## Posts

### Ordinal numbers, transfinite induction, and Zorn's Lemma

Hello from Cambridge, Massachusetts! I've just arrived this past week from California, as August 31 marks my last day as a PhD student at Stanford and September 1 my first day as a postdoctoral researcher at MIT. I've been on a significantly reduced work schedule during the move, so rather than working directly on research problems, I've been taking time to bone up some fundamental math skills. This has involved, mostly, an exploration into functional analysis. One of the fundamental theorems of functional analysis is the Hahn-Banach theorem , which is proved using a very standard application of Zorn's lemma . During my reading, though, I realized that I actually had no idea how Zorn's lemma is proved. I've applied it many times in math courses, but the underpinnings of the lemma elude me. Trying to figure out the proof led me down a pretty deep rabbit hole — it turns out Zorn's lemma is basically a simplified packaging of a general technique called "tr

### Hilbert spaces of Majorana fermions

When we talk about fermions in quantum mechanics, we talk about two kinds: Dirac and Majorana. Both of these are supposed to have the property that if I create a fermion of type $1$ and then a fermion of type $2,$ the resulting state will be related by a minus sign to the state where I create a fermion of type $2$ and then a fermion of type $1.$ But there is a decision to make as to what should happen if we try to create two type-$1$ fermions. Dirac fermions are defined by the property that if you try to create two type-$1$ fermions, the state is completely annihilated to the zero vector. Majorana fermions are defined by the property that if you try to create two type-$1$ fermions, they annihilate one another and leave the total state unchanged. These properties are summarized by saying that the algebras of operators that create and annihilate the two different types of fermions should be different. The Dirac fermions have creation and annihilation operators $a_j, a_j^{\dagger}$ satisf

### Canonical (anti)commutation relations from the path integral

I like the path integral. The question of whether the path integral or canonical quantization is "more fundamental" is purely ideological, and impossible to answer. On the one hand, canonical quantization is rigorously defined in a much wider setting than the path integral; on the other hand, canonical quantization introduces "order ambiguities" in quantizing certain operators, and requires renormalization to define operators like $\phi(0)^2,$ while both of these issues are automatically avoided in the path integral. These advantages make me, personally, a path integral devotee. One big advantage of canonical quantization over the path integral, however, is that it automatically gives you the canonical (anti)commutation relations between field operators and their conjugate momenta; in fact, this is basically the definition of canonical quantization: it gives a prescription for defining canonical (anti)commutators in terms of a bracket on the phase space of the clas