We obtain the integral formulae for computing the tetrads and metric components in Riemann normal coordinates and the Fermi coordinate system of an observer in arbitrary motion. Our approach admits an essential enlarging of the range of validity of these coordinates. The results obtained are applied to the geodesic deviation in the field of a weak plane gravitational wave and the computation of the plane-wave metric in Fermi normal coordinates.

The relationship is established between quantum phase transitions and complex geometric phases for open quantum systems governed by a non-Hermitian effective Hamiltonian with accidental crossing of the eigenvalues. In particular, the geometric phase associated with the ground state of the one-dimensional dissipative Ising model in a transverse magnetic field is evaluated, and it is demonstrated that the related quantum phase transition is of the first order.

We study theoretically the noise-assisted quantum exciton (electron) transfer (ET) in bio-complexes consisting of a single-level electron donor and an acceptor which has a complicated internal structure, and is modeled by many electron energy levels. Interactions are included between the donor and the acceptor energy levels and with the protein-solvent noisy environment. Different regions of parameters are considered, which characterize (i) the number of the acceptor levels, (ii) the acceptor 'band-width', and (iii) the amplitude of noise and its correlation time. Under some conditions, we derive analytical expressions for the ET rate and efficiency. We obtain equal occupation of all levels at large times, independent of the structure of the acceptor band and the noise parameters, but under the condition of nondegeneracy of the acceptor energy levels. We discuss the multi-scale dynamics of the acceptor population, and the accompanying effect of quantum coherent oscillations. We also demonstrate that for a large number of levels in the acceptor band, the efficiency of ET can be close to 100%, for both downhill and uphill transitions and for sharp and flat redox potentials.

In the framework of nonassociative geometry a unified description of continuum and discrete spacetime is proposed. In our approach at the Planck scales the spacetime is described as a so-called diodular discrete structure which at large spacetime scales "looks like" a differentiable manifold. After a brief review of foundations of nonassociative geometry, we discuss the nonassociative smooth and discrete de Sitter spacetimes.PACS numbers: 04.20.Gz; 02.40.Hw Numerous attempts to construct the quantum theory of gravitation and to understand the structure of spacetime has not been successful so far and the problem is still open (for recent reviews see: [1][2][3][4][5]). In general one should distinguish two strategies beyond the common treatment:• Quantize a classical structure and then restore it as some kind of the classical limit of the quantum theory.• Regard the classical structure as one being emerged from the other theory.The second strategy may require a revision of the quantum theory itself in a way that that in quantum theory of gravitation the standard concept of spacetime must be replaced at the Planck scales by some kind of the discrete structure [1].In this paper we propose in the framework of nonassociative geometry [6-10] a new approach to a classical and discrete structure of spacetime, which provides the unified description of continuum and discrete spacetime. The corresponding construction may be described as follows. In a neighborhood of an arbitrary point on a manifold with an affine connection one can introduce the geodesic local loop, which is uniquely defined by means of the parallel translation of geodesics along geodesics [12][13][14]. The family of local loops constructed in this way uniquely defines the space with affine connection, but not every family of geodesic loops on a manifold defines an affine connection; there exist some relations between the loops at distinct points. Taking into account that the local loop sructure admits an additional operation, namely, the multiplication of the point by scalar, and a vector space structure induced by means of the exponential mapping; one can express the above mentioned relations by means of some algebraic identities. This leads to the notion of odule and the so-called geoodular structure.A geoodular covering of the affinely connected manifolds, consisiting of the odular covering with some additional algebraic identities, contains complete information about the manifold and allows us to reconstruct it. If we take an arbitrary smooth geoodular covering, then it uniquely generates an affine connection, whose geoodular covering coincides with the initial one. Introducing the left invariant diodular metric, one obtains the Riemannian (pseudoRiemannian) geodiodular manifold. This implies that a smooth geoodular (Riemannian/pseudo-Riemannian) manifold is an affinely connected (Riemannian/pseudo-Riemannian) manifold being described in another language and there is the equivalence of the corresponding categories [15,16]. Ignoring the smoothness, ...