Reciprocal relativity of noninertial frames and the quaplectic group

Newtonian mechanics has the concept of an absolute inertial rest frame. Special relativity eliminates the absolute rest frame but continues to require the absolute inertial frame. General relativity solves this for gravity by requiring particles to have locally inertial frames on a curved position-time manifold. The problem of the absolute inertial frame for other forces remains. We look again at the transformations of frames on an extended phase space with position, time, energy and momentum degrees of freedom. Under nonrelativistic assumptions, there is an invariant symplectic metric and a line element dt^2. Under special relativistic assumptions the symplectic metric continues to be invariant but the line elements are now -dt^2+dq^2/c^2 and dp^2-de^2/c^2. Max Born conjectured that the line element should be generalized to the pseudo- orthogonal metric -dt^2+dq^2/c^2+ (1/b^2)(dp^2-de^2/c^2). The group leaving these two metrics invariant is the pseudo-unitary group of transformations between noninertial frames. We show that these transformations eliminate the need for an absolute inertial frame by making forces relative and bounded by b and so embodies a relativity that is 'reciprocal' in the sense of Born. The inhomogeneous version of this group is naturally the semidirect product of the pseudo-unitary group with the nonabelian Heisenberg group. This is the quaplectic group. The Heisenberg group itself is the semidirect product of two translation groups. This provides the noncommutative properties of position and momentum and also time and energy that are required for the quantum mechanics that results from considering the unitary representations of the quaplectic group.Comment: Substantial revision, Publicon LaTe

Viscoelastic Properties of Crystals

We examine the question of whether fluids and crystals are differentiated on the basis of their zero frequency shear moduli or their limiting zero frequency shear viscosity. We show that while fluids, in contrast with crystals, do have a zero value for their shear modulus, in contradiction to a widespread presumption, a crystal does not have an infinite or exceedingly large value for its limiting zero frequency shear viscosity. In fact, while the limiting shear viscosity of a crystal is much larger than that of the liquid from which it is formed, its viscosity is much less than that of the corresponding glass that may form assuming the liquid is a good enough glass former.Comment: 13 pages, 3 figure

Towards Integrated Environmental Management: A Reconnaissance of State Statutes

15 p. ; 28 cmhttps://scholar.law.colorado.edu/books_reports_studies/1060/thumbnail.jp

A Modality-Specific Feedforward Component of Choice-Related Activity in MT

The activity of individual sensory neurons can be predictive of an animal\u27s choices. These decision signals arise from network properties dependent on feedforward and feedback inputs; however, the relative contributions of these inputs are poorly understood. We determined the role of feedforward pathways to decision signals in MT by recording neuronal activity while monkeys performed motion and depth tasks. During each session, we reversibly inactivated V2 and V3, which provide feedforward input to MT that conveys more information about depth than motion. We thus monitored the choice-related activity of the same neuron both before and during V2/V3 inactivation. During inactivation, MT neurons became less predictive of decisions for the depth task but not the motion task, indicating that a feedforward pathway that gives rise to tuning preferences also contributes to decision signals. We show that our data are consistent with V2/V3 input conferring structured noise correlations onto the MT population

Representations of the Canonical group, (the semi-direct product of the Unitary and Weyl-Heisenberg groups), acting as a dynamical group on noncommuting extended phase space

The unitary irreducible representations of the covering group of the Poincare group P define the framework for much of particle physics on the physical Minkowski space P/L, where L is the Lorentz group. While extraordinarily successful, it does not provide a large enough group of symmetries to encompass observed particles with a SU(3) classification. Born proposed the reciprocity principle that states physics must be invariant under the reciprocity transform that is heuristically {t,e,q,p}->{t,e,p,-q} where {t,e,q,p} are the time, energy, position, and momentum degrees of freedom. This implies that there is reciprocally conjugate relativity principle such that the rates of change of momentum must be bounded by b, where b is a universal constant. The appropriate group of dynamical symmetries that embodies this is the Canonical group C(1,3) = U(1,3) *s H(1,3) and in this theory the non-commuting space Q= C(1,3)/ SU(1,3) is the physical quantum space endowed with a metric that is the second Casimir invariant of the Canonical group, T^2 + E^2 - Q^2/c^2-P^2/b^2 +(2h I/bc)(Y/bc -2) where {T,E,Q,P,I,Y} are the generators of the algebra of Os(1,3). The idea is to study the representations of the Canonical dynamical group using Mackey's theory to determine whether the representations can encompass the spectrum of particle states. The unitary irreducible representations of the Canonical group contain a direct product term that is a representation of U(1,3) that Kalman has studied as a dynamical group for hadrons. The U(1,3) representations contain discrete series that may be decomposed into infinite ladders where the rungs are representations of U(3) (finite dimensional) or C(2) (with degenerate U(1)* SU(2) finite dimensional representations) corresponding to the rest or null frames.Comment: 25 pages; V2.3, PDF (Mathematica 4.1 source removed due to technical problems); Submitted to J.Phys.

World-line Quantisation of a Reciprocally Invariant System

We present the world-line quantisation of a system invariant under the symmetries of reciprocal relativity (pseudo-unitary transformations on ``phase space coordinates" $(x^\mu(\tau),p^\mu(\tau))$ which preserve the Minkowski metric and the symplectic form, and global shifts in these coordinates, together with coordinate dependent transformations of an additional compact phase coordinate, $\theta(\tau)$). The action is that of free motion over the corresponding Weyl-Heisenberg group. Imposition of the first class constraint, the generator of local time reparametrisations, on physical states enforces identification of the world-line cosmological constant with a fixed value of the quadratic Casimir of the quaplectic symmetry group $Q(D-1,1)\cong U(D-1,1)\ltimes H(D)$, the semi-direct product of the pseudo-unitary group with the Weyl-Heisenberg group (the central extension of the global translation group, with central extension associated to the phase variable $\theta(\tau)$). The spacetime spectrum of physical states is identified. Even though for an appropriate range of values the restriction enforced by the cosmological constant projects out negative norm states from the physical spectrum, leaving over spin zero states only, the mass-squared spectrum is continuous over the entire real line and thus includes a tachyonic branch as well

Multispecies virial expansions

We study the virial expansion of mixtures of countably many different types of particles. The main tool is the Lagrange–Good inversion formula, which has other applications such as counting coloured trees or studying probability generating functions in multi-type branching processes. We prove that the virial expansion converges absolutely in a domain of small densities. In addition, we establish that the virial coefficients can be expressed in terms of two-connected graphs

Water Dynamics at Protein Interfaces: Ultrafast Optical Kerr Effect Study

The behavior of water molecules surrounding a protein can have an important bearing on its structure and function. Consequently, a great deal of attention has been focused on changes in the relaxation dynamics of water when it is located at the protein surface. Here we use the ultrafast optical Kerr effect to study the H-bond structure and dynamics of aqueous solutions of proteins. Measurements are made for three proteins as a function of concentration. We find that the water dynamics in the first solvation layer of the proteins are slowed by up to a factor of 8 in comparison to those in bulk water. The most marked slowdown was observed for the most hydrophilic protein studied, bovine serum albumin, whereas the most hydrophobic protein, trypsin, had a slightly smaller effect. The terahertz Raman spectra of these protein solutions resemble those of pure water up to 5 wt % of protein, above which a new feature appears at 80 cm–1, which is assigned to a bending of the protein amide chain