Term Paper Topics Part 2

Here are the rest of the topics with brief descriptions.

6) The Spin/Statistics Connection

States involving multiple identical particles involve a fundamental inability to distinguish between the particles. In three-dimensional space, it turns out that a state involving multiple identical particles must be symmetric or antisymmetric under particle swaps. Particles that swap symmetrically are called bosons, while particles that swap antisymmetrically are fermions. As we have seen, particles are partly categorized by their spin, and it turns out that relativity imposes the condition that integer spin particles are bosonic while half-integer spin particles are fermionic. The relationship between particle swapping properties and particle spin is called the spin-statistics connection. This term paper should explore why the connection is true at both a conceptual and more mathematical level.

7) The Spinor Helicity Formalism

There is a curious feature of four-dimensional spacetime: 4-vectors can be decomposed into pairs of two-component spinors. Over the last several years researchers have used this relationship to discover new formulations of four-dimensional quantum field theory and much more efficient methods for calculating scattering amplitudes and probabilities. This term paper explores the basic elements of this relationship and how it can be used to formulate theories of massless particles.

8) Supersymmetry

There are fermions and there are bosons. Supersymmetry refers to a whole class of symmetry transformations that exchange the one particle type with the other. You can show that any fermion-type particle can be described in terms of a supersymmetric world-line. On the other hand, supersymmetry relating different particle types in a four-dimensional theory has shaped theoretical and experimental research programs for decades. There are many directions that this term paper topic can go. I would expect any such paper to provide an elementary description of supersymmetry at the quantum level and explore some of the generic implications.

9) Bosonic Strings

The theory of bosonic strings is one of the simplest generalizations of particle-based quantum field theory. This term paper should explore the relationship between the world-line formalism of standard quantum field theory and the worldsheet approach to strings. In the course of doing so, it would be great if the paper could provide some arguments for why (a) the theory requires 26 dimensions for consistency and (b) why nevertheless, the existence of a tachyonic state renders the theory problematic.

Term Paper Topics Part 1

Below is a partial list the term paper topics for class with some very brief thoughts about how the topic connects to the course:

1) Lagrangian formulation of QM/QFT and Path Integrals

We’ve been describing quantum theory (and eventually we will describe QFT) from a Hamiltonian perspective. An alternate way to formulate things is using Lagrangians, which is convenient when developing a quantum theory from an already known formulation of a classical theory (quantization). The paper I’m envisioning here would basically explore this alternate formulation.

2) History of QFT

QFT was not built by any one individual. It was built over a period of years with some of the key steps occurring even as people like Schrodinger, Heisenberg, and Dirac were struggling with the formulation of non-relativistic quantum mechanics. The scope of possibilities for this topic is pretty broad. One way or another, I’d like there to be some discussion of technical issues that folks ran into in the early development.

3) Feynman Diagrams

Feynman diagrams are the bread-and-butter tools that folks use when organizing their calculations for scattering amplitudes. I’d expect this paper to at least describe where Feynman diagrams come from (i.e. how the basic diagrams come from certain terms in the Lagrangian density of a theory) and the basic rules for combining them and translating them into a mathematical expression. You can even use such diagrams to do ordinary integrals, which may be a nice toy-example to explicitly lay out in your paper.

4) The Standard Model

In our course, we’ve spent a great deal of time discussing the combination of relativity with the principles of quantum theory. Among the key topics has been how the non-interacting single-particle theories work when the particles possess non-zero spin. The Standard Model of particle physics is built from these basic ingredients. At minimum, I’d expect a description of the different particle types and how they fit together into the framework of the Standard Model.

5) Linearized Gravity from QFT

What happens when you write down a quantum theory consisting of a massless spin 2 particle? Well, if you’re careful about extending the Hilbert space to maintain manifest Lorentz symmetry and then determining the constraints that identify the physical subspace of states you discover gravity—almost. In fact, you are forced to write down the linear approximation to the equations of general relativity. I would expect this term paper basically describes this construction in detail.

Mistakes and Muddles and Notes Update (10/11/19)

Last class (on 10/10/19) I got a bit confused during what I expected would be a simple justification for the Lorentz transformation of the spacetime momentum operator. As it turns out, the calculation was telling me that there was an inconsistency in the formulas that I wrote down!

Image result for picard facepalm meme

The issue I ran into is connected to the difference between a passive and an active transformation. You can think of an active transformation as taking our particle and boosting it so that it has a different momentum in our reference frame. By contrast, you can think of a passive transformation as boosting an observer who initially is in our frame to a new inertial frame. Now, given a particle in a state of definite momentum p, if you boost the particle then you with your original momentum operator will measure the particle’s momentum to be Λp. If you also consider a boosted observer who moves along with the boosted particle then they should observe the original momentum p. Therein lies the error I made!

I’ve decided to punish my students for the error of my ways and I’ve added an exercise (4.7) to my notes!

Reading Assignment 1: (Ch. 0)

The assignment was to read chapter 0 of our textbook (QFT for the Gifted Amateur) and then to hand in a note with the top two topics that you would like to review from that chapter.

The winning topics were Lorentz/Poincaré symmetry (lumping together related things like 4-vectors, tensors, etc…) and Fourier transformations. These are great choices!

There were also some questions inspired by the subsection “What is a field?” I think it’d be great to talk about these questions, but I’m going to save that for another time, or perhaps for a write-up on the course blog.

In our review class today (1/26/18), I’ll do some Fourier transform review, with an eye toward applying it to QFT in particular. I’ll be fairly light on the underlying intuition, but you can find a wonderful discussion that should really help here:

https://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/

By the way, I can’t recommend betterexplained.com highly enough! The people putting that site together have done a marvelous job.

I will do some review of special relativity, probably during next week’s review class (Fri, Feb 2). That said, one question/topic request had to do with choices of metric signature. The two competing choices are the East Coast convention \((-,+,+,+)\) and the West Coast convention \((+,-,-,-)\). People get pretty passionate about which one is the “correct” choice. My sympathies lie with the East Coast convention (I’m a New York City boy, after all!), mainly because I think it’s simpler to deal with one minus sign than three (it also extends more easily to higher spatial dimensions since no new signs are introduced).

Despite my personal preferences, its common for particle physics and QFT books to use the West Coast convention. I think that if there is a reason other than just blind tradition, then it is likely due to the fact that the 4-momentum squared is proportional to rest-mass squared in this convention rather than negative rest-mass squared. Folks probably find that negative sign annoying, and if you’re doing lots of calculations involving four-momenta, it may be more practical to go this route.

For an amusingly high-minded argument for the superiority of the East Coast convention, take a look at:

http://www.math.columbia.edu/~woit/wordpress/?p=7773

Most folks agree that these things are called conventions since there isn’t a truly correct choice between them. But Woit makes a good case for going with the East Coast convention that goes beyond simple convenience.