Showing papers on "Quantum geometry published in 2005"
TL;DR: In this paper, an application of loop quantum cosmology to homogeneous systems, which removes classical singularities, is presented, where the main effects are introduced into effective classical equations, which allow one to avoid the interpretational problems of quantum theory.
Abstract: Quantum gravity is expected to be necessary in order to understand situations in which classical general relativity breaks down. In particular in cosmology one has to deal with initial singularities, i.e., the fact that the backward evolution of a classical spacetime inevitably comes to an end after a finite amount of proper time. This presents a breakdown of the classical picture and requires an extended theory for a meaningful description. Since small length scales and high curvatures are involved, quantum effects must play a role. Not only the singularity itself but also the surrounding spacetime is then modified. One particular theory is loop quantum cosmology, an application of loop quantum gravity to homogeneous systems, which removes classical singularities. Its implications can be studied at different levels. The main effects are introduced into effective classical equations, which allow one to avoid the interpretational problems of quantum theory. They give rise to new kinds of early-universe phenomenology with applications to inflation and cyclic models. To resolve classical singularities and to understand the structure of geometry around them, the quantum description is necessary. Classical evolution is then replaced by a difference equation for a wave function, which allows an extension of quantum spacetime beyond classical singularities. One main question is how these homogeneous scenarios are related to full loop quantum gravity, which can be dealt with at the level of distributional symmetric states. Finally, the new structure of spacetime arising in loop quantum gravity and its application to cosmology sheds light on more general issues, such as the nature of time.
811 citations
TL;DR: In this article, the authors provided detailed evidence for the claim that nonperturbative quantum gravity, defined through state sums of causal triangulated geometries, possesses a large-scale limit in which the dimension of spacetime is four and the dynamics of the volume of the universe behaves semiclassically.
Abstract: We provide detailed evidence for the claim that nonperturbative quantum gravity, defined through state sums of causal triangulated geometries, possesses a large-scale limit in which the dimension of spacetime is four and the dynamics of the volume of the universe behaves semiclassically. This is a first step in reconstructing the universe from a dynamical principle at the Planck scale, and at the same time provides a nontrivial consistency check of the method of causal dynamical triangulations. A closer look at the quantum geometry reveals a number of highly nonclassical aspects, including a dynamical reduction of spacetime to two dimensions on short scales and a fractal structure of slices of constant time.
435 citations
TL;DR: In this paper, the quantum hydrodynamic model for charged particle systems is extended to the cases of nonzero magnetic fields, and conditions for equilibrium in ideal quantum magnetohydrodynamics are established.
Abstract: The quantum hydrodynamic model for charged particle systems is extended to the cases of nonzero magnetic fields. In this way, quantum corrections to magnetohydrodynamics are obtained starting from the quantum hydrodynamical model with magnetic fields. The importance of the quantum corrections is described by a parameter H which can be significant in dense astrophysical plasmas. The quantum magnetohydrodynamic model is analyzed in the infinite conductivity limit. The conditions for equilibrium in ideal quantum magnetohydrodynamics are established. Translationally invariant exact equilibrium solutions are obtained in the case of the ideal quantum magnetohydrodynamic model.
426 citations
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20 May 2005
TL;DR: In this article, the Bohmian route to the Hydrodynamic Equations and the Phase Space Route to the hydrodynamic equations are discussed. But the authors focus on the properties of Quantum Trajectory Dynamics.
Abstract: to Quantum Trajectories.- The Bohmian Route to the Hydrodynamic Equations.- The Phase Space Route to the Hydrodynamic Equations.- The Dynamics and Properties of Quantum Trajectories.- Function and Derivative Approximation on Unstructured Grids.- Applications of the Quantum Trajectory Method.- Adaptive Methods for Trajectory Dynamics.- Quantum Trajectories for Multidimensional Dynamics.- Approximations to the Quantum Force.- Derivative Propagation Along Quantum Trajectories.- Quantum Trajectories in Phase Space.- Mixed Quantum-Classical Dynamics.- Topics in Quantum Hydrodynamics: The Stress Tensor and Vorticity.- Quantum Trajectories for Stationary States.- Challenges and Opportunities.
307 citations
TL;DR: The loop quantum gravity approach as mentioned in this paper is a variant of the canonical approach, the oldest being quantum geometrodynamics where the fundamental configuration variable is the three-metric Loop quantum gravity has developed from a new choice of canonical variables introduced by Abhay Ashtekar in 1986, the new configuration variable being a connection defined on a three-manifold Instead of the connection itself, the loop approach employs a non-local version in which the connection is integrated over closed loops This is similar to the Wilson loops used in gauge theories.
