Progress

Well sorry for not posting for a while anyway progress of sorts has been made on M338 I got my first TMA back and did reasonably well but lost a few marks for taking short cuts. Also there was some difficulty in notation hopefully the wrinkles will be ironed out in time for the exam. Have to admit some of this course seems like wading  through treacle and it's difficult to see the point of a lot of it. Still just keep carrying on I know a knowledge of topology is an essential pre-requisite if one wants to understand things like the singularity theorems of General Relativity proved by Penrose and Hawking so mustn't get too disheartened.

Also finished the first assignment for MS324 which was on the whole a joy to do I'm definitely an Applied mathematician. The questions were essentially a review of mathematical techniques one should be aware of with one or two wrinkles thrown in. The first question was a question on Partial Fractions and an integration based on taking the limit of the integral to infinity. This involved some tricky manipulations of Logartihms but I think I got there. The second question was a straightforward and pleasurable solution of an inhomogeneous second order differential equation. The third question was a straightforward but tedious question on multivariable calculus and finding the stationary points. What made this question a bit tedious was the fact that the nature of the points varied for a wide range of conditions and you had to be careful not to miss out any combinations. The final question was on the deformation of springs more of a modelling question than a mathematics question I think I got most of it correct but I admit the very last part seemed badly phrased so I'm not sure if I got it correct or not. Still it's only 3 marks so should be on target for a good stab at this.

Both courses haven't really got started yet I really hope I can penetrate the fog of definition and theorem that is plaguing the topology units but at the minute I really can't see the wood for the trees. Conversely I know that the first real block of MS324 is on one of my favourite topics namely the wave equation and it's solutions I and feel reasonably confident I can polish off the next TMA in the next two weeks. The problem is it would be like a kid who eats his dessert before his main course instead of leaving it to last but what the heck.

As a final point at the risk of overloading people I've found yet another book on mathematics as applied to physics.

http://www.amazon.co.uk/Mathematics-Classical-Quantum-Physics-Dover/dp/048667164X/ref=sr_1_1?ie=UTF8&qid=1330880090&sr=8-1

Although quite old it covers all the branches of mathematics (apart from Topology) and their direct relevance to physics. It has quite an accessible introduction to the Hilbert space formulation of quantum mechanics stresses the key analogy between geometric and vector spaces which is so important to physics as well as having lots of concrete and challenging examples. I intend to try at least some of the questions which involve the separation of variables of pde's in spherical coordinate systems something not really covered in MS324 but which plays an important part in both classical and quantum physics. Anyway nice to see a book covering both classical and quantum physics in the same volume. A bit less abstract than say Szekres

http://www.amazon.co.uk/Course-Modern-Mathematical-Physics-Differential/dp/0521829607/ref=sr_1_sc_2?s=books&ie=UTF8&qid=1330880446&sr=1-2-spell

but not just another recipe book, Szekres and Byron complement each other nicely in my honest opinion.

  

General Relativity and Neutron Stars

Hi I've been trying to consolidate some of my physics Notes that I wrote (over 10 years ago How frightening) on general relativity. I've put a couple on my dropbox public folder

A general review of General Relativity

http://dl.dropbox.com/u/16049029/GR_rev.pdf

And an application to Neutron Stars

http://dl.dropbox.com/u/16049029/CDF_Neutron_star.pdf

This gives the background and then goes through the original calculation that Oppenheimer and Volkoff did just before the Start of the second world war. I wrote it as at the time there did not seem to be a clear account of the actual calculation in the public literature. I was also gratified to find a misprint in the original paper.

I wrote this stuff as at work I used to convene  a 'Failed physicists' society our work has lots of people who did PhD's and even post doctorates in physics or in my case a humble MSc but were brutally ejected from Academia. It arose out of a sense of frustration that the popular literature gave no hint of the actual maths and physics behind key caclulations or that people were doing calculations for themselves but had no means of sharing them with other people.   Anyway in the hope that other people find this useful I have posted the results here.

