A common feature of the extended phase space of gauge theory, the crossed product of quantum theory, and quantum reference frames (QRFs) is the adjoining of degrees of freedom followed by a constraining procedure for the resulting total system. Building on previous work, we identify non-commutative or quantum geometry as a mathematical framework which unifies these three objects. We first provide a rigorous account of the extended phase space, and demonstrate that it can be regarded as a classical principal bundle with a Poisson manifold base. We then show that the crossed product is a trivial quantum principal bundle which both substantiates a conjecture on the quantization of the extended phase space and facilitates a relational interpretation. Combining several crossed products with possibly distinct structure groups into a single object, we arrive at a novel definition of a quantum orbifold. We demonstrate that change of frame maps within the quantum orbifold corresponds to quantum gauge transformations, which are QRF preserving maps between crossed product algebras. Finally, we conclude that the quantum orbifold is equivalent to the G-framed algebra proposed in prior work, thereby placing systems containing multiple QRFs squarely in the context of quantum geometry.

United States

Shadi Ali Ahmad

Wissam Chemissany

Marc S. Klinger

Robert G. Leigh

Relational quantum geometry

We establish an equivalence between non-isometry of quantum codes and state dependence of operator reconstruction, and discuss implications of this equivalence for holographic duality. Specifically, we define quantitative measures of non-isometry and state dependence and describe bounds relating these quantities. In the context of holography we show that, assuming known gravitational path integral results for overlaps between semiclassical states, non-isometric bulk-to-boundary maps with a trivial kernel are approximately isometric and bulk reconstruction approximately state-independent. In contrast, non-isometric maps with a non-empty kernel always lead to state-dependent reconstruction. We also show that if a global bulk-to-boundary map is non-isometric, then there exists a region in the bulk which is causally disconnected from the boundary. Finally, we conjecture that, under certain physical assumptions for the definition of the Hilbert space of effective field theory in AdS space, the presence of a global horizon implies a non-isometric global bulk-to-boundary map.

Stefano Antonini

United Kingdom

Belgium

Vijay Balasubramanian

Ning Bao

ChunJun Cao

Non-isometry, state dependence and holography

All physical observations are made relative to a reference frame, which is a system in its own right. If the system of interest admits a group symmetry, the reference frame observing it must transform commensurately under the group to ensure the covariance of the combined system. We point out that the crossed product is a way to realize quantum reference frames from the bottom-up; adjoining a quantum reference frame and imposing constraints generates a crossed product algebra. We provide a top-down specification of crossed product algebras and show that one cannot obtain inequivalent quantum reference frames using this approach. As a remedy, we define an abstract algebra associated to the system and symmetry group built out of relational crossed product algebras associated with different choices of quantum reference frames. We term this object the <math><mi>G</mi></math>-framed algebra, and show how potentially inequivalent frames are realized within this object. We comment on this algebra’s analog of the classical Gribov problem in gauge theory, its importance in gravity where we show that it is relevant for semiclassical de Sitter and potentially beyond the semiclassical limit, and its utility for understanding the frame dependence of physical notions like observables, density states, and entropies.

Quantum reference frames from top-down crossed products

We study the connections between three quantities that can be used as diagnostics for quantum chaos, i.e., the out-of-time-order correlator (OTOC), Loschmidt echo (LE), and complexity. We generalize the connection between OTOC and LE for infinite dimensions and extend it for higher-order OTOCs and multi-fold LEs. Novel applications of this intrinsic relation are proposed. We also investigated the relationship between a specific circuit complexity and LE by using the inverted oscillator model and made a conjecture about their relationship. These relationships signal a deeper connection between these three probes of quantum chaos.

India

Arpan Bhattacharyya

South Africa

S. Haque

Bin Yan

Towards the web of quantum chaos diagnostics

We discuss, from a quantum information perspective, recent proposals of Maldacena, Ryu, Takayanagi, van Raamsdonk, Swingle, and Susskind that spacetime is an emergent property of the quantum entanglement of an associated boundary quantum system. We review the idea that the informational principle of minimal complexity determines a dual holographic bulk spacetime from a minimal quantum circuit U preparing a given boundary state from a trivial reference state. We describe how this idea may be extended to determine the relationship between the fluctuations of the bulk holographic geometry and the fluctuations of the boundary low-energy subspace. In this way we obtain, for every quantum system, an Einstein-like equation of motion for what might be interpreted as a bulk gravity theory dual to the boundary system.

Germany

Tobias Osborne

Holographic fluctuations and the principle of minimal complexity

We provide a general algorithm for constructing the holographic dictionary for any asymptotically locally Lifshitz background, with or without hyperscaling violation, and for any values of the dynamical exponents z and θ , as well as the vector hyperscaling violating exponent [1, 2], that are compatible with the null energy condition. The analysis is carried out for a very general bottom up model of gravity coupled to a massive vector field and a dilaton with arbitrary scalar couplings. The solution of the radial Hamilton-Jacobi equation is obtained recursively in the form of a graded expansion in eigenfunctions of two commuting operators [3], which are the appropriate generalization of the dilatation operator for non scale invariant and Lorentz violating boundary conditions. The Fefferman-Graham expansions, the sources and 1-point functions of the dual operators, the Ward identities, as well as the local counterterms required for holographic renormalization all follow from this asymptotic solution of the radial Hamilton-Jacobi equation. We also find a family of exact backgrounds with z &gt; 1 and θ &gt; 0 corresponding to a marginal deformation shifting the vector hyperscaling violating parameter and we present an example where the conformal anomaly contains the only z = 2 conformal invariant in d = 2 with four spatial derivatives.

Spain

Ioannis Papadimitriou

Lifshitz holography: the whole shebang

We present a general algorithm for constructing the holographic dictionary for Lifshitz and hyperscaling violating Lifshitz backgrounds for any value of the dynamical exponent z and any value of the hyperscaling violation parameter θ compatible with the null energy condition. The objective of the algorithm is the construction of the general asymptotic solution of the radial Hamilton–Jacobi equation subject to the desired boundary conditions, from which the full dictionary can be subsequently derived. Contrary to the relativistic case, we find that a fully covariant construction of the asymptotic solution for running non-relativistic theories necessitates an expansion in the eigenfunctions of two commuting operators instead of one. This provides a covariant but non-relativistic grading of the expansion, according to the number of time derivatives.

Generalized dilatation operator method for non-relativistic holography

We investigate the recently proposed possibility of chaotic inflation with respect to the imaginary part of the field T in a supersymmetric embedding of the Starobinsky model. We show that the stage of rapid expansion driven by ImÂ T in this model ends almost instantly, and the subsequent stages of inflation are driven by the real part of the field T, as in the standard Starobinsky model. Thus, the Starobinsky model and its supersymmetric generalizations remain disfavored by the recent BICEP2 data.

Renata Kallosh

Andrei Linde

Bert Vercnocke

Is imaginary Starobinsky model real?

A possibility of journeys through antigravity has recently been proposed, with the suggestion that Weyl-invariant extension of scalars coupled to Einstein gravity allows for an unambiguous classical evolution through cosmological singularities in anisotropic spacetimes. We compute the Weyl invariant curvature squared and find that it blows up for the proposed anisotropic solution both at the Big Crunch as well as at the Big Bang. Therefore the cosmological singularities are not resolved by uplifting Einstein theory to a Weyl invariant model.