Sunday, 29 January 2012

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. 
   

Friday, 27 January 2012

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


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


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.

Tuesday, 10 January 2012

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.

Monday, 9 January 2012

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.
 

Thursday, 29 December 2011

Hume on Scepticism Part II

Greetings to my fellow readers and bloggers. I hope you all had a good Christmas, I managed to watch Lohengrin an intriguing mix of styles between the Ring cycle and Parsifal. Indeed Lohengrin is in fact Parsifal's son. Which may raise a few eyebrows as Parsifal in the opera devoted to him has commited himself to celibacy. Still consistency was never a strong point of mythology. Those who don't know Lohengrin should get to know it.  Act II is the best, the music looking forward to the Ring Cycle. Anyway this is not the main point of the post, which is to finish off my examination of Hume's scepticism.

Recall that Hume is arguing for a mitigated scepticism which recognises the limits of Human Knowledge, we saw in the first post on this that one of the benefits of Hume's scepticism is that it would avoid dogmatism. The other benefit Hume sees is that it will act as a brake on speculative philosophy by acknowledging that there are limits to what philosophy or any branch of human knowledge can achieve and we should take care to recognise speculation for what it is and concentrate on problems which we can actually solve.

It seems to me that Hume's lesson needs to be taken on board in a number of areas not least in some developments of physics. There is an interesting tension, on the one hand given Heisenberg's uncertainty principle  and Godel's theorem. we seem to be acknowledging that even our most rigorous methods of obtaining knowledge about the world have severe limitations, something which Hume would be delighted to
know.

On the other hand some aspects of modern physics do seem to border on the realms of speculation to which
there can be no testable consequences. I refer of course to the speculation that seems rife in quantum physics such as  Many Worlds theories and also that attempts to quantise gravity seem to be inherently untestable.

Readers of earlier posts will know that I tend to view claims that the wavefunction in quantum physics as a real field with a healthy degree of scepticism. The simplest explanation of it's role is as a mathematical device for generating the probabilites of quantum effects to occur. Thus when a measurement occurs the wave function does not physically collapse, all that happens is that one of the possibilities is realised. This pragmatic and slightly unromantic view of the situation cuts out all the speculation associated with what for want of a better word is the mystical or science fiction interpretation of quantum mechanics. The Many worlders would agree with me that the wavefunction does not physically collapse, but they would argue (if I understand it correctly) that because the wavefunction is a real field every time a measurement is made the wavefunction splits creating another universe, but of course we have no means of communication with that other universe.
(How convenient),   If that is one of the prices one has to pay for seeing the wavefunction as a real field then I'm afraid it's not worth it. By definition such a hypothesis is untestable and therefore cannot be good science as it is 'Not even wrong'.

A similar criticism could be made of the current attempts to quantise gravity. Indeed such a criticism has been made of superstring theory by Peter Woit.


I first read this book about 6 years ago. In it he argues that superstring theory is inherently untestable as the methodology used there does not lead to anything which can be tested in the lab. For example it's not clear that it does contain the standard model of particle physics as a lower limit. In this it is quite different from say quantum physics or relativty which do reduce to Newtonian physics in the limit. Woit rightly in my opinion criticises the amount of money that has been spent on such research.
Woit has devoted a blog which challenges the claims of those who would over exaggerate the importance
of areas of research such as superstrings.


And I recommend it to my readers as an antidote to all the hype about modern physics one consistently gets in the media.

Part of the problem, it would seem is that superstrings and the mystical/science fiction interpretation of quantum mechancis seem to tap into some part of the human pschye for a modern form of mysticism. Well fine, but don't claim that this is real science and certainly don't expect the taxpayer to fund your lifestyle.

Why not admit that superstrings or many worlds theories are speculations, that quantum physics far from being a mystery is in fact a well developed tool for predicting the behaviour of atomic, molecular and nuclear systems with applications to astrophysics, particle physics, chemistry and other fields. There is more than enough awe and wonder to be found in the day to day applications of science to solve currently intractable problems and given that one can go out and test such applications I would have thought more rewarding.     

I'll let Hume have the last word, here he is referring to the typical 'proofs' current in his time, that the Supreme Being is actively causing everything (even my finger movements over the keyboard as I type this post).

"Though the chain of arguments, which conduct to it, were ever so logical, there must arise a strong suspicion, if not absolute assurance, that it has carried us quite beyond the reach of our faculties, when it leads to conclusions so extraordinary, and so remote from common life and experience. We are got into fairy land."

It seems to me that talk of many worlds or us living in a world of 10,11 or 26 dimensions is just as likely to have led us into fairy land as anything else.

    

Saturday, 24 December 2011

A Defence of Liberal Secularism for Xmas

Ok well as promised here is my defence of liberal secularism, call it Chris's Christmas Sermon if you wish. Today is Christmas Eve, and no doubt this evening and tomorrow, there will be many churches and other religious instittuions, denouncing liberal secularism, as it has led to a decline in moral standards as exemplified by the toleration of behaviour of the rioters last Summer. The true meaning of Christmas being obscured by the rampant consumerism of the past months. Moral relativism, leading to a disrespect for authorities and tolerating any form of behaviour, even that which casues harm to other people. There will no doubt be an attack on multiculturalism and the usual guff that councils up and down the land are banning the word Christmas for fear of causing offence to other religions. Finally, the claim, that without a belief in God there can be no basis for morality. We should remember, so the argument goes,  as the Prime Minister has recently stated that the foundations for our morality lie in Christianity and we as a nation should return to it.

In what follows I want to challenge 4  basic premises that form the basis of such beliefs these are

1) A belief in God is necessary for morality.
2) The basis for western morality lies with Christianity.
3) Liberal secularism is the same as moral relativism hence liberal secularists can have no values.
4) Non believers in God must be living impoverished lives as they are 'spiritually dead'.

