Showing papers on "Quantum geometry published in 2010"
TL;DR: The fact that one can associate thermodynamic properties with horizons brings together principles of quantum theory, gravitation and thermodynamics and possibly offers a window to the nature of quantum geometry as mentioned in this paper.
Abstract: The fact that one can associate thermodynamic properties with horizons brings together principles of quantum theory, gravitation and thermodynamics and possibly offers a window to the nature of quantum geometry. This review discusses certain aspects of this topic, concentrating on new insights gained from some recent work. After a brief introduction of the overall perspective, sections 2 and 3 provide the pedagogical background on the geometrical features of bifurcation horizons, path integral derivation of horizon temperature, black hole evaporation, structure of Lanczos-Lovelock models, the concept of Noether charge and its relation to horizon entropy. Section 4 discusses several conceptual issues introduced by the existence of temperature and entropy of the horizons. In section 5 we take up the connection between horizon thermodynamics and gravitational dynamics and describe several peculiar features which have no simple interpretation in the conventional approach. The next two sections describe the recent progress achieved in an alternative perspective of gravity. In section 6 we provide a thermodynamic interpretation of the field equations of gravity in any diffeomorphism invariant theory and in section 7 we obtain the field equations of gravity from an entropy maximization principle. The last section provides a summary.
835 citations
TL;DR: In this article, it was shown that the Fisher information metric, both classical and quantum, can be described by means of the Hermitian tensor on the manifold of pure states.
111 citations
TL;DR: In this paper, the authors give six arguments that the Planck scale should be viewed as a fundamental minimum or boundary for the classical concept of spacetime, beyond which quantum effects cannot be neglected and the basic nature of space-time must be reconsidered.
Abstract: We give six arguments that the Planck scale should be viewed as a fundamental minimum or boundary for the classical concept of spacetime, beyond which quantum effects cannot be neglected and the basic nature of spacetime must be reconsidered. The arguments are elementary, heuristic, and plausible and as much as possible rely on only general principles of quantum theory and gravity theory. The main goal of the paper is to give physics students and nonspecialists an awareness and appreciation of the Planck scale and the role it should play in present and future theories of quantum spacetime and quantum gravity.
109 citations
TL;DR: An analysis is reported demonstrating that quantum gravity corrections to quantum electrodynamics have a quadratic energy dependence that result in the electric charge vanishing at high energies, a result known as asymptotic freedom.
Abstract: Quantum electrodynamics describes the interactions of electrons and photons. Electric charge (the gauge coupling constant) is energy dependent, and there is a previous claim that charge is affected by gravity (described by general relativity) with the implication that the charge is reduced at high energies. However, that claim has been very controversial and the matter has not been settled. Here I report an analysis (free from the earlier controversies) demonstrating that quantum gravity corrections to quantum electrodynamics have a quadratic energy dependence that result in the electric charge vanishing at high energies, a result known as asymptotic freedom.
97 citations
TL;DR: In this paper, the authors describe an attempt to describe quantum gravity as a path integral over geometries, which is a non-renormalizable quantum field theory when the dimension of spacetime is four.
Abstract: A major unsolved problem in theoretical physics is to reconcile the classical theory of general relativity with quantum mechanics. These lectures will deal with an attempt to describe quantum gravity as a path integral over geometries. Such an approach has to be non-perturbative since gravity is a non-renormalizable quantum field theory when the dimension of spacetime is four.
90 citations
TL;DR: In this article, the authors follow the Feynman procedure to obtain a path integral formulation of loop quantum cosmology starting from the Hilbert space framework, which is well defined and provides intuition for the differences between loop quantum Cosmology and the Wheeler-DeWitt theory.
Abstract: We follow the Feynman procedure to obtain a path integral formulation of loop quantum cosmology starting from the Hilbert space framework. Quantum geometry effects modify the weight associated with each path so that the effective measure on the space of paths is different from that used in the Wheeler-DeWitt theory. These differences introduce some conceptual subtleties in arriving at the WKB approximation. But the approximation is well defined and provides intuition for the differences between loop quantum cosmology and the Wheeler-DeWitt theory from a path integral perspective.
86 citations
TL;DR: In this article, the geometry of quantum adiabatic evolution and quantum phase transition has been elucidated, and a Riemannian metric tensor has been proposed to describe the geodesic equations in the vicinity of a quantum critical point.
