On the Volume-Preserving Procrustes Problem

This paper is intended to study the volume-preserving procrustes problem arising from practical areas. The corresponding solution should satisfy a matrix equation which is solved by the singular value decomposition method. Some further results are also given to characterize the solution. Using these results, a numerical algorithm is introduced and some numerical results are provided to illustrate the effectiveness of the algorithm. Key words volume-preserving - procrustes problems - singular value decomposition MSC2000 65F30 - 65K10 Project supported by NNSFC (Grant No. 10371076), E-Institutes of Shanghai Municipal Education Commission (Grant No. N. E03004)

Volume-preserving Fields and Reeb Fields on 3-manifolds

Volume-preserving field X on a 3-manifold is the one that satisfies LxΩ = 0 for some volume Ω. The Reeb vector field of a contact form is of volume-preserving, but not conversely. On the basis of Geiges-Gonzalo＇s parallelization results, we obtain a volume-preserving sphere, which is a triple of everywhere linearly independent vector fields such that all their linear combinations with constant coefficients are volume-preserving fields. From many aspects, we discuss the distinction between volume-preserving fields and Reeb-like fields. We establish a duality between volume-preserving fields and h-closed 2-forms to understand such distinction. We also give two kinds of non-Reeb-like but volume-preserving vector fields to display such distinction.

Generating Functions for Volume-preserving Mappings and Hamilton-Jacobi Equations for Source-free Dynamical Systems

The general generating function theory for volume-preserving mappings and Hamilton-Jacobi theory for source-free systems are developed.

Equivalent Conditions of Dominated Splitting for Volume-Preserving Diffeomorphisms

We discuss the equivalent conditions of dominated splitting for conservative diffeomorphisms in C^1 topology.

SHADOWING,EXPANSIVENESS AND SPECIFICATION FOR C^1-CONSERVATIVE SYSTEMS

We prove that a Cl-generic volume-preserving dynamical system （diffeomor- phism or flow） has the shadowing property or is expansive or has the weak specification property if and only if it is Anosov. Finally, as in [10, 27], we prove that the Cl-robustness, within the volume-preserving context, of the expansiveness property and the weak specifica- tion property, imply that the dynamical system （diffeomorphism or flow） is Anosov.

Scattering Cross-Sections in Quantum Gravity—The Case of Matter-Matter Scattering

Viewing gravitational energy-momentum PG<sup style='margin-left:-7px;'>μ as equal by observation, but different in essence from inertial energy-momentum PI<sup style='margin-left:-7px;'>μ naturally leads to the gauge theory of volume-preserving diffeomorphisms of a four-dimensional inner space. To analyse scattering in this theory, the gauge field is coupled to two Dirac fields with different masses. Based on a generalized LSZ reduction formula the S-matrix element for scattering of two Dirac particles in the gravitational limit and the corresponding scattering cross-section are calculated to leading order in perturbation theory. Taking the non-relativistic limit for one of the initial particles in the rest frame of the other the Rutherford-like cross-section of a non-relativistic particle scattering off an infinitely heavy scatterer calculated quantum mechanically in Newtonian gravity is recovered. This provides a non-trivial test of the gauge field theory of volume-preserving diffeomorphisms as a quantum theory of gravity.

Conservation of Gravitational Energy-Momentum and Inner Diffeomorphism Group Gauge Invariance

Viewing gravitational energy momentum as equal by observation, but different in essence from inertial energy-momentum requires two different symmetries to account for their independent conservations—spacetime and inner translation invariance. Gauging the latter a generalization of non-Abelian gauge theories of compact Lie groups is developed resulting in the gauge theory of the non-compact group of volume-preserving diffeomorphisms of an inner Minkowski space M4. As usual the gauging requires the introduction of a covariant derivative, a gauge field and a field strength operator. An invariant and minimal gauge field Lagrangian is derived. The classical field dynamics and the conservation laws for the new gauge theory are developed. Finally, the theory’s Hamiltonian in the axial gauge is expressed by two times six unconstrained independent canonical variables obeying the usual Poisson brackets and the positivity of the Hamiltonian is related to a condition on the support of the gauge fields.

A Renormalizable Theory of Quantum Gravity: Renormalization Proof of the Gauge Theory of Volume Preserving Diffeomorphisms

Inertial and gravitational mass or energy momentum need not be the same for virtual quantum states. Separating their roles naturally leads to the gauge theory of volume-preserving diffeomorphisms of an inner four-dimensional space. The gauge-fixed action and the path integral measure occurring in the generating functional for the quantum Green functions of the theory are shown to obey a BRST-type symmetry. The related Zinn-Justin-type equation restricting the corresponding quantum effective action is established. This equation limits the infinite parts of the quantum effective action to have the same form as the gauge-fixed Lagrangian of the theory proving its spacetime renormalizability. The inner space integrals occurring in the quantum effective action which are divergent due to the gauge group’s infinite volume are shown to be regularizable in a way consistent with the symmetries of the theory demonstrating as a byproduct that viable quantum gauge field theories are not limited to finite-dimensional compact gauge groups as is commonly assumed.

General Relativity as the Classical Limit of the Renormalizable Gauge Theory of Volume Preserving Diffeomorphisms

The different roles and natures of spacetime appearing in a quantum field theory and in classical physics are analyzed implying that a quantum theory of gravitation is not necessarily a quantum theory of curved spacetime. Developing an alternative approach to quantum gravity starts with the postulate that inertial energy-momentum and gravitational energy-momentum need not be the same for virtual quantum states. Separating their roles naturally leads to the quantum gauge field theory of volume-preserving diffeomorphisms of an inner four-dimensional space. The classical limit of this theory coupled to a quantized scalar field is derived for an on-shell particle where inertial energy-momentum and gravitational energy-momentum coincide. In that process the symmetry under volume-preserving diffeomorphisms disappears and a new symmetry group emerges: the group of coordinate transformations of four-dimensional spacetime and with it General Relativity coupled to a classical relativistic point particle.

Conservation of Gravitational Energy Momentum and Renormalizable Quantum Theory of Gravitation

Viewing gravitational energy-momentum as equal by observation, but different in essence from inertial energymomentum naturally leads to the gauge theory of volume-preserving diffeomorphisms of an inner Minkowski space which can describe gravitation at the classical level. This theory is quantized in the path integral formalism starting with a non-covariant Hamiltonian formulation with unconstrained canonical field variables and a manifestly positive Hamiltonian. The relevant path integral measure and weight are then brought into a Lorentz- and gauge-covariant form allowing to express correlation functions—applying the De Witt-Faddeev-Popov approach—in any meaningful gauge. Next the Feynman rules are developed and the quantum effective action at one loop in a background field approach is renormalized which results in an asymptotically free theory without presence of other fields and in a theory without asymptotic freedom including the Standard Model (SM) fields. Finally the BRST apparatus is developed as preparation for the renormalizability proof to all orders and a sketch of this proof is given.

A Note on the Stability of Geodesics on Diffeomorphism Groups with One-side Invariant Metric

In this note,we consider the stability of geodesics on volume-preserving diffeomorphism groups with one-side invariant metric.We showed that for non-Beltrami fields on a three-dimensional compact manifold,there does not exist Eulerian stable flow which is Lagrangian exponential unstable.We noticed that a stationary flow corresponding to the KdV equation can be Eulerian stable while the corresponding motion of the fluid is at most exponentially unstable.