Skip to main content
eScholarship
Open Access Publications from the University of California

Scholarly Works (11 results)

  • Thesis
  • Peer Reviewed
UC Santa Barbara Electronic Theses and Dissertations (2023)

The gravitational path integral has long served as a crucial tool in deciphering mysteries within quantum gravity. In recent years, studies of the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence have offered many valuable insights into comprehending those mysteries, and many fruitful results have been yielded from utilizing the gravitational path integral within the framework of AdS/CFT.

This dissertation is devoted to studying certain aspects of the gravitational path integral, discussing its relation with gravitational entropies, spacetime geometries, and its algebraic aspects. We explore contexts from Euclidean to Lorentz signature, from holographic theories to general theories, with a goal of understanding quantum gravity in the real world.

In Part I, we discuss the fixed-(HRT)-area states in the gravitational path integral. The fixed-area states are holographic states where the area of the Hubeny-Rangamani-Takayanagi (HRT) surface, the holographic dual of entanglement entropy for a region in the boundary CFT, is constrained to a small window when prepared by the gravitational path integral. The study of those fixed-area states helps understand quantum gravity beyond the leading semiclassical order. We first show that by decomposing a general holographic state into fixed-area states, an important subleading correction appears to the entanglement entropy near phase transitions. Then we explore the intrinsic spacetime geometries of fixed-area states under Lorentz-signature time evolution.

In Part II, we study saddle-point geometries of the real-time gravitational path integral, in the context of computing holographic Renyi entropies. Unlike their Euclidean counterparts, these real-time saddles necessarily have complex metrics, giving an example where the saddle point is off the original contour of integration. We first present the formalism of this setup, illustrating the relevant variational problem, and features of the complex saddles. Then we demonstrate explicitly the structure of those saddles by showing examples in low dimensions by direct calculation. We also find that it is possible to deform the original integration contour to pass through saddles of this kind constructed in two-dimensional Jackiw-Teitelboim gravity. Finally, we show that the existence of these saddles results in a consequence which is necessary for unitarity to hold in quantum gravity.

In Part III, we take a step towards explaining the origin of gravitational entropies, by utilizing the mathematical tool of von Neumann algebras. In particular, we give an explanation of the HRT formula purely from the bulk perspective, without making any reference to holography. This is done by constructing Hilbert spaces and von Neumann algebras from boundary conditions of the gravitational path integral with several natural axioms. The von Neumann algebras we find from this construction allows us to define a notion of entropy, which matches the HRT formula in the semiclassical limit. One of the axioms we assume which is crucial for the construction of von Neumann algebras -- the trace inequality, is proven in the semiclassical limit, and it leads to novel positivity conjectures for the gravitational action.

