Skip to main content

Statement of purpose

Right now I'm a fourth year PhD student in theoretical physics, working at the interface of quantum information and quantum gravity. Many of the subjects I end up learning for my research lack good introductory references. The physics subjects are often explained in research papers that were written decades ago in now-outdated notation and terminology; the math subjects are explained in textbooks for mathematicians that mostly lack physical intuition. For aspiring physicists like me, it can be helpful to have concepts that are well-understood by experts re-interpreted and re-explained in concise, pedagogical terms.

While learning new math and physics subjects for my research, I often end up writing detailed "explainers" for myself that I think fit this niche. This blog will serve mostly as a repository for these explainers. I'll post explainers here as I write them in the hopes that they might be useful to other researchers trying to penetrate formidable subjects. At the very least, it'll be helpful for me to have this blog as a database I can pull from when a younger graduate student asks for references on a subject. Post types will likely include:

  • Detailed notes on fundamental math/physics concepts;
  • Summaries of interesting papers I read;
  • Explainers of my own papers, if I think they'll be helpful;
  • Videos and PDF notes of talks I give, both pedagogical and research-focused.

The selection of topics will depend entirely on what things I become interested in learning. My perspective is very geometric; I could accurately be called a "mathematical physicist," and my explainers prioritize visual or geometric intuition over computational power. I hope these notes will be helpful for people who think like I do.

Comments

Popular posts from this blog

Unbounded generators of unitary groups

Stone's theorem tells us when a one-parameter unitary group has a self-adjoint generator. If $U(t)$ is a group --- i.e., it satisfies $U(t_1 + t_2) = U(t_1) U(t_2)$ --- then we can write $U(t) = e^{i H t}$ for some unbounded $H$ if and only if we have $\lim_{t \to t_0} U(t) |\psi \rangle = U(t_0) |\psi\rangle$ for every $|\psi\rangle$ in Hilbert space. This is the condition that $U(t)$ is strongly continuous . What if we have a unitary group with multiple parameters? Within any one-parameter subgroup we should be able to find a generator. But is there a connection between the generators of different subgroups? Concretely, imagine we have a two-parameter subgroup that forms a representation of $\mathbb{R}^2.$ I.e., assume we have $$U(t_1 + t_2, s_1 + s_2) = U(t_1, s_1) U(t_2, s_2).$$ Assume further that the map from $\mathbb{R}^2$ to unitary operators is strongly continuous. Then for any one-parameter subgroup we have a generator, in particular, we have $$U(t,0) = e^{i H_1 t}$$ and...

Envelopes of holomorphy and the timelike tube theorem

Complex analysis, as we usually learn it, is the study of differentiable functions from $\mathbb{C}$ to $\mathbb{C}$. These functions have many nice properties: if they are differentiable even once then they are infinitely differentiable; in fact they are analytic, meaning they can be represented in the vicinity of any point as an absolutely convergent power series; moreover at any point $z_0$, the power series has radius of convergence equal to the radius of the biggest disc centered at $z_0$ which can be embedded in the domain of the function. The same basic properties hold for differentiable functions in higher complex dimensions. If $\Omega$ is a domain --- i.e., a connected open set --- in $\mathbb{C}^n$, and $f : \Omega \to \mathbb{C}^n$ is once differentiable, then it is in fact analytic, and can be represented as a power series in a neighborhood of any point $z_*$, i.e., we have an expression like $$f(z) = \sum a_{k_1 \dots k_n} (z_1 - z_*)^{k_1} \dots (z_n - z_*)^{k_n}.$$ The ...

Ward Identities

Ward identities are one of the most fundamental tools for studying quantum field theory, and they're encountered in almost any quantum field theory course. You've almost certainly encountered them before, so why should I bother writing about them? Simply put: despite learning how to derive Ward identities for the first time more than 5 years ago (in my first quantum field theory class, as an undergraduate at UChicago), I didn't really understand why they were important until quite recently. This is a product of my own unique research path — I haven't ever done any research in pure QFT, working instead mostly in quantum information and classical geometry, which means I haven't ever had to really understand what's going on under the hood in field theory. I don't think this oversight is so uncommon, so I'm putting together some basic thoughts on Ward identities in this post. So, what is a Ward identity? On its face, it's an equation that tells you how ...