Conference: Higher structures in quantum field theory and string theory

This week I am attending a conference on higher structures in quantum field theory and string theory. It’s an event that I have been excited about since the new year. So far there have been some very nice talks, with interesting ideas and calculations presented.

There is the expression about going down a rabbit hole. In the world of mathematical concepts and fundamental physics, it is easy to get excited about an especially stimulating talk and follow down several rabbit holes. I’m trying to stay especially focused on presentations that are more directly related with my current research, but sometimes the excitment and sense of interest in the discussion topic becomes too strong! This afternoon I am looking forward to Bob Knighton speak on an exact AdS/CFT correspondence and Fiona Seibold talk about integrable deformations of superstrings. The rest of the week should also be a lot of fun.

Meanwhile, in the background I’ve been working on my PhD research (even though I don’t formally start until 1 August) and some double sigma model stuff. I’m hoping to also have my next post on categorical products, duality, and universality finished, which, as it is currently drafted, also talks a bit more about M-theory motivations but I may save this part for a detailed entry of its own.

Teaching myself Quantum Field Theory and String Theory

I’ve been a reader in String Theory (ST) for some time. And since my formal admission to university, I have been revising what I had learned and also expanding deeper into ST, whilst satisfying other formal university course requirements. From my understanding, the university is currently considering ways in which my degree may be fast tracked, so that I can continue unabated with my post-grad/post-doc. studies. This would enable me to more wholly focus on research. Meanwhile, in teaching myself ST, I have picked up both volumes of Polchinski’s “String Theory”, a great and much celebrated work. Although I haven’t been able to dedicate much focused time, with many other things pulling at my attention, in a couple short late night sittings I have powered through the first two chapters and enjoyed every moment. I can also honestly say that string theory is the most difficult and challenging thing I have taken up so far in life. It is in no way easy, but it is immensely enjoyable and I find stringy ideas motivating. (David Tong’s lecture notes are also superb, and I highly recommend that any interested reader also engage with these).

As alluded, I have grown frustrated with more of my time being consumed and taken, especially as I must continue to undergo the formal university process, and I am trying my best to adapt. If I had things my way I would sit and devour Polchinski in a week and then move on to some other books on my list, with topics in scattering amplitudes and elsewhere requiring urgent attention. But such is the nature of the current circumstance.

In that I have also taught myself and am actively extending my knowledge in Quantum Field Theory, I am currently reviewing a few QFT texts alongside my Polchinski readings. Whilst perhaps unorthodox (?), I have found it fruitful to go through both simultaneously. These activities have happily coexisted with my active thinking on matters to do with the string landscape and the swampland conjectures, compactifying all of this with other studies in particle physics, cosmology and mathematics and whatever else that I currently consider worthwhile. (For anyone interested in QFT, there are a number of books that I recommend, some that I am also eager to read through. These include the likes of Schwartz’s “Quantum Field Theory and the Standard Model”, Steven Weinberg’s “The Quantum Theory of Fields”, and Klauber’s “Quantum Field Theory”. Of course there are also some classics that have gone unstated, which anyone interested should read).

Now that the winter break is here, I would like to cross several of these activities off my list in the coming weeks. This is my primary aim, and then I can continue to advance toward some other things. Unfortunately, the winter break also means a break from my sessions with Prof. Padilla, to whom I am grateful for spending time with me and talking with me about all sorts of subjects. We discuss stringy things and also lots of cool cosmology-related theories and concepts. He has also been guiding my self-studies in ST and QFT and in other areas. I am also continuing to sketch some potential PhD theses, encircling different concerns and potentially fruitful research projects. Nima Arkani-Hamed’s works, among others, has been a lasting source of inspiration.

***

On another note, I have been encouraged to write more personal blogs and so I thought I would write my first personal post about some of the studies I am actively pursuing. In time, these sorts of posts will become more refined as I understand more how to write them and generally why one might communicate in this way. With my Asperger’s, such engagements and communication don’t come naturally.