As for the maths courses I had my first tutorial on topology last week with Alan and it was good to meet up with Neil. Alan's tutorials are always useful in that they clear up ambiguities, for example when defining a set which covers all of the Real numbers should we include infinity. The answer is no so an interval R is the open set (-Infinity, Infinity) and not [-Infinity, Infinity]. Despite this procrastination has meant that I haven't yet tidied up the TMA for final send off. Still I'm not the only one in this position    

http://learn.open.ac.uk/mod/oublog/view.php?user=108287

Must be something to do with the gloomy and rather oppressive winter in Edinburgh

Also have been racing ahead with the MS324 course jumping straight into one of my favourite topics of Applied maths namely solutions to partial differential equations by separation of variables. For this topic unlike Topology or pure maths I feel like I'm coming home, or to use a crude expression 'like a pig in shit'.

Topics like this remind me why I got interested in maths in the first place. I love going to the pub, notepad in hand doing such problems and I find they just flow. Guess that means I'll always be an Applied mathematician/theoretical physicist, rather than a Pure mathematician/ Mathematical physicsist. I'll put another post explaining the difference.

Finally last week I got my essay back that I did for the Edinburgh university course on Hume. It was essentially a synthesis of ideas that I explored on my posts about Hume. I was pleased to get 85% and my tutor seemed impressed. He said he didn't have much to add except some criticism of some stylistic features. I tend when laying out the points of a philosophical argument to list the points separately (see some of my earlier posts). Anyway thats another 10 points towards my philosophy education I have 100 so far, and I look forward to the Philosophy of Art course in the Summer given by the same tutor Dr John Gordan.

As far as further studies go,  It looks like I could have a unique opportunity to gain more than the usual 120 points for philosophy via the OU. The course I did A211 has been replaced with a more traditional course which is not an excluded combination with A211,

http://www3.open.ac.uk/study/undergraduate/course/a222.htm

Also the current third level course AA308

http://www3.open.ac.uk/study/undergraduate/course/aa308.htm

is due to be replaced with one entitled 'Key  Questions in Philosophy' in October 2014 but given the way the OU works with replacement courses I suspect it wont be until 2015 till such a course appears on the horizon. So it looks like I could do the other second level philosophy course, AA308 and the new course by 2015 which would mean getting a total of 240 points in philosophy  before embarking on postgraduate work.
That would suit my other ambitions by then I would hope to be well into the Maths MSc and have finished my compositon courses.

So must stop procrastinating and write up the Topology TMA this week and do the first TMA for MS324.

M338 and MS324 progress so far

Just a short post here, I've more or less finished the first TMA for M338. The first question seems apparently straightforward. A skectch of a simple function, specifying the subsets over a specified interval, and then finding the image set of a function which isn't quite an inverse as the function is many valued. Whilst straightforward it's probably a warm up getting us used to the specification of continuous functions in terms of the behavior of the  inverse of open sets of a function. Sort of backwards but seems fundamental in topology. 

The second question is a standard proof that  a composite function involving trigonometric functions is continuous. 

Then another question involving proving that a function is continuous on the real line using the epsilon delta definition of continuity. This is then combined with the first new stuff namely the extension of continuity to Euclidean space namely a vector space of N dimensions. We live in three dimensional space but it is of course possible to conceive of N dimensional space. The definitions of continuity in N dimensional space involve the Pythagorean distance between two points instead of the modulus function. Anyway much of the same ideas to continuity apply to N dimensional space with this distance function. The questions seem relatively straightforward but as with all pure maths it's important to learn the phraseology and make sure all aspects are specified. It is tricky to get this correct and I find it takes two or three goes to get it right. Sometimes it feels more like writing an essay rather than doing maths anyway it's all part of getting in to how Pure mathematicians think. 

So it's a relief to get back to 'real' maths with MS324. So far the website hasn't opened but I've been working my way through unit 0.   This is a revision unit to get one up to speed the first part covers Integration and differential equations. The second part concerns multi valued calculus and a bit of vector calculus leading up to the divergence theorem. So far so good, I've not found the problems too difficult and I think I stand a good chance of 'winging' this one namely just concentrating on the TMA's and past exam papers as a light relief to all the heavy stuff on M338. There are end of section questions and I will see if I can do those first before tackling the main units. I can't wait till I get the TMA's which should be available once the website opens tomorrow. 

