Quantum electrodynamics

• classical mechanics describes large and slow objects
• special relativity describes large and fast objects (they are getting close to the speed of light, so we have to consider relativity)
• classical quantum mechanics describes small and slow objects.
• QED describes objects that are both small and fast

• in special relativity, we have the mass-energy equivalence
• in quantum mechanics, thinking along the time-energy uncertainty principle, $\Delta E\sim \frac{1}{\Delta t}$

• photon field
• electromagnetism field

• Lamb shift: by far the most famous one
• hyperfine structure TODO confirm
• anomalous magnetic dipole moment of the electron

• www.youtube.com/watch?v=Ix-3LQhElvU Anomalous Magnetic Moment Of The Electron | One Loop Quantum Correction | Quantum Electrodynamics by Dietterich Labs (2019)

A Relativistic Quantum Theory of a Finite Number of Particles is Impossible.

• physics.stackexchange.com/questions/101307/dirac-equation-in-qft-vs-relativistic-qm
• physics.stackexchange.com/questions/44188/what-is-the-relativistic-particle-in-a-box/44309#44309 says:

  By several reasons explained in textbooks, the Dirac equation is not a valid wavefunction equation. You can solve it and find solutions, but those solutions cannot be interpreted as wavefunctions for a particle

• physics.stackexchange.com/questions/64206/why-is-the-dirac-equation-not-used-for-calculations
• www.physicsforums.com/threads/is-diracs-equation-still-useful-after-qed-is-developed.663994/

• $F$ is the electromagnetic tensor

• Dirac Lagrangian, which only describes the "inertia of bodies" part of the equation
• the electromagnetic interaction term $\frac{1}{4{\mu }_0}{F}_{\mu \nu }{F}^{\mu \nu }$, which describes term describes forces

• by "Lagrangian" we mean Lagrangian density
• the generalized coordinates of the Lagrangian are fields

• spacetime symmetry: reaches the derivation of the Dirac equation, but has no interactions
• add the $U(1)$ internal symmetry to add interactions, which reaches the full equation

• local symmetries of the Lagrangian imply conserved currents. $U(1)$ gives conserved charges.
• OK now. We want a local $U(1)$ symmetry. And we also want:
  • Dirac equation: quantum relativistic Newton's laws that specify what forces do to the fields
  • electromagnetism: specifies what causes forces based on currents. But not what it does to masses.
  Given all of that, the most obvious and direct thing we reach a guess at the quantum electrodynamics Lagrangian is Video "Deriving the qED Lagrangian by Dietterich Labs (2018)"

• www.youtube.com/watch?v=pCQ9GIqpGBI&t=1663s mentions this idea first came about from Hermann Weyl.
• youtu.be/pCQ9GIqpGBI?t=2827 mentions that in that case the curvature is given by the electromagnetic tensor.

• www.youtube.com/watch?v=qtf6U3FfDNQ Symmetry and Quantum Electrodynamics (The Standard Model Part 1) by ZAP Physics (2021)
• www.youtube.com/watch?v=OQF7kkWjVWM The Symmetry and Simplicity of the Laws of Nature and the Higgs Boson by Juan Maldacena (2012). Meh, also too basic.

• physics.stackexchange.com/questions/144607/are-all-positrons-electrons-traveling-back-in-time
• www.quora.com/Is-a-positron-just-an-electron-going-backwards-in-time?share=1

• Advanced quantum mechanics by Freeman Dyson (1951)

We shall not develop straightaway a correct theory including many particles. Instead we follow the historical development. We try to make a relativistic quantum theory of one particle, find out how far we can go and where we get into trouble.

• Part 1: is saying "photons exist"
• Part 2: is amazing, and describes how photons move as a sum of all possible paths, not sure if it is relativistic at all though, and suggests that something is minimized in that calculation (the action)
• Part 3: is where he hopelessly tries to explain the crucial part of how electrons join the picture in a similar manner to how photons do.
  He does make the link to light, saying that there is a function $P(A,B)$ which gives the amplitude for a photon going from A to B, where A and B are spacetime events.
  And then he mentions that there is a similar function $E(A,B)$ for an electron to go from A to B, but says that that function is too complicated, and gives no intuition unlike the photon one.
  He does not mention it, but P and E are the so called propagators.
  This is likely the path integral formulation of QED.
  On Quantum Mechanical View of Reality by Richard Feynman (1983) he mentions that $E$ is a Bessel function, without giving further detail.
  And also mentions that:

  $$\begin{gathered} E=f(1,2,m) \\ P=f(1,2,0) \end{gathered}$$

  (1)

  where m is basically a scale factor. such that both are very similar. And that something similar holds for many other particles.
  And then, when you draw a Feynman diagram, e.g. electron emits photon and both are detected at given positions, you sum over all the possibilities, each amplitude is given by:

  $c\times E(A,D)\times E(D,B)\times P(B,C)$

  (2)

  summed over all possible $D$ Spacetime points.
  This is basically well said at: youtu.be/rZvgGekvHes?t=3349 from Quantum Mechanical View of Reality by Richard Feynman (1983).
  TODO: how do electron velocities affect where they are likely to end up? $E(A,D)$ suggests the probability only depends on the spacetime points.
  Also, this clarifies why computations in QED are so insane: you have to sum over every possible point in space!!! TODO but then how do we calculate anything at all in practice?
• Part 4: known problems with QED and thoughts on QCD. Boring.