There are many other things I would like to talk about, such as Jackson’s book on electrodynamics. It is a book I have yet to read, although I have already studied the contents. So I am also looking forward to spending time with that book over the break. I would also like to write about it when time permits. It may sound odd, but I first learned Special Relativity through its connection to electrodynamics. Other books I am eager to engage with are Hooft’s Yang-Mills and Nima Arkani-Hamed et al. title on the positive Grassmannian and scattering amplitudes.

Finally, I’ve been encouraged to think about making more maths and physics videos. Making maths and physics lectures or tutorials is a way for me to practice communicating, and I generally find it to be quite fun. At first I wanted my youtube channel to be a complete tour, capturing all of mathematics and physics, so naturally I decided to start from the basics and build. My plan was to also incorporate history – such as the history maths and important mathematical concepts. But that quickly became boring and I have trouble with keeping focused on things not directly related to my active research and study interests, so I sort of lost motivation. But in that I’ve been encouraged to continue making lecture videos and perhaps instead reformat my channel, the idea is that I start with a series on QFT or ST (I can talk about how I’ve taught myself these subjects, how I continue to approach them, and also offer lectures for anyone else interested in learning). I would also like to write a series of blog posts, when time permits, where we could even just begin with introducing notation and history and then deepen from there.

However, I am still not certain what approach I would prefer to take for such a series of lectures and posts, whether it is best for wider audience to start with the RQM entry or the path integral formalism.

In any case, I certainly have lots of notes that I will aim to post on here in the future.

Bye for now.

 

 

 

Klein-Gordon equation

Klein–Gordon equation

[latexpage]

The other day I was thinking about the Klein-Gordon equation, otherwise known as the Klein–Fock–Gordon equation. I had to use it for something, and afterwards I found myself thinking of its derivation. So, for fun, let’s derive it!

Klein Gordon equation traveling wave plot5

[For the reader not familiar with this equation, it is a relativistic wave equation related to the Schrödinger equation. You can find a very general entry into its use and importance here. I also found some lecture slides that go over its derivation in more detail than my own treatment here].

***

In that we are dealing here with relativistic quantum mechanics, in the following derivation of the Klein–Fock–Gordon equation we’re going to employ the Einstein energy-momentum relation. Also, to simplify things let’s take the invoke the standard convention for the units in which $hbar = c =1$. This will allow us to not have to focus on $hbar$ and $c$ terms that appear throughout. It’s not so much lazy, just practical.

We start with the Schrödinger equation in natural units,

[ i frac{partial{d}psi}{partial{t}t} + frac{1}{2m} nabla^2 psi + Vpsi = 0 ]

As we continue to set the stage, let’s now also remind ourselves of the equation for the nonrelativistic energy of a free particle (this will be important for reasons that will become clear in a moment),

[ E= frac{vec{p}}{2m} ]

Now that we have our equation written, we can proceed. The key here is that we’re going to want to quantise this nonrelativistic equation. The result we get is $hat{H} psi = E psi$, where $hat{H} = T + V =  frac{p^2}{2m} + V$.

(Recall: the inner product of momentum (lorentz invariant) gives, $p^{mu}p_{mu} = m^2c^2 = frac{epsilon^2}{c^2}=mid{p}^smid$).

Taking the quantum mechanical operators, we get:

[ E rightarrow i hbar frac{partial}{partial t} ]

[ P rightarrow -i hbar vec{nabla} ]

With these operators defined, the purpose for doing this is because a natural attempt is to want to use the Einstein energy-momentum relation. We can now turn our attention to this relation, which one may already know takes the form:

[ hat{E}^2 = p^2 +m^2 ]

Substituting for our quantum operators, we get something immediately looking like this:

[ (i hbar frac{partial}{partial t})^2 = (-i hbar vec{nabla})^2 + m^2 ]

Or, one will sometimes see this written as $(frac{E}{C})^2 = p^2 + m^2c^2$, then substituting for $E$ and $vec{p}$ you will find $(frac{hbar}{c} frac{partial}{partial t})^2 phi = (hbar nabla – m^2 C^2) /phi$. But because we’ve set $hbar = c =1$ our approach is slightly different. In our case, when we simply expand the brackets and simplify,

[ – frac{partial^2 phi}{partial t^2} = m^2 phi – nabla^2 phi  (*) ]

This is essentially the Klein-Gordon equation. However, we have some conflicts in notation. Really, we want our equation above to be in four-vector notation. So, what do we do? I’ve already given some hints in the graphic I posted above (a sketch from my notebook).