    Cover page: Unraveling Quantum Gravity through the Gravitational Path Integral: Geometries, Entropies, and Algebras
    • Article
    • Peer Reviewed
    UC Santa Barbara Previously Published Works (2021)
    Abstract : The Hubeny-Rangamani-Takayanagi surface γHRT computing the entropy S(D) of a domain of dependence D on an asymptotically AdS boundary is known to be causally inaccessible from D. We generalize this gravitational result to higher replica numbers n > 1 by considering the replica-invariant surfaces (aka ‘splitting surfaces’) γ of real-time replica-wormhole saddle-points computing Rényi entropies Sn(D) and showing that there is a sense in which D must again be causally inaccessible from γ when the saddle preserves both replica and conjugation symmetry. This property turns out to imply the Sn(D) to be independent of any choice of any Cauchy surface ΣD for D, and also that the Sn(D) are independent of the choice of boundary sources within D. This is a key hallmark of unitary evolution in any dual field theory. Furthermore, from the bulk point of view it adds to the evidence that time evolution of asymptotic observables in quantum gravity is implemented by a unitary operator in each baby universe superselection sector. Though we focus here on pure Einstein-Hilbert gravity and its Kaluza-Klein reductions, we expect the argument to extend to any two-derivative theory who satisfies the null convergence condition. We consider both classical saddles and the effect of back-reaction from quantum corrections.
      Creative Commons 'BY' version 4.0 license
      • Article
      • Peer Reviewed
      UC Santa Barbara Previously Published Works (2020)
      Abstract : Recent results suggest that new corrections to holographic entanglement entropy should arise near phase transitions of the associated Ryu-Takayanagi (RT) surface. We study such corrections by decomposing the bulk state into fixed-area states and conjecturing that a certain ‘diagonal approximation’ will hold. In terms of the bulk Newton constant G, this yields a correction of order O(G−1/2) near such transitions, which is in particular larger than generic corrections from the entanglement of bulk quantum fields. However, the correction becomes exponentially suppressed away from the transition. The net effect is to make the entanglement a smooth function of all parameters, turning the RT ‘phase transition’ into a crossover already at this level of analysis. We illustrate this effect with explicit calculations (again assuming our diagonal approximation) for boundary regions given by a pair of disconnected intervals on the boundary of the AdS3 vacuum and for a single interval on the boundary of the BTZ black hole. In a natural large-volume limit where our diagonal approximation clearly holds, this second example verifies that our results agree with general predictions made by Murthy and Srednicki in the context of chaotic many-body systems. As a further check on our conjectured diagonal approximation, we show that it also reproduces the O(G−1/2) correction found Penington et al. for an analogous quantum RT transition. Our explicit computations also illustrate the cutoff-dependence of fluctuations in RT-areas.
        Cover page: Probing phase transitions of holographic entanglement entropy with fixed area states
        • Article
        • Peer Reviewed
        UC Santa Barbara Previously Published Works (2019)
        Abstract : The AdS/CFT understanding of CFT entanglement is based on HRT surfaces in the dual bulk spacetime. While such surfaces need not exist in sufficiently general spacetimes, the maximin construction demonstrates that they can be found in any smooth asymptotically locally AdS spacetime without horizons or with only Kasner-like singularities. In this work, we introduce restricted maximin surfaces anchored to a particular boundary Cauchy slice C ∂ . We show that the result agrees with the original unrestricted maximin prescription when the restricted maximin surface lies in a smooth region of spacetime. We then use this construction to extend the existence theorem for HRT surfaces to generic charged or spinning AdS black holes whose mass-inflation singularities are not Kasner-like. We also discuss related issues in time-independent charged wormholes.
          Cover page: Restricted Maximin surfaces and HRT in generic black hole spacetimes
          • Article
          • Peer Reviewed
          UC Santa Barbara Previously Published Works (2021)
          Abstract : We generalize the Gao-Jafferis-Wall construction of traversable two-sided wormholes to multi-boundary wormholes. In our construction, we take the background spacetime to be multi-boundary black holes in AdS3. We work in the hot limit where the dual CFT state in certain regions locally resembles the thermofield double state. Furthermore, in these regions, the hot limit makes the causal shadow exponentially small. Based on these two features of the hot limit, and with the three-boundary wormhole as our main example, we show that traversability between any two asymptotic regions in a multi-boundary wormhole can be triggered using a double-trace deformation. In particular, the two boundary regions need not have the same temperature and angular momentum. We discuss the non-trivial angular dependence of traversability in our construction, as well as the effect of the causal shadow region.
            Cover page: Traversability of multi-boundary wormholes
            • Article
            • Peer Reviewed
            UC Santa Barbara Previously Published Works (2024)
