The actions of the Hamiltonian constraint onto the members of the extended knot families {φi}2^2, {φi}3^4 and {φi}4^6, and the check of their invariance under the Mandelstam identities are given in the extended loo...The actions of the Hamiltonian constraint onto the members of the extended knot families {φi}2^2, {φi}3^4 and {φi}4^6, and the check of their invariance under the Mandelstam identities are given in the extended loop representation of loop quantum gravity.展开更多
This paper presents a new theory of gravity, called here Ashtekar-Kodama (AK) gravity, which is based on the Ashtekar-Kodama formulation of loop quantum gravity (LQG), yields in the limit the Einstein equations, and i...This paper presents a new theory of gravity, called here Ashtekar-Kodama (AK) gravity, which is based on the Ashtekar-Kodama formulation of loop quantum gravity (LQG), yields in the limit the Einstein equations, and in the quantum regime a full renormalizable quantum gauge field theory. The three fundamental constraints (hamiltonian, gaussian and diffeomorphism) were formulated in 3-dimensional spatial form within LQG in Ashtekar formulation using the notion of the Kodama state with positive cosmological constant Λ. We introduce a 4-dimensional covariant version of the 3-dimensional (spatial) hamiltonian, gaussian and diffeomorphism constraints of LQG. We obtain 32 partial differential equations for the 16 variables E<sub>mn</sub> (E-tensor, inverse densitized tetrad of the metric) and 16 variables A<sub>mn</sub> (A-tensor, gravitational wave tensor). We impose the boundary condition: for large distance the E-generated metric g(E) becomes the GR-metric g (normally Schwarzschild-spacetime). The theory based on these Ashtekar-Kodama (AK) equations, and called in the following Ashtekar-Kodama (AK-) gravity has the following properties. • For Λ = 0 the AK equations become Einstein equations, A-tensor is trivial (constant), and the E-generated metric g(E) is identical with the GR-metric g. • When the AK-equations are developed into a Λ-power series, the Λ-term yields a gravitational wave equation, which has only at least quadrupole wave solutions and becomes in the limit of large distance r the (normal electromagnetic) wave equation. • AK-gravity, as opposed to GR, has no singularity at the horizon: the singularity in the metric becomes a (very high) peak. • AK-gravity has a limit scale of the gravitational quantum region 39 μm, which emerges as the limit scale in the objective wave collapse theory of Gherardi-Rimini-Weber. In the quantum region, the AK-gravity becomes a quantum gauge theory (AK quantum gravity) with the Lie group extended SU(2) = ε-tensor-group(four generators) as gauge group and a corresponding covariant derivative. • AK quantum gravity is fully renormalizable, we derive its Lagrangian, which is dimensionally renormalizable, the normalized one-graviton wave function, the graviton propagator, and demonstrate the calculation of cross-section from Feynman diagrams.展开更多
Projected Runge-Kutta （R-K） methods for constrained Hamiltonian systems are proposed. Dynamic equations of the systems, which are index-3 differential-algebraic equations （DAEs） in the Heisenberg form, are establi...Projected Runge-Kutta （R-K） methods for constrained Hamiltonian systems are proposed. Dynamic equations of the systems, which are index-3 differential-algebraic equations （DAEs） in the Heisenberg form, are established under the framework of Lagrangian multipliers. R-K methods combined with the technique of projections are then used to solve the DAEs. The basic idea of projections is to eliminate the constraint violations at the position, velocity, and acceleration levels, and to preserve the total energy of constrained Hamiltonian systems by correcting variables of the position, velocity, acceleration, and energy. Numerical results confirm the validity and show the high precision of the proposed method in preserving three levels of constraints and total energy compared with results reported in the literature.展开更多
This paper focuses on studying Lie symmetries and conserved quantities of discrete nonholonomic Hamiltonian systems. Firstly, the discrete generalized Hamiltonian canonical equations and discrete energy equation of no...This paper focuses on studying Lie symmetries and conserved quantities of discrete nonholonomic Hamiltonian systems. Firstly, the discrete generalized Hamiltonian canonical equations and discrete energy equation of nonholonomic Hamiltonian systems are derived from discrete Hamiltonian action. Secondly, the determining equations and structure equation of Lie symmetry of the system are obtained. Thirdly, the Lie theorems and the conservation quantities are given for the discrete nonholonomic Hamiltonian systems. Finally, an example is discussed to illustrate the application of the results.展开更多
In this paper, an implicit symmetry constraint is calculated and its associated binary nonlinearization of the Lax pairs and the adjoint Lax pairs is carried out for the modified Korteweg-de Vries (mKdV) equation. Aft...In this paper, an implicit symmetry constraint is calculated and its associated binary nonlinearization of the Lax pairs and the adjoint Lax pairs is carried out for the modified Korteweg-de Vries (mKdV) equation. After introducing two new inde-pendent variables, we find that under the implicit symmetry constraint, the spatial part and the temporal part of the mKdV equation are decomposed into two finite-dimensional systems. Furthermore we prove that the obtained finite-dimensional systems are Hamiltonian systems and completely integrable in the Liouville sense.展开更多