Abstract: We elucidate the geometry of quantum adiabatic evolution By minimizing the deviation from adiabaticity, we find a Riemannian metric tensor underlying adiabatic evolution Equipped with this tensor, we identify a unified geometric description of quantum adiabatic evolution and quantum phase transitions that generalizes previous treatments to allow for degeneracy The same structure is relevant for applications in quantum information processing, including adiabatic and holonomic quantum computing, where geodesics over the manifold of control parameters correspond to paths which minimize errors We illustrate this geometric structure with examples, for which we explicitly find adiabatic geodesics By solving the geodesic equations in the vicinity of a quantum critical point, we identify universal characteristics of optimal adiabatic passage through a quantum phase transition In particular, we show that in the vicinity of a critical point describing a second-order quantum phase transition, the geodesic exhibits power-law scaling with an exponent given by twice the inverse of the product of the spatial and scaling dimensions
83 citations
TL;DR: In this paper, the meaning of geometrical constructions associated to loop quantum gravity states on a graph is discussed and a simple relation between these and Regge geometries is derived.
Abstract: We discuss the meaning of geometrical constructions associated to loop quantum gravity states on a graph. In particular, we discuss the "twisted geometries" and derive a simple relation between these and Regge geometries.
79 citations
TL;DR: In this article, the asymptotics of the SU(2) 15j-symbol are obtained using coherent states for the boundary data, and the resulting formula is interpreted in terms of the Regge action of the geometry of a 4-simplex in fourdimensional Euclidean space.
Abstract: The asymptotics of the SU(2) 15j-symbol are obtained using coherent states for the boundary data. The geometry of all nonsuppressed boundary data is given. For some boundary data, the resulting formula is interpreted in terms of the Regge action of the geometry of a 4-simplex in four-dimensional Euclidean space. This asymptotic formula can be used to derive and extend the asymptotics of the spin foam amplitudes for quantum gravity models. The relation of the SU(2) Ooguri model to these quantum gravity models and their continuum Lagrangians is discussed.
76 citations
TL;DR: In this article, the quantum group spin-foam model is defined and a finite partition function on a fixed triangulation is given, which is a spinfoam quantization of discrete gravity with a cosmological constant.
Abstract: We study the quantum group deformation of the Lorentzian EPRL spin-foam model. The construction uses the harmonic analysis on the quantum Lorentz group. We show that the quantum group spin-foam model so defined is free of the infra-red divergence, thus gives a finite partition function on a fixed triangulation. We expect this quantum group spin-foam model is a spin-foam quantization of discrete gravity with a cosmological constant.
76 citations
TL;DR: In this paper, the authors present a class of physically acceptable solutions for a general commutation relation without directly solving the corresponding generalized Schrodinger equations, and exhibit the effect of the deformed algebra on the energy spectrum.
Abstract: Various candidates of quantum gravity such as string theory, loop quantum gravity and black hole physics all predict the existence of a minimum observable length which modifies the Heisenberg uncertainty principle to the so-called generalized uncertainty principle (GUP) This approach results from the modification of the commutation relations and changes all Hamiltonians in quantum mechanics In this paper, we present a class of physically acceptable solutions for a general commutation relation without directly solving the corresponding generalized Schrodinger equations These solutions satisfy the boundary conditions and exhibit the effect of the deformed algebra on the energy spectrum We show that this procedure prevents us from doing equivalent but lengthy calculations
TL;DR: In this paper, an extension of the usual axioms of quantization in which classical Nambu-Poisson structures are translated to n-Lie algebras at quantum level is described.
Abstract: We investigate the geometric interpretation of quantized Nambu-Poisson structures in terms of noncommutative geometries. We describe an extension of the usual axioms of quantization in which classical Nambu-Poisson structures are translated to n-Lie algebras at quantum level. We demonstrate that this generalized procedure matches an extension of Berezin-Toeplitz quantization yielding quantized spheres, hyperboloids, and superspheres. The extended Berezin quantization of spheres is closely related to a deformation quantization of n-Lie algebras, as well as the approach based on harmonic analysis. We find an interpretation of Nambu-Heisenberg n-Lie algebras in terms of foliations of R^n by fuzzy spheres, fuzzy hyperboloids, and noncommutative hyperplanes. Some applications to the quantum geometry of branes in M-theory are also briefly discussed.
TL;DR: In this article, an extension of the usual axioms of quantization in which classical Nambu-Poisson structures are translated to n-Lie algebras at quantum level is described.