In four-vectors, one will already likely be familiar with the idea of having one time component and three space components. We can write this as $X^{mu} = (X^o, bar{X})$, where $bar{X}$ is just short for our x, y, and z components.

Now, with that sorted, we need to think about our four-gradient, which we can write as $partial_{mu} = (frac{partial}{partial t}, vec{nabla})$. Taking the dot product, or, in other words, taking the Einstein summation convention into consideration, $partial_{mu}partial^{mu} = partial_{mu} g^{mu nu} partial_{nu}$, where one might recognise $g^{mu nu}$ as our metric.  Understanding the metric convention here has the signature (+ – – -), we come to the following:

[ partial_{mu}partial^{mu} = partial_{mu} g^{mu nu} partial_{nu} implies (frac{partial}{partial t} nabla) begin{pmatrix}
1 & 0 \
0 & bar{-1}
end{pmatrix}
begin{pmatrix}
frac{partial}{partial t} \
bar{nabla}
end{pmatrix} ]

which, after performing standard matrix multiplication, comes out to

[ frac{partial}{partial t} nabla begin{pmatrix}
frac{partial}{partial t} \
bar{nabla}
end{pmatrix} = frac{partial^2}{partial t^2} – bar{nabla}^2]

So, we have (invoking the d’Alembertian at the end)

[ partial^{mu} partial_{nu} = frac{partial^2}{partial t^2} – bar{nabla}^2 implies Box = frac{partial^2}{partial t^2} – bar{nabla}^2 ]

As we approach our final result, recognise that what we have now looks very much like (*). That’s because we have arrived at a representation for our d’Alembertian, and we need to perform a substitution. All that is then left is some algebra and we’re done! Here is what I mean: let us now return to our previous equation, (*), rearrange it so that we can substitute directly for our d’Alembertian:

[ – frac{partial^2 phi}{partial t^2} = m^2 phi – nabla^2 phi ]

[ – frac{partial^2 phi}{partial t^2} + nabla^2 phi – m^2 phi = 0 ]

[ frac{partial^2 phi}{partial t^2} – nabla^2 phi + m^2 phi = 0 ]

Sub for the d’Alembert operator,

[ Box^2 phi + m^2 phi = 0 ]

[ implies (Box^2 + m^2) phi = 0 ]

And here is the version of the Klein-Gordon equation you will see in many texts, except it does not include $hbar$ and $c$. In that case one will often see it written as, $(Box + frac{m^2c^2}{hbar})phi = 0$ where, again, $Box = (frac{1}{x} partial t)^2 – nabla $.

Concluding remarks

What is really cool about this equation is that you can find plane wave solutions to it relatively easily. The caveat being that the plane wave is a solution to the Klein-Gordon equation so long that energy and momentum follows Einstein’s relation. This last comment provides a hint for further study, should the inquisitive reader immediately think of connections with GR and RQM.

I’ll also save some discussion on some of the problems, or limitations, pertaining to the KG equation for another time. For now it is just nice to appreciate the result – an attempt at relativistic quantum mechanics!

In a following post, I will show alternative way (there are a few) to derive the KG equation which is much more terse or abrupt. I think it is useful to know.

Higgs boson

The Standard Model: Elementary Particles

Since the time of at least ancient Greece, human beings have tried to understand the fundamental constituents of all matter in the universe. Indeed, the idea that matter is comprised of discrete units dates back to many ancient cultures. For example, the etymology of the word “atom” can be traced back to the ancient Greek philosophers Leucippus and Democritus. References to the concept of atoms – the indivisible – has also been found in ancient Indian philosophical texts.

Today, we have a very good idea of the fundamental constituents of all matter in the universe. But we’re still missing some pieces of the puzzle.  That is to say, the picture is not yet complete. While the ancient Greeks speculated on the existence of atoms, primarily by way of philosophical and theological reasoning (as opposed to science-based methods), modern science has significantly advanced our understanding of the fundamental constituents of all matter. Whether we are close to discovering a “theory of everything” is debatable and uncertain. What we can say for certain, however, is that we already have an exceptionally successful theory which explains almost all experimental results and has precisely predicted a wide variety of phenomena, including new particles like the recently discovered Higgs boson. It is known as the Standard Model of Particle Physics.

