"BROOKLYN" QUANTUM MEETUP ANTISYMMETRIZED

In order to get a better handle on second quantization, it'll be helpful to consider another example: fermionic second quantization. Before in our double oscillator spin construction, we employed bosonic second quantization, where the harmonic oscillators which count the number of particles in a given state can have any number of excitations, and which describes a theory of a variable number of permutation symmetric particles. Fermionic second quantization is actually simpler than bosonic: there can be at most one particle in a given state, and we end up with a theory of a variable number of permutation antisymmetric particles. Watch out though! The creation and annihilation operators obey different commutation relations than in the bosonic case. Then, we'll discuss the spin statistics theorem, which suggests that in important theories, half integer spins should be quantized as fermions and integer spins should be quantized as bosons. We'll play with a suggestive model due to Berry and Robbins, which has the added benefit of employing the oscillator representation of spin, and so we'll practice our bosonic second quantization skills at the same time!