論文

The renormalization group method has been adapted to the analysis of the long-time behavior of the nonlinear partial differential equation and has demonstrated its power in the study of critical phenomena of gravitational collapse. In the present work we apply the renormalization group to the Einstein equation in cosmology and carry out a detailed analysis of renormalization group flow in the vicinity of the scale invariant fixed point in the spherically symmetric and inhomogeneous dust filled universe model.

Hamiltonian structures for spatially compact locally homogeneous vacuum universes are investigated, provided that the set of dynamical variables contains the Teichmuller parameters, parameterizing the purely global geometry. One of the key ingredients of our arguments is a suitable mathematical expression for quotient manifolds, where the universal cover metric carries all the degrees of freedom of geometrical variations, i.e., the covering group is fixed. We discuss general problems concerned with the use of this expression in the context of general relativity, and demonstrate the reduction of the Hamiltonians for some examples. For our models, all the dynamical degrees of freedom in Hamiltonian view are unambiguously interpretable as geometrical deformations, in contrast to the conventional open models. (C) 1997 American Institute of Physics.

A complete description of dynamics of compact locally homogeneous universes is given, which, in particular, includes explicit calculations of Teichmuller deformations and careful counting of dynamical degrees of freedom, We regard each of the universes as a simply connected four-dimensional space-time with identifications by the action of a discrete subgroup of the isometry group. We then reduce the identifications defined by the space-time isometries to ones in a homogeneous section, and find a condition that such spatial identifications must satisfy. This is essential for explicit construction of compact homogeneous universes. Some examples are demonstrated for Bianchi II, VI0, VII0, and I universal covers. (C) 1997 American Institute of Physics.

To confirm the scenario of critical behavior in gravitational collapse in a previous paper we carry out a rather complete analysis for perfect fluids with pressure proportional to density, in a wide range of the adiabatic index γ. In particular, the uniqueness of the relevant mode around a fixed point is established by a Lyapunov analysis. This shows that critical phenomena occur not only for the radiation fluid but also for perfect fluids with (Formula presented) We also analyze the stability of other self-similar solutions and also discuss the universality class. © 1999 The American Physical Society.

We investigate the nature of asymptotically de Sitter space-times containing a black hole. We show that if the matter fields satisfy the dominant energy condition and cosmic censorship holds in the considered spacetime, the area of the cosmological event horizon for an observer approaching a future timelike infinity does not decrease; i.e., the second law is satisfied. We also show under the same conditions that the total area of the black hole and the cosmological event horizon, a quarter of which is the total Bekenstein-Hawking entropy, is less than 12 pi/Lambda, where Lambda is the cosmological constant. The physical implications are also discussed.

We present a general framework for finding the time-optimal evolution and the optimal Hamiltonian for a quantum system with a given set of initial and final states. Our formulation is based on the variational principle and is analogous to that for the brachistochrone in classical mechanics. We reduce the problem to a formal equation for the Hamiltonian which depends on certain constraint functions specifying the range of available Hamiltonians. For some simple examples of the constraints, we explicitly find the optimal solutions.

Extending our previous work on time-optimal quantum state evolution [Carlini , Phys. Rev. Lett. 96, 060503 (2006)], we formulate a variational principle for finding the time-optimal realization of a target unitary operation, when the available Hamiltonians are subject to certain constraints dictated either by experimental or by theoretical conditions. Since the time-optimal unitary evolutions do not depend on the input quantum state this is of more direct relevance to quantum computation. We explicitly illustrate our method by considering the case of a two-qubit system self-interacting via an anisotropic Heisenberg Hamiltonian and by deriving the time-optimal unitary evolution for three examples of target quantum gates, namely, the swap of qubits, the quantum Fourier transform, and the entangler gate. We also briefly discuss the case in which certain unitary operations take negligible time.

© 2019 IOP Publishing Ltd. A classical string whose world sheet shares a one-dimensional symmetry with the spacetime is called cohomogeneity-one (C1). We propose C1-string integrability, i.e. integrability of all C1 strings in the spacetime, as a class of hidden symmetry of a spacetime. The C1 string may probe symmetry which is not probed by a particle. We present a simple, systematic procedure for finding constants of motion of a C1 string and examining C1 string integrability of a spacetime. We apply the framework to some physically important spacetimes such as AdS5, AdS5 × S5, and AdS5 × Tp,q. C1 strings and C1-string integrability may be useful for a prior examination of general string integrability of a highly symmetric spacetime since, among the examples above, all C1 string integrable spacetimes are string-integrable, and the previously obtained chaotic string solutions in those or other spacetimes are of class C1.

