Quantum Field Theory

Brian Skinner’s Picture Book on QFT (PDF here)

If each point in a field — the gravitational field, the electron field, and so on — is to have local phase freedom (historically called gauge freedom), such that it can undergo gauge transformation by changing its phase without any global change in phase — such that it is a purely local variable, and not a global one affecting all points in the field — then some method for informing its neighbors as to the free decision it has made is needed. If you did not have a method of communicating phase information from point to point then the field would just be a collection of uncoordinated, solipsistic points. The communication is key to keeping each point informed as to what every other point is doing. It allows the points in the field to communicate with one another.

As a purely practical matter, students of electromagnetism “choose a phase” (meaning choose a gauge) when working a problem — e.g. the Coulomb phase or the Lorentz phase (see p.14 of Moriyasu). “When an electromagnetic field is present, a different choice of phase at each point in space can then be accommodated easily by interpreting the potential Aμ as a connection which relates phases at different points.” Moriyasu p.17.

Internal spaces for phase factors. Following Schwichtenberg p.83-6, in QFT it is understood that “above” or “inside”(though it’s really not spatiotemporally located; “above” and “inside” are just spatiotemporal metaphors) each point on a field (say the electron field, or the photon field) is a complex unit circle, CUC, as defined by Euler’s formula

$e^{i\theta}=cos\theta+isin\theta$ .

This little invisible CUC is a local internal space called that point’s “phase factor” and something called local gauge freedom (where, crazily, for historical reasons, “gauge” really means “phase”, so that we are really talking about phase freedom) allows that each individual point in the field can set it’s “phase factor” to anything it likes, meaning to any value of theta ( $\theta$ ).

Algebraically you write this like so:

$\psi \rightarrow \psi'=e^{i\theta}\psi$

which just means you can take some wavefunction $psi$ and multiply it by Euler’s formula with the theta set to any value you like and you get the same wavefunction, just with a different phase.

It is often helpful to “spell this out” more completely by imagining that the wavefunction is a real function called $R(t,\vec{x})$ (e.g, only uses real numbers) that is multiplied by a complex phase factor called $e^{i\psi(t,\vec{x})}$ . This is written algebraically like this:

$\psi(t,\vec{x})=R(t,\vec{x})e^{i\psi(t,\vec{x})}$

What is sometimes said is that this phase is “unphysical” meaning it doesn’t alter the behavior of the electron per se. But it does affect how electrons interact, so that “unphysical” has a pretty limited meaning.

Electromagnetic potential

The electromagnetic four-potential

$A_{\mu}=(\phi,\mathbf{A)}$

is comprised of two quantitites: $\phi$ , a scalar called the eletric potential which in neurobiology we call voltage, and $\mathbf{A}$ , the magnetic potential, which is a vector (not a scalar). Thus the “electro” and the “magnetic” in electromagnetic map to two different mathematical objects, one a scalar, one a vector.

The electric and magnetic fields are made out of these potentials, with it being important to understand the distinction between a field (lots of points) and a potential (one point). The magnetic field is $\mathbf{B}$

$\mathbf{B}=\mathbf{\nabla\times A}$

and the electric field is $\mathbf{E}$

$\mathbf{E}=-\nabla\phi-\frac{\partial\mathbf{A}}{\partial t}$

Where we can define the three spatial coordinates of each field as

$\mathbf{E}=(E_{1,}E_{2,}E_{3})$

and

$\mathbf{B}=(B_{1,}B_{2,}B_{3})$

we can produce the electromagnetic field strength tensor in covariant, $F_{\mu v}$ , and contravariant, $F^{\mu v}$ , which look like this

$F^{\mu\nu}=\left[\begin{array}{cccc} 0 & -E_{1} & -E_{2} & -E_{3}\\ E_{1} & 0 & -B_{3} & B_{2}\\ E_{2} & B_{3} & 0 & -B_{1}\\ E_{3} & -B_{2} & B_{1} & 0 \end{array}\right]$

and

$F_{\mu\nu}=\left[\begin{array}{cccc} 0 & E_{1} & E_{2} & E_{3}\\ -E_{1} & 0 & -B_{3} & B_{2}\\ -E_{2} & B_{3} & 0 & -B_{1}\\ -E_{3} & -B_{2} & B_{1} & 0 \end{array}\right]$

where, as a sidenote, index lowering is achieved like this: $F_{\mu v}=\eta_{\mu\alpha}F^{\alpha\beta}\eta_{\beta\nu}$

It is often more pragmatic to not write out all these E’s and B’s and so you often see it just stipulated by way of

$F_{\mu\nu}=\left[\begin{array}{cccc} F_{00} & F_{01} & F_{02} & F_{03}\\ F_{10} & F_{11} & F_{12} & F_{13}\\ F_{20} & F_{21} & F_{22} & F_{23}\\ F_{30} & F_{31} & F_{32} & F_{33} \end{array}\right]$

Fiber bundles

1. Readings

Forbes whirlwind common-person tour of field theory

DrPhysicsA on Youtube Series on Particle Physics, which features gauge theory prominently Note that if you scroll along on the bottom you can skip to the formulae you really need explained. QED begins here.

Gauge bosons, Gauge theory, in wikipedia

Electromagnetic gauge theory

physicsdetective.com

Ranked in order of usefulness to a total novice

#1 Kuhlmann, Meinard (2020) Stanford Encyclopedia of Philosophy article on QFT. PDF version. A good place to start to get a lot of history and context.

#2 Schwichtenberg, Jacob. Physics from finance: introduction to fiber bundles and gauge theory (Amazon Kindle version) Nice little drawings, sometimes too simple, sometimes too fast, but definitely a good “first pass” through the material to get the basic idea.

#3 Moriyasu, K. Elementary Primer for Gauge Theory. Same on internet archive. Same on Amazon Kindle. My personal favorite. As a novice I have found this to be the most useful per unit time spent for introductory information on QFT. But higher level than the next, below.

Chapter 2 is historical but useful
Chapter 3 is where he starts to dig into the gauge theory

#4 Auyang, Sunny (1995) How is quantum field theory possible? (OUP) A work of philosophy more than physics, not the best place to start if you are trying to get the main idea.

#5 Torre Why do we need QFT and What Is a Photon?

#6 de Wit Introduction to Gauge Theory and the Standard Model

#7 Scholarpedia article

#8 Sean Carroll video on Gauge symmetry (over 1h long!)

At minute 16:50 or so he talks about why we have a photon field — to gauge the electron’s phase. He gets into the idea that the photon field is necessitated by the electron field very briefly bc it seems very obvious to him.

Electromagnetic potential

Fiber bundles

1. Readings

Share this:

Like this: