MMath research project

While in my final year of undergraduate study I researched the symmetries behind the Standard Model of Particle Physics under the supervision of Dr Madalena Lemos. The project as a whole stemmed from her proposal to unify symmetries and particles. You might think that this would be a very particle physics-heavy project (and it was!), but underneath the physics lies some very beautiful maths in the form of group theory.

I chose this project because it has been a long curiosity of mine to figure what the universe is made from and how everything interacts with each other at a fundamental level. Once I found out that beneath all of particle physics is nothing but abstract algebra, I was hooked. I am eternally grateful for Dr Andreas Braun who introduced me to this concept during his course on the Geometry of Mathematical Physics.

My dissertation will be published here as a series of lecture notes in due course. A question you might want answering in the meantime, though, is what actually is the Standard Model?

The Standard Model of Particle Physics is often described as the fundamental theory of Nature. It is, to date, the most successful description of particle physics the modern world has to offer. In a physical sense it is a theory that describes all known fundamental particles and their interactions in terms of relativistic quantum field theories, that can be (and have been) verified by experiment. This fact alone makes the Standard Model seem extremely complicated—and that would be a correct judgement; the Lagrangian for this theory contains at least 74 terms when expanded in the way as in my report.

From a mathematical point of view, one can understand the beginnings of the Standard Model as being reliant on the idea that the spacetime we live in obeys certain symmetries. The core aim of the report is to motivate the construction of the Standard Model from the first principles of symmetry, which is known to be the underlying proponent of group theory.

If it were not for these symmetries, the work of science would have to be redone in every new laboratory and in every passing moment. —Stephen Weinberg

Humankind has always been fascinated by what stuff is made up of and the question dates back millennia. In our history there have been many schools of thought as how to describe matter with intuition. One initially successful idea led by Leucippus and Democritus proposed that matter is discrete, is made up of indivisible parts, named atoms. These were later, more aptly named particles, which would be held together by and interact via various forces. This theory carried us through classical physics and allowed for the discoveries of various particles, famoulsy the electron by Thompson, the proton by Rutherford and the neutron by Chadwick, etc.

Another successful approach didn’t come until many centuries after, this being that matter could be continuously distributed; that there exist fields permeating space that carry information such as mass, momentum, energy and spin. It is hard to pinpoint the origins of this idea exactly since many physicists in the early 20th century were working on this idea, but one notable person to highlight would be Louis de Broglie. In 1924, he proposed that electrons (and therefore all ‘particles’ of matter) experienced wave-like behaviour. This paved the way for the earliest theories of quantum mechanics, with Schrödinger in 1926 giving us his equation for how the probability amplitude of a particle’s wavefunction changes over time, and then with Dirac in 1928 who aimed for a relativistic wave equation that was specific to fermions like the electron. It was then thought these wavefunctions propagate through fields, which began the study of quantum field theory.

One other note is that this description of wave-particle duality has its origins in light. Newton had claimed light showed frequency-like properties and therefore would appear in discrete amounts, but this was contradictory to what Huygens and Young believed in a similar time frame, which was that the effects of light mimic that of pebbles thrown in water; ripple-like patterns emerge and their interference increases light intensity. Further experiments (such as ones involving the photoelectric effect and double-slit experiment) verified that light exhibits both properties. Although light may naturally exist as waves, its energy is quantised in packets, as particles, called ‘photons’. In was then proposed in theories developed by Faraday and Maxwell that light itself carries the electromagnetic force, and that interactions between charged particles are due to the exchange of photons. In this sense we have that the forces between particles are themselves particles, which could alternatively be described via fields.

As it turns out, both the particle and field descriptions of matter and forces are correct. But also neither description is correct. In Leonard Susskind’s lecture series on the Standard Model he says ‘‘there are subtleties in the study of quantum physics that satisfy both ideas, but leave some bits of either one unsatisfied’’. It has therefore been desired for a long time for there to exist a natural connection between particles, fields and forces that unifiy all of the theories into one, beautiful theory of everything.