Cambridge MAFFS

How does one make sure one's mathematical skills are up to date and relevant?  One way would be to aim to do past papers from relevant mathematics courses once a year as they come out.  Such a set of papers and resources for study of these papers is provided by the Cambridge University mathematics department. This post collects together a few resources for self study of the  Cambridge Mathematical Methods courses for Scientists (NST). I am continually amazed at the generosity of the Cambridge Maths site in offering to the public these resources

The past papers  are collated here. 

https://www.maths.cam.ac.uk/undergradnst/pastpapers

 The syllabus for both years is given here 

https://www.maths.cam.ac.uk/undergradnst/files/misc/NSTschedules.pdf

There are two courses one for first year students and one for second year science students who want to study mathematics in more depth. 

The recommended text book is Riley Benson and Hobson 

https://www.amazon.co.uk/Mathematical-Methods-Physics-Engineering-Comprehensive/dp/0521679710

Which has a solution manual which covers the odd problems in great detail so ideal for self study 

https://www.amazon.co.uk/Student-Solution-Mathematical-Methods-Engineering/dp/0521679737/ref=pd_lpo_14_t_0/262-4198048-8655769?_encoding=UTF8&pd_rd_i=0521679737&pd_rd_r=c25e3e37-5d85-425d-ae5f-e2b662f55a36&pd_rd_w=5uBAB&pd_rd_wg=upWCR&pf_rd_p=da0677f5-a47b-4543-8b54-10be576b8f26&pf_rd_r=2N0058F97ARMT25NE1M6&psc=1&refRID=2N0058F97ARMT25NE1M6

Investing in these two books would cover most mathematical methods that a physicist is likely to need in their careers. However it would probably take a good few years to do all the examples in Riley Benson and Hobson so it is more for reference than anything. 

Buried in the vaults of the Cambridge Mathematics department are the following resources 

Answers to first year past papers can be found from this website 

https://www.robinson.cam.ac.uk/iar1/teaching/index.html#nst1a_maths

Unfortunately I have not been able to find any lecture notes or the example sheets. However if you have Riley Bence and Hobson you should be able to find the techniques that you don't know already covered there. However having the answers to the exam questions is obviously helpful 

Year II is much better served the lectures occur in three parts 

The first part is covered by a set of lectures from  Dr Simon Cowley who from his web site seems quite an engaging person

http://www.damtp.cam.ac.uk//user/sjc1/index.html 

Lecture notes for the various courses he has taught over the years can be downloaded from here 

http://www.damtp.cam.ac.uk//user/sjc1/teaching/

The ones of main interest are the notes for the first part of the Cambridge NST Part IB 

http://www.damtp.cam.ac.uk//user/sjc1/teaching/

This covers things such as vector calculus, matrices partial differential equations Fourier Transforms and solution of differential equations,  by series including a survey of Legendre Polynomials, Green's Function and a bit on real analysis (for physicists not mathematicians 😅). This is more or less equivalent to MS224 and the mathematical methods part of MST326 but also includes Green's Functions and goes into more detail about the solution of differential equations by Series solution especially the Frobenius Method. 

Notes for the second lecture course in Part IB are given by Dr Hunt 

https://www.damtp.cam.ac.uk/user/reh10/lectures/

Scroll down to the Mathematical Methods part 

He also gives the examples sheets that were part of the course and hints for their solution 

This second part covers variational principles, solutions to Poisson's equation and a summary of the basic techqniques of Complex Analysis again taught as Physicists would use them rather than a Pure Mathematician. There is a bit of overlap with the OU course MS327 but neither MST326 or MS327 cover inhomogeneous partial differential equations in any depth. 

