Stefan Leichenauer

Light-sheets and AdS/CFT

One may ask whether the conformal field theory (CFT) restricted to a subset b of the anti-de Sitter (AdS) boundary has a well-defined dual restricted to a subset H(b) of the bulk geometry. The Poincare patch is an example, but more general choices of b can be considered. We propose a geometric construction of H. We argue that H should contain the set C of causal curves with both endpoints on b. Yet H should not reach so far from the boundary that the CFT has insufficient degrees of freedom to describe it. This can be guaranteed by constructing a superset L of H from light-sheets off boundary slices and invoking the covariant entropy bound in the bulk. The simplest covariant choice is L=L^+∩L^-, where L^+ (L^-) is the union of all future-directed (past-directed) light-sheets. We prove that C = L, so the holographic domain is completely determined by our assumptions: H = C = L. In situations where local bulk operators can be constructed on b, H is closely related to the set of bulk points where this construction remains unambiguous under modifications of the CFT Hamiltonian outside of b. Our construction leads to a covariant geometric renormalization-group flow. We comment on the description of black hole interiors and cosmological regions via AdS/CFT.

Null Geodesics, Local CFT Operators and AdS/CFT for Subregions

We investigate the nature of the AdS/CFT duality between a subregion of the bulk and its boundary. In global AdS/CFT in the classical G_N=0 limit, the duality reduces to a boundary value problem that can be solved by restricting to one-point functions of local operators in the conformal field theory (CFT). We show that the solution of this boundary value problem depends continuously on the CFT data. In contrast, the anti–de Sitter (AdS)-Rindler subregion cannot be continuously reconstructed from local CFT data restricted to the associated boundary region. Motivated by related results in the mathematics literature, we posit that a continuous bulk reconstruction is only possible when every null geodesic in a given bulk subregion has an endpoint on the associated boundary subregion. This suggests that a subregion duality for AdS-Rindler, if it exists, must involve nonlocal CFT operators in an essential way.

Quantum focusing conjecture

We propose a universal inequality that unies the Bousso bound with the classical focussing theorem. Given a surface that need not lie on a horizon, we dene a nite generalized entropy Sgen as the area of in Planck units, plus the von Neumann entropy of its exterior. Given a null congruence N orthogonal to , the rate of change of Sgen per unit area denes a quantum expansion. We conjecture that the quantum expansion cannot increase along N. This extends the notion of universal focussing to cases where quantum matter may violate the null energy condition. Integrating the conjecture yields a precise version of the Strominger-Thompson Quantum Bousso Bound. Applied to locally parallel light-rays, the conjecture implies a Quantum Null Energy Condition: a lower bound on the stress tensor in terms of the second derivative of the von Neumann entropy. We sketch a proof of this novel relation in quantum eld

Proof of the quantum null energy condition

We prove the quantum null energy condition (QNEC), a lower bound on the stress tensor in terms of the second variation in a null direction of the entropy of a region. The QNEC arose previously as a consequence of the quantum focusing conjecture, a proposal about quantum gravity. The QNEC itself does not involve gravity, so a proof within quantum field theory is possible. Our proof is somewhat nontrivial, suggesting that there may be alternative formulations of quantum field theory that make the QNEC more manifest. Our proof applies to free and super-renormalizable bosonic field theories, and to any points that lie on stationary null surfaces. An example is Minkowski space, where any point p and null vector ka define a null plane N (a Rindler horizon). Given any codimension-2 surface Σ that contains p and lies on N, one can consider the von Neumann entropy Sout of the quantum state restricted to one side of Σ. A second variation Sout′′ can be defined by deforming Σ along N, in a small neighborhood of p with area A. The QNEC states that ⟨Tkk(p)⟩≥ℏ2πlimA→0Sout′′/A.

Holographic proof of the quantum null energy condition

We use holography to prove the quantum null energy condition (QNEC) at leading order in large N for CFTs and relevant deformations of CFTs in Minkowski space which have Einstein gravity duals. Given any codimension-two surface Σ which is locally stationary under a null deformation in the direction k at the point p, the QNEC is a lower bound on the energy-momentum tensor at p in terms of the second variation of the entropy to one side of Σ: ⟨Tkk⟩≥S′′/2πh. In a CFT, conformal transformations of this inequality give results which apply when Σ is not locally stationary. The QNEC was proven previously for free theories, and taken together with our result this provides strong evidence that the QNEC is a true statement about quantum field theory in general.

A geometric solution to the coincidence problem, and the size of the landscape as the origin of hierarchy

Weinbergs seminal prediction of the cosmological constant relied on a provisional method for regulating eternal inflation which has since been put aside. We show that a modern regulator, the causal patch, improves agreement with observation, removes many limiting assumptions, and yields additional powerful results. Without assuming necessary conditions for observers such as galaxies or entropy production, the causal patch measure predicts the coincidence of vacuum energy and present matter density. Their common scale, and thus the enormous size of the visible Universe, originates in the number of metastable vacua in the landscape.

Eternal inflation predicts that time will end

Present treatments of eternal inflation regulate infinities by imposing a geometric cutoff. We point out that some matter systems reach the cutoff in finite time. This implies a nonzero probability for a novel type of catastrophe. According to the most successful measure proposals, our galaxy is likely to encounter the cutoff within the next 5x10{sup 9} years.

Star formation in the multiverse

We develop a simple semianalytic model of the star formation rate as a function of time. We estimate the star formation rate for a wide range of values of the cosmological constant, spatial curvature, and primordial density contrast. Our model can predict such parameters in the multiverse, if the underlying theory landscape and the cosmological measure are known.

Saturating the holographic entropy bound

The covariant entropy bound states that the entropy,

Limits on New Physics from Black Holes

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