Hi reading the various forums there seem to be a lot of people wanting to at least learn about if not do an MSc or even a Phd in particle physics or Cosmology. Eg this guy who I've just come across in the past week or so and who has done me and other fellow bloggers the honour of calling me a fellow Jedi (even though I do not really believe in Jedism but the compliment is noted).

http://openuniversityrocketscientist.blogspot.com/

His ambition is admirable and I hope he makes it but unfortunately the Open University courses good as they are will only take you so far. I'm going to have some harsh words to say about the physics (or lack of physics courses) available on the Open University. In particlar S207, having downloaded an exam paper I was quite shocked at the level of some the questions. In particular the lack of mathematics and very little use of calculus. They seemed to be set at a level which was no more than A level and not even that. Where for example is any stuff on AC circuits. Some of my fellow bloggers, currently doing MST209, will have spent a large amount of time learning how to solve problems involving simple harmonic oscillators a really key part of learning the relevance of calculus to physical problems. However what is even more amazing is the realisation that the same part of maths also applies to electrical circuits for me that was a real testimony to the power of maths when I came across it in the 6th form. It seems to me that the Open University has missed a trick here. Again from what I can tell the thermodynamics part will not really stress the maths behind it, for example the Maxwell relations which provide an interesting use of partial differential equations. OK maybe I'm being a bit unfair but those who like maths and do that course must be feeling really frustrated.

So is there a remedy? I would say yes first of all t'Hooft a guy who did much to revitalise the use of quantum field theory in the development of the Standard model of particle physics has put together a web site which forms a structured way of getting most of the background to become a theoretical physicist

http://www.staff.science.uu.nl/~hooft101/theorist.html

But of course there is no formal assessment and how can you know you are up there with the best of them, one way to check and again fill in the gaps is to follow the Cambridge Maths department which gives out the example sheets for both their pure maths and theoretical physics courses and also the exams. If you look under the radar you can also find lecture notes on the courses.

http://www.damtp.cam.ac.uk/user/examples/

Of particular interest are the courses in differential equations and mathematical methods for both the specialists and the scientists, If you can do the exams and the problems on the sheets you know you are up there, if not then you know what you have to aspire to. To know what modern particle physics looks like the courses on particle physics and symmetries in the Cambridge part III cover the basics

http://www.damtp.cam.ac.uk/user/examples/indexP3.html

If you can get onto part III and do well you'll have made it and my advice to my Jedi friend is to work a path aided by the Open university to see how feasible it is to get on such a course. Imperial College and Kings College offer suitable alternatives.

In the mean time those who want to do say some of the mathematical physics courses at the OU but do not necessarily want to do S207 to get some physical background could start here

http://www.amazon.co.uk/Theoretical-Concepts-Physics-Alternative-Reasoning/dp/052152878X/ref=sr_1_1?ie=UTF8&qid=1308687397&sr=8-1

Of course no reccomendation on physics books at undergraduate level would be complete without mentioning the Feynman lectures on physics.

http://www.amazon.co.uk/Feynman-Lectures-Physics-boxed-set/dp/0465023827/ref=sr_1_1?ie=UTF8&qid=1308687506&sr=8-1

But these are more of a dip into both books do suffer from the defect of not having problems associated with them but this is easily remedied by following the Cambridge example sheets. A more tradtional (and a bit boring book is Halliday and Resnick)

http://www.amazon.co.uk/Principles-Physics-David-Halliday/dp/0470561580/ref=sr_1_1?s=books&ie=UTF8&qid=1309091597&sr=1-1

Obviously the OU courses in electromagnetism quantum mechanics and arguably the relativistic Universe or at least volume 1 which covers the basics of general relativity, are essential. As is MST209 and either one or both of the fluid mechanics course and the partial differential equations course which is going to supersede the current waves, diffusion and variational principles course. The latter would be better as it includes a section on Lagrangian and Hamilonian mechanics which is more abstract than that in MST209 and forms the starting point for quantum field theory. Longair has a chapter on it but again in order to master the problem solving skills attempting some of the Cambridge problems is probably essential. Also if like my Jedi friend he wants to do the MSc in Maths or at least keep his options open you need M208 and M337 as essential parts.