Finally the third part covers group theory and representation theory for physicists including an analysis of multi-mode oscillations. MS327 covers multi-mode oscillations but do not relate it to group theory. 

https://www.damtp.cam.ac.uk/user/examples/N23L.pdf

The examples sheets for the whole year are also given scroll down to the bottom 

Ok  I aim to have generated a complete set of solutions to the 2019 papers by the end of the year. This is a nice way IMHO to extend ones mathematical ability and keep it topped up. Overall the topics covered here in terms of mathematical knowledge overlap with MS224, MS327 and MST326 but also extend it to cover topics such as

1) Solution by Series of Differential equations including the Frobenius Method

2) Green's Functions and their application to Partial Differential equations 

3) Complex Analysis (although to be fair if you do M337 you will do far more than is given here) 

4) More detail on Special Functions including Legendre Polynomials 

5) Group Theory 

Also I suspect the questions will be harder than the OU exam questions 

Anyway I hope this post will encourage you to extend and revise your mathematics skills. 

De Moivre's Theorem and Trig Identities

 Hi Guys and Gals Hope you have a great New Years eve party if you are having one. Hopefully 2021 will see the end of Covid and life can get back to normal. IMHO the restrictions have been unnecessarily punitive and the shut down of the pubs, theatres and concert halls has been totally unnecessary anyway I'm not here to gripe about the restrictions laid down by Mrs Oliver Cromwell (aka Nicola Sturgeon) up in Scotland during the lockdown. I have been amusing myself to see if I can do some Cambridge Maths past papers set for their Natural science students and available here 

https://www.maths.cam.ac.uk/undergradnst/pastpapers/2019

In doing the 2019 1st paper I came across this question which at first sight looks nigh impossible the question was 

Use De Moivre's theorem to write out the expression 

$$ 8 cos 6\theta + 15sin 4\theta sin 2\theta $$ in terms of $ cos \theta$ and $ sin \theta $ 

However I discovered for myself a simple method to write out the expansion of $cos (n\theta)$ or $sin (n\theta) $ Almost instantly from Pascal's triangle 

First let us recall De Moivre's theorem it states that 

$$exp(in\theta) = [exp(i\theta)]^n $$ using 

$$ exp(in\theta) = cos(n\theta) + i sin(n\theta) $$ 

it is seen that 

$$cos(n\theta) = Re [cos(\theta)  + i sin(\theta)]^n $$ and 

$$sin(n\theta) = Im [cos(\theta) + i sin(\theta)]^n $$ 

Ok that looks really messy how do I remember the coefficients in a binomial expansion and how do I pick out the Real and Imaginary parts of each coefficient and all in about 5-10 mins which is all I have time to answer the question

Well a short cut is as follows Recall that the binomial coefficients in a binomial expansion are given by the rows of Pascal's triangle . Namely 

                                n = 0                                 1

                                n  = 1                              1     1

                                n  = 2                         1     2    1

                                n  =  3                     1     3     3    1

                                n  =  4                   1     4     6     4    1

                                n =  5                1     5    10   10   5     1

                               n   = 6            1    6    15     20    15  6      1

where each number in a given row is given by adding the two numbers above it and the coefficient corresponds to the following expansion 

$$ (a + b)^n = C_1  a^n + C_2  a^{n-1} b + C_3 a^{n-2} b^2 + ...........C_{n-1}.a b^{n-1} + C_n b^n$$

The coefficients are just the corresponding element in the nth row of Pascal's triangle. So for example 

$$(a + b)^5 = a^5 + 5 a^4 b + 10 a^3 b^2 + 10 a^2 b^3 + 5 a^1 b^4 + b^5 $$

Now to apply this to De Moivre's theorem we would have 

$$[(cos(\theta) + i sin(\theta)]^n = C_1 cos^n (\theta) + C_2 cos ^{n-1}(\theta) (i sin(\theta)) + $$

                                                   $$   C_3 cos^{n-2}(\theta) (i sin(\theta))^2 + ..... $$

So the odd coefficients in the expansion will correspond to the real part and the even coefficients will correspond to the imaginary part of the expansion, Furthermore as $i^2 = -1$ and $i^4 = +1$ the coefficients will alternate in sign 

So this gives us a nice quick method of writing out $cos(n\theta)$ and $sin(n\theta)$ in terms of $cos(\theta)$ and $sin(\theta)$

