Lecture 23: A Toy Model of Black Hole Complementarity

In this lecture, we discuss a simple and precise model of black hole complementarity based on arXiv:1603.02812.  The formula that we wish to establish is:

f(x) = P(f(y_1) ... f(y_N))

where f(x) is a simple "local" field operator, and the points y_1, ... y_N are all spacelike separated from x and P is a "very complicated" polynomial.Nothing about this formula is special to black-holes, and the simple example we consider to establish the formula is empty anti-de Sitter space.

We take x to be a point in the center of AdS, and the points y_i to lie in an annulus defined by cot(T/2) < r < \infty. Clearly, all y_i are spacelike to x. Using a version of the Reeh-Schlieder theorem, we can write

f(x) = \sum_{i, j} X_i |0> <0| X_j^*

where X_i is a simple polynomial in the operators f(y_i). Nothing here violates locality, and we could have written down a similar formula in any quantum field theory.

What is special about gravity is that the projector on the vacuum that appears in the middle can be written as

|0> <0| = e^{-\alpha H}

where H is the Hamiltonian and \alpha is a suitably large number. But the Hamiltonian is purely a boundary term.  So it can be written in terms of the f(y_i) and the projector itself can be approximated by a very complicated polynomial, Q, that depends only on the operators f(y_i). This leads to a formula

f(x) = \sum_{i, j} X_i Q  X_j^*

which explicitly realizes the idea of complementarity!

Lecture 23 notes