Having mastered the classical background and mathematical techniques one can go on to study Non relativistic quantum mechanics A good reference is Modern Quantum Mechanics by Sakurai

http://www.amazon.co.uk/Modern-Quantum-Mechanics-J-Sakurai/dp/0321503368/ref=sr_1_2?ie=UTF8&qid=1309090554&sr=8-2#_

and this would be a good supplement to the OU course.

Ok you don't have to do everything in Sakurai but in order to supplement say the quantum mechanics course taught at the OU you would probably need to understand how it is applied especially scattering theory. The material in the applications of quantum mechanics sheets

http://www.damtp.cam.ac.uk/user/examples/D19a.pdf

gives an idea of what you should know. You can probably ignore solid state physics (Squalid State physics as some famous physicists have called it). Many of the Standard books on quantum physics contain this material. and it would be a useful supplement to anyone considering doing the OU course in quantum mechanics. In order to solve 'real' problems an understanding of how to solve differential equations by series and a familiarity with the so called special functions Bessel Functions, Legendre Polynomials, Spherical Harmonics is necessary. Also for some specialist topics such as scattering theory some familiarity with Complex Integration especially contour integration is necessary. The Cambridge maths methods courses cover this material and some of it is covered in the open university course. Mathematical Methods and Fluid mechanics, but if you can't wait that long then get a good book on mathematical methods such as Benson et al

http://www.amazon.co.uk/Mathematical-Methods-Physics-Engineering-Comprehensive/dp/0521679710/ref=sr_1_1?s=books&ie=UTF8&qid=1309091303&sr=1-1

or "Work through" the Cambridge examples sheets especially those on mathematical methods and Complex methods. The notes for these are freely available.

One mathematical technique not really covered by any of the Open university course which plays an important part in Advanced physics especially quantum field theory is the technique of Green's functions and the Dirac delta function the Cambridge lecture notes in mathematical methods cover this and probably develop more facility.

However even that is not enough, one thing that is tragic in the OU course provision is the lack of any application of Group theory to physical problems this book by Jones gives a good overview starting with finite groups and their application to molecular spectra and then going to give a brief overview of symmetry principles and their application to particle physics.

http://www.amazon.co.uk/Feynman-Lectures-Physics-boxed-set/dp/0465023827/ref=sr_1_1?ie=UTF8&qid=1308687506&sr=8-1

Finally an understanding of special relativity especially 4 vector formalism is needed and it's application to Maxwell's equations. This is covered briefly in Longair You should be then in a good position to embark on understanding particle physics this is also covered in the Relativistic Universe.

For a good uptodate first book on particle physics the book by Perkins is recommended

http://www.amazon.co.uk/Introduction-Energy-Physics-Donald-Perkins/dp/0521621968/ref=sr_1_1?s=books&ie=UTF8&qid=1308688487&sr=1-1

So when you have completed the 3rd level physics and maths courses and can't get onto an MSc or just want to know about the subject at graduate level what next. There are three levels, one is a basic understanding of relativistic particle physics at first order of perturbation theory this can be got either (and although it's expensive would be my first recomendation) Halzen and Martin

http://www.amazon.co.uk/Quarks-Leptons-Introductory-Particle-Physics/dp/0471887412/ref=sr_1_1?s=books&ie=UTF8&qid=1308688806&sr=1-1

It's direct gets you calculating straight away and you will learn a lot about how actual particle physics calculations are done. Even though quantum field theory is the foundation of particle physics it's quite surprising when it comes down to it, how little of it is needed in practical calculations. Once one has the Feynman rules, then that is all one needs, to be able to calculate scattering cross sections, If you like quantum field theory is the scaffolding which can be thrown away once you've got there. So I would argue the first step to acquire in particle physics (and remember I'm a failed one, so I'm probably talking rubbish) is facility in calculating Feynman diagrams and an appreciation of the basic phenomenology.