For a given n work out the corresponding row in Pascal's triangle then the expansions of $cos(n \theta)$ and $sin(n \theta)$ are as follows 

$$ cos(n\theta) = C_1 cos^n(\theta) - C_3 cos^{n-2}(\theta) sin^2(\theta) + $$

$$ C_5 cos^{n-4}(\theta) sin^4(\theta)+ ...$$

$$ sin(n\theta) = C_2 cos^{n-1}(\theta) sin(\theta) - C_4 cos^{n-3}sin^3(\theta)+ .... $$ 

Hence given a horrendous expression involving $cos(n\theta)$ and $sin(n\theta)$ the expansions of these functions in individual powers of $cos (\theta)$ or $sin(\theta)$ can be written down by sight 

So for our problem we have 

$$sin(2\theta) = 2 sin(\theta) cos(\theta) $$ as the second row of pascals triangle is 1 2 1 and the even coefficient is 2 

$$sin(4\theta) = 4 cos^3(\theta) sin(\theta) - 4 cos(\theta) sin^3(\theta) $$ as the fourth row of Pascal's triangle is 1 4 6 4 1 and the even terms are 4 and 4 

Finally 

$$cos(6\theta) = cos^6(\theta) - 15 cos^4(\theta) sin^2(\theta) + 15 cos^2(\theta) sin^4(\theta) - sin^6(\theta) $$ 

as the 6th row of Pascal's triangle is 1 6 15 20 15  6 1 and the odd numbered coefficients are 1 15 15 and 1 

So 

$$8 cos(6\theta) =8[ cos^6(\theta) - 15 cos^4(\theta) sin^2(\theta) + 15 cos^2(\theta) sin^4(\theta) -$$ $$sin^6(\theta) ]$$ 

and 

$$15sin4\theta sin 2\theta=15 [4 cos^3(\theta)sin(\theta)-4cos(\theta)sin^3(\theta) ] $$

$$ \times 2 sin(\theta)cos(\theta)$$

This simplifies to 

$$ 15 \times  8 [cos^4(\theta)sin^2(\theta) - cos^2(\theta) sin^4(\theta)] $$ 

and it is seen that the mixed terms in $cos(\theta)$ and $sin(\theta)$ cancel leaving 

$$ 8 cos 6\theta + 15sin 4\theta sin 2\theta = 8(cos^6(\theta) -sin^6(\theta)) $$

So next time you are faced with a seemingly horrendous expression involving $sin(n\theta)$ or $cos(n\theta)$ just remember Pascal's triangle and De Moivre's theorem.

Hope every one reading this has a great new year I will write another post over the weekend summarising my plans for 2021 and hopefully achieving them :) 

Foundations of Quantum Mechanics Norsen a review

Hi Sorry I haven't posted for a while (over 2 years) I have not been as focused as I should have been so apologies. I hope you are all surviving the COVID lock down about which the least said the better and the sooner it is over the best for all of us.

I found myself reviewing a book on the Foundations of quantum mechanics for Amazon which is fine as far as it goes but I do think it is fundamentally flawed

The book is

https://link.springer.com/book/10.1007/978-3-319-65867-4

and you can download it for free for now via the springer website which is really helpful anyway here are my thoughts on the book

On the whole this is a pretty good survey of the foundations of quantum mechanics and the alleged problems associated with it's Interpretation and the ways out of it. However it is arguable that all one needs to interpret quantum mechanics successsfully are the Born Rule and the link between the eigenvalues of the Schrodinger equation and the energy levels of the system under consideration

If the Born rule is taken seriously, then it implies that the solution to Schrodinger's equation, the so called wave function is in effect the square root of a probability density function or probability amplitude. However unlike most square roots of a probability density functions, in some cases this may be a complex function and one can only get a measurable quantity by taking the modulus squared of the solution to Schrodinger's equation. If there is more than one possibility then to get the overall probability for a given situation one must add theprobability amplitudes for the two or more possibilities together before taking the modulus squared of the wavefunction to get the overall probability density function for the situation. As these can be complex functions then there is interference between the two or more possibilities and this interference is seen in many quantum systems. All of this is explained (or described) in Feynman's lectures on quantum mechanics, which surprisingly the author does not mention.