   

More books for a budding Physicist the (relatively) easy way

Hi Prompted by Keiths question on my recent blog as to whether or not there was one text covering supersymmetry, quantum field theory for some one with MST209 and M208 under their belt. The quick answer is not really apart from the route I mapped out in my two reading lists for General relativity and particle physicists.



However a more straightforward route to get a glimpse would be to take advantage of the following texts, as before the best way to get a backgound in classical physics prior to study of quantum physics is via Longair's book

http://www.amazon.co.uk/Theoretical-Concepts-Physics-Alternative-Reasoning/dp/052152878X

If one wanted some background on Maxwell's equations one could go also get hold of

http://www.amazon.co.uk/Students-Guide-Maxwells-Equations/dp/0521701473/ref=sr_1_1?s=books&ie=UTF8&qid=1327684577&sr=1-1

Then one would be ready to start on quantum physics and general relativity and a good overview but which skimps on calculational detail is given by the 'Demystified series' so in order of reading

Quantum Mechanics

http://www.amazon.co.uk/Quantum-Mechanics-Demystified-David-Mcmahon/dp/0071455469/ref=sr_1_1?s=books&ie=UTF8&qid=1327684714&sr=1-1

Relativity

http://www.amazon.co.uk/Relativity-Demystified-David-Mcmahon/dp/0071455450/ref=sr_1_2?s=books&ie=UTF8&qid=1327684746&sr=1-2

(This book does actually have quite an accessible introduction to modern coordinate free differential geometry)

Then Quantum Field theory

http://www.amazon.co.uk/Quantum-Field-Theory-Demystified-Mcmahon/dp/0071543821/ref=sr_1_4?s=books&ie=UTF8&qid=1327684746&sr=1-4

This is good as an overview but it really skimps on calculational detail.

Then you can try Superstings if thats your bag (It certainly isn't mine !!)

http://www.amazon.co.uk/String-Theory-Demystified-David-Mcmahon/dp/0071498702/ref=sr_1_5?s=books&ie=UTF8&qid=1327684746&sr=1-5

I have it on my shelf and whilst I can understand the maths I'm still mystified as to the relevance of superstrings to anything but worth having to get a glimpse of what all the fuss is about.

Finally and I wasn't aware of this there is one on supersymmetry (which I've just ordered)

http://www.amazon.co.uk/Supersymmetry-DeMYSTiFied-Patrick-Labelle/dp/0071636412/ref=sr_1_1?s=books&ie=UTF8&qid=1327685098&sr=1-1

 The one thing that the series doesn't cover is applications of group theory to physics so my recommendation is still Jones

http://www.amazon.co.uk/Groups-Representations-Physics-H-F-Jones/dp/0750305045/ref=sr_1_1?s=books&ie=UTF8&qid=1327685218&sr=1-1

This could profitably be read after the quantum mechanics book.

Added 28th JAn 2012

I forgot to mention the other field of current research namely quantum information theory and quantum computing, the best starting point is Mermin's lecture notes and they could be started straight away (Although the quantum mechanics book could be read beforehand)

http://people.ccmr.cornell.edu/~mermin/qcomp/CS483.html

I do find it remarkable that such a productive field of current research makes no mention of wavefunctions the distinction between classical computing and quantum computing seems to be the extension to q-bits rather than c-bits being the fact that q-bits are complex probability amplitudes rather than just binary numbers.



Those who want the real McCoy however are referred to my other reading lists

http://chrisfmathsphysicsmusic.blogspot.com/2011/10/general-relativity-reading-list.html

http://chrisfmathsphysicsmusic.blogspot.com/2011/06/reading-list-for-budding-particle.html

What type of mathematician/physicist are you? Part II

In a post I wrote many months ago I asked the question what type of mathematician are you ? I have been having various debates on the Fora about this, Including one with Nilo

http://mathematics-diary.blogspot.com/2011/12/what-is-origin-of-mathematics.html?showComment=1325325044840#comment-c4230731908399424425

The big divide is between those who think (like Nilo) that we discover maths and those like myself that we use maths in order to make sense of the world around us. On my view I have no problem with seeing maths essentially as a construction like a work of  Art or music. It's relationship with the real world being an accident rather than some god given laws. Indeed when one applies mathematics to physics there will always be a gap. The Simple Harmonic Oscillator only works well when applied to small amplitudes. If one applies group theory to particle physics it only really works if we assume the particles have the same mass but of course they don't. Nevertheless it is still productive to see how far this can be taken.

Anyway an interesting article I came across in the Physics Arxiv concerns a debate between 3 physicists/mathematicans which delineates three views

http://arxiv.org/abs/physics/0510188

1) The Fundamentalist view articulated by Max Tegmark who seems to be prominent on the TV more and more. This essentially on the extreme view that Max Tegmark asserts that to every mathematical concept there is a corresponding  element of reality if not in this universe then in another universe. It has it's roots in Platonism and many mathematicians have been attracted to Platonism such as G H Hardy and in our own time Penrose and Marcus Du Satoy are prominent examples although they probably would not be as extreme as Tegmark. For them as I have commented before mathematics is about uncovering the deep nature of reality so that whenever a mathematical theorem is deduced no matter how trivial something fundamental is discovered about the nature of reality. On this view mathematics seems to be akin to a new form of mysticism and it is interesting that most TV programmes which talk about maths always seem to have people who express the Platonist view. Very rarely if at all are other views

2) The mystical view represented by Peter Hut which claims that in 500 years time the laws of physics and the principles of mathematics will have changed beyond recognition so that we are only getting glimpses of what is really the case. I'll leave you to read the article to find out more

3) Finally the secular view represented by Marcus Alford and which I would identify with. For us secularists mathematics is essentially a construction of the human mind which we impose on the world around us in order to make sense of it. There are only a limited subset of mathematical concepts which are applicable to the world around us and there will always be a gap between the real world and the mathematics that is used to understand it. For example (and this is the point I was making to Nilo) the symmetries of particle physics assume that all particles have the same mass which they don't. Nevertheless they are useful in classifying particles and making some order out of the apparent chaos. Again not all mathematics has to apply to the real world but one shouldn't dismiss out of hand their success when it occasionally works. This is not to denigrate maths at all but to see it as one amongst many of humanities greatest achievements.

Going back to Hume's distinction between relationships between ideas and matters of fact mathematics concerns itself with relationship between ideas and not necessarily matters of fact. On the one hand there is this pristine pefect world in the realm of ideas as exemplified by mathematics and on the other hand there is the messy world which aided by mathematics and our knowledge of the laws of physics we can begin to make a little sense of.  But we have to acknowledge that there is no route to absolute truth even if humanity occasionally wins some battles.

Visions of a failed physicist

Apologies if this post seems a bit incoherent but it may help to understand why I call myself a failed physicist as opposed to a moderately succsessful one. OK here is some context I was brutally ejected from paradise when it was made plain that despite my showing a moderate ability to pass exams on quantum field theory and group theory that there was no way I was ever going to get a grant to pursue particle physics beyond my modest MSc, This was despite the fact that I got more marks in the quantum field theory exam than other of my contempories who had already got a full time gramt. Whatever happened to them I dont know but given that their names are not seen as being leading lights in theoretical physics then I guess their career petered out eventually and they are now (just as I am) makong modest contributions to the world of industry, finance or comerce. So what was the point anyway despite the bitterness one has to acknowledge that the 'system' is extremely ruthless and that better people than me have failed to get to the holy grail of a permanemt lectureship and so despite the scars which still sting after 30 years one has to grow up and get used to it.

Anyway those who have read my blogs and contributions to the OU fora will realise that I am deeply concerned about the way modern physics has developed. So at the risk of sticking my neck out I will tty and turm the clock back 30 years when the garden seemed rosy and the nagging feeling (of which I have no means of verifying)  that something was missed back in those early years. Also I will stick my neck out (and I've had a few pints so apologies for my boldness) and state some constraints that I feel that provide some basic ground rules for any theory of physics should adhere if they are to be credible.

