What Leads To Factorization Problem? How Half Wormholes Can Fix It? (Maths / Cosmology / Quantum)

Wormholes not only play a key role in understanding the nonperturbative physics of quantum black holes, for instance: the eternal traversable wormhole; the long-time behavior of the spectral form factor and correlation functions, the Page curve etc. but also, it leads to puzzles, in particular the factorization problem. Imagine two decoupled boundary systems in the AdS/CFT context, labelled L and R. From the boundary perspective, if one evaluates a partition function in the combined system the result is just the product of the results for the two component systems:

It factorizes. But, if the bulk calculation of ZLR includes a wormhole linking L and R then superficially at least ZLR ≠ ZLZR. It fails to factorize. Some of the phenomena recently explained by wormholes, in particular the spectral form factor and squared matrix elements, are described by decoupled boundary systems and so the wormhole explanation give rise to a factorization puzzle.

But, you can remove this factorization puzzle by averaging the L and R systems over the same ensemble, denoted by (·), with the help of the SYK model. The factorization puzzle solves because (ZLZR) need not to be same as (ZL) (ZR). And infact this link between wormholes and ensembles is not a new one, it dates back to the 1980s. However, it has been recently applied in AdS/CFT context.

We can create a new form of factorization puzzle in such ensembles by asking what happens to the wormholes connecting decoupled systems when we focus on just 1 element of the ensemble. Now, Phil Saad and colleagues addressed this question in the SYK model where instead of averaging the L and R systems they picked a fixed set of couplings between the fermions.

These pictures represent saddle points of the SYK path integral, associated to the sketched bulk topology by the pattern of correlation. As the wormhole contribution is self-averaging, they have depicted it with a small red “x” to indicate the small amount of randomness. The half-wormhole contributions are not self-averaging, so they have depicted them as “half” of a wormhole with a jagged red boundary to indicate the large amount of randomness. They have included a red line linking the pair of half-wormholes on the LHS, to remind them that the LR collective fields are present, but set to zero, distinguishing this contribution from the unlinked pair of half wormholes on the RHS. © Phil Saad et al.

After averaging over fermion couplings, SYK model has a collective fields called G and Σ, that sometimes has “wormhole” solutions. Phil Saad and colleagues studied the fate of these wormholes when the couplings are fixed.

Working mainly in a simple model, they found that the wormhole saddles persist, and the dependence on the couplings is weak. The wormhole is “self-averaging”. But, that new saddles also appear elsewhere in the integration space, which they interpret as “half-wormholes.” The half-wormhole contributions depends sensitively on the particular choice of couplings.

Finally, they showed that, the half-wormholes are crucial for factorization (or restore factorization) of decoupled systems with fixed couplings, but they vanish after averaging, leaving the non-factorizing wormhole behind.

Reference: Phil Saad, Stephen H. Shenker, Douglas Stanford, and Shunyu Yao, “Wormholes without averaging”, Arxiv, pp. 1-34, 2021.

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