Abstract: The most difficult unsolved problem in fundamental theoretical physics is the consistent implementation of the gravitational interaction into a quantum framework, which would lead to a theory of quantum gravity Although a final answer is still pending, several promising attempts do exist Despite the general title, this book is about one of them - loop quantum gravity This approach proceeds from the idea that a direct quantization of Einstein's theory of general relativity is possible In contrast to string theory, it presupposes that the unification of all interactions is not needed as a prerequisite for quantum gravity Usually one divides theories of quantum general relativity into covariant and canonical approaches Covariant theories employ four-dimensional concepts in its formulation, one example being the path integral approach Canonical theories start from a classical Hamiltonian version of the theory in which spacetime is foliated into spacelike hypersurfaces Loop quantum gravity is a variant of the canonical approach, the oldest being quantum geometrodynamics where the fundamental configuration variable is the three-metric Loop quantum gravity has developed from a new choice of canonical variables introduced by Abhay Ashtekar in 1986, the new configuration variable being a connection defined on a three-manifold Instead of the connection itself, the loop approach employs a non-local version in which the connection is integrated over closed loops This is similar to the Wilson loops used in gauge theories Carlo Rovelli is one of the pioneers of loop quantum gravity which he started to develop with Lee Smolin in two papers written in 1988 and 1990 In his book, he presents a comprehensive and competent overview of this approach and provides at the same time the necessary technical background in order to make the treatment self-contained In fact, half of the book is devoted to 'preparations' giving a detailed account of Hamiltonian mechanics, quantum mechanics, general relativity and other topics According to the level of the reader, this part can be skipped or studied as interesting material on its own The penetrating theme of the whole book (its leitmotiv) is background independence In non-gravitational theories, dynamical fields are formulated on a fixed background spacetime that plays the role of an absolute structure in the theory In general relativity, on the other hand, there is no background structure - all fields are dynamical This was a confusing point already during the development of general relativity and led Albert Einstein in 1913 erroneously to give up general covariance before recognizing his error and presenting his final correct field equations that are of course covariant This story is instructive, circling around the famous 'hole problem', and is told in detail in Rovelli's book Its solution is that points on a bare manifold do not make sense in physics; everything, including the gravitational field, is dragged around by a diffeomorphism - there is just no background available, only the fields exist In loop quantum gravity, physical space (called 'quantum geometry') itself is formed by loop-like quantum states: a suitable orthonormal basis is provided by spin-network states (a spin-network is a graph with edges and nodes, where spins are assigned to the edges), and the quantum geometry is a superposition of such states Time and space in the usual sense have disappeared In the second half of his book, Rovelli discusses at length the major successes of this approach First of all, the formalism yields a unique kinematical Hilbert space for the quantum states obeying the Gauss and diffeomorphism constraints The situation with the Hamiltonian constraint is more subtle The need for a Hilbert-space structure in quantum gravity is, however, not discussed After all, the Hilbert-space structure in quantum mechanics is tied to the presence of an external time and the conservation of probability with respect to this external time But in quantum gravity there is no background structure, in particular no external time Secondly, there exist two important operators that are connected, respectively, with area and volume in the classical limit These operators have a discrete spectrum and thus provide elementary 'quanta' of area and volume This gives a vague hint of a discrete structure at the Planck scale, about which there were speculations for many decades In spite of these promising results, loop quantum gravity is still far away from a physical theory This is also reflected in this volume where the technical treatment prevails and where physical applications are relegated to about 20 pages These applications deal with quantum cosmology and black holes The part on loop quantum cosmology summarizes briefly recent results about a possible singularity avoidance and a new mechanism for inflation These results are not derived from loop quantum gravity but from imposing the discrete structure of the full theory directly on the quantum cosmological models The part on black holes discusses the derivation of the Bekenstein-Hawking entropy from counting the number of relevant spin-network states Since the theory contains a free parameter (the 'Barbero-Immirzi parameter'), the best one can do is to determine this parameter by demanding that the result be the Bekenstein-Hawking entropy The book does not yet contain the results of recent papers, published in 2004, that correct the earlier entropy calculations presented here From the new value of the Barbero-Immirzi parameter, the appealing connection with quasi-normal modes, as discussed in the book, may be lost The book concludes with a brief discussion of the major open issues Among these are the following: a well-defined and physically sensible semiclassical limit, the precise form of the Hamiltonian, the role of unification (most of the work in loop quantum gravity deals only with pure gravity) and, last but not least, the issue of quantitative and testable predictions Whether loop quantum gravity will become a physical theory is not clear Nor is this clear for string theory or any other approach However, loop quantum gravity provides a fascinating line of research and has much conceptual appeal The present volume gives both an introduction and a review of this approach, making it suitable for advanced students as well as experts It is certainly of interest for the readers of Classical and Quantum Gravity
226 citations
TL;DR: This book surveys proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adiabatic algorithms, quantum-mechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, Malament-Hogarth spacetimes, quantum gravity, closed timelike curves, and "anthropic computing".