The author does a good job in describing the ignorance interpretation of Born, Einstein and Schrodinger which essentially claims that the wave function is not a physical entity but more a reflection of possible outcomes on measurement. He also goes onto show problems with interpreting the wave function as something physical. However for reasons which he does not really clarify goes onto reject the ignorance interpretation. This is despite the fact that it avoids all the problems with the collapse of the wavefunction, if taken literally as a physical phenomenon and one does not need to invoke spooky action at a distance to explain the violation of the Bell inequalities. Indeed the latter half of the book is the most disappointing, having shown the problems with interpreting the wave function as a physical entity he then embraces a literal interpretation of Bohm's version including the rather tendentious claim that in situations such as the Aspect experiment, really does involve a causal influence from one part of the system to another at speeds faster than the speed of light.

This is not really plausible despite the claims of many people in the popular literature. From the ignorance interpretation, the explanation is really quite simple. Prior to making a measurement observer A will have a 50% of measuring a photons or electrons spin in one direction but does not know which, so his wave function is constructed in such a way to reflect this 50% chance. However after his measurement he will know that his spin is in one direction, and he instanteanously knows that another observer will measure the spin of the other particle to be in the opposite direction. But until B makes his or her measurement B will not know the outcome of his measurement/ There is no signalling, the other particles spin was set at the moment of emission due to the law of conservation of angular momentum and to deny otherwise is to make a mockery of the conservation of angular momentum which plays such a major part in understanding particle physics today. As there are no hidden variables A or B will never know on a given measurement what he or she will measure, only that if a sufficient number of measurements are made 50% of the time A will measure the spins to be up and 50% of the time they will down and vice versa for B, It does puzzle me greatly that this simple solution is claimed to be ruled out by Bell's analysis by many books including this one. But does this interpretation of the wave function for situations actually violate the quantum mechanical predictions. of course not as it is using the wave function that makes the correct prediction.

 Looking at the assumptions behind the proof of Bell's inquality as many people do, doesn't shed any light on why the quantum mechanical explanation works as we know Bell is analysing the phenomenon in a classical manner. Despite the overblown rhetoric by Bell and others, all he managed to show was that the joint probability distribution for the particles spins could not be separated into a product of the probability distributions of the individual particles spins. But it is well known in statistics that if a probability distribution for a multivariable system cannot be factorised into a product of the individual variables probability distribution functions then the variables are not independent. But we know this for the particles spin components as the total spin must obey the conservation of angular momentum. If there were signalling as Bell claims why doesn't any signalling (a Green's function say) term appear in the actual maths which predicts the measured correlation. 

Bell himself at the end of his 1964 paper claimed to have shown that it is only if one wishes to go beyond a purely statistical intepretation of quantum mechanics then superluminal signalling must be involved in situations such as the Aspect experiment. This may or may not be true, however if one wishes to avoid invoking superluminal signalling then one is stuck with a statistical interpretation to which there has been no experimental refutation

For this reason I can only give the book 3 stars despite the fact that it is extremely well written and probably is one of the most accessible accounts of the various positions informing the current debate today

I am currently working on completion of my calculation of neutrino proton scattering which was the final piece of the jigsaw in establishing the quark model and I will hopefully publish my results by the end of June. Even though no one has yet liberated a quark the results from these experriments carried out in the early 1970's gave evidence that quarks had fractional charge which is really quite amazing. 

Also I have some thoughts on the two slit experiment which I will publish soon. One of the things that most accounts miss is that the characteristic interference pattern is in the far field, However in the near field there is no interference, thus the idea that a photon or electron passes through both slits simultaneously is clearly incorrect. 

