Sunday, 28 January 2018

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 

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. 

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 

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 

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. 

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