In doing so I will draw on an Early dialogue by Plato called the Euthyphro dialogue and Mill's Essay on Liberty.

To take the first premise, that a belief in God is necessary for morality. This was shown to be fallacious by Plato in an Early dialogue which he attributes to Socrates on his way to his trial for impeiety. He gets talking to a young man Eutyhphro (hence the name of the dialogue) who is also on his way to prosecute his father for murder. This leads to a general discussion on the nature of piety which for the Greeks was considered synomonous with morality or Ethics. Socrates poses the dillema, Is morality or ethics, good initself  or is it  only good because God (or the Gods) have ordained them. If the former, then obviously one does not need God, or a belief in God, as a  foundation for  our ethical principles. If the latter, then what guarantee do we have that the commandments of God are in fact good. To say that God, defines goodness is to beg the question, especially as in some of the texts God is seen ordering the Israelites to smite other nations, or more pertinately in todays climate, it is the duty of Islamists to wage violent Jihad on the decadent West. Given the tendency of religious institutions to want to impose a theocracy on non believers, then they must presumably think that Ethical principles are good in themselves, but then you don't need a belief in God to justify them. It is quite remarkable that a dialogue written 2,400 years ago asks such a pertinent question. There is no justification for the view that ethical or moral principles require a belief in God.

I come to the second argument that our ethical views are based in Christianity. I would argue that 400 years before Christianity, the foundations of our ethical perspective was given by the Greek philosophers especially Plato and Aristotle. The difference between the Greeks and Faith based views of ethics is that whereas for the most part faith based ethics says simply these principles were ordained by God and are enshrined in the texts. (As if one could simply write down a system of ethics, valid for eternity 2000 years ago and simply apply it to another age, without regard to context or the changes in society) Philosophers would say that you must provide a good argument for your case.  Instead of basing our ethics on religious texts I would suggest that a study of the main philosophical ethical texts is a far better way to learn ethical principles. The benefits will be  learnng how to argue for your case, rather just 'It's in the Bible, Koran or whatever religious text therefore it must be correct'. Also a certain modesty (cf Yesterdays post) and an ability to revise ones position, when confronted with something that doesn't quite fit the usual ethical percepts. To give a trivial example, whilst we can probably agree, that in general lying is a bad thing, it would be a brave man  who when asked 'Does my bum look big in this?' by his girlfriend said yes. Obviously there are more serious cases does one want to reveal where ones friends are to an enemy who wants to kill them for example.  Anyway the point is that it was the Greeks who invented philosophical ethics 400 years before Christianity was even heard of and it is those principles which forms the basis of our society not Christianity.

This brings me to my third point the claim that  liberal secularists have no values. Anyone who claims this has obviously not read Mill's essay on liberty. There Mill introduces a very simple principle namely that any lifestyle belief or cultural practice is fine provided it does not harm any other person. This is a two edged sword, on the one hand diversity in life is to be encouraged and welcomed. On the other hand if cultural practices do cause harm to other people then those practices are to be condemned. Thus for example liberal secularists who follow Mill in the harm principle would have no problem in, attacking for example genital mutilation and it is not cultural imperialism to do so.  A further pertinent point that Mill makes is that offence is no harm. Religious people who are often vociferous in banning plays which criticise religion or make fun of it often claim that they are offended by this. Well fair enough you have a right to express your view but it is not your right to close down plays by threat of violence, neither is it your right to seek to kill those who write books criticising your religion. You want to attack liberals for having no morals or values, well then learn to accept criticism yourself. 

Far from liberals having no values I would argue that liberals possess a set of core values, firstly the harm principle as enunciated above, but furthermore we believe that a diversity of lifestyles is good for society we also believe that discrimination against people on the basis of ethnicity, gender and sexual orientation is also a bad thing. If your religious institution or way of life maintains discrimination against women or gay people then it's about time you changed your position. Furthermore if you are a public servant and you discriminate against ethnic minorities, women or gay people then you deserve to lose your job. The nub of the battle between liberals and the Pope for example isn't that liberals don't have ethical values whereas instituitionalised religion does. It is that there is a real clash of values and in the case of institutions which refuse to let women or gay people rise to the top based on their merits such behaviour is essentially racist, misoygnistic and bigoted. The more you denounce certain ways of life which you don't approve of, then the more you will be justly criticised and the more you just show your institutionalised bigotry. I leave it to you to ponder, whether or not such attitudes are consistent with the teachings of a Galilean peasant, who says you should love your neighbour as yourself.  

To my final point the alleged lack of fullfilment of non religious people as they are spiritually dead. I would argue that this is clearly not the case. The secular world has much to offer in terms of fulfillment, I prefer to use the word aesthetic rather than spiritual. Here it is abundantly obvious that the Arts have much to offer here. I would say from personal experience that for exampel the symphonies of Mahler, the operas of Wagner, the novels of Tolstoy, Dostoevsky and the paintings of Picasso have far more to offer than a few hymns or carols and the  continual regurgitation of the same old text week in week out. For other people it may simply be enjoying a good walk in the country or just cultivating friendships and family. Also one must not forget the joy that the study of the sciences and mathematics can give.

So, I long for the day when this world tolerates other lifestyles, when people can rise to the peak of their abilities, without being discriminated against, because of their gender, ethnicity or sexual orientation. I long for the day when people don't harm anyone else. I also long for the day when religious institutions live up to their founders principles instead of indulging in institutional bigotry as most of them seem to, 

Have a good Christmas Winter Solstice, excuse to get drunk whatever way you want to call it

Best wishes Chris