In the mean time keep sane 

Deep Inelastic Scattering Part 1

Hi everyone and Happy Easter

I am pleased to report that I have finished the first part of my review of Deep Inelastic scattering which describes the early experiments at Stanford which led to first real evidence that quarks existed and were not just bookeeping devices to classify elementary particles with.

During the late 1960's experiments at the newly built Stanford Linear Accelerator showed that at high energies the results of measurement of the inelastic scattering of electrons off protons could be interpreted as the electron scattering off point like spin 1/2 particles which Feynman called partons.

Further work showed that these partons could be identified with the quarks of Gell Mann and Zweig also that whilst the spin half partons could account for half of the protons momentum distribution there was evidence of other neutral partons these later came to be identified with the gluons of quantum chromodynamics (QCD) the current theory of the strong interaction;

The notes for those interested are given below as usual I have tried to give full derivations of the main equations. I also enclose my own analysis of the data from the SLAC experiments

https://drive.google.com/open?id=13AWUTddACpf5c0lewnMj_I59_C1ebUX0

However despite the success of the parton model, it wasn't until scattering experiments involving neutrinos took place that the full identification of the partons with the quarks was able to be made. This will be the topic of the next set of notes in this series which hopefully will be completed by Mid Summer

Stephen Hawking RIP

Most of the readers of this blog will be saddened by the death of Stephen Hawking earlier last week a good obituary by his first colleague Roger Penrose can be found here

https://www.theguardian.com/science/2018/mar/14/stephen-hawking-obituary

Also his Adams Prize essay which deals with his earlier (and in my mind most significant work ) where he along with Penrose established the fact that Classical General relativity had to have inevitable singularities associated with it. Penrose had established this for black holes and stellar collapse, Hawking along with a coworker Ellis was able to establish this for cosmology.


An introduction to the essay is given here

https://www.epj.org/images/stories/news/2014/10.1140--epjh--e2014-50014-x.pdf

And the actual essay itself is here

https://www.epj.org/images/stories/news/2014/10.1140--epjh--e2014-50013-6.pdf

The essay was later extended into a book

https://www.amazon.co.uk/Structure-Space-Time-Cambridge-Monographs-Mathematical/dp/0521099064

Whilst the work is undoubtedly important most physicists and cosmologists  will probably find this approach totally alien to their background as it assumes a knowledge of topology and coordinate free differential geometry. I fall into this category myself. and whilst I bought the book have never really understood it. It along with Von Neumann's book on the Foundations of quantum mechanics must deserve the title of one of the most incomprehensible books on physics ever written.


It unfortunately does not seem that easy to get the requisite background either. As the pure mathematicians way of thinking proof lemma proof does not lend itself easily to those who want to get stuck in and calculate say the Riemann tensor for a given geometry and solve the resulting equations.

At least the essay does not seem as intimidating as the book, but it would take a long time to get the necessary background to understand it. I don't as yet have the pre-requisites to understand the pre-requistes :). Who knows one day I might get this background but it's a long way off and I want to concentrate on more accessible calculations.



Anyway despite not being in a position to understand Hawking's or even Penrose's work one cannot but feel sad that Hawking is no longer with us. I would add at the risk of being churlish, the claim in the media that Hawking is the greatest physicist since Einstein is typical of unjustified hype. Feynman for example is surely greater. One could make a case that Hawking and Penrose and all the people such as Ed Witten working on superstrings, whilst udoubtedly great mathematicians are not physicists in that they have not actually established a new fact about nature. In that respect the developers of the Standard model of particle physics such as Weinberg, Salam, t'Hooft and Veltmann have achieved more from a physics point of view, than all the mathematicians working on quantum gravity will ever do. Of course given the nature of quantum gravity, it is highly unlikely that any empirical fact will emerge from it that one can measure. It is highly unlikely that quantum gravitational corrections to say the decay rate of the neutron or the magnetic moment will ever lead to anything measurable. Thus by choosing to concentrate on a subject in which it is highly unlikely that any empirical evidence will arise to verify the claims, then it must be said that Hawking was not really a physicist.

