Fermions: a tale of two minus signs
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Lecture Description

Leonard Susskind introduces quantum field theory and its connection to quantum harmonic oscillators. Gravity aside, quantum field theory offers the most complete theoretical description of our universe.

Professor Susskind presents the quantum mechanics of multi-particle systems, and demonstrates that fermions and bosons are distinguished by the two possible solutions to the wave function equation when two particles are swapped. When two particles are swapped, the boson wave function equation has a phase factor of +1 whereas the fermion equations has a phase factor of -1. For fermions, this results in a wave function with zero probability for two particles to be in the same state, thus demonstrating the exclusion principle. On the other hand, bosons prefer to be in the same state. This is what makes a photon (boson) laser possible, but an electron (fermion) laser impossible.

The spin variable is required to allow two electrons to occupy the same state in an atom. Electrons are fermions which have half-integer spins. This implies that a rotation of the angular momentum by 2π will result in a phase change by -1. This implies that the identity operation for fermions is not a rotation by 2π, but rather a rotation by 4π, and that a rotation by 2π can be offset or canceled by a swap of two particles. This is the tale of 2 minus signs.

Recorded on October 28, 2013.

Topics:
- Bosons
- Fermions
- Spin statistics
- Permutation groups
- Solitons

Course Description

This course will explore the various types of quantum systems that occur in nature, from harmonic oscillators to atoms and molecules, photons, and quantum fields. Students will learn what it means for an electron to be a fermion and how that leads to the Pauli exclusion principle. They will also learn what it means for a photon to be a boson and how that allows us to build radios and lasers. The strange phenomenon of quantum tunneling will lead to an understanding of how nuclei emit alpha particles and how the same effect predicts that cosmological space can “boil.” Finally, the course will delve into the world of quantum field theory and the relation between waves and particles.