We present a general theoretical framework for finding the time-optimal unitary evolution of the quantum systems when the Hamiltonian is subject to arbitrary constraints. Quantum brachistochrone (QB) is such a framework based on the variational principle, whose drawback is that it only deals with equality constraints. While inequality constraints can be reduced to equality ones in some situations, they usually cannot, especially when a drift field, an uncontrollable part, is present in the Hamiltonian. We first develop a framework based on Pontryagin's maximum principle (MP) in order to deal with inequality constraints as well. The new framework contains QB as a special case, and their detailed correspondence is given. Second, we address the problem of singular controls, which satisfy MP trivially so as to cause a trouble in determining the optimal protocol. To overcome this difficulty, we derive an additional necessary condition for a singular protocol to be optimal by applying the generalized Legendre-Clebsch condition. Third, we discuss general relations among the drift, the singular controls, and the inequality constraints. Finally, we demonstrate how our framework and results work in some examples. We also discuss the physical meaning of singular controls.

We construct a four-dimensional spacetime using a three-dimensional contact manifold equipped with a degenerate metric. The degenerate metric is set to be compatible with the contact structure. The compatibility condition is defined in this paper. Our construction yields a Ricci tensor of a particularly simple form, which leads to a solution of the Einstein equation with a null dust and cosmic strings. The solution includes two arbitrary functions: the energy density of the null dust and the number density of the cosmic strings. When there exist the cosmic strings, the spacetime is of Petrov type D. Otherwise, the spacetime is conformally flat. For some simple matter densities, we examine the Einstein equation in detail.

For a class of translation-invariant free-fermion systems including those with uniform nearest-neighbor hopping) on a d-dimensional L×â »×L hypercubic lattice, we prove that, starting from an arbitrary pure initial state, the system equilibrates with respect to the coarse-grained density within a timescale of order L. This scaling is optimal, since there exist initial states whose equilibration requires time of order L. Our result establishes O(L) as the equilibration timescale, as is expected in normal macroscopic systems with a conserved quantity, such as total number of particles.

We investigate a class of spatially compact inhomogeneous spacetimes. Motivated by Thurston's geometrization conjecture, we give a formulation for constructing spatially compact composite spacetimes as solutions for the Einstein equations. Such composite spacetimes are built from the spatially compact locally homogeneous vacuum spacetimes which have two commuting local Killing vector fields and are homeomorphic to torus bundles over the circle by gluing them through a timelike hypersurface admitting a homogeneous spatial torus spanned by the commuting local Killing vector fields, We also assume that the matter which will arise from the gluing is compressed on the boundary, i.e, we take the thin-shell approximation. By solving the junction conditions, we can see dynamical behaviour of the connected (composite) spacetime. The Teichmuller deformation of the torus can also be obtained. We apply our formalism to a concrete model. The relation to the torus sum of 3-manifolds and the difficulty of this problem are also discussed.

Recently many people have discussed the possibility that the universe is hyperbolic and was in an inflationary phase at an early stage. Under these assumptions, it is shown that the universe cannot have compact hyperbolic time slices. Though the universal covering space of the universe has a past Cauchy horizon and can be extended analytically beyond it, the extended region has densely many points which correspond to singularities of the compact universe. The result is essentially attributed to the ergodicity of the geodesic flow on a compact negatively curved manifold. The relationship with the strong cosmic censorship conjecture also is discussed.

We formulate and investigate the simplest version of time-optimal quantum computation theory (TO-QCT), where the computation time is defined by the physical one and the Hamiltonian contains only one- and two-qubit interactions. This version of TO-QCT is also considered as optimality by sub-Riemannian geodesic length. The work has two aims: One is to develop a TO-QCT itself based on a physically natural concept of time, and the other is to pursue the possibility of using TO-QCT as a tool to estimate the complexity in conventional gate-optimal quantum computation theory (GO-QCT). In particular, we investigate to what extent is true the following statement: Time complexity is polynomial in the number of qubits if and only if gate complexity is also. In the analysis, we relate TO-QCT and optimal control theory (OCT) through fidelity-optimal computation theory (FO-QCT); FO-QCT is equivalent to TO-QCT in the limit of unit optimal fidelity, while it is formally similar to OCT. We then develop an efficient numerical scheme for FO-QCT by modifying Krotov’s method in OCT, which has a monotonic convergence property. We implemented the scheme and obtained solutions of FO-QCT and of TO-QCT for the quantum Fourier transform and a unitary operator that does not have an apparent symmetry. The former has a polynomial gate complexity and the latter is expected to have an exponential one which is based on the fact that a series of generic unitary operators has an exponential gate complexity. The time complexity for the former is found to be linear in the number of qubits, which is understood naturally by the existence of an upper bound. The time complexity for the latter is exponential in the number of qubits. Thus, both the targets seem to be examples satisfyng the preceding statement. The typical characteristics of the optimal Hamiltonians are symmetry under time reversal and constancy of one-qubit operation, which are mathematically shown to hold in fairly general situations.