            Abstract : The AdS/CFT correspondence states that certain conformal field theories are equivalent to string theories in a higher-dimensional anti-de Sitter space. One aspect of the correspondence is an equivalence of density matrices or, if one ignores normalizations, of positive operators. On the CFT side of the correspondence, any two positive operators A, B will satisfy the trace inequality Tr(AB) ≤ Tr(A)Tr(B). This relation holds on any Hilbert space $$ \mathcal{H} $$ H and is deeply associated with the fact that the algebra B($$ \mathcal{H} $$ H ) of bounded operators on $$ \mathcal{H} $$ H is a type I von Neumann factor. Holographic bulk theories must thus satisfy a corresponding condition, which we investigate below. In particular, we argue that the Euclidean gravitational path integral respects this inequality at all orders in the semi-classical expansion and with arbitrary higher-derivative corrections. The argument relies on a conjectured property of the classical gravitational action, which in particular implies a positive action conjecture for quantum gravity wavefunctions. We prove this conjecture for Jackiw-Teitelboim gravity and we also motivate it for more general theories.
              Creative Commons 'BY' version 4.0 license
              • Article
              • Peer Reviewed
              UC Santa Barbara Previously Published Works (2018)
              We study the "complexity equals volume" (CV) and "complexity equals action" (CA) conjectures by examining moments of of time symmetry for $\rm AdS_3$ wormholes having $n$ asymptotic regions and arbitrary (orientable) internal topology. For either prescription, the complexity relative to $n$ copies of the $M=0$ BTZ black hole takes the form $\Delta C = \alpha c \chi $, where $c$ is the central charge and $\chi$ is the Euler character of the bulk time-symmetric surface. The coefficients $\alpha_V = -4\pi/3$, $\alpha_A = 1/6 $ defined by CV and CA are independent of both temperature and any moduli controlling the geometry inside the black hole. Comparing with the known structure of dual CFT states in the hot wormhole limit, the temperature and moduli independence of $\alpha_V$, $\alpha_A$ implies that any CFT gate set defining either complexity cannot be local. In particular, the complexity of an efficient quantum circuit building local thermofield-double-like entanglement of thermal-sized patches does not depend on the separation of the patches so entangled. We also comment on implications of the (positive) sign found for $\alpha_A$, which requires the associated complexity to decrease when handles are added to our wormhole.
                Cover page: Holographic complexity is nonlocal
                • Article
                • Peer Reviewed
                UC Santa Barbara Previously Published Works (2022)
                Abstract : The concept of fixed-area states has proven useful for recent studies of quantum gravity, especially in connection with gravitational holography. We explore the Lorentz-signature spacetime geometry intrinsic to such fixed-area states in this paper. This contrasts with previous treatments which focused instead on Euclidean-signature saddles for path integrals that prepare such states. We analyze general features of fixed-area state geometries and construct explicit examples. The spacetime metrics are real at real times and have no conical singularities. With enough symmetry the classical metrics are in fact smooth, though more generally their curvatures feature power-law divergences along null congruences launched orthogonally from the fixed-area surface. While we argue that such divergences are not problematic at the classical level, quantum fields in fixed-area states feature stronger divergences. At the quantum level we thus expect fixed-area states to be well-defined only when the fixed-area surface is appropriately smeared.
                  Creative Commons 'BY' version 4.0 license
                  • Article
                  • Peer Reviewed
                  UC Davis Previously Published Works (2021)
                  We continue the study of real-time replica wormholes initiated in arXiv:2012.00828. Previously, we had discussed the general principles and had outlined a variational principle for obtaining stationary points of the real-time gravitational path integral. In the current work we present several explicit examples in low-dimensional gravitational theories where the dynamics is amenable to analytic computation. We demonstrate the computation of R enyi entropies in the cases of JT gravity and for holographic two-dimensional CFTs (using the dual gravitational dynamics). In particular, we explain how to obtain the large central charge result for subregions comprising of disjoint intervals directly from the real-time path integral.
                    Cover page: Real-time gravitational replicas: Low dimensional examplesCreative Commons 'BY-NC-ND' version 4.0 license
                    • Article
                    • Peer Reviewed
                    UC Davis Previously Published Works (2021)
                    This work is the first step in a two-part investigation of real-time replica wormholes. Here we study the associated real-time gravitational path integral and construct the variational principle that will define its saddle-points. We also describe the general structure of the resulting real-time replica wormhole saddles, setting the stage for construction of explicit examples. These saddles necessarily involve complex metrics, and thus are accessed by deforming the original real contour of integration. However, the construction of these saddles need not rely on analytic continuation, and our formulation can be used even in the presence of non-analytic boundary-sources. Furthermore, at least for replica- and CPT-symmetric saddles we show that the metrics may be taken to be real in regions spacelike separated from a so-called ‘splitting surface’. This feature is an important hallmark of unitarity in a field theory dual.
                      Cover page: Real-time gravitational replicas: formalism and a variational principle
                      Top