Abstract: We investigate the geometric interpretation of quantized Nambu–Poisson structures in terms of noncommutative geometries. We describe an extension of the usual axioms of quantization in which classical Nambu–Poisson structures are translated to n-Lie algebras at quantum level. We demonstrate that this generalized procedure matches an extension of Berezin–Toeplitz quantization yielding quantized spheres, hyperboloids, and superspheres. The extended Berezin quantization of spheres is closely related to a deformation quantization of n-Lie algebras as well as the approach based on harmonic analysis. We find an interpretation of Nambu–Heisenberg n-Lie algebras in terms of foliations of Rn by fuzzy spheres, fuzzy hyperboloids, and noncommutative hyperplanes. Some applications to the quantum geometry of branes in M-theory are also briefly discussed.
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TL;DR: In this paper, the authors give a standard introduction to loop quantum gravity, from the ADM variables to spin network states, and include a discussion on quantum geometry on a fixed graph and its relation to a discrete approximation of general relativity.
Abstract: We give a standard introduction to loop quantum gravity, from the ADM variables to spin network states. We include a discussion on quantum geometry on a fixed graph and its relation to a discrete approx imation of general relativity.
TL;DR: In quantum gravity, the face amplitude is simpler than originally thought as discussed by the authors, and the structure of the boundary Hilbert space and the condition that amplitudes behave appropriately under compositions determine the Face amplitude of a spinfoam theory.
Abstract: The structure of the boundary Hilbert space and the condition that amplitudes behave appropriately under compositions determine the face amplitude of a spinfoam theory. In quantum gravity the face amplitude turns out to be simpler than originally thought.
TL;DR: In this paper, a covariant spin-foam formulation of quantum gravity has been developed, characterized by a kinematics which appears to match well the one of canonical loop quantum gravity.
Abstract: A covariant spin-foam formulation of quantum gravity has been recently developed, characterized by a kinematics which appears to match well the one of canonical loop quantum gravity. In this paper we reconsider the implementation of the constraints that defines the model. We define in a simple way the boundary Hilbert space of the theory, introducing a slight modification of the embedding of the SU(2) representations into the ones. We then show directly that all constraints vanish on this space in a weak sense. The vanishing is exact (and not just in the large quantum number limit). We also generalize the definition of the volume operator in the spin-foam model to the Lorentzian signature and show that it matches the one of loop quantum gravity, as in the Euclidean case.
TL;DR: In this article, the weak value deflection given in an article by Dixon et al. was derived both quantum mechanically and classically, including diffraction effects, and the mathematical details omitted in that article owing to space constraints.
Abstract: We derive the weak value deflection given in an article by Dixon et al. [P. B. Dixon et al. Phys. Rev. Lett. 102 173601 (2009)] both quantum mechanically and classically, including diffraction effects. This article is meant to cover some of the mathematical details omitted in that article owing to space constraints.
TL;DR: In this article, the volume operator in the covariant spin-foam formulation of quantum gravity has been derived, characterized by a kinematics which appears to match well the one of canonical loop quantum gravity.
Abstract: A covariant spin-foam formulation of quantum gravity has recently been developed, characterized by a kinematics which appears to match well the one of canonical loop quantum gravity. In particular, the geometrical observable giving the area of a surface has been shown to be the same as the one in loop quantum gravity. Here we discuss the volume observable. We derive the volume operator in the covariant theory and show that it matches the one of loop quantum gravity, as does the area. We also reconsider the implementation of the constraints that define the model: we derive in a simple way the boundary Hilbert space of the theory from a suitable form of the classical constraints and show directly that all constraints vanish weakly on this space.
TL;DR: In this article, the authors compare three-dimensional loop quantum gravity to the combinatorial quantisation formalism based on the Chern-Simons formulation for 3D Lorentzian and Euclidean gravity with vanishing cosmological constant and derive explicit expressions for the action of these quantum groups on the space of cylindrical functions associated with graphs.
Abstract: We relate three-dimensional loop quantum gravity to the combinatorial quantisation formalism based on the Chern-Simons formulation for three-dimensional Lorentzian and Euclidean gravity with vanishing cosmological constant. We compare the construction of the kinematical Hilbert space and the implementation of the constraints. This leads to an explicit and very interesting relation between the associated operators in the two approaches and sheds light on their physical interpretation. We demonstrate that the quantum group symmetries arising in the combinatorial formalism, the quantum double of the three-dimensional Lorentz and rotation group, are also present in the loop formalism. We derive explicit expressions for the action of these quantum groups on the space of cylindrical functions associated with graphs. This establishes a direct link between the two quantisation approaches and clarifies the role of quantum group symmetries in three-dimensional gravity.