Abstract: Can NP-complete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adiabatic algorithms, quantum-mechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, Malament-Hogarth spacetimes, quantum gravity, closed timelike curves, and "anthropic computing." The section on soap bubbles even includes some "experimental" results. While I do not believe that any of the proposals will let us solve NP-complete problems efficiently, I argue that by studying them, we can learn something not only about computation but also about physics.
225 citations
TL;DR: In this article, it was pointed out that the standard quantum geometry formalism with spin one-half at each puncture is not consistent with the area law with logarithmic corrections.
Abstract: Various approaches to black hole entropy yield the area law with logarithmic corrections, many involving a coefficient $1/2$, and some involving $3/2$ It is pointed out here that the standard quantum geometry formalism with spin one-half at each puncture is not consistent with $3/2$
174 citations
TL;DR: Gibbons et al. as discussed by the authors characterized the convex hull of stabilizer states and showed that negativity of Wigner functions is necessary for exponential speedup in pure-state quantum computation.
Abstract: Gibbons et al. [Phys. Rev. A 70, 062101 (2004)] have recently defined a class of discrete Wigner functions $W$ to represent quantum states in a finite Hilbert space dimension $d$. I characterize the set ${C}_{d}$ of states having non-negative $W$ simultaneously in all definitions of $W$ in this class. For $d\ensuremath{\leqslant}5$ I show ${C}_{d}$ is the convex hull of stabilizer states. This supports the conjecture that negativity of $W$ is necessary for exponential speedup in pure-state quantum computation.
128 citations
TL;DR: In this article, it was shown that the particle-wave duality can be resolved amicably by assuming space-time to be a fuzzy K3 manifold, similar to that of E-Infinity theory.
Abstract: Starting from the two-slit experiment we show that the so-called particle–wave duality could be resolved amicably by assuming space–time to be a fuzzy K3 manifold akin to that of E-Infinity theory. Subsequently, we show how many of the fundamental constants of nature such as the electromagnetic fine structure as well as the quantum gravity coupling may be deduced from the topology and geometry of the space–time manifold.
126 citations
TL;DR: A tentative implementation of a technique for defining and computing -point functions in the context of a background-independent gravitational quantum field theory in a perturbatively finite model defined using spin foam techniques in thecontext of loop quantum gravity is constructed.
Abstract: We devise a technique for defining and computing n-point functions in the context of a background-independent gravitational quantum field theory. We construct a tentative implementation of this technique in a perturbatively finite model defined using spin foam techniques in the context of loop quantum gravity.
115 citations
TL;DR: In this article, the authors construct physical semiclassical states annihilated by the Hamiltonian constraint operator in the framework of loop quantum cosmology as a method of systematically determining the regime and validity of the semiclassic limit of the quantum theory.
Abstract: We construct physical semiclassical states annihilated by the Hamiltonian constraint operator in the framework of loop quantum cosmology as a method of systematically determining the regime and validity of the semiclassical limit of the quantum theory. Our results indicate that the evolution can be effectively described using continuous classical equations of motion with nonperturbative corrections down to near the Planck scale below which the Universe can only be described by the discrete quantum constraint. These results, for the first time, provide concrete evidence of the emergence of classicality in loop quantum cosmology and also clearly demarcate the domain of validity of different effective theories. We prove the validity of modified Friedmann dynamics incorporating discrete quantum geometry effects which can lead to various new phenomenological applications. Furthermore the understanding of semiclassical states allows for a framework for interpreting the quantum wave functions and understanding questions of a semiclassical nature within the quantum theory of loop quantum cosmology.
