had a copy no great surprises. The unfortunate thing is that this year the Open University have decided to issue the TMA's on line only. As a consequence despite reading Brannan's book I feel I haven't really started. However as the Introductory units seem to be a basic revision of Unit D of MST221 (without the tedious RSA coding thank goodness) it should be fairly straightforward. The first unit is on curve sketching so I suspect the first question will be a 'Tedious but Straightforward' exercise of systematically finding the asymptotes, stationary points etc of a function which will probably be a rational polynomial of some form or other with at least two zero's. The rest of the Introductory units cover Complex Numbers, proof by Induction (fine if you don't have to do it under exam pressure) and other related proof stuff. Also a first look at the number system in preparation for the analysis units. Block 1 covers the first part of group theory and goes further than MST221. Having used quite a lot of group theory in my physics studies it will be interesting to compare the 'Pure maths approach' with the physicist's approach.

Physicist's are usually given a quick run through of the main definitions then its straight into representation theory (basically reducing all group operations to matrices) with the aim of applying it first to things like molecular vibrations, then when particle physics is studied its continuous group theory with an intuitive approach to Lie groups.

Then a somewhat tenuous link (IMHO) is made to grouping particles by their various quantum numbers. It starts off OK given that the mass of the neutron and proton are roughly the same so it makes sense to see them as identical apart from their charge. This symmetry is called isospin. Then when it comes to grouping various mesons and baryons together there is quite a convincing decuplet and octet which Gell Mann found could be considered simplified using a basis state of three particles known as quarks. Mesons essentially being seen as a combination of a quark and an antiquark and baryons being a combination of 2 quarks with one antiquark. If you postulate as Gell Mann did that quarks have a charge of either 1/3 the charge of an electron or two thirds all the particle states can in principle be constructed from these basic building blocks. This led to the postulation of SU(3) symmetry. However amongst these islands of connection with reality there are a  vast combination of possible quark states which do not correspond to any particle.

Furthermore since Gell Mann's day we now know there are 6 quarks instead of the original three. In the standard model of particle physics these are grouped in pairs allegedly of similar mass. Whilst this is OK for the so called first generation, the mass discrepancy for the second generation is quite significant, and by the time you get to the third generation it really seems incredible to group the 5th and 6th quarks together on the basis of their mass. The current masses along with their ratios are as follows

1          u (up)              6 MeV 
2          d (down)       10 MeV   ratio d/u   = 1.67
3         s (Strange)    0.25 GeV
4         c (Charm)     1.2 GeV    ratio   c/s = 4.8
5         b (bottom)    4.3 GeV
6        t  (top)        180 GeV     ratio   t/b  = 180/4.3 = 41.8

It may well be that this current grouping is incorrect, hopefully when data starts coming from the Large Hadron Collider that the top quark is seen as the lower mass of another grouping of quarks, in which case
much of the underlying symmetry of the current Standard model will have to be rewritten. I really  hope something new will come along to shake us out of our current complacency.

Hm I seem to have digressed. For those who want a physicists approach to group theory I refer to the excellent notes given by Peter Osborne for Cambridge Part III students

The notes and example sheets are for the course 3P2 Particles and Symmetries. I hope to make a reasonable attempt to read these notes and make a decent stab at some of the example sheets over the next year to complement the Pure maths aspects of group theory that I will be getting from M208.

For those who don't have the background in quantum mechanics then sections 1 and section 5 of the undergraduate course on the principles of  quantum mechanics gives a quick overview.

Some of you reading this blog will be aware than Duncan and I have tried one or two of the more basic undergraduate problems from the dynamics sheets to do with rotational motion and I hope to continue to work my way through some of the other sheets. Particularly Mathematical Methods, Differential equations and Complex Methods. However I also want to reach for the sky and try my hand at the particle symmetries course.


  1. Chris, you always -know- things! :-) I wasn't aware of the fact that the OU no longer includes printed TMAs in the course-packs. Does this mean that TMA's will be shipped x-weeks prior to cut-off or all in once? - M208 is supposedly all about proofs but there wasn't a single 'Show that' question in the exam. M208 will not be a problem for you ( what is, I wonder ) but if you want another distinction then practice on pressure and speed. I intend to do ( part of ) TMAs in an exam-like situation, but improve before shiment, of course.

  2. They are only going to be accessible from the course website which for my courses opens on 17th Jan. I hope my performance in M208 reflects your confindence in me. From what I can tell I do find it worrying that you aren't expected to memeorise or even demonstrate your understanding of proofs. Ok that might enable you to do reasonably well in the TMA's and exam maybe even get a distinction. But sooner or later especially when I want to branch out to independent study. I want to be able to understand all the gaps that are in textbooks and be able to tackle the exercises in the books just rushing through M208 and polishing up the exam technique doesn't really aid understanding and may lead to a false confidence in ones abilities. Thats why it's necessary to compare OU material with other sources eg Cambridge. I suspect however that as the year drags on I'll be tempted to take shortcuts.

  3. The OU courses are very, very good in explaining difficult topics. I really mean this. I studied The Counting Theorem ( M208 stuff ) in various books, but only after M208 I really got it. - Every Coin Has Two Sides: A plus is that you go well prepared to the OU exam. A minus is that mathematics books, in general, seem more difficult than the mathematics you did at the OU. An optical illusion since good mathematicians write easy books and bad ( read: lazy ) ones write impossible to get through books.

  4. Yes I agree I've Brannan's book on Analysis really helpful and glad I've put the time in still lots of gaps to fill in but at least I've got an overview and feel I can tackle the analysis parts of the TMA's reasonably well. I'm looking forward to the group theory. Analysis is for the head but group theory seems for the heart.