TL;DR: In this paper, the authors focus on the behaviour of a quantum field defined on a manifold containing regions of different signatures and show how the problem of quantum fields exposed to finite regions of Euclidean-signature (Riemannian) geometry has similarities with the quantum barrier penetration problem.
Abstract: Within the framework of either Euclidean (functional integral) quantum gravity or canonical general relativity the signature of the manifold is a priori unconstrained. Furthermore, recent developments in the emergent spacetime programme have led to a physically feasible implementation of (analogue) signature change events. This suggests that it is time to revisit the sometimes controversial topic of signature change in general relativity. Specifically, we shall focus on the behaviour of a quantum field defined on a manifold containing regions of different signature. We emphasize that regardless of the underlying classical theory, there are severe problems associated with any quantum field theory residing on a signature-changing background. (Such as the production of what is naively an infinite number of particles, with an infinite energy density.) We show how the problem of quantum fields exposed to finite regions of Euclidean-signature (Riemannian) geometry has similarities with the quantum barrier penetration problem. Finally we raise the question as to whether signature change transitions could be fully understood and dynamically generated within (modified) classical general relativity, or whether they require the knowledge of a theory of quantum gravity.
01 Apr 2010
TL;DR: The Causal Sets approach as mentioned in this paper is an attempt to construct a quantum theory of gravity starting with a fundamentally discrete spacetime. But it may have to abandon certain assumptions we were making, in particular the concept of spacetime as a continuum substrate.
Abstract: In order to construct a quantum theory of gravity, we may have to abandon certain assumptions we were making. In particular, the concept of spacetime as a continuum substratum is questioned. Causal Sets is an attempt to construct a quantum theory of gravity starting with a fundamentally discrete spacetime. In this contribution we review the whole approach, focusing on some recent developments in the kinematics and dynamics of the approach.
TL;DR: In this article, the authors show how information geometry, the natural geometry of discrete probability distributions, can be used to derive the quantum formalism based on three elementary features of quantum phenomena, namely complementarity, measurement simulability, and global gauge invariance.
Abstract: In this paper, we show how information geometry, the natural geometry of discrete probability distributions, can be used to derive the quantum formalism. The derivation rests upon three elementary features of quantum phenomena, namely complementarity, measurement simulability, and global gauge invariance. When these features are appropriately formalized within an information geometric framework, and combined with a novel information- theoretic principle, the central features of the finite-dimensional quantum formalism can be reconstructed.
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28 Jun 2010TL;DR: In this article, the authors introduce engineers to quantum mechanics and the world of nanostructures, enabling them to apply the theories to numerous nanostructure problems, including uncertainty relations, the Schrodinger equation, perturbation theory, and tunneling.
Abstract: The properties of new nanoscale materials, their fabrication and applications, as well as the operational principles of nanodevices and systems, are solely determined by quantum-mechanical laws and principles. This textbook introduces engineers to quantum mechanics and the world of nanostructures, enabling them to apply the theories to numerous nanostructure problems. The textbook covers the fundamentals of quantum mechanics, including uncertainty relations, the Schrodinger equation, perturbation theory, and tunneling. These are then applied to a quantum dot, the smallest artificial atom, and compared to hydrogen, the smallest atom in nature. Nanoscale objects with higher dimensionality, such as quantum wires and quantum wells, are introduced, as well as nanoscale materials and nanodevices. Numerous examples throughout the text help students to understand the material.
TL;DR: In this article, the authors extended the quantum theory to horizons of arbitrary shape and showed that the Hilbert space obtained by quantizing the full phase space of all generic horizons with a fixed area is identical to that originally found in spherical symmetry.
Abstract: Isolated horizons model equilibrium states of classical black holes. A detailed quantization, starting from a classical phase space restricted to spherically symmetric horizons, exists in the literature and has since been extended to axisymmetry. This paper extends the quantum theory to horizons of arbitrary shape. Surprisingly, the Hilbert space obtained by quantizing the full phase space of all generic horizons with a fixed area is identical to that originally found in spherical symmetry. The entropy of a large horizon remains one quarter its area, with the Barbero-Immirzi parameter retaining its value from symmetric analyses. These results suggest a reinterpretation of the intrinsic quantum geometry of the horizon surface.
TL;DR: By employing a quantum hydrodynamic model for bounded three-component quantum plasmas, two kinds of streaming instabilities due to ion streaming and dust streaming are studied in this article.
