Time-dependent linear Hamiltonian systems and quantum mechanics

In this Letter, a theorem on time-dependent linear Hamiltonian systems is recalled and its connection with the Schrödinger equation is discussed. The kernel of the evolution operator of such quantum systems is computed. Furthermore, the Lewis and Riesenfeld theory for systems with many degrees of freedom is generalized.     

Quantum geometry and quantum mechanics of integrable systems

Quantum integrable systems and their classical counterparts are considered. We show that the symplectic structure and invariant tori of the classical system can be deformed by a quantization parameter ħ to produce a new (classical) integrable system. The new tori selected by the ħ-equidistance rule represent the spectrum of the quantum system up to O(ħ ^∞) and are invariant under quantum dynamics in the long-time range O(ħ ^−∞). The quantum diffusion over the deformed tori is described. The analytic apparatus uses quantum action-angle coordinates explicitly constructed by an ħ-deformation of the classical action-angles.     

Bures geometry of the three-level quantum systems

We compute — using a formula of Dittmann — the Bures metric tensor (g) for the eight-dimensional state space of three-level quantum systems, employing a newly developed Euler angle-based parameterization of the 3 ×3 density matrices. Most of the individual metric elements (g_ij) are found to be expressible in relatively compact form, many of them in fact being exactly zero.     

Riemann-Cartan-Weyl quantum geometry,I: Laplacians and supersymmetric systems

In this first article of a series dealing with the geometry of quantum mechanics, we introduce the Riemann-Cartan-Weyl (RCW) geometries of quantum mechanics for spin-0 systems as well as for systems of nonzero spin. The central structure is given by a family of Laplacian (or D'Alembertian) operators on forms of arbitrary degree associated to the RCW geometries. We show that they are conformally equivalent with the Laplacian operators introduced by Witten in topological quantum field theories. We show that the Laplacian RCW operators yield a supersymmetric system, in the sense of Witten, and study the relation between the RCW geometries and the symplectic structure of loop space. The RCW family of Laplacians are the infinitesimal generators of diffusion processes on nondegenerate space-times of systems of arbitrary spin.     

Quantum geometry and quantum mechanics of integrable systems. II

For a generic quantum integrable system, we describe the asymptotics of the eigenstate density and of the trace of the evolution operator in all orders of the quantization parameter. This is done by using quantum symplectic geometry, which makes the given quantum system to be equivalent to a deformed classical system with arbitrary accuracy with respect to the quantization parameter. The asymptotics is explicitly given via the deformed symplectic form, deformed Liouville-Arnold tori, and deformed Maslov class.     

Time-dependent variational principle for quantum lattices

We develop a new algorithm based on the time-dependent variational principle applied to matrix product states to efficiently simulate the real- and imaginary-time dynamics for infinite one-dimensional quantum lattices. This procedure (i) is argued to be optimal, (ii) does not rely on the Trotter decomposition and thus has no Trotter error, (iii) preserves all symmetries and conservation laws, and (iv) has low computational complexity. The algorithm is illustrated by using both an imaginary-time and a real-time example.     

Inflation from quantum geometry

Quantum geometry predicts that a universe evolves through an inflationary phase at small volume before exiting gracefully into a standard Friedmann phase. This does not require the introduction of additional matter fields with ad hoc potentials; rather, it occurs because of a quantum gravity modification of the kinetic part of ordinary matter Hamiltonians. An application of the same mechanism can explain why the present day cosmological acceleration is so tiny.     

Time-dependent exactly solvable models for quantum computing

A time-dependent periodic Hamiltonian admitting exact solutions is applied to construct a set of universal gates for a quantum computer. The time evolution matrices are obtained in an explicit form and used to construct logic gates for computation. A way of obtaining an entanglement operator is discussed, too. The method is based on transformation of soluble time-independent equations into time-dependent ones by employing a set of special time-dependent transformation operators. The text was submitted by the authors in English.     

Consistent discretization and loop quantum geometry

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.     

Time-dependent density functional theory for quantum transport

The rapid miniaturization of electronic devices motivates research interests in quantum transport. Recently time-dependent quantum transport has become an important research topic. Here we review recent progresses in the development of time-dependent density-functional theory for quantum transport including the theoretical foundation and numerical algorithms. In particular, the reducedsingle electron density matrix based hierarchical equation of motion, which can be derived from Liouville–von Neumann equation, is reviewed in details. The numerical implementation is discussed and simulation results of realistic devices will be given.     

The quantum geometry of spin and statistics

Both, spin and statistics of a quantum system can be seen to arise from underlying (quantum) group symmetries. We show that the spin–statistics theorem is equivalent to a unification of these symmetries. Besides covering the Bose–Fermi case we classify the corresponding possibilities for anyonic spin and statistics. We incorporate the underlying extended concept of symmetry into quantum field theory in a generalised path integral formulation capable of handling general braid statistics. For bosons and fermions the different path integrals and Feynman rules naturally emerge without introducing Grassmann variables. We also consider the anyonic example of quons and obtain the path integral counterpart to the usual canonical approach.     

Time-dependent mechanical symmetries and extended Hamiltonian systems

Time-dependent mechanical symmetries are discussed in the framework of an extended Hamiltonian system. The Lie-algebraic structure of the time-dependent symmetry is made clear by introducing an extended Poisson bracket. Moreover, the relationship between the symmetry algebras of the classical and the quantum system is established.     

Time-dependent variational method for sine-Gordon quantum field theory

The sine-Gordon model in 1+1 dimensions is studied within the Schrödinger framework for field theory. In particular we evaluate the effective potential and examine the finiteness ofm(t), the soliton mass, for allt.     

Time-dependent quantum mechanical approach to reactive scattering and related processes

The time-dependent quantum mechanical approach has emerged as a powerful and a practical computational tool for studying a variety of gas-phase chemical problems in recent years. In this report, we discuss the various developments that have made this possible with special emphasis on methodology and application to reactive scattering, photo-excitation processes and gas-surface interaction.     

Noncommutative geometry of the quantum clock

We introduce a model of noncommutative geometry that gives rise to the uncertainty relations recently derived from the discussion of a quantum clock. We investigate the dynamics of a free particle in this model from the point of view of doubly special relativity and discuss the geodesic motion in a Schwarzschild background.     

The geometry of generalized quantum logics

Let II be a quantum logic; by this we mean an orthocomplemented, orthomodular, partially ordered set. We assume that II carries a sufficiently large collection of states (probability measures). Then, is embedded as a base for the cone of a partially ordered normed spaceL and II is also embedded in the dual order-unit Banach spaceL ^*. We consider conditions on the pairs (, II) and (L,L ^*) that guarantee that II is a dense subset of the extreme points of the positive part of the unit ball ofL ^*. We demonstrate a connection of these conditions in noncommutative measure theory. The assumptions made here are far weaker than the assumptions of the traditional quantum mechanical formalisms and also apply to situations quite different from quantum mechanics. Finally, we show the connections of this theory to the well-known models of quantum mechanics and classical measure theory.