In this post I want to focus primarily on introducing the three categories or groups of elementary particles that comprise the current version of the Standard Model. In the process, I also want to discuss some of them neat characteristics of these particles. In future posts I will then expand on the details and elaborate on the theoretical complexities and nuances. We’ll also dive deeper into the physics and explore the maths.

***

Let’s start with some basic definition. What distinguishes an elementary particle from an atom or a molecule? Inasmuch that the ancient Greeks were concerned with atoms as fundamental indivisible objects, we can say today that atoms are not fundamental in the sense of elementary particles. An atom is the smallest constituent unit of ordinary matter; but it is also divisible, comprised of subatomic particles. On the other hand, an elementary or fundamental particle is indivisible – that is, an elementary particle is not composed of anything more basic.

One helpful way to think of the elementary particles of the Standard Model is by creating a sort of hierarchical picture. We take as our starting point the existence of matter. For example, many readers will already be familiar with how the stuff of everyday life is made of molecules which are then made of atoms. But as was already noted, atoms are divisible – they are composite objects made of smaller divisible parts: namely neutrons, protons and electrons.

Moreover, this is the basic picture one will learn in introductory science class: we have atoms, around which orbit electrons (electron cloud). The nucleus of an atom is then comprised of protons and neutrons. All three of these – electrons, protons and neutrons – are what we call fermions (they have odd half-integer spin). For the purposes of this post, it is assumed that one is already comfortable with this level of subatomic particles, such that one has sufficient knowledge of the electric force; the relative charges of electrons, protons and neutrons; and thus some sense of the physics of how equal charges repel. For example, one should be comfortable with the idea that inside the nucleus of an atom two or more protons would repel and fly apart if it were not for the strong nuclear force that holds protons and neutrons together. And the strong nuclear force is just that, very strong. So it takes a lot of energy to overcome it and to therefore break the nuclei apart.

It is worth noting that any reference to size in the hunt for ‘smaller’ and ‘smaller’ particles is a bit deceiving. When one thinks of particles, they might be inclined to think of something like a very tiny subatomic marble. But we’re not speaking of objects with such well-defined size and hardened surface, as one might relate in their mind with basic objects of everyday life. Particles indeed have particle-like properties in that they do carry discrete amounts of energy. But they also have wave-like properties, and when we observe and study elementary particles in great detail we find their wave-like properties.

So when we break the nucleus apart, and when we break open (in a manner of speaking) the separated protons and neutrons, we find even smaller constituent units. For instance, when we smash protons together in a particle collider, we find that they are made of even smaller particles known as quarks. More technically, they are comprised of up quarks and down quarks. More technically still, the recipe for a proton is two up quarks and one down quark. As for neutrons, they are made of two down quarks and one up quark!

What about electrons? Electrons are curious in that we still have no observable evidence that they are composite units (made up of smaller particles). The thing about discovering new particles is that it takes a lot of energy. (There is a terrific point of discussion that could be raised here in relation to Einstein’s famous equation, E=mc^2, which I’ve mentioned previously and will certainly discuss in more detail in a later post). Think of it this way: at the Large Electron-Positron Collider electrons are moving so fast that they can experience as much as 100 000 times an increase in mass. Although they are smashed at 99.999999999% the speed of light, we have seen no sign of anything smaller.

 

This means that quarks and electrons are two elementary particles. But in various experiments other elementary particles have also been found.

Of the matter particles, we have two categories: quarks which we already talked about and leptons. Together, these are also called fermions. There are 6 quarks: up, down, charm, strange, top, and bottom. There are also 6 leptons: the electron, electron neutrino, muon, muon neutrino, tau, and tau neutrino. Three of these leptons – electron, muon and tau – are also known as charged leptons. The other three types of leptons are, in fact, three types of neutrino. Neutrinos are neat in that, in many textbooks, they are considered massless. But there is good evidence to suggest neutrinos have mass, only that the mass is very tiny (smaller than electrons, for example). Another quirky feature of the neutrinos is how they tend to not interact with other particles. Moreover, one will occasionally read an account of neutrinos as being ‘shy’. Neutrinos are so abundant that some estimates suggest ~100 trillion of them pass through your body every second. But the tendency for neutrinos to not interact with other particles means that, as they collide with Earth for example, most of them will just pass directly through and emerge unscathed.

