We now come to the ﬁrst hard part of the class. We’ll try to learn what insights we can gain from the equation governing relativistic quantum mechanics.
- 5.1: The Diﬀerence Between Relativistic and Non-Relativistic Quantum Mechanics
- One of the key points in particles physics is that special relativity plays a key rôle. As you all know, in ordinary quantum mechanics we ignore relativity. Of course people attempted to generate equations for relativistic theories soon after Schrödinger wrote down his equation. There are two such equations, one called the Klein-Gordon and the other one called the Dirac equation.
- 5.2: Antiparticles
- Both the Klein-Gordon and the Dirac equation have a really nasty property. Since the relativistic energy relation is quadratic, both equations have, for every positive energy solution, a negative energy solution. We don’t really wish to see such things, do we? Energies are always positive and this is a real problem. The resolution is surprisingly simple, but also very profound – It requires us to look at the problem in a very different light.
- 5.3: QED - Photon Couples to e⁺e⁻
- We know that electrons and positrons have charge and thus we need to include electrodynamics in the relativistic quantum theory of the electron. That is even more clear when we take into account that an electron and positron can annihilate by emitting two photons (the well-known 511 keV lines).
- 5.4: Fluctuations of the Vacuum
- The great problem is in understanding the meaning of virtual particles. Suppose we are studying the vacuum state in QED. We wish to describe this vacuum in terms of the states of no positrons, electrons and photons (the naive vacuum). Since these particles interact we have short-lived states where e+e− pairs, and photons, and .... appear for a short while and disappear again. This is also true for real particles.
- 5.5: Inﬁnities and Renormalization
- When we introduce α and e in our theory these we use the measured value of the charge of an electron – which is a solution to the full theory, not to the artificial problem with all vacuum fluctuations turned of. Renormalization is the mathematical procedure that express all our answers in physically sensible (measurable) quantities. A theory (such as QED) is called renormalizable if we can make all expressions finite by re-expressing them in a finite number of physical parameters.
- 5.6: The Predictive Power of QED
- It is hard to say that a theory has predictive power without comparing it to experiment, so let me highlight a few successes of QED.
Thumbnail: One of the Feynman diagrams for an electron-electron scattering.