Author: dkagan

Quanta Magazine: All Of Physics From Symmetries of Nature

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During our QFT course this semester we’ve seen how a relatively minimal set of initial postulates about the structure of spacetime and of course the non-commutative nature of quantum observables leads almost inevitably to the framework of quantum field theory. A recent article in Quanta Magazine describes some of the remarkable results going back to the 1960s indicating that the mere combination of spin and relativistic structure predicts almost all of the particle dynamics we observe in Nature.

We’ve directly played with some of these ideas: When we define the theory of a massless spin-1 particle, we more-or-less are forced to introduce gauge invariance and a set of dynamical equations that have the same mathematical form as Maxwell’s equations. Massless spin-2 particles similarly lead to the infinitesimal or linearized form of Einstein’s equation for general relativity.

What’s truly amazing is that similar arguments reveal that the simplest interactions you can add to these free theories are once again more-or-less restricted to the ones we observe. If you want interacting spin-1 particles you are forced to introduce multiple species of them and–poof!–the theoretical structure that emerges is Yang-Mills theory, which is a generalization of Maxwell theory that describes the self interactions of gluons and very high energy weak bosons. Do the same analysis for spin-2 and you discover that the massless spin-2 particle must be unique and has to couple universally to the energy-momentum of other matter or fields.

Incidentally, these results largely explain why string theory takes the form that it does. Structures like gravity, Yang-Mills theory, and so on basically are forced on the low-energy effective approximations to the full theory, though string theory adds additional constraints that require all of these fields to be present (and others that I haven’t mentioned as well). String theory is a much more restrictive framework than general quantum field theory.

Term Paper Topics Part 2

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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

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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)

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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!

More On Spinors

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I came across a very nice paper that goes into good detail about spinors in 3D space and 3+1 dimensional spacetime. I should add however that in other spatial dimensions certain features of this explanation don’t work (I don’t think)—basically for the same reason that you can only define the cross-product of two vectors in three-dimensions.

https://arxiv.org/pdf/1312.3824.pdf

 

 

Notes Update 9/20/19; Fun SR Facts

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I’ve posted an update to the notes with some reorganization of the section on quantum symmetry. Specifically, I included a subsection describing the momentum state eigenbasis and the Fourier relation between position and momentum bases.

I came across a fun article the other day that highlights something that is rather surprising about how length contraction would appear in the real world. Usually when I’ve described length contraction, we imagine it from the side in a rather two-dimensional sort of way. We draw a picture of a 2D car with the car’s image squeezed along the direction of its motion. This is fine for getting the idea across, but it doesn’t actually reflect what we’d really see. You see, objects are three dimensional and light from different parts of the object would reach us at different times. It turns out that as a result of this, length contraction would actually appear to be something like a rotated version of the object.

For more details and some very helpful graphics that explain what’s going on, check out this article from Physics World:

https://physicsworld.com/a/the-invisibility-of-length%E2%80%AFcontraction/

 

Homework Problems and Notes Update: 9/10/19

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I noticed some egregious typos in my notes, so I’ve updated the file.

This semester, the first homework assignment is going to be based on our discussion of rotations in 2D. There were a few claims I made without proof (exercise numbers are from my notes.):

1) Prove that the rotated vector components really are supposed to be what I claimed. (Ex. 2.1.)

2) Prove that the length of the vector is unchanged by rotations. (Ex. 2.2.)

3) 2D rotations commute. Note that this is not true for higher dimensions. (Ex. 2.3.)

We didn’t discuss this in class, but there are an infinite number of complex representations of the 2D rotation group. You can work these out by mapping the 2D vectors to complex numbers. For example, you can let the x-basis vector map to 1 and the y-basis vector map to i. After doing that, any other 2D vector maps to a complex number. A fun exercise is to see that our vector rotations correspond to multiplication by a phase exp(i theta). (Ex. 2.4.)

 

QFT, Fall 2019!

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Welcome to the 2019 Fall Semester, and my introduction to quantum field theory course. The course has been completely redeveloped from the ground-up since the first time I taught it.

For standard course information, click on the “QFT – Welcome” link at the top of the page, or just click on the link below:

QFT Welcome Page

Feel free to follow the links in the sidebar on the right of the page. Course notes will be regularly updated. As of right now, you can find the introduction and the sections on special relativity and quantum mechanics. Exercises are interspersed throughout the text to help you practice or to alert you to non-trivial facts that I might state but expect you to check!