TL;DR: In this article, the authors reformulate spin networks in terms of harmonic oscillators and show how the holographic degrees of freedom of the theory are described as matrix models, making a link with non-commutative geometry and to look at the issue of the semiclassical limit of loop quantum gravity from a new perspective.
Abstract: Loop quantum gravity defines the quantum states of space geometry as spin networks and describes their evolution in time. We reformulate spin networks in terms of harmonic oscillators and show how the holographic degrees of freedom of the theory are described as matrix models. This allows us to make a link with non-commutative geometry and to look at the issue of the semiclassical limit of loop quantum gravity from a new perspective. This work is thought of as part of a bigger project of describing quantum geometry in quantum information terms.
TL;DR: A proper counting of states for black holes in the quantum geometry approach shows that the dominant configuration for spins are distributions that include spins exceeding one-half at the punctures as discussed by the authors.
TL;DR: In this paper, a modified Heisenberg uncertainty principle is referred as gravitational uncertainty principle (GUP) in literatures, which has some novel implications on various domains of theoretical physics such as string theory, quantum geometry, loop quantum gravity and black hole physics.
Abstract: String theory, quantum geometry, loop quantum gravity and black hole physics all indicate the existence of a minimal observable length on the order of Planck length. This feature leads to a modification of Heisenberg uncertainty principle. Such a modified Heisenberg uncertainty principle is referred as gravitational uncertainty principle(GUP) in literatures. This proposal has some novel implications on various domains of theoretical physics. Here, we study some consequences of GUP in the spirit of Quantum mechanics. We consider two problem: a particle in an one-dimensional box and momentum space wave function for a "free particle". In each case we will solve corresponding perturbational equations and compare the results with ordinary solutions.
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TL;DR: In this paper, the authors proposed a method of unifying quantum mechanics and gravity based on quantum computation, where fundamental processes are described in terms of pairwise interactions between quantum degrees of freedom.
Abstract: This paper proposes a method of unifying quantum mechanics and gravity based on quantum computation. In this theory, fundamental processes are described in terms of pairwise interactions between quantum degrees of freedom. The geometry of space-time is a construct, derived from the underlying quantum information processing. The computation gives rise to a superposition of four-dimensional spacetimes, each of which obeys the Einstein-Regge equations. The theory makes explicit predictions for the back-reaction of the metric to computational `matter,' black-hole evaporation, holography, and quantum cosmology.
TL;DR: In this paper, the detailed theory of the geometry of quantum type I horizons and the calculation of their entropy can be generalized to type II, and the leading term in entropy of large horizons is again given by 1/4th of the horizon area for the same value of the Barbero-Immirzi parameter.
Abstract: Equilibrium states of black holes can be modelled by isolated horizons. If the intrinsic geometry is spherical, they are called type I while if it is axi-symmetric, they are called type II. The detailed theory of the geometry of quantum type I horizons and the calculation of their entropy can be generalized to type II, thereby including arbitrary distortions and rotations. The leading term in entropy of large horizons is again given by 1/4th of the horizon area for the same value of the Barbero–Immirzi parameter as in the type I case. Ideas and constructions underlying this extension are summarized.
TL;DR: Following a general program, space-like singularities in spherically symmetric quantum geometry, as well as other inhomogeneous models, are shown to be absent and one sees how the classical reduction from infinitely many kinematical degrees of freedom to only one physical one, the mass, can arise.
Abstract: Spherically symmetric space-times provide many examples for interesting black hole solutions, which classically are all singular. Following a general program, spacelike singularities in spherically symmetric quantum geometry, as well as other inhomogeneous models, are shown to be absent. Moreover, one sees how the classical reduction from infinitely many kinematical degrees of freedom to only one physical one, the mass, can arise, where aspects of quantum cosmology such as the problem of initial conditions play a role.
TL;DR: In this article, it was shown that for a compact Abelian group G endowed with a continuous length function l and a sequence (Hn)n∈N of closed subgroups of G converging to G for the Hausdorff distance induced by l, C*G^,σ is the quantum GromovÕ-HausdÕ limit of any sequence C*Hn,σnn ∈N for the natural quantum metric structures and when the lifts of σn to G^ converge pointwise to σ.
TL;DR: The "consistent discretization" approach to general relativity is applied, leaving the spatial slices continuous and one ends up with a theory that has as physical space what is usually considered the kinematical space of loop quantum geometry, given by diffeomorphism invariant spin networks endowed with appropriate rigorously defined diffeomorphicism invariants measures and inner products.