If that book seems to expensive then an alternative is these by Aitichson and Hey

http://www.amazon.co.uk/Gauge-Theories-Particle-Physics-Relativistic/dp/0750308648/ref=sr_1_fkmr0_2?ie=UTF8&qid=1308689343&sr=1-2-fkmr0

http://www.amazon.co.uk/Gauge-Theories-Particle-Physics-Electroweak/dp/0750309504/ref=sr_1_fkmr0_1?ie=UTF8&qid=1308689343&sr=1-1-fkmr0

However it has the disadvantage of being in two volumes and the problems do not have solutions whereas Halzen and Martin does.

OK so thats stage 1 a basic mastery of phenomenology an understanding of what a Feynman diagram is and the ability to calculate Feynman diagrams at the first order of perturbation. Also an understanding of the basic symmetries of the Standard model of particle physics and it's Lagrangian.

Stage 2 is then learning how to do second order calculations and here quantum field theory is a must again taking a pragmatic view here rather than getting bogged down in formalism I would recommend Peskin and Schroeder (In my day it was Bjorken and Drell). Here there is a combination of both quantum field theory, and its application to concrete problems, One really useful feature is at the end of each main section there are three research problems. I guess my level is about here which is where my MSc took me but I never did any calculations at the level of the research projects in the book, I do hope to do them eventually as they would be a real achievement and certainly my Jedi friend should aspire to do them if he wants to be taken seriously.

Finally after a good basic understanding of Peskin and Schroder has been achieved then one can move onto the deeper books such as those by Weinberg

http://www.amazon.co.uk/Quantum-Theory-Fields-Foundations-v/dp/0521670535/ref=sr_1_2?s=books&ie=UTF8&qid=1308690055&sr=1-2

http://www.amazon.co.uk/Quantum-Theory-Fields-Modern-Applications/dp/0521670543/ref=pd_bxgy_b_text_b

http://www.amazon.co.uk/Quantum-Theory-Fields-Supersymmetry-v/dp/0521670551/ref=pd_bxgy_b_img_c

But these are really deep books more for reference, I should also mention Zee's book which gives a concise but still rigorous view of quantum field theory and could probably be read before embarking on Peskin and Schroeder

http://www.amazon.co.uk/Quantum-Field-Theory-Nutshell-Second/dp/0691140340/ref=sr_1_1?s=books&ie=UTF8&qid=1308690646&sr=1-1

You will note I've not said anything about Superstring theory the best comment I can make on that is from Perkins

"At the present time string theory is under rapid development. Nobody yet knows, and probably will not know for some years, whether it has any relevance to the real world"

Ok I hope this post hasn't seemed pretentious, it strikes me that there is little of this information freely available and I hope I've sketched a feasible path for someone to follow who wants to understand particle physics. At the risk of pontificating more I'll sketch out a similar path for a theoretical Astrophysicist to follow in a not too distant post.

## Tuesday, 21 June 2011

### Next year's courses

Well I've commited myself to the following courses next year

M338 Topology and M336 Groups and Geometry and given that I don't want to take on to much that will be it. However in October 2012 the switch over to October starts will occur and so there wont be any let up as I'll then be straight into Fluid mechanics and Maths methods and Logic and Number theory.

I will probably do two courses from Geoffrey Klempner's Pathways modules Starting in October of this year

http://www.philosophypathways.com/programs/index.html

Namely metaphysics and philosophy of language, as far as I can tell these are equivalent to typical undergraduate courses in this area. It's the only feasible way of getting the requisite background, I estimate on top of the maths that will work out at about 6 months per course. Assessment is by 5 1000 word essays per course I would have preferred 2000 but one can't have everything I suppose. Then I may or may not squeeze in the 3rd level OU course thought and experience alongside Fluid mechanics and Number theory or leave it a year whilst embarking on the OU MSc in Maths. I have a long term aim to supplement my retirement years by becoming a part time tutor in maths via the OU and I want to be there by the time I'm 60 having moved to part time work at least thats the hope. So despite my earlier misgivings about not doing the MSc I feel I would be stupid not to, I will also make a concrete attempt to see if Edinburgh University will have me in their department of language and logic so I can pursue my dream of investigating the link between contemporary debates in philosophy of language and their application or not to quantum mechanics, But I need to finish the OU courses first.

M338 Topology and M336 Groups and Geometry and given that I don't want to take on to much that will be it. However in October 2012 the switch over to October starts will occur and so there wont be any let up as I'll then be straight into Fluid mechanics and Maths methods and Logic and Number theory.

