Canonically Relativistic Quantum Mechanics: Representations of the Unitary Semidirect Heisenberg Group, U(1,3) *s H(1,3)

Born proposed a unification of special relativity and quantum mechanics that placed position, time, energy and momentum on equal footing through a reciprocity principle and extended the usual position-time and energy-momentum line elements to this space by combining them through a new fundamental constant. Requiring also invariance of the symplectic metric yields U(1,3) as the invariance group, the inhomogeneous counterpart of which is the canonically relativistic group CR(1,3) = U(1,3) *s H(1,3) where H(1,3) is the Heisenberg Group in 4 dimensions and "*s" is the semidirect product. This is the counterpart in this theory of the Poincare group and reduces in the appropriate limit to the expected special relativity and classical Hamiltonian mechanics transformation equations. This group has the Poincare group as a subgroup and is intrinsically quantum with the Position, Time, Energy and Momentum operators satisfying the Heisenberg algebra. The representations of the algebra are studied and Casimir invariants are computed. Like the Poincare group, it has a little group for a ("massive") rest frame and a null frame. The former is U(3) which clearly contