Nima Lashkari

A bstract: We write down the global Hamiltonian of conformal field theory (CFT) in finite volume in terms of the modular Hamiltonian of the vacuum reduced to a local ball-shaped region, and use it to propose local approximations to the global Minkowski Hamiltonian in quantum field theory (QFT). The proposed Hamiltonians are motivated by the operator-algebraic property of nuclearity. They are constructed from the characteristic operators of inclusions of algebras and can be viewed as regulators of the modular Hamiltonian of local algebras of QFT.

Fliss [.] extended the exact renormalization group (ERG) to arbitrary wave functionals in quantum field theory. Applying this formalism, we show that a Lindblad master equation gives the ERG flow of density matrices. The Lindbladian consists of a “Hamiltonian” term, which is the sum of a scaling and a coarse-graining (disentangling) operator, and a dissipative term with absorption and emission rates for each momentum mode. We consider as examples the flow of Gaussian states and the perturbative ground state of λ ϕ 4 theory and highlight the role of the dissipative terms in generating accurate contributions to the flow of couplings at nontrivial order in λ . Integrating the Lindblad master equation, we find that a finite ERG flow of density matrices is described by a quantum channel. It follows from the data processing inequality that any distinguishability measure of states is an ERG monotone.

A bstract: We compute the modular flow and conjugation of time interval algebras of conformal Generalized Free Fields (GFF) in 0 + 1-dimensions in the vacuum. For non-integer scaling dimensions, for general time intervals, the modular conjugation and the modular flow of operators outside the algebra are non-geometric. This is because they involve a Generalized Hilbert Transform (GHT) that treats positive and negative frequency modes differently. However, the modular conjugation and flows viewed in the dual bulk AdS ₂ are local, because the GHT geometrizes as the local antipodal symmetry transformation that pushes operators behind the Poincaré horizon. These algebras of conformal GFF satisfy a Twisted Modular Inclusion and a Twisted Modular Intersection property. We prove the converse statement that the existence of a (twisted) modular inclusion/intersection in any quantum system implies a representation of the (universal cover of) conformal group PSL(2 , ℝ), respectively. We discuss the implications of our result for the emergence of AdS ₂ geometries in large N theories without a large gap. Our result applied to higher dimensional eternal AdS black holes explains the emergence of two copies of PSL(2 , ℝ) on future and past Killing horizons.

In quantum gravity, the gravitational path integral involves a sum over topologies, representing the joining and splitting of multiple universes. To account for topology change, one is led to allow the creation and annihilation of closed and open universes in a framework often called third quantization or universe field theory. We argue that since topology change in gravity is a rare event, its contribution to exponentially late-time physics is universally described by a Poisson process. This universal Poisson process is responsible for the plateau in the multi-boundary generalization of the spectral form factor at late times. In the Fock space of closed baby universes, this universality implies that the statistics of the total number of baby universes is captured by a coherent state. Allowing for the creation of asymptotic open universes calls for a noncommutative generalization of a Poisson process. We propose such an operator algebraic framework, called Poissonization, which takes as input the observable algebra and a (unnormalized) state of a quantum system and outputs a von Neumann algebra of a many-body theory represented on its symmetric Fock space. Physically, Poissonization is a generalization of the coherent state vacua of bipartite quantum systems or matrix quantum mechanics. The multi-boundary correlators of the Marolf-Maxfield toy model of baby universes, and the sum over bordisms of closed-open 2D topological quantum field theory, are entirely captured by Poissonization. In the Jackiew-Teitelboim gravity, the universal Poisson process reproduces the late-time correlators in the $τ$-scaling limit.

The local two-dimensional Poincaré algebra near the horizon of an eternal AdS black hole, or in proximity to any bifurcate Killing horizon, is generated by the Killing flow and outward null translations on the horizon. In holography, this local Poincaré algebra is reflected as a pair of unitary flows in the boundary Hilbert space whose generators under modular flow grow and decay exponentially with a maximal Lyapunov exponent. This is a universal feature of many geometric vacua of quantum gravity. To explain this universality, we show that a two-dimensional Poincaré algebra emerges in any quantum system that has von Neumann subalgebras associated with half-infinite modular time intervals (modular future and past subalgebras) in a limit analogous to the near-horizon limit. In ergodic theory, quantum dynamical systems with future or past algebras are called quantum K-systems. The surprising statement is that modular K-systems are always maximally chaotic. Interacting quantum systems in the thermodynamic limit and large N theories above the Hawking-Page phase transition are examples of physical theories with future/past subalgebras. We prove that the existence of (modular) future/past von Neumann subalgebras also implies a second law of (modular) thermodynamics and the exponential decay of (modular) correlators. We generalize our results from the modular flow to any dynamical flow with a positive generator and interpret the positivity condition as quantum detailed balance.

The exact renormalization group (ERG) is a powerful tool for understanding the formal properties of field theories. By adapting generalized ERG schemes to the flow of wave functionals, we obtain a large class of continuous unitary networks, a special case of which includes a class of Gaussian continuous multiscale renormalization Ansätze (cMERAs). The novel feature of these generalized wave functional ERG schemes is allowing for modifications of the dispersion relation, which drastically changes the entanglement structure of the ultraviolet states. Through our construction, we demonstrate that cMERA can be derived from a more fundamental “microscopic” principle, which amounts to the usual RG principle of path integral independence, suitably adapted to quantum states of the field theory. The establishment of such a principle may provide a path forward for exploring cMERA beyond the free field regime, and for understanding the nature of entanglement renormalization intrinsically in the continuum.

A manifestation of the black hole information loss problem is that the two-point function of probe operators in a large Anti-de Sitter black hole decays in time, whereas, on the boundary CFT, it is expected to be an almost periodic function of time. We point out that the decay of the two-point function (clustering in time) holds important clues to the nature of observable algebras, states, and dynamics in quantum gravity.We call operators that cluster in time “mixing” and explore the necessary and sufficient conditions for mixing. The information loss problem is a special case of the statement that in type I algebras, there exists no mixing operators. We prove that, in a thermofield double state (KMS state), if mixing operators form an algebra (close under multiplication), the resulting algebra must be a von Neumann type III₁ factor. In other words, the physically intuitive requirement that all nonconserved operators should exponentially mix is so strong that it fixes the observable algebra to be an exotic algebra called a type III₁ factor. More generally, for an arbitrary out-of-equilibrium state of a general quantum system (von Neumann algebra), we show that if the set of operators that mix under modular flow forms an algebra, it is a type III₁ von Neumann factor.In a theory of Generalized Free Fields (GFF), we show that if the two-point function clusters in time, all operators are mixing, and the algebra is a type III₁ factor. For example, in 𝒩 = 4 SYM, above the Hawking-Page phase transition, clustering of the single trace operators implies that the algebra is a type III₁ factor, settling a recent conjecture of Leutheusser and Liu. We explicitly construct the C^∗-algebra and von Neumann subalgebras of GFF associated with time bands and, more generally, open sets of the bulk spacetime using the HKLL reconstruction map.