I will probably do two courses from Geoffrey Klempner's Pathways modules Starting in October of this year

http://www.philosophypathways.com/programs/index.html

Namely metaphysics and philosophy of language, as far as I can tell these are equivalent to typical undergraduate courses in this area. It's the only feasible way of getting the requisite background, I estimate on top of the maths that will work out at about 6 months per course. Assessment is by 5 1000 word essays per course I would have preferred 2000 but one can't have everything I suppose. Then I may or may not squeeze in the 3rd level OU course thought and experience alongside Fluid mechanics and Number theory or leave it a year whilst embarking on the OU MSc in Maths. I have a long term aim to supplement my retirement years by becoming a part time tutor in maths via the OU and I want to be there by the time I'm 60 having moved to part time work at least thats the hope. So despite my earlier misgivings about not doing the MSc I feel I would be stupid not to, I will also make a concrete attempt to see if Edinburgh University will have me in their department of language and logic so I can pursue my dream of investigating the link between contemporary debates in philosophy of language and their application or not to quantum mechanics, But I need to finish the OU courses first.

## Saturday, 18 June 2011

### Woo Hoo Finished TMA04 for M208

Glad to say I've finished TMA04 for M208. This was the first analysis block and all that reading of Brannan's book paid off. The TMA covered the following topics

1) Proof of various inequalities and finding the greatest lower bound of a function

2) Testing varying sequences for convergence

3) Testing varying series for convergence

4) Testing to see if various functions are continuous at a given point

Have to say I found it fairly straightforward, not that analysis is, but the TMA essentially was a run through of various tests which are relatively straightforward to apply. The main problem, as with M337, is to make sure every detail is included, thus for example when proving a sequence made up of a whole sequence of terms it's not enough to divide by the dominant term and then caclulate the basic limit you have to include such phrases as because

$$ {1/n},{1/n^3} $$ etc are basic null sequences, by the Combination rules it can be seen that the sequence converges to a and so forth. All of which is so easy to forget and I guess if I do such questions in the exam I'll lose a few marks for not including every detail. The only way to guarantee success is to look carefully at the way answers to the examples in the course units have been structured and learn it off almost by rote.

On the other hand even the answers given in the course units miss out essential details, for example they quite often to forget to mention that cos(x) or sin(x) is only bounded if

$$ x \epsilon R$$ where R is the set of real numbers, Or for example a term like

$$c=\sqrt{\frac{a}{b-m'}}$$ which is often encountered when trying to find the least upper bound or greatest lower bound of a sequence is only real if b-m' > 0. Thus an appeal to the Archimedean princple namely that if $$a \epsilon R$$, there is an integer n such that n > a only works if the term c is real. Again a point not stressed by the units.

Coo getting more pedantic than the M208 answers must be learning something

Seriously though all this at first sight pedantry is essential if one wants to be sure of ones conclusion, for example as my mate Neil has pointed out it took mathematicians over 2000 years to resolve Zeno's paradox properly.

As a final point it's a bit worrying that the TMA's only appear to be scratching the surface still I suppose to avoid complaints that the TMA's are to hard I suppose needs must. I can always get a sanity check by looking at the Cambridge Maths examples sheets for analysis.

Anyway a day off from TMA questions tomorrow who hoo.

1) Proof of various inequalities and finding the greatest lower bound of a function

2) Testing varying sequences for convergence

3) Testing varying series for convergence

4) Testing to see if various functions are continuous at a given point

Have to say I found it fairly straightforward, not that analysis is, but the TMA essentially was a run through of various tests which are relatively straightforward to apply. The main problem, as with M337, is to make sure every detail is included, thus for example when proving a sequence made up of a whole sequence of terms it's not enough to divide by the dominant term and then caclulate the basic limit you have to include such phrases as because

$$ {1/n},{1/n^3} $$ etc are basic null sequences, by the Combination rules it can be seen that the sequence converges to a and so forth. All of which is so easy to forget and I guess if I do such questions in the exam I'll lose a few marks for not including every detail. The only way to guarantee success is to look carefully at the way answers to the examples in the course units have been structured and learn it off almost by rote.

