Invariant subspaces of clustering operators

A class of clustering operators is defined which is a generalization of a transfer matrix of a Gibbs lattice field with an exponential decay of correlations. It is proved that for small values of the clustering operator has invariant subspaces which are similar tok-particle subspaces of the Fock space. The restriction of the clustering operator onto these subspaces resembles the operator exp(-H _k, whereH _k is thek- particle Schrödinger Hamiltonian in nonrelativistic quantum mechanics. The spectrum of eachH _k,k1, is contained in the interval (C ₁^k,C ₂^k). These intervals do not intersect with each other.     

Invariant subspaces of clustering operators. II

Clustering operators, when restricted tok-particle invariant subspaces, are shown still to cluster.     

Invariant differential operators for non-compact Lie groups: Summary of su(4,4) multiplets

The present paper is part of the project of systematic construction of invariant differential operators of noncompact semisimple Lie algebras. Here we give a summary of all multiplets containing physically relevant representations including the minimal ones for the algebra su(4, 4). Due to the recently established parabolic relations the results are valid also for the algebras sl(8, R) and su*(8)     

Invariant differential operators for non-compact Lie groups: The main su(n, n) cases

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebras su(n, n). Our choice of these algebras is motivated by the fact that for n = 2 this is the conformal algebra of 4-dimensional Minkowski space-time. Furthermore for general n these algebras belong to a narrow class of algebras, which we call “conformal Lie algebras”, which have very similar properties to the conformal algebras of n ²-dimensional Minkowski space-time. We give the main multiplets of indecomposable elementary representations for n = 2, 3, 4, including the necessary data for all relevant invariant differential operators.     

几类粒子密度算符和粒子流密度算符的讨论

从粒子密度和粒子流密度出发,分析了几类相关的算符,讨论了它们的性质与相互关系;从具体的物理意义出发,明确了合适的粒子密度算符和粒子流密度算符的表达形式.     

Invariant amplitudes for pion electroproduction

The invariant amplitudes for pion electroproduction on the nucleon are evaluated by dispersion relations at constant t with MAID as input for the imaginary parts of these amplitudes. In the threshold region these amplitudes are confronted with the predictions of several low-energy theorems derived in the soft-pion limit. In general agreement with chiral perturbation theory, the dispersive approach yields large corrections to these theorems because of the finite pion mass.     

Invariant tensors for simple groups

The forms of the invariant primitive tensors for the simple Lie algebras A_l, B_l, C_l, and D_l are investigated. A new family of symmetric invariant tensors is introduced using the non-trivial cocycles for the Lie algebra cohomology. For the A_l algebra it is explicitly shown that the generic forms of these tensors become zero except for the l primitive ones and that they give rise to the l primitive Casimir operators. Some recurrence and duality relations are given for the Lie algebra cocycles. Tables for the 3- and 5-cocycles for su(3) are su(4) are also provided. Finally, new relations involving the d and f su(n) tensors are given.     

Scaling Equation for Invariant Measure

An iterated function system (IFS) is constructed. It is shown that the invariant measure of IFS satisfies the same equation as scaling equation for wavelet transform (WT). Obviously, IFS and scaling equation of WT both have contraction mapping principle.     

Invariant Measures for Cherry Flows

We investigate the invariant probability measures for Cherry flows, i.e. flows on the two-torus which have a saddle, a source, and no other fixed points, closed orbits or homoclinic orbits. In the case when the saddle is dissipative or conservative we show that the only invariant probability measures are the Dirac measures at the two fixed points, and the Dirac measure at the saddle is the physical measure. In the other case we prove that there exists also an invariant probability measure supported on the quasi-minimal set, we discuss some situations when this other invariant measure is the physical measure, and conjecture that this is always the case. The main techniques used are the study of the integrability of the return time with respect to the invariant measure of the return map to a closed transversal to the flow, and the study of the close returns near the saddle.     

Equivalent angular momentum operators for second quantised operators

Formula useful for replacing second quantised operators by angular momentum operators are derived and commented on in the light of recent results on exchange interactions.     

Invariant Velocities for Two-Dimensional Minkowski Space-Time

It is shown that in two-dimensional space-time only the velocities parallel to the velocities of moving reference frames may be invariant with respect to Lorentz transformations.     

Invariant metric for nonlinear symplectic maps

In this paper, we construct an invariant metric in the space of homogeneous polynomials of a given degree (≥3). The homogeneous polynomials specify a nonlinear symplectic map which in turn represents a Hamiltonian system. By minimizing the norm constructed out of this metric as a function of system parameters, we demonstrate that the performance of a nonlinear Hamiltonian system is enhanced.     

Additive Invariant Functionals for Dynamical Systems

We consider the problem of defining completely a class of additive conservation laws for the generalized Liouville equation whose characteristics are given by an arbitrary system of first-order ordinary differential equations. We first show that if the conservation law, a time-invariant functional, is additive on functions having disjoint compact support in phase space, then it is represented by an integral over phase space of a kernel which is a function of the solution to the Liouville equation. Then we use the fact that in classical mechanics phase space is usually a direct product of physical space and velocity space (Newtonian systems). We prove that for such systems the aforementioned representation of the invariant functionals will hold for conservation laws which are additive only in physical space; i.e., additivity in physical space automatically implies additivity in the whole phase space. We extend the results to include non-degenerate Hamiltonian systems, and, more generally, to include both conservative and dissipative dynamical systems. Some applications of the results are discussed.     

Invariant simultaneity

The concept of the relativity of simultaneity, as a consequence of the Lorentz transformations, is shown to be an unproven inference based on the implicit idea that simultaneity can be determined only on the basis of synchroneity. The Lorentz transformations do imply, by interpretive inference, a relativity of synchroneity whereby a moving system of synchronized clocks appears to be nonsynchronous, with constant nonzero time invariant phases among the clocks that depend only on their relative fixed distances from each other. It does not seem to have been recognized that such an array of uniformly running nonsynchronous clocks, described as isochronous, can also lead to the unambiguous determination of simultaneity. The important significance of the relative temporal phases, namely the relativity of synchroneity, entailed by the Lorentz transformations is that certain alleged logical inconsistencies, asserted by both proponents and opponents of the special theory of relativity, can be readily resolved. The relativity of synchroneity does, however, raise certain other consequences that merit attention and careful consideration.