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 The physics subjects are often explained in research papers that were written decades ago in now-outdated notation and terminology; the math subjects are explained in textbooks for mathematicians that mostly lack physical intuition.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.

Labels:

Comments

Post a Comment

Popular posts from this blog

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 ...
Post a Comment
Read more

Pick functions and operator monotones

Any time you can order mathematical objects, it is productive to ask what operations preserve the ordering. For example, real numbers have a natural ordering, and we have $x \geq y \Rightarrow x^k \geq y^k$ for any odd natural number $k$. If we further impose the assumption $y \geq 0,$ then order preservation holds for $k$ any positive real number. Self-adjoint operators on a Hilbert space have a natural (partial) order as well. We write $A \geq 0$ for a self-adjoint operator $A$ if we have $$\langle \psi | A | \psi \rangle \geq 0$$ for every vector $|\psi\rangle,$ and we write $A \geq B$ for self-adjoint operators $A$ and $B$ if we have $(A - B) \geq 0.$ Curiously, many operations that are monotonic for real numbers are not monotonic for matrices. For example, the matrices $$P = \frac{1}{2} \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$$ and $$Q = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$$ are both self-adjoint and positive, so we have $P+Q \geq P \geq 0$, but a str...
Post a Comment
Read more

Stone's theorem

 Stone's theorem is the basic result describing group-like unitary flows on Hilbert space. If the map $t \mapsto U(t)$ is continuous in a sense we will make precise later, and each $U(t)$ is a unitary map on a Hilbert space $\mathcal{H},$ and we have $U(t+s)=U(t)U(s),$ then Stone's theorem asserts the existence of a (self-adjoint, positive definite, unbounded) operator $\Delta$ satisfying $U(t) = \Delta^{it}.$ This reduces the study of group-like unitary flows to the study of (self-adjoint, etc etc) operators. Quantum mechanically, it tells us that every group-like unitary evolution is generated by a time-independent Hamiltonian. This lets us study very general symmetry transformations in terms of Hamiltonians. The standard proof of Stone's theorem, which you'll see if you look at Wikipedia , involves trying to make sense of a limit like $\lim_{t \to 0} (U(t) - 1)/t$. However, I have recently learned of a beautiful proof of Stone's theorem that works instead by stud...
Post a Comment
Read more