Research Notes

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

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

My posting frequency has decreased since grad school, since while I'm spending about as much time learning as I always have, much more of my pedagogy these days ends up in papers. But I've given a few pedagogically-oriented talks recently that may be of interest to the people who read this blog. I gave a mini-course on "the algebraic approach" at Bootstrap 2024. The lecture notes can be found here , and videos are available here . The first lecture covers the basic tools of algebraic quantum field theory; the second describes the Faulkner-Leigh-Parrikar-Wang argument for the averaged null energy condition in Minkowski spacetime; the third describes recent developments on the entropy of semiclassical black holes, including my recent paper with Chris Akers . Before the paper with Chris was finished, I gave a general overview of the "crossed product" approach to black hole entropy at KITP. The video is available here . The first part of the talk goes back in ti

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

 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