Abstract: By employing a quantum hydrodynamic model for bounded three-component quantum plasmas, two kinds of streaming instabilities due to ion-streaming and dust-streaming are studied. For this purpose, the dispersion relation for the bounded wave in quantum electron-ion-dust plasmas is obtained by carrying out a normal mode analysis. The results of theoretical analysis and numerical simulation show that the streaming speeds, the boundary conditions, and the quantum parameters strongly influence the instabilities. In particular, it is found that the boundary effect and the quantum effect are closely coupled, i.e., the quantum effect depends on the geometry parameters.
TL;DR: In this article, the first written exposition of some ideas (announced in a previous survey) on an approach to quantum gravity based on Tomita-Takesaki modular theory and A. Connes non-commutative geometry aiming at the reconstruction of spec- tral geometries from an operational formalism of states and categories of observables in a covariant theory is presented.
Abstract: This paper contains the first written exposition of some ideas (announced in a previous survey) on an approach to quantum gravity based on Tomita-Takesaki modular theory and A. Connes non-commutative geometry aiming at the reconstruction of spec- tral geometries from an operational formalism of states and categories of observables in a covariant theory. Care has been taken to provide a coverage of the relevant background on modular theory, its applications in non-commutative geometry and physics and to the detailed discussion of the main foundational issues raised by the proposal.
TL;DR: In this article, a group field theory for 4D quantum gravity is presented, which incorporates the constraints that give gravity from BF theory and has quantum amplitudes with the explicit form of simplicial path integrals for first-order gravity.
Abstract: We present a new group field theory for 4D quantum gravity. It incorporates the constraints that give gravity from BF theory and has quantum amplitudes with the explicit form of simplicial path integrals for first-order gravity. The geometric interpretation of the variables and of the contributions to the quantum amplitudes is manifest. This allows a direct link with other simplicial gravity approaches, like quantum Regge calculus, in the form of the amplitudes of the model, and dynamical triangulations, which we show to correspond to a simple restriction of the same.
TL;DR: In this article, the authors explore the possibility of using quantum walks on graphs to find structural anomalies, such as extra edges or loops, on a graph, focusing on star graphs, whose edges are like spokes coming out of a central hub.
Abstract: We explore the possibility of using quantum walks on graphs to find structural anomalies, such as extra edges or loops, on a graph. We focus our attention on star graphs, whose edges are like spokes coming out of a central hub. If there are $N$ spokes, we show that a quantum walk can find an extra edge connecting two of the spokes or a spoke with a loop on it in $O(\sqrt{N})$ steps. We initially find that if all except one of the spokes have loops, the walk will not find the spoke without a loop, but this can be fixed if we choose the phase with which the particle is reflected from the vertex without the loop. Consequently, quantum walks can, under some circumstances, be used to find structural anomalies in graphs.
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TL;DR: The Fubini-Study metric as mentioned in this paper is a complex tensor with the real part serving as the Riemannian metric that measures the ''quantum distance'' and the imaginary part being the Berry curvature.
Abstract: Geometric Quantum Mechanics is a novel and prospecting approach motivated by the belief that our world is ultimately geometrical. At the heart of that is a quantity called Quantum Geometric Tensor (or Fubini-Study metric), which is a complex tensor with the real part serving as the Riemannian metric that measures the `quantum distance', and the imaginary part being the Berry curvature. Following a physical introduction of the basic formalism, we illustrate its physical significance in both the adiabatic and non-adiabatic systems.
TL;DR: In this paper, the authors derived the effective Hamiltonian for a quantum system constrained to a submanifold (the constraint manifold) of configuration space (the ambient space) in the asymptotic limit, where the restoring forces tend to infinity.
Abstract: We derive the effective Hamiltonian for a quantum system constrained to a submanifold (the constraint manifold) of configuration space (the ambient space) in the asymptotic limit, where the restoring forces tend to infinity. In contrast to earlier works, we consider, at the same time, the effects of variations in the constraining potential and the effects of interior and exterior geometry, which appear at different energy scales and, thus, provide a complete picture, which ranges over all interesting energy scales. We show that the leading order contribution to the effective Hamiltonian is the adiabatic potential given by an eigenvalue of the confining potential well known in the context of adiabatic quantum waveguides. At next to leading order, we see effects from the variation of the normal eigenfunctions in the form of a Berry connection. We apply our results to quantum waveguides and provide an example for the occurrence of a topological phase due to the geometry of a quantum wave circuit (i.e., a closed quantum waveguide).