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 domain in which the series converges is a *polydisc*, i.e., a product of $n$ discs for each variable individually. A version of Cauchy's integral formula holds for integration over the boundary of a polydisc:

$$f(z) = \frac{1}{(2 \pi i)^{n}} \int d\zeta_1 \dots d\zeta_n \frac{f(\zeta_1, \dots, \zeta_n)}{(\zeta_1 - z_1) \dots (\zeta_n - z_n)}.$$

There is one very important difference between complex analysis in one variable versus complex analysis in several variables, and it is a difference that has remarkable consequences for the structure of quantum field theory. In single-variable complex analysis, many important questions are questions of analytic continuation: given an analytic function $f$ defined in a domain $\Omega,$ does there exist a larger domain $\tilde{\Omega}$ in which $f$ has a unique analytic continuation?

In one variable, this is always a question about the function. **In several variables, it is sometimes a question about the domain. **More precisely, in $\mathbb{C}^n$ with $n \geq 2,$ there exist pairs of domains $\Omega \subsetneq \tilde{\Omega}$ such that every function $f$ analytic in $\Omega$ admits a unique analytic continuation to $\tilde{\Omega}.$ When this happens, $\tilde{\Omega}$ is called an **envelope of holomorphy** for $\Omega.$

We will see an explicit example momentarily, but let us first consider the physical consequences of this extraordinary fact. In mathematical approaches to quantum field theory, the fundamental objects of interest are field expectation values in a state $\omega,$

$$\langle \phi(x_1) \dots \phi(x_n) \rangle_{\omega}.$$

In $d$-dimensional Minkowski spacetime, these correlation functions can be thought of as complex-valued functions of $(\mathbb{R}^d)^n.$ (They are in fact distributions, not functions, but this will not be so important for us right now.)

Under certain natural assumptions --- Lorentz invariance and the Hamiltonian being bounded below --- one can often show that the correlation function $\langle \phi(x_1) \dots \phi(x_n) \rangle_\omega$ admits an analytic continuation to some domain in $(\mathbb{C}^d)^n.$ Because of the phenomenon of envelopes of holomorphy, it often suffices to begin with a very modest analytic continuation of the correlation function, then to invoke general principles to show that it must be continuable to a much larger domain. Analytic functions are highly constrained, so simply knowing that correlation functions are analytic in some particular domain is enough to have consequences for the physics of quantum fields.

The purpose of this post is to explain one such physical consequence, which is the **timelike tube theorem** proved by Borchers. Recently, a generalization due to Strohmaier and Witten has attracted interest due to an apparent connection with bulk reconstruction in holographic quantum gravity. We will content ourselves with the simplest version of the timelike tube theorem, which concerns a timelike segment in two-dimensional Minkowski spacetime with a scalar quantum field. Generalizations to higher-dimensional Minkowski, to more general timelike curves, or to non-scalar fields, are just matters of bookkeeping.

Consider a causal diamond with spatial extent $x \in [-1, 1]$ and time extent $t \in [-1, 1]$, sketched in the figure above. Consider also the vertical timelike segment $x=0, t=[-1, 1],$ which we will call $\gamma.$ The timelike tube theorem says that field operators anywhere in the diamond can be approximated arbitrarily well by field operators in an arbitrarily small neighborhood of $\gamma.$ An equivalent formulation is that if an operator $O$ commutes with every field operator $\phi(x)$ for $x$ in an arbitrarily small neighborhood of $\gamma,$ then $O$ must commute with every field operator $\phi(x)$ in the full causal diamond.

In the rest of this post, we will learn some basic facts about envelopes of holomorphy, then prove the version of the timelike tube theorem explained above. I learned about envelopes of holomorphy from Vladimirov's excellent book (now available from Dover under a slightly different name), and about the timelike tube theorem by plugging Borchers's original German-language paper line-by-line into Google translate.

**Prerequisites: **Complex analysis and basic quantum field theory.

__Table of Contents__

## 1. A simple envelope of holomorphy

*does*have an analytic continuation to the disc $|z| < 1.$ By Cauchy's integral formula, for $|z| < 3/4,$ the analytic continuation must be given by the contour integral

**does**give an analytic function in $|z| < 3/4,$ and by moving the contour outwards, we can use it to define an analytic function in the full disc $|z| < 1.$ The thing that goes wrong is that this newly defined analytic function is not necessarily an analytic continuation of the original function $f$! For example, given $f(z) = 1/z,$ it is a straightforward exercise to check that the contour integral evaluates to zero everywhere. This is an analytic function in the disc, but it does not continue the function we started with.

*full filling-in of that circle to a disk is in the annulus*. In other words, as we drag the contour around the "hole" in the annulus, eventually its interior is contained in the annulus. Because $f$ is analytic in the annulus, Cauchy's integral formula tells us that the left- and right-hand sides of the above equation agree in this regime. Since these functions are analytic, they agree everywhere, and we have given an explicit formula showing how to analytically continue any function in the annulus to a function in the full ball.

## 2. The timelike tube theorem

with

$$c^2 = (x_0 - 1)^2 - t_0^2.$$

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