There are a few other interesting things about leptons. The etymology of the word, lepton, is derived from the Ancient Greek word ‘leptos’, meaning small. The first lepton discovered was the electron, the smallest of the three particles we typically associate with an atom. But what is curious is how, of the three charged leptons, the muon and tau are much heavier than the electron. Despite this difference, they each have the same basic properties. So a remaining question that remains is why are there three? What role or purpose does the muon and tau serve?

Putting this question aside,  you will have done well to notice that, as pictured above, the 6 quarks and the 6 leptons are also categorised by generation. The first generation, which is comprised of up quarks, down quarks, electron, and electron neutrino – these are the lightest and most stable particles.

There is a lot more to be said about all of these elementary particles, but it is important we leave room for the third category: bosons. The interesting thing about bosons is that there are two groups. First, we have gauge bosons. The cool thing about gauge bosons is that they are force-carriers. In other words, one can think of them as mediators for three of the four fundamental forces: strong, weak, and electromagnetic interactions. It turns out that there are 4 gauge bosons: gluon, photon, Z boson and W boson (the Z and W bosons transmit the weak force). These can also be described as vector bosons.

This leaves the famous Higgs boson. The Higgs is a scalar boson. We will talk more deeply about the Higgs in a future post (as with most other things presented here). Meanwhile, one of the truly fascinating properties about the Higgs boson has us return to a consideration of the Z and W bosons mentioned above. Before the predicted Higgs was observed at CERN, a problem existed with the gauge invariant theory for the weak force. The theory predicted that the photon – the gauge boson responsible for transmitting the electromagnetic force – and the W and Z bosons should have zero mass. While the photon is massless, the W and Z bosons are shown to have mass. So the following question for particle physicists emerged: what was giving these particles their mass? The answer is the Higgs boson. Indeed, it is the task of the Higgs to give the W and Z bosons their mass. But the Higgs field also helps explain why other fundamental constituents of matter, such as leptons and quarks, have mass.

With the inclusion of the Higgs, these particles are then the elementary particles of the Standard Model, which, as far as we know, have no substructure and are therefore pointlike. The up quark, down quark, electron and electron neutrino seem the most essential of the elementary matter particles, with the other 8 suspected to have played a key role in the early universe. Each one of the particles in the Standard Model also has an antiparticle. Antiparticles are something we’ll talk about more in latter post. Meanwhile, it is worth concluding with a brief introduction to the four fundamental forces.

***

One of the great accomplishments represented by the Standard Model is how three of the four fundamental forces are included under the unifying principle known as the gauge principle. This is also why the Standard Model is described as a gauge theory. As a demonstrably proven and astonishingly successful theory, it brings together things like: a) Quantum Chromodynamics (QCD), which is the single theoretical framework of gauge theory in which three of the four fundamental forces are represented; b) Weinberg-Salam theory, which is a unified field theory that brings together the electromagnetic and weak nuclear interactions, or forces, in what we describe as “electroweak” interactions; c) finally, we also have Quantum Electrodynamics (QED). Behind all of these theories are many important theoretical concepts, such as gauge symmetry as well as other things like the concept of quantum fields. Symmetry plays an important role in particle physics, and the study of symmetry has even led to the production of new theories in the field. But, again, I’ll bracket these discussions for separate entries.

To close, the Standard Model is a theory that successfully describes fundamental particles and how they interact. More deeply, it describes a quark-lepton picture of matter and the quantum theory of the fundamental forces. These fundamental forces are the weak nuclear force, the strong nuclear force, electromagnetism and gravity. However, gravity is not included in the Standard Model. This exclusion refers to the issues relating quantum mechanics and the general theory of relativity. There are also other issues with the Standard Model, which suggest that our picture is not yet complete and that a physics beyond may be waiting to be discovered.

 

 

 

Amplituhedron

The Amplituhedron

The amplituhedron – a newly discovered geometrical object that enlivens the imagination. As a student in the area of theoretical particle physics, it has energised my curiousity in a very unique and notable way.

I am absolutely fascinated with this object, so much so that it has been a week with some sleepless nights. Fueled mostly by museli, vegetables and countless cups of tea, each day has merged into the next, interspersed with very few breaks. I suppose in general such study patterns are the norm of my existence. But in this present moment of day to day existence, if not donating time to unavoidable school related demands and on preparing for exams, much of my day is completely and totally consumed by the study of the amplituhedron and the physics related to it.

It is not that I have just discovered this geometrical object or just learned of its discovery. In fact, I originally came across it several months ago. But I was so busy with other things, my interest was more gentle. Over that time I dabbled in and out, thinking mostly of the broad consequences, particularly in relation to the idea of spacetime potentially being emergent.

More recently, I have listened to almost every lecture by Nima Arkani-Hamed that I could find on Youtube and elsewhere on the web. Correlating, too, with my reading of Penrose (I am interested in Twistor Theory), I have begun to dig into numerous papers, such as Arkani-Hamed and Jaroslav Trnka’s original 2013 paper introducing the amplituhedron, as well as their 2017 paper (with Hugh Thomas) on unwinding the amplituhedron in binary.

Admittedly, my twistor maths skills and my understanding of Grassmannians and of projective space are not yet at the level they need to be for me to truly understand and theoretically encircle the object and the physics in relation to it. Likewise, my understanding of planar N = 4 supersymmetric Yang–Mills theory is currently weak, and I am only starting to scratch the surface of the basics. This includes just how, or why, the amplituhedron represents a solution for Super Yang-Mills.

amplituhedron
Drawing of amplituhedron by Nima Arkani-Hamed

But I think my interest is such that I could see my masters being focused on the amplituhedron in some way. There is still time, and things could change. But the very broad ideas associated with Arkani-Hamed’s (et al.) work are just so exciting. Perhaps in the near future I’ll write more about what makes it so exciting. For me, it is much more than just the possibility of the idea of how we might be able to simplify our calculations of scattering amplitudes. Certainly, this is one of the popular advertisements for the theory and that is understandable. For anyone who has studied, likes to draw, or understands Fenyman diagrams as a common way to calculate scattering amplitudes in quantum field theory, it can be incredibly grueling and quite a grind. Things become so complicated, and the amount of diagrams increases so much with the increase in the amount of loops, that not only does it become difficult to make accurate calculations but, at least for me, it suggests something deeper is missing. Something is not quite right. But what?

It almost seems that with Fenyman diagrams, we’re picking up or obtaining a glimpse of something about nature – a fragment of a more total picture. Or, to borrow a line from David Skinner, it very much seems like we’re picking up pieces or looking at shards of a broken Ming vase.

Think of the scattering of gluons. Something so simple, such as two gluons colliding to produce four less energetic gluons – to calculate the amplitude in using the textbook approach by way of Feynman diagrams, this would involve 220 diagrams. We’re talking tens of thousands of terms – pages and pages!

In that the amplituhedron might simplify these calculations, this evokes in me a sense of curiosity that the suspicion of something being missing might be true. It is at least worth thinking about and pursuing, to whatever end. But it’s also the idea of the reformulation of the whole of QFT, and things like how one can arrive at the same equation for the loop amplitudes without spacetime, gauge symmetry, quantum mechanics and the use of the path integral. It is really just so very cool.

From the perspective of my current understanding, the geometry itself is rooted in energy-momentum space, with the amplitude being the volume of the amplituhedron. Again, quite amazing to think about. If I am right in my understanding, unitarity and locality are also not completely discarded, nor are they required; instead, they are seen as being emergent (along with spacetime and QM). Again, fascinating.

The ideas may be in their infancy, and there may be lots of speculative impulses, but the entire theory is incredibly intriguing nonetheless.

My plan of attack is first to continue working toward studying and more deeply understanding the mathematics of Twistor Theory, and then also Grassmanians and so on.

I’m sure more posts will be made along the way.