On the other hand even the answers given in the course units miss out essential details, for example they quite often to forget to mention that cos(x) or sin(x) is only bounded if

$$ x \epsilon R$$ where R is the set of real numbers, Or for example a term like

$$c=\sqrt{\frac{a}{b-m'}}$$ which is often encountered when trying to find the least upper bound or greatest lower bound of a sequence is only real if b-m' > 0. Thus an appeal to the Archimedean princple namely that if $$a \epsilon R$$, there is an integer n such that n > a only works if the term c is real. Again a point not stressed by the units.

Coo getting more pedantic than the M208 answers must be learning something

Seriously though all this at first sight pedantry is essential if one wants to be sure of ones conclusion, for example as my mate Neil has pointed out it took mathematicians over 2000 years to resolve Zeno's paradox properly.

As a final point it's a bit worrying that the TMA's only appear to be scratching the surface still I suppose to avoid complaints that the TMA's are to hard I suppose needs must. I can always get a sanity check by looking at the Cambridge Maths examples sheets for analysis.

Anyway a day off from TMA questions tomorrow who hoo.

## Monday, 13 June 2011

### Back on Track M337 TMA02

After a fairly intensive weekend managed to get back on track with Complex Variables and have just posted off the 2nd TMA

The topics covered were

1) Complex integration

2) Cauchy's theorem

3) Taylor series

4) Laurent series

These form the hard core of any Complex Variable theory course. The questions covered the following

1) Simple integration along two different paths of a function showing that the two integrals were not the same

and hence that the function was not entire. Then a couple of complex integrals which can be treated in a similar manner to ordinary integrals

2) 1st part a relatively straightforward application of Cauchy's Integral and derivative formula to integrate a function

2nd Part (Probably the hardest) a question to really stretch your understanding of Liouville's theorem which states that a function which is entire over the whole of the complex plane and bounded must be constant, Some parts had me really scratching my head here but I think I got there in the end (time will tell).

3) Exercises on Taylor series, not much different from stuff in MS221 but the conditions for convergence etc involve the |z|

Then another tricky question on the uniqueness theorem which states that if two functions can be shown to identical at a limit point on a region which is a subset of C then they are identical all over C. I think this is a warm up exercise for the topic of analytical continuation

4) Exercises on Laurent series and an application of the Residue formula

PArt of the problem is learning how to phrase the proofs and assumptions and that can only come with practice. I would love to be able to write phrases such as

"This function is analytic apart from the points a1, b1 and c1 where it has poles of order 2 and 3 at b1 and c1 and an essential singularity at a1, Thus on the contour C1 we can integrate the function but not on C2 ..." naturally, before diving into the calculation but that will not happen immediately.

So how do I feel not sinking but missing out huge chunks of theory and it's not coming naturally just yet, but slowly but surely it's sinking in. So my predictions for the course a grade 2 is feasible but I wouldn't feel confident about a distinction

The pressure is not really off yet as I have just over a week to do my next TMA for M208 as this is real analysis it will be interesting to see if all that work I did reading Brannan in between MS221 and M208 has paid off. I have to say that whilst analysis is interesting it doesn't grab me in the way that group theory does,

"Analysis is for the head but group theory is for the heart"

Looking forward to getting to the 2nd part of group theory for M208.

The topics covered were

1) Complex integration

2) Cauchy's theorem

3) Taylor series

4) Laurent series

These form the hard core of any Complex Variable theory course. The questions covered the following

1) Simple integration along two different paths of a function showing that the two integrals were not the same

and hence that the function was not entire. Then a couple of complex integrals which can be treated in a similar manner to ordinary integrals

2) 1st part a relatively straightforward application of Cauchy's Integral and derivative formula to integrate a function

2nd Part (Probably the hardest) a question to really stretch your understanding of Liouville's theorem which states that a function which is entire over the whole of the complex plane and bounded must be constant, Some parts had me really scratching my head here but I think I got there in the end (time will tell).

