## Posts

### 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

### Zeta function regularization, heat kernels, and one-loop determinants

In the path integral formulation of quantum field theory, a free theory is one whose Euclidean action can be expressed as $$S[\phi] = \frac{1}{2} \int_{\mathcal{M}} \boldsymbol{\epsilon} \phi L \phi, \tag{1}$$ where $\phi$ is some tensor field and $L$ is a self-adjoint differential operator whose spectrum is bounded below. For example, if we make $\phi$ a real scalar field and $L$ the differential operator $$L = m^2 - \nabla^2,$$ then $S[\phi]$ is the action of a free scalar with mass $m.$ It is useful to have general techniques for studying the path integrals of free field theories, not only because free theories serve as an important testing ground for interesting physical ideas, but also because in the semiclassical limit, any quantum field theory can be approximated as free. In particular, if we consider some Euclidean quantum field theory defined by the path integral $$\int \mathcal{D} \phi e^{- \frac{1}{\hbar} S[\phi]},$$ then in the limit $\hbar \rightarrow 0$ the path integral