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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

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

Vector integration

Lately I've been thinking a lot about algebras of operators acting on a Hilbert space, since they provide an extremely useful tool for thinking about locality in quantum field theory. I'm working on a review article about Tomita-Takesaki modular theory to supplement my recent review on the type classification of von Neumann algebras . The core object of study in Tomita-Takesaki theory is a one-parameter group of unitary operators $\Delta^{it},$ generated by a single positive (often unbounded) operator $\Delta.$ In physics, the Tomita-Takesaki unitaries furnish a "hidden thermodynamic symmetry" of a physical state. A lot of interesting physics and mathematics can be learned by studying the analytic structure of the function $z \mapsto \Delta^{z}$ for generic complex $z,$ or of the function $z \mapsto \Delta^{z} |\psi\rangle$ for generic complex $z$ and some fixed state $|\psi\rangle.$ But in order to do this, we need to understand how to do complex analysis for operat