Abstract: We apply the ``consistent discretization'' approach to general relativity leaving the spatial slices continuous. The resulting theory is free of the diffeomorphism and Hamiltonian constraints, but one can impose the diffeomorphism constraint to reduce its space of solutions and the constraint is preserved exactly under the discrete evolution. One ends up with a theory that has as physical space what is usually considered the kinematical space of loop quantum geometry, given by diffeomorphism invariant spin networks endowed with appropriate rigorously defined diffeomorphism invariant measures and inner products. The dynamics can be implemented as a unitary transformation and the problem of time explicitly solved or at least reduced to a numerical problem. We exhibit the technique explicitly in ($2+1$)-dimensional gravity.
TL;DR: This Letter considers the Bianchi I model, both the vacuum case and the addition of a cosmological constant, and shows using generating function techniques that only the zero solution satisfies constraints on the evolution equation.
Abstract: Loop quantum cosmology, the symmetry reduction of quantum geometry for the study of various cosmological situations, leads to a difference equation for its quantum evolution equation. To ensure that solutions of this equation act in the expected classical manner far from singularities, additional restrictions are imposed on the solution. In this Letter, we consider the Bianchi I model, both the vacuum case and the addition of a cosmological constant, and show using generating function techniques that only the zero solution satisfies these constraints. This implies either that there are technical difficulties with the current method of quantizing the evolution equation, or else loop quantum gravity imposes strong restrictions on the physically allowed solutions.
TL;DR: In this article, the theoretical foundations of quantum similarity measures and the varied usefulness of the enveloping mathematical structure are described, starting with the definition of tagged sets, continuing with matrix products, matrix signatures, and vector semispaces, and finally, the construction and structure of quantum density functions become clear and facilitate entry into the description of quantum object sets.
Abstract: This work presents a schematic description of the theoretical foundations of quantum similarity measures and the varied usefulness of the enveloping mathematical structure. The study starts with the definition of tagged sets, continuing with inward matrix products, matrix signatures, and vector semispaces. From there, the construction and structure of quantum density functions become clear and facilitate entry into the description of quantum object sets, as well as into the construction of atomic shell approximations (ASA). An application of the ASA is presented, consisting of the density surfaces of a protein structure. Based on this previous background, quantum similarity measures are naturally constructed, and similarity matrices, composed of all the quantum similarity measures on a quantum object set, along with the quantum mechanical concept of expectation value of an operator, allow the setup of a fundamental quantitative structure- activity relationship (QSPR) equation based on quantum descriptors. An application example is presented based on the inhibition of photosynthesis produced by some naphthyridinone derivatives, which makes them good herbicide candidates. © 2004 Wiley
TL;DR: In this article, a model of Riemannian loop quantum cosmology with a self-adjoint quantum scalar constraint is presented, and the physical Hilbert space is constructed using refined algebraic quantization.
Abstract: In this paper we present a model of Riemannian loop quantum cosmology with a self-adjoint quantum scalar constraint. The physical Hilbert space is constructed using refined algebraic quantization. When matter is included in the form of a cosmological constant, the model is exactly solvable and we show explicitly that the physical Hilbert space is separable, consisting of a single physical state. We extend the model to the Lorentzian sector and discuss important implications for standard loop quantum cosmology.
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TL;DR: In this paper, the authors investigate the possibility that a background independent quantum theory of gravity is not a theory of quantum geometry and provide a way for global spacetime symmetries to emerge from a background-independent theory without geometry.
Abstract: We investigate the possibility that a background independent quantum theory of gravity is not a theory of quantum geometry. We provide a way for global spacetime symmetries to emerge from a background independent theory without geometry. In this, we use a quantum information theoretic formulation of quantum gravity and the method of noiseless subsystems in quantum error correction. This is also a method that can extract particles from a quantum geometric theory such as a spin foam model.
TL;DR: The recent efforts to formulate and study a mode-coupling approach to real-time dynamic fluctuations in quantum liquids and extensions of the theory to supercooled and glassy states where quantum fluctuations compete with thermal fluctuations are reviewed.
Abstract: We review our recent efforts to formulate and study a mode-coupling approach to real-time dynamic fluctuations in quantum liquids. Comparison is made between the theory and recent neutron scattering experiments performed on liquid ortho-deuterium and para-hydrogen. We discuss extensions of the theory to supercooled and glassy states where quantum fluctuations compete with thermal fluctuations. Experimental scenarios for quantum glassy liquids are briefly discussed.