In a recent paper we have studied the Hamiltonian and path integral quantizations of the conformally gauge-fixed Polyakov D1 brane action in the instant-form of dynamics using the equal world-sheet time framework on t...In a recent paper we have studied the Hamiltonian and path integral quantizations of the conformally gauge-fixed Polyakov D1 brane action in the instant-form of dynamics using the equal world-sheet time framework on the hyperplanes defined by the world- sheet time . In the present work we quantize the same theory in the equal light-cone world-sheet time framework, on the hyperplanes of the light-front defined by the light-cone world-sheet time , using the standard constraint quantization techniques in the Hamiltonian and path integral formulations. The light-front theory is seen to be a constrained system in the sense of Dirac, which is in contrast to the corresponding case of the instant-form theory, where the theory remains unconstrained in the sense of Dirac. The light-front theory is seen to possess a set of twenty six primary second-class contraints. In the present work Hamiltonian and path integral quantizations of this theory are studied on the light-front.展开更多
Recently we have studied the instant-form quantization (IFQ) of the conformally gauge-fixed Polyakov D1 brane action with and without a scalar dilaton field using the Hamiltonian and path integral formulations in the ...Recently we have studied the instant-form quantization (IFQ) of the conformally gauge-fixed Polyakov D1 brane action with and without a scalar dilaton field using the Hamiltonian and path integral formulations in the equal world-sheet time framework on the hyperplanes defined by the world- sheet time σ0=τ=constant . The light-front quantization (LFQ) of this theory without a scalar dilaton field has also been studied by us recently. In the present work we study the LFQ of this theory in the equal light-cone world-sheet time framework, on the hyperplanes of the light-front defined by the light-cone world-sheet time σ+=τ+σ=constant , using the Hamiltonian and path integral formulations. The light-front theory is seen to be a constrained system in the sense of Dirac. The light-front theory is seen to possess a set of twenty seven primary second-class contraints. In the present work Hamiltonian and path integral quantizations of this theory are studied on the light-front.展开更多
文摘The actions of the Hamiltonian constraint onto the members of the extended knot families {φi}2^2, {φi}3^4 and {φi}4^6, and the check of their invariance under the Mandelstam identities are given in the extended loop representation of loop quantum gravity.
文摘This paper presents a new theory of gravity, called here Ashtekar-Kodama (AK) gravity, which is based on the Ashtekar-Kodama formulation of loop quantum gravity (LQG), yields in the limit the Einstein equations, and in the quantum regime a full renormalizable quantum gauge field theory. The three fundamental constraints (hamiltonian, gaussian and diffeomorphism) were formulated in 3-dimensional spatial form within LQG in Ashtekar formulation using the notion of the Kodama state with positive cosmological constant Λ. We introduce a 4-dimensional covariant version of the 3-dimensional (spatial) hamiltonian, gaussian and diffeomorphism constraints of LQG. We obtain 32 partial differential equations for the 16 variables E<sub>mn</sub> (E-tensor, inverse densitized tetrad of the metric) and 16 variables A<sub>mn</sub> (A-tensor, gravitational wave tensor). We impose the boundary condition: for large distance the E-generated metric g(E) becomes the GR-metric g (normally Schwarzschild-spacetime). The theory based on these Ashtekar-Kodama (AK) equations, and called in the following Ashtekar-Kodama (AK-) gravity has the following properties. • For Λ = 0 the AK equations become Einstein equations, A-tensor is trivial (constant), and the E-generated metric g(E) is identical with the GR-metric g. • When the AK-equations are developed into a Λ-power series, the Λ-term yields a gravitational wave equation, which has only at least quadrupole wave solutions and becomes in the limit of large distance r the (normal electromagnetic) wave equation. • AK-gravity, as opposed to GR, has no singularity at the horizon: the singularity in the metric becomes a (very high) peak. • AK-gravity has a limit scale of the gravitational quantum region 39 μm, which emerges as the limit scale in the objective wave collapse theory of Gherardi-Rimini-Weber. In the quantum region, the AK-gravity becomes a quantum gauge theory (AK quantum gravity) with the Lie group extended SU(2) = ε-tensor-group(four generators) as gauge group and a corresponding covariant derivative. • AK quantum gravity is fully renormalizable, we derive its Lagrangian, which is dimensionally renormalizable, the normalized one-graviton wave function, the graviton propagator, and demonstrate the calculation of cross-section from Feynman diagrams.
