Information, Geometry, and Physics Seminar

Understanding the limit of infinitely large discrete systems is notoriously hard and one often resorts to methods of effective field theory that try to describe the large scale structure of the system by the introduction of an effective description that approximates "low energy/large distance" observables in terms of quantum field theories. While it is possible to perform phenomenological studies in such terms, a more microscopic and formal description of the infinitely large system is difficult to obtain and precisely how the quantum field theory emerges from it is often left implicit. In particular, it is unclear what properties of the discrete system can lead to relativistic invariance of the effective description. In the context of AdS/CFT two related ideas appeared: First, that one should think of the (from the boundary perspective) emergent radial direction as a renormalization group (RG) flow that encodes long distance physics of the boundary as degrees of freedom that live deep inside the bulk and building on this idea, that the mechanism by which the bulk is encoded in the boundary is in terms of an quantum error correcting code, where degrees of freedom that live deep inside the "entanglement wedge" (the region dual to a boundary region) are well protected against errors that take place in the complementary boundary region. A perspective on the renormalization group that grew out of these considerations is that one can think of the renormalization group as an approximate quantum error correcting code which encodes long distance physics in the Hilbert space of UV degrees of freedom and protects them against errors that arise from these UV degrees of freedom. In this talk, I will describe how one can combine these ideas using tensor networks that are believed to implement RG flows, to obtain an explicit and well defined description of the infinitely large network in terms a net of von Neumann algebras that describe subsystems of the limiting system. I will argue that from this point of view, there is little hope for the HaPPY code to give rise to a CFT as a limiting description and what qualitative difference networks implementing the MERA have that might allow for the possibility of an effective description in terms of a quantum field theory. If time permits I will comment on a relationship between the type classification of von Neumann algebras and how our work gives rise to the idea that, to prepare states that look like vacua of quantum field theories, it is necessary that they have a high degree of "magic", a quantum computational resource that makes these states hard to simulate by classical computers. This talk is based on https://arxiv.org/abs/2504.00096