TL;DR: In this paper, the authors use statistical geometry as a way to quantify the uncertainties in the correspondence between discrete spacetime structures at "quantum scales" and classical geometries at large scales.
Abstract: This paper is a contribution to the development of a framework, to be used in the context of semiclassical canonical quantum gravity, in which to frame questions about the correspondence between discrete spacetime structures at “quantum scales” and continuum, classical geometries at large scales. Such a correspondence can be meaningfully established when one has a “semiclassical” state in the underlying quantum gravity theory, and the uncertainties in the correspondence arise both from quantum fluctuations in this state and from the kinematical procedure of matching a smooth geometry to a discrete one. We focus on the latter type of uncertainty, and suggest the use of statistical geometry as a way to quantify it. With a cell complex as an example of discrete structure, we discuss how to construct quantities that define a smooth geometry, and how to estimate the associated uncertainties. We also comment briefly on how to combine our results with uncertainties in the underlying quantum state, and on their use when considering phenomenological aspects of quantum gravity.
TL;DR: The model of a quantum computer with N qubits on a quantum space background, which is a fuzzy sphere with n = 2N elementary cells, can be viewed as the minimal model for quantum gravity.
Abstract: We argue that the model of a quantum computer with N qubits on a quantum space background, which is a fuzzy sphere with n = 2N elementary cells, can be viewed as the minimal model for quantum gravity. In fact, it is discrete, has no free parameters, is Lorentz-invariant, naturally realizes the holographic principle, and defines a subset of punctures of spin networks' edges of loop quantum gravity labelled by spins j = 2N-1-½. In this model, the discrete area spectrum of the cells, which is not equally spaced, is given in units of the minimal area of loop quantum gravity (for j = 1/2), and provides a discrete emission spectrum for quantum black holes. When the black hole emits one string of N bits encoded in one of the n cells, its horizon area decreases of an amount equal to the area of one cell.
TL;DR: A coupled quantum drift‐diffusion Schrodinger–Poisson model for stationary resonant tunneling simulations in one space dimension is proposed and the coupling of the two models is realized by assuming the continuity of the electron and current densities at the interface points.
Abstract: A coupled quantum drift‐diffusion Schrodinger–Poisson model for stationary resonant tunneling simulations in one space dimension is proposed. In the ballistic quantum zone with the resonant quantum barriers, the Schrodinger equation is solved. Near the contacts, where collisional effects are assumed to be important, the quantum drift‐diffusion model is employed. The quantum drift‐diffusion model was derived by a quantum moment method from a collisional Wigner equation by Degond et al. [J. Statist. Phys., 118 (2005), pp. 625–665]. The derivation yields an $O(\hbar^4)$ approximation of the equilibrium Wigner function which is used as the “alimentation function” in the mixed‐state formula for the electron and current densities at the interface. The coupling of the two models is realized by assuming the continuity of the electron and current densities at the interface points. Current‐voltage characteristics of a one‐dimensional tunneling diode are numerically computed. The results are compared to those from t...
TL;DR: In this paper, it was shown that discretization of spacetime naturally suggests discretisation of Hilbert space itself, and that in a universe with a minimal length, no experiment can exclude the possibility that Hilbert space is discrete.
TL;DR: In this paper, a new model of quantum walk on a one-dimensional momentum space that includes both discrete jumps and continuous drift is considered, and its time evolution has two stages; a Markov diffusion followed by localized dynamics.
Abstract: We consider a new model of quantum walk on a one-dimensional momentum space that includes both discrete jumps and continuous drift. Its time evolution has two stages; a Markov diffusion followed by localized dynamics. As in the well known quantum kicked rotor, this model can be mapped into a localized one-dimensional Anderson model. For exceptional (rational) values of its scale parameter, the system exhibits resonant behavior and reduces to the usual discrete time quantum walk on the line.
TL;DR: A structure called a causal site to use as a setting for quantum geometry, replacing the underlying point set, has an interesting categorical form, and a natural “tangent 2-bundle,” analogous to the tangent bundle of a smooth manifold.
Abstract: We propose a structure called a causal site to use as a setting for quantum geometry, replacing the underlying point set. The structure has an interesting categorical form, and a natural “tangent 2-bundle,” analogous to the tangent bundle of a smooth manifold. Examples with reasonable finiteness conditions have an intrinsic geometry, which can approximate classical solutions to general relativity. We propose an approach to quantization of causal sites as well.