Having said that of course there is still room for understanding the underlying mathematical structure of a subject and it would appear that Hawking did this brilliantly with General Relativity it is such a shame however that his ideas will probably only be understood by a select few individuals. Anyway with the death of Hawking we have lost one of the major players in the field and the world is a poorer place without him. At least he will now know how the universe works 😃it just a pity he wont be able to tell us.

Added 3rd April 2018 

There is in fact an overview of Hawking and Penroses work on singularities which was a set of lectures that the two gave in the 1990's. Hawkings lectures are contained here

https://arxiv.org/abs/hep-th/9409195


And the full set is here

https://press.princeton.edu/titles/9165.html

This is certainly more informative than Hawkings popular books such his notorious 'Brief History of Time' or his later one 'The Grand Design'

Anyway reading the lectures and the book should give the average physicist who knows a smattering of General relativity a bit of an idea of the singularities associated with General relativity but it still would not be a substitute for the Adams Prize Essay or the actual book itself.

3 bloggers

Just a short post to highlight 3 blogs which are useful as a guide to what is going on in the world of physics today.

The first is run by Peter Woit

http://www.math.columbia.edu/~woit/wordpress/

Peter Woit first came to attention after writing his book Not Even Wrong debunking the pretensions of superstring theory.

https://www.amazon.co.uk/Not-Even-Wrong-Continuing-Challenge/dp/0224076051

His blog amongst other things continues to fight the battle. Most recently attacking the move by some superstring theorists to remove the concept of falsifiability as a criteria for assessing physical theories especially by people such as Sean Carroll.

http://www.math.columbia.edu/~woit/wordpress/?p=9938


This blog has been going for a while now

A more recent blog which shares the same aims is run by Sabine Hossenfelder

http://backreaction.blogspot.co.uk/

One of her key beliefs is that we may have to live with the apparent ugliness of the Standard model as it seems that at present we have no clues as to what lies beyond it


(Yippee I can focus on the standard model and cosmology) and ignore supersymmetry, grand unified theories and of course superstrings 😊 means  I might actually understand physics before I die)

She has a book coming out in the early summer

https://www.amazon.co.uk/Lost-Math-Beauty-Physics-Astray/dp/0465094252/ref=sr_1_1?s=books&ie=UTF8&qid=1518383480&sr=1-1&refinements=p_27%3ASabine+Hossenfelder

Which I intend to buy

As an antidote (and quite amusing if you can get beyond the way in which he attacks his critics or anyone who disagrees with him)  is the blog by Lubos Motl

https://motls.blogspot.co.uk/

Unlike Peter Woit or Sabine Hossenfelder he defends quite vigorously those who work in superstring theory and regards the above two blogs of being antiscientific and science haters. An opinion which I do not shate. However once one gets beyond the name calling and ad hominen attacks he does put the case for research in superstring theory quite eloquently and so is worth reading an eloquent defence is here

https://motls.blogspot.co.uk/2004/10/beauty-of-string-theory.html

However the most interesting posts are where Lubos defends the orthodox interpretations of quantum mechanics and explains that all the foundations were developed by the founders especially by the Born Interpretation.

Here is Lubos debunking the idea that the violations of the Bell Inequalities involve superluminal communication

https://motls.blogspot.co.uk/2017/09/why-vanishing-commutators-imply-theres.html

and there are plenty more where that came from 😊

So three blogs to keep in touch with developments in physics and I do find Lubos's attacks on his critics quite amusing

Calculations for 2018 and beyond

This may be a bit ambitious but I thought I would outline the key calculations that I would like to do in both Cosmology and particle physics over the next few years. Now that I do not have the distraction of the Open University to deal with I can hopefully concentrate on these calculations (We'll see)

I have grouped them by year and topic and I aim to do at least 4 calculations a year

2018 Particle Physics

In the calculations that follow I shall take a fairly intuitive approach to the derivation of the Feynman rules and avoid as far as possible any attempt to justify the calculations rigorously from quantum field theory. The aim is to understand the actual calculations, for that purpose all that is needed is relativistic particle physics and Fermi's golden rule. 

