New Geometric Formalism for Gravity Equation in Empty Space

In this paper, a complex daor field which can be regarded as the square root of space-time metric is proposed to represent gravity. The locally complexified geometry is set up, and the complex spin connection constructs a bridge between gravity and SU(1,3) gauge field. Daor field equations in empty space are acquired, which are one-order differential equations and not conflict with Einstein's gravity theory.Comment: 20 pages, to appear in Int. J. Mod. Phys.

A Unifying Approach to Quaternion Adaptive Filtering: Addressing the Gradient and Convergence

A novel framework for a unifying treatment of quaternion valued adaptive filtering algorithms is introduced. This is achieved based on a rigorous account of quaternion differentiability, the proposed I-gradient, and the use of augmented quaternion statistics to account for real world data with noncircular probability distributions. We first provide an elegant solution for the calculation of the gradient of real functions of quaternion variables (typical cost function), an issue that has so far prevented systematic development of quaternion adaptive filters. This makes it possible to unify the class of existing and proposed quaternion least mean square (QLMS) algorithms, and to illuminate their structural similarity. Next, in order to cater for both circular and noncircular data, the class of widely linear QLMS (WL-QLMS) algorithms is introduced and the subsequent convergence analysis unifies the treatment of strictly linear and widely linear filters, for both proper and improper sources. It is also shown that the proposed class of HR gradients allows us to resolve the uncertainty owing to the noncommutativity of quaternion products, while the involution gradient (I-gradient) provides generic extensions of the corresponding real- and complex-valued adaptive algorithms, at a reduced computational cost. Simulations in both the strictly linear and widely linear setting support the approach

Unification of Gravitation, Gauge Field and Dark Energy

This paper is composed of two correlated topics: 1. unification of gravitation with gauge fields; 2. the coupling between the daor field and other fields and the origin of dark energy. After introducing the concept of ``daor field" and discussing the daor geometry, we indicate that the complex daor field has two kinds of symmetry transformations. Hence the gravitation and SU(1,3) gauge field are unified under the framework of the complex connection. We propose a first-order nonlinear coupling equation of the daor field, which includes the coupling between the daor field and SU(1,3) gauge field and the coupling between the daor field and the curvature, and from which Einstein's gravitational equation can be deduced. The cosmological observations imply that dark energy cannot be zero, and which will dominate the doom of our Universe. The real part of the daor field self-coupling equation can be regarded as Einstein's equation endowed with the cosmological constant. It shows that dark energy originates from the self-coupling of the space-time curvature, and the energy-momentum tensor is proportional to the square of coupling constant \lambda. The dark energy density given by our scenario is in agreement with astronomical observations. Furthermore, the Newtonian gravitational constant G and the coupling constant \epsilon of gauge field satisfy G= \lambda^{2}\epsilon^{2}.Comment: 24 pages, revised version; references added; typos correcte

A quantum mechanical version of Price's theorem for Gaussian states

This paper is concerned with integro-differential identities which are known in statistical signal processing as Price's theorem for expectations of nonlinear functions of jointly Gaussian random variables. We revisit these relations for classical variables by using the Frechet differentiation with respect to covariance matrices, and then show that Price's theorem carries over to a quantum mechanical setting. The quantum counterpart of the theorem is established for Gaussian quantum states in the framework of the Weyl functional calculus for quantum variables satisfying the Heisenberg canonical commutation relations. The quantum mechanical version of Price's theorem relates the Frechet derivative of the generalized moment of such variables with respect to the real part of their quantum covariance matrix with other moments. As an illustrative example, we consider these relations for quadratic-exponential moments which are relevant to risk-sensitive quantum control.Comment: 11 pages, to appear in the Proceedings of the Australian Control Conference, 17-18 November 2014, Canberra, Australi

A Unified Stochastic Formulation of Dissipative Quantum Dynamics. I. Generalized Hierarchical Equations

We extend a standard stochastic theory to study open quantum systems coupled to generic quantum environments including the three fundamental classes of noninteracting particles: bosons, fermions and spins. In this unified stochastic approach, the generalized stochastic Liouville equation (SLE) formally captures the exact quantum dissipations when noise variables with appropriate statistics for different bath models are applied. Anharmonic effects of a non-Gaussian bath are precisely encoded in the bath multi-time correlation functions that noise variables have to satisfy. Staring from the SLE, we devise a family of generalized hierarchical equations by averaging out the noise variables and expand bath multi-time correlation functions in a complete basis of orthonormal functions. The general hiearchical equations constitute systems of linear equations that provide numerically exact simulations of quantum dynamics. For bosonic bath models, our general hierarchical equation of motion reduces exactly to an extended version of hierarchical equation of motion which allows efficient simulation for arbitrary spectral densities and temperature regimes. Similar efficiency and exibility can be achieved for the fermionic bath models within our formalism. The spin bath models can be simulated with two complementary approaches in the presetn formalism. (I) They can be viewed as an example of non-Gaussian bath models and be directly handled with the general hierarchical equation approach given their multi-time correlation functions. (II) Alterantively, each bath spin can be first mapped onto a pair of fermions and be treated as fermionic environments within the present formalism.Comment: 31 pages, 2 figure