基金Project supported by the National Natural Science Foundation of China(No.11432010)the Doctoral Program Foundation of Education Ministry of China(No.20126102110023)+2 种基金the 111Project of China(No.B07050)the Fundamental Research Funds for the Central Universities(No.310201401JCQ01001)the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University(No.CX201517)
文摘Projected Runge-Kutta （R-K） methods for constrained Hamiltonian systems are proposed. Dynamic equations of the systems, which are index-3 differential-algebraic equations （DAEs） in the Heisenberg form, are established under the framework of Lagrangian multipliers. R-K methods combined with the technique of projections are then used to solve the DAEs. The basic idea of projections is to eliminate the constraint violations at the position, velocity, and acceleration levels, and to preserve the total energy of constrained Hamiltonian systems by correcting variables of the position, velocity, acceleration, and energy. Numerical results confirm the validity and show the high precision of the proposed method in preserving three levels of constraints and total energy compared with results reported in the literature.
基金The project supported by National Natural Science Foundation of China under Grant No. 10371023 and Shanghai Shuguang Project of China under Grant No. 02SG02
基金supported by the National Natural Science Foundations of China (Grant No. 11072218)the Natural Science Foundation of Zhejiang Province of China (Grant No. Y6110314)
文摘This paper focuses on studying Lie symmetries and conserved quantities of discrete nonholonomic Hamiltonian systems. Firstly, the discrete generalized Hamiltonian canonical equations and discrete energy equation of nonholonomic Hamiltonian systems are derived from discrete Hamiltonian action. Secondly, the determining equations and structure equation of Lie symmetry of the system are obtained. Thirdly, the Lie theorems and the conservation quantities are given for the discrete nonholonomic Hamiltonian systems. Finally, an example is discussed to illustrate the application of the results.
文摘In this paper, an implicit symmetry constraint is calculated and its associated binary nonlinearization of the Lax pairs and the adjoint Lax pairs is carried out for the modified Korteweg-de Vries (mKdV) equation. After introducing two new inde-pendent variables, we find that under the implicit symmetry constraint, the spatial part and the temporal part of the mKdV equation are decomposed into two finite-dimensional systems. Furthermore we prove that the obtained finite-dimensional systems are Hamiltonian systems and completely integrable in the Liouville sense.
文摘In a recent paper we have studied the Hamiltonian and path integral quantizations of the conformally gauge-fixed Polyakov D1 brane action in the instant-form of dynamics using the equal world-sheet time framework on the hyperplanes defined by the world- sheet time . In the present work we quantize the same theory in the equal light-cone world-sheet time framework, on the hyperplanes of the light-front defined by the light-cone world-sheet time , using the standard constraint quantization techniques in the Hamiltonian and path integral formulations. The light-front theory is seen to be a constrained system in the sense of Dirac, which is in contrast to the corresponding case of the instant-form theory, where the theory remains unconstrained in the sense of Dirac. The light-front theory is seen to possess a set of twenty six primary second-class contraints. In the present work Hamiltonian and path integral quantizations of this theory are studied on the light-front.
文摘Recently we have studied the instant-form quantization (IFQ) of the conformally gauge-fixed Polyakov D1 brane action with and without a scalar dilaton field using the Hamiltonian and path integral formulations in the equal world-sheet time framework on the hyperplanes defined by the world- sheet time σ0=τ=constant . The light-front quantization (LFQ) of this theory without a scalar dilaton field has also been studied by us recently. In the present work we study the LFQ of this theory in the equal light-cone world-sheet time framework, on the hyperplanes of the light-front defined by the light-cone world-sheet time σ+=τ+σ=constant , using the Hamiltonian and path integral formulations. The light-front theory is seen to be a constrained system in the sense of Dirac. The light-front theory is seen to possess a set of twenty seven primary second-class contraints. In the present work Hamiltonian and path integral quantizations of this theory are studied on the light-front.
