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. 

The Rosenbluth Scattering Cross Section

Hi everyone and a Happy new year

I am pleased to report that I have completed my work on the calculation of the Rosenbluth Cross section. This is a calculation of the elastic scattering cross section for an electron off a proton. It is one of the standard calculations left as an exercise for the student in many books on quantum field theory. However the actual calculation is far from straightforward and difficult to find so I have written it out in full

I have enclosed a link to the document here (hopefully there are not too many typos)

https://drive.google.com/file/d/1qA2keWm01SSdoU84jQx-HqeOZroaQYF3/view?usp=sharing

I have also enclosed a brief account of the first measurements of the cross section by Hofstadter at a proto type of what would eventually become the Standford Linear Accelerator. (SLAC). I was able to reproduce one of the early graphs from this work.

One slightly unusual feature of this work is that it does not require the use of quantum field theory and a simple derivation of the Feynman rules based on a relativistic generalisation of Fermi's Golden rule is all that is needed. The first half of the notes describes how the Feynman rules for QED can be derived. Once one has the Feynman rules then one can go onto to derive expressions for the scattering cross section or particle decay rate of whatever process you are interested in, in terms of the modulus squared of the matrix element. This stage is the same as in other allegedly more rigorous methods based on quantum field theory. It is an interesting question as to whether or not quantum field theory is as necessary to particle physics as it is sometimes made out.

As a final note the Rosenbluth cross section is expressed in terms of the electromagnetic and magnetic form factors of the proton which can only be determined from the experimental data. Once one has the form factors one can go on to derive expressions for the mean radius of the proton. This work after suitable extensions was carried out was shown to be compatible with other estimates based on the Lamb shift calculated from electron hydrogen spectroscopy. One would have thought that that would have been the end of the matter but in 2010 a series of experiments based on muon spectroscopy has come up with a radius of the proton that is about 4% smaller. Thus the proton has shrunk in size. This is shown below by this whimsical cartoon from the New York times showing the proton worried about it's shrinking waistline

There is a flurry of activity currently taking place to try and resolve the discrepancy, One collaboration (MUSE) is going to look at the scattering cross section from muon proton scattering to see if the discrepancy can be resolved. Details of which are given here 

http://www.physics.rutgers.edu/~rgilman/elasticmup/

It is of interest that calculations performed almost 70 years ago are still of relevance today. 

When I next revisit this work (don't hold your breath 😏) I shall look at the inelastic scattering of electons from protons. This led to  the somewhat surprising fact that at high energies the proton could be considered as made up of point like constituents which at the time were called partons. These were late identified as the quarks of Gell Mann (and others) and led to our current understanding of the strong interaction namely QCD. Hopefully by this time next year I will have finished describing this work. The current work provides the requisite background and I hope eventually to provide my followers with a working knowledge of the standard model of particle physics with nothing more than a smattering knowledge of relativistic quantum physics and Fermi's golden rule instead of the more arcane and somewhat mysterious knowledge of quantum field theory in it's many incarnations be it canonical formalism or the path Integral formalism.