Research Notes

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...

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...

(Thank you to Alex May for catching an error in the first version of this post!) I had to learn Fano's inequality for a quantum information project, but found the proof as it currently exists on Wikipedia extremely confusing, largely because it uses a lot of jargon about "messages" and "estimates" that, in my opinion, obscures the underlying mathematics. So here's a simple proof. I'll probably try to edit the Wikipedia proof after I post this. Heuristically speaking, Fano's inequality is a bound on how successfully I can guess the output of a random process $X$ if I can't access $X$ directly but can access a different random process $Y$ that is correlated with $X.$ To state the bound precisely, we'll need a bit of notation. A finite random variable , henceforth just a "random variable," is a finite set $\mathcal{X}$ together with a map $p_X : \mathcal{X} \rightarrow [0, 1]$ satisfying $$\sum_{x \in \mathcal{X}} p_X(x) = 1.$$ The fu...

I came to physics research through general relativity, where the stress energy tensor plays a very important role, and where it has a single unambiguous meaning as the functional derivative of the theory with respect to metric perturbations. In flat-space quantum field theory, some texts present the stress tensor this way, while some present the stress tensor as a set of Noether currents associated with spatial translations. These definitions are usually presented as being equivalent, or rather, equivalent up to the addition of some total derivative that doesn't affect the physics. However, this is not actually the case. The two stress tensors differ by terms that can be made to vanish classically, but that have an important effect in the quantum theory. In particular, the Ward identities of the two different stress tensors are different. This has caused me a lot of grief over the years, as I've tried to compare equations between texts that use two different definitions of the...