1) Deep Inelastic scattering part 1 (By end March )

    This concentrates on the early experiments at Stanford carried out in the late 1960's which concentrated on the inelastic scattering of electrons from protons. These experiments showed that the proton could be considered as made up of point like constituents of spin 1/2 initially called partons but conjectured to be the quarks of Gell Mann also that there were other non charge like  constitutents present which were later identified with the carriers of the Strong Interaction in a manner similar to that of photons in the electromagnetic interaction. These are called gluons.

2) Deep Inelastic scattering part 2 (By end June )

The development of the parton model and the structure of the proton was further clarified by scattering of neutrino's off the proton, These experiments were able to distinguish between quarks and anti-quarks and gave evidence that the partons had fractional charge thus strengthing the identification of the partons with the static quark model of Gell Mann and also thevgluons. A brief overview of the weak interaction will also be given.

3) The Lagrangian of the Standard model (End of 2018)

Taken together 1 and 2 give evidence for the development of our modern theory of the strong Interaction namely quantum chromodynamics, Also the fact that the weak interaction involves interactions between quarks and leptons. Concurrently with the work outlined above the idea that electromagnetism, the weak interaction and the strong interaction could be see as a gauge theory became prominent. However in order to correctly account for the masses of the carriers of the weak interaction the Higgs mechanism had to be invoked. All this will be outlined also it will be pointed out that when it comes to quantising the theory, the beautiful symmetry of Gauge theories is no longer present, mainly becasuse the propagators for the photons and the gluons are ill defined classically, However it is possible to correctly account for the quantisation rules by invoking an extended symmetry called BRS symmetry (Which I have mentioned before 

https://www.blogger.com/blogger.g?blogID=7897235423812277683#editor/target=post;postID=8991275106266049243;onPublishedMenu=template;onClosedMenu=template;postNum=0;src=postname )

This involves the introduction of ghost particles Normally in most quantum field theory books these are introduced in a highly convoluted manner using path Integrals when by imposing the BRS symmetries right from the start it is possible to obtain the correct quantisation procedure right from the start. Amazing (or at least I think so 😂). It will be shown in a fairly informal manner how to write down the appropriate Feynman rules for the Standard Model 

2019 "The year of the loop"

The calculations above have so far only dealt with the first order of perturbation theory the so called classical level. However relativistic particle physics only becomes interesting when one goes beyond the tree level to the so called loop level as the Feynman diagrams involve loops these calculations established two amazing facts 

a) Quantum electrodynamics 

For quantum electrodynamics, the corrections to the anamolous magnetic moment of the proton first carried out by Schwinger, and even more amazing the Lamb shift. It was these two calculations that put quantum electrodynamics calculations on the map. However until Non Abelian theories were developed it was not clear how to do extend quantum field theory to other interactions such as the weak and the strong interaction

b) The Asymptotic Freedom of the Strong Interaction. 

Prior to about 1973 attempts to apply quantum field theory to the strong interaction were stymied as it was not clear that perturbation theory could be applied in a satisfactory manner. However a remarkable property of Non Abelian gauge  theories showed that at high energies the coupling constant decreased thus making it feasible to apply perturbation theory to the strong interaction. This calculation (which is quite long to say the least) will show how this works at the one loop level. 

2020 and beyond Radiative Corrections to particle physics calculations 

I would hope after the basics of loop calculations has been mastered in 2019 to demonstrate how real calculations at the one loop level are performed. For starters I would like to attempt the 2 research projects in Peskin and Schroeder. 

https://www.amazon.co.uk/Introduction-Quantum-Theory-Frontiers-Physics/dp/0201503972

The first project at the end of the first section  calculates the scattering cross section for electron positron annhilation and involves the handling of of  High energy Divergences (Ultra Violet) and Infra Red Divergences which miraculously cancel.

Then the culmination of my calculations in Quantum Field theory will be the last project in Peskin and Schroeder chapter which is a summary of the predictions of the decay rates of the Higg's boson.