OK when I did my MSc back in 1978-1980 the big things were gauge theories, the establishment of the standard model of particle physics and extensions which involved either supersymmetry or attempts to incorporate the established symmetry of SU(3)xSU(2)xU(1) in terms of a higher symmetry such as SU(5).
It would be a slightly cynical view (Albeit one I hold to) to claim that as far as particle physics is oncerned that there has been no real development since then apart from finding out specific values of masses of particles such as the top quark that we didn'nt know. Anyway the undoubtable succes of the standard model of particle physics in accounting for all the phenomenon that we experience at paticle acelerators at CERN or FERMILAB cannot be disputed.

So this gives me my first boundary condition of any new theory of physics to replicate our current success that it must show in a fairly obvious way that the Lagrangian of the Standard model of particle phyiscs is a part of it. (This is not true of any theory involving Superstrings)

Secondly there has to be a link with experimental results for any quantum field theory based on a Lagrangian this is fairly routine (albeit quite technically complicated) in that for a Lagrangian based on 4 diminsional space time it is relatively straightforward to calculate a scattering matrix for a given process (this seems not to be the case for any theory of superstrings or loop gravity (any experts please correct me if I'm wrong)).

Now here is the crux of my vision, when one quantizes a classical Lagrangian such as those for the strong interaction if one wants to make consistent calculations one has to introduce ghost particles and these ghost particles at first sight have to be introduced ad hoc. Neverheless once one has done this the symmetry of the strong interaction namely an SU(3) symmetry is seen as part of a deeper symetry called BRST symmetry.
which mixes fermions and bosons together

http://en.wikipedia.org/wiki/BRST_quantization

Simulataneously as these symmetries were being discovered developments of supersymmetry which also combines fermion and bosonic symmetries together were being explored

http://en.wikipedia.org/wiki/Supersymmetry

The natural extension of these techniques to relativity led to the development of a theory of gravity called
supergravity in which the quantisation of General relativity led to the coupling of the graviton a spin 2 particle to a spin 3/2 particle equation. (NB the usual claim that superstrings are the only natural way that predicts the graviton as is sometimes claimed is false any means of turning general relativity into a quantum field theory will involve the prediction of a spin 2 particle not just superstrings).
Unfortunately (and here is the failure) attempts to treat gravity as just another quantum field theory was inconcliusive (but then despite the hype superstrings are just inconclusive) so here is my statememt of faith

Supersymmetry on it's own is clearly inadequate
on the other hand Supersymmetry combined with BRST symmetrty a symmetry which seems a fundamental
symmetry of any quantisation procedure could lead to something fundamental I don't know what nor do I have any real clues as to what it might involve.

On the other hand abandoning the most empricically successful theory we have namely Lagrangian quantum field theory in favour of something like superstrings seems misguided given the empirical success of quantum ffield theory via the standard model of particle physics. And maybe just maybe a combination of BRST symmetry and supersymmetry a priori might just work.

M338 Topology First Impressions

Well the website for M338 Topology has opened although I still have to receive the materials. Still it gave me a chance to look at the units ahead it's like a sandwich really Abstract Topology, Geometric Topology and then more Abstract Topology. The first block is a brisk review of continuity on the Real line including the epsilon delta definition of continuity before extending it to Euclidean spaces. This is a prelude to metric spaces and then properties of open and closed sets culminating in a general definition of continuity based on open sets. The first TMA (10%) is based solely on the first unit of block A and doesn't have to be done until  Feb 24th (although it will soon catch up with me). My strategy for this course is to try and do it in small chunks one or two subsections per day and instead of just focusing on what the TMA questions ask and working backwards. I want to try and do the units more systematically than I have done in previous courses. Well thats the intention anyway. I think Topology will be one of the hardest courses I've done so far but the point is not to get too overwhelmed and to do  a little per day rather than trying to cram it all at once. I reckon I can do 1/2 a unit per week which should keep me on target. One thing that will handicap this course is the fact that there are relatively few past exam papers. Anyway looking forward to the challenge and can't wait to get the actual books.