One theory—or many?

Take a fixed quantum theory and evaluate its partition function on two disconnected boundary components. With independent copies and no extra coupling, the answer factorizes. That is the ordinary expectation.

Now look at a gravitational path integral. It may include connected geometries joining those boundaries. A wormhole contribution seems to know about both components at once.

What, then, is the boundary quantity that gravity is computing?

Statistics offers a natural sentence

For random variables, the average of a product need not equal the product of the averages. Their difference is a connected correlation. Schematically,

can be nonzero even when each individual member of the ensemble defines an ordinary factorizing theory. The same random choice of theory enters both factors.

This gives wormholes a plausible statistical vocabulary. A connected bulk contribution might encode a connected correlation of boundary data rather than a failure of quantum mechanics inside a single boundary theory.

The word “might” is doing necessary work. A suggestive structure is the start of an interpretation, not its complete definition.

Calculations that make the idea concrete

Cotler and Jensen computed an AdS₃ gravitational path integral on the topology of a torus times an interval and extracted spectral correlations, including near-threshold behavior suggestive of random-matrix structure.^[1] Their torus-wormhole calculation should not be casually described as an ordinary smooth real Einstein saddle; off-shell aspects of the gravitational prescription matter.

Chandra, Collier, Hartman, and Maloney subsequently developed an ensemble of large-central-charge CFT data, constrained by bootstrap information, which reproduces a variety of semiclassical gravitational results.^[2] This extends the statistical picture beyond a loose analogy.

There are also settings where an averaging measure is supplied explicitly. Narain CFTs provide one such laboratory.^[3] But the existence of a well-defined average in that family does not automatically supply the measure needed for pure Einstein gravity. The theories and the bulk constructions are different.

“An ensemble” of what?

The phrase can conceal several targets. We might be averaging exact, complete CFTs. We might have a statistical model of selected operator dimensions and OPE coefficients. We might only reproduce a particular family of observables in a controlled asymptotic regime.

These are all potentially useful. They are not the same claim.

To see the difference, imagine receiving a table of proposed means and correlations for several random quantities. Before declaring that a probability distribution exists, you must check that those numbers can be moments of one positive measure. A covariance matrix, for example, cannot have a negative quadratic form on an allowed test vector. Passing that test still does not settle every higher-moment requirement.

For CFTs the challenge is richer, because each sample is supposed to satisfy its own consistency conditions. Randomly selecting plausible dimensions and coefficients does not automatically produce crossing symmetry, unitarity, or a complete theory.

The question became more precise

The statistical perspective does not make the search for a boundary definition disappear. It tells us to specify a different possible object with equal care. What is sampled? Which observables are matched? At what precision? What is exact, and what is a large-\(c\) approximation?

That is progress even before the ultimate measure is in hand. We have replaced a slogan with a set of tests.

One difficulty remains especially practical: how do we compute sufficiently rich bulk amplitudes to test these ideas? A modern answer returns to the old themes of topology, boundary states, and gluing—but with Virasoro conformal blocks at the center of the machinery.