Other calculations and experiments leading to say the discovery of the W and Z bosons and the top quark may follow.

If I were to tackle these purely by myself then I would probably get discouraged and give up fortunately there are many sources on the internet where clues as to how the calculations are done can be found. Indeed the first project is described in some detail in Schwartz's book 

https://www.amazon.co.uk/Quantum-Field-Theory-Standard-Model/dp/1107034736

So I won't just be on my own. 

Concurrently with the Quantum Field theory caclulations I want to look at Cosmology in particular the Peebles calculation 

Homogeneous Cosmology (2018 to 2020) 

The aim of these set of calculations is to reproduce the calculations of Peebles who predicted the correct ratio of Hydrogen to Helium abundances in the early universe. This involved a synthesis of ideas from Fermi's theory of the weak interaction, Cosmological solutions to Einstein's Field equations, relativistic statistical physics and nuclear reaction physics. He and other people were able to predict the correct abundances of the light elements and it is my aim over the next two years to finally finish the work I started on this over 10 years ago 

Interlude Numerical solutions to Differential Equations (June 2018) 

In order to reproduce Peebles calculation it is necessary to have a robust numerical code which solves differential equations. The standard workhorse for most scientific work is the 4th order Runge Kutta Method and an investigation and derivation of the method will be given along with some examples showing the dependence of the accuracy of the solution on step size will be given.

Classical Cosmology and the Concordance Model (End 2018) 

This calculation will show how General Relativity can be used to derive the Friedmann equations and I have already completed this part, (and a heart breaking calculation it was too 😢 ) however I have yet to show how the current model of the universe involving  matter, dark matter and dark energy explains the acceleration of the universe and it is possible for a particular combination of matter, dark matter and dark energy it is possible to estimate the age of the universe and other parameters that cosmologists are interested in.  The code developed above will be used to calculate the present age of the universe and also demonstrate the rather surprising conclusion that the Galaxies are actually moving away from us at speeds greater than the speed of light. 

Relativistic Statistical Physics  (2019)

As a prerequisite to calculating the Abundances of the light elements of the early universe it is necessary to derive expressions for the number density, the entropy and the pressure of the universe as a function of time. This involves expressions not usually found in undergraduate text books on statistical physics, but again a judicious internet search will uncover details usually glossed over. The culmination of this stage will be a code which calculates these properties as a function of the temperature of the universe. 

Calculation of the light element abundance in the universe  (2020)

Using estimates of the likely nuclear reactions taking place in the early universe Peebles was able to estimate how the plasma of electrons, neutrinos protons and neutrons were able to combine to give the current ratio of Hydrogen and Helium currently observed. As this contradicted the ideas of people such as Hoyle and Bondi who thought that the remnants of stellar explosions could account for this abundance and Peebles ratio was shown to be correct this put the big bang on the map. Peebles early work just concentrated on a few reaction pathways and it will be the aim of the first part to simply reproduce these calculations. However over the years a sophisticated understanding of about 90 reactions was added to improve the accuracy of the calculation. This work is summarised in two reports by Kawano at Fermi Lab 

https://ntrs.nasa.gov/search.jsp?R=19920015920

He also wrote a code Nuc123.for which I managed to down load a while back which calculates the abundances of the light elements and I hope eventually to update his code to something a bit more modern such as MatLab.

After the work on homogeneous cosmology if I have enough energy left I will look at inhomogenous cosmology with the aim of understanding the anisotropies of the Cosmic microwave background. As ideas about this are still speculative (although some people would say they are not) then I won't be too concerned if I don't complete this work soon. The above calculations should be more than enough to understand how current ideas in particle physics and cosmology relate to the world around us. Fortunately given the internet it is a lot easier for a lone worker outside academia to understand the calculations in some detail and I hope that even though the work is not original putting all this together in some coherent form that is understandable for those who have an undergraduate degree in either physics or maths, will still be useful for those who want to understand contemporary physics.

Needless to say I shall probably not look at music or philosophy in any great depth until this work is completed that can come later.