3) Exercises on Taylor series, not much different from stuff in MS221 but the conditions for convergence etc involve the |z|

Then another tricky question on the uniqueness theorem which states that if two functions can be shown to identical at a limit point on a region which is a subset of C then they are identical all over C. I think this is a warm up exercise for the topic of analytical continuation

4) Exercises on Laurent series and an application of the Residue formula

PArt of the problem is learning how to phrase the proofs and assumptions and that can only come with practice. I would love to be able to write phrases such as

"This function is analytic apart from the points a1, b1 and c1 where it has poles of order 2 and 3 at b1 and c1 and an essential singularity at a1, Thus on the contour C1 we can integrate the function but not on C2 ..." naturally, before diving into the calculation but that will not happen immediately.

So how do I feel not sinking but missing out huge chunks of theory and it's not coming naturally just yet, but slowly but surely it's sinking in. So my predictions for the course a grade 2 is feasible but I wouldn't feel confident about a distinction

The pressure is not really off yet as I have just over a week to do my next TMA for M208 as this is real analysis it will be interesting to see if all that work I did reading Brannan in between MS221 and M208 has paid off. I have to say that whilst analysis is interesting it doesn't grab me in the way that group theory does,

"Analysis is for the head but group theory is for the heart"

Looking forward to getting to the 2nd part of group theory for M208.

## Wednesday, 8 June 2011

### Galois Theory Lie Groups and distractions

Here I am 1 week and 2 weeks away from TMA deadlines and I should be getting my head down trying to polish them off. However I got distracted a couple of weeks ago by a link someone posted on one of the forums to a book on Lie Groups Physics and Geometry, Chapters of which are accessible here.

http://einstein.drexel.edu/~bob/LieGroups.html

The first chapter in particular caught my attention as it gave an accessible account of Galois Theory. As most reading this blog will know Galois theory arose out of the attempt to find a formula similar to those available for quadratic equations, cubic equations and quartic equations. It was one of the many strands leading into our current group theory. Most books on abstract algebra eg Artin leave Galois theory till the end with the implication that you have to master all sorts of arcane subjects such as rings, fields, field extensions etc before you can get the gist of what is going on. Of course Galois himself was unaware of such theories and it is doubtful if he could pass an exam in a subject of which he is the founder. It is refreshing to find an account which tells you the point before launching into theorem after theorem. Of course once one has read such a chapter one can then go onto the more formal stuff. I was slightly distracted and probably missed the point by a transformation which applies to the quadratic equation

$$x^2+px+c=0$$ if we make the transformation x -> -x-p then a quick substitution shows that the equation doesn't change. I had hoped that one could find global invariants of the cubic equation

$$x^3+px^2+qx+c=0$$

which if we write as

$$(x-r1)(x-r2)(x-r3)=0$$ where r1,r2 and r3 are the three roots of the equation we can obtain expressions for p,q and c in terms of the roots Namely.

c = -(r1 r2 r3)

p = -(r1+r2+r3)

q = r1r2 + r1r3 +r2r3

So you might think that you should be able to find a transformation similar to the quadratic equation transformation which leaves the equation invaraint. I thought by expanding two of the brackets in the root equation eg

$$(x-r1)(x^2-(r2+r3)x-r2r3)$$ that you could apply the quadratic transformation

x -> -x - (r2+r3)

and lo and behold recover the original cubic equation. I wasted hours trying to find invariants but not for the first time in my life I was up the garden path. because of course you have transform the other bracket so x-r1 -> -x-(r1+r2+r3) which when expanded does not lead you back to the original cubic. However it was fun whilst the illusion lasted I felt alive and joyful and it was a refreshing change to do some maths for its own sake not simply because yet another TMA question had to be answered. A bit of further reading showed me that Galois was concerned with symmetries between the roots rather than global symmetries of the equation itself. From the above expressions for p.q and c all sorts of invariants can be constructed also note that permuting the roots leaves the above expressions invariants. Thus there is a close connection between group theory and something as mundane as solving polynomial equations.

My experience was a lesson in being overoptimistic, just as a bad chess player (which I am) will spend hours developing an attack, only to miss the obvious fact that his queen is under threat and the priority should be to move the queen out of danger. Just as my priority should be to finish my two TMA's. I take heart that my mate Neil who is a good chess player and is revising for a computing exam had a similar distraction

http://neilanderson.freehostia.com/thoughts/degree/m257/

Anyway along with all the other distractions I've got an interest in Galois theory and have ordered this book by Cox which Nilo recommends

http://www.amazon.co.uk/Galois-Theory-Pure-Applied-Mathematics/dp/0471434191/ref=sr_1_1?ie=UTF8&s=books&qid=1307564746&sr=8-1

It seems accessible in that it starts from the motivation of the solution of quadratic, cubic and quartic equations

Also a good set of notes available on the net amongst many is here

http://www.warwick.ac.uk/~masda/MA3D5/Galois.pdf

So yet another distraction but one thats fun hopefully I've got it out of my system and can concentrate on the TMA's over the next two weeks.

http://einstein.drexel.edu/~bob/LieGroups.html

The first chapter in particular caught my attention as it gave an accessible account of Galois Theory. As most reading this blog will know Galois theory arose out of the attempt to find a formula similar to those available for quadratic equations, cubic equations and quartic equations. It was one of the many strands leading into our current group theory. Most books on abstract algebra eg Artin leave Galois theory till the end with the implication that you have to master all sorts of arcane subjects such as rings, fields, field extensions etc before you can get the gist of what is going on. Of course Galois himself was unaware of such theories and it is doubtful if he could pass an exam in a subject of which he is the founder. It is refreshing to find an account which tells you the point before launching into theorem after theorem. Of course once one has read such a chapter one can then go onto the more formal stuff. I was slightly distracted and probably missed the point by a transformation which applies to the quadratic equation

$$x^2+px+c=0$$ if we make the transformation x -> -x-p then a quick substitution shows that the equation doesn't change. I had hoped that one could find global invariants of the cubic equation

$$x^3+px^2+qx+c=0$$

which if we write as

$$(x-r1)(x-r2)(x-r3)=0$$ where r1,r2 and r3 are the three roots of the equation we can obtain expressions for p,q and c in terms of the roots Namely.

c = -(r1 r2 r3)

p = -(r1+r2+r3)

q = r1r2 + r1r3 +r2r3

So you might think that you should be able to find a transformation similar to the quadratic equation transformation which leaves the equation invaraint. I thought by expanding two of the brackets in the root equation eg

$$(x-r1)(x^2-(r2+r3)x-r2r3)$$ that you could apply the quadratic transformation

x -> -x - (r2+r3)

and lo and behold recover the original cubic equation. I wasted hours trying to find invariants but not for the first time in my life I was up the garden path. because of course you have transform the other bracket so x-r1 -> -x-(r1+r2+r3) which when expanded does not lead you back to the original cubic. However it was fun whilst the illusion lasted I felt alive and joyful and it was a refreshing change to do some maths for its own sake not simply because yet another TMA question had to be answered. A bit of further reading showed me that Galois was concerned with symmetries between the roots rather than global symmetries of the equation itself. From the above expressions for p.q and c all sorts of invariants can be constructed also note that permuting the roots leaves the above expressions invariants. Thus there is a close connection between group theory and something as mundane as solving polynomial equations.

My experience was a lesson in being overoptimistic, just as a bad chess player (which I am) will spend hours developing an attack, only to miss the obvious fact that his queen is under threat and the priority should be to move the queen out of danger. Just as my priority should be to finish my two TMA's. I take heart that my mate Neil who is a good chess player and is revising for a computing exam had a similar distraction

http://neilanderson.freehostia.com/thoughts/degree/m257/

Anyway along with all the other distractions I've got an interest in Galois theory and have ordered this book by Cox which Nilo recommends

http://www.amazon.co.uk/Galois-Theory-Pure-Applied-Mathematics/dp/0471434191/ref=sr_1_1?ie=UTF8&s=books&qid=1307564746&sr=8-1

It seems accessible in that it starts from the motivation of the solution of quadratic, cubic and quartic equations

Also a good set of notes available on the net amongst many is here

http://www.warwick.ac.uk/~masda/MA3D5/Galois.pdf

So yet another distraction but one thats fun hopefully I've got it out of my system and can concentrate on the TMA's over the next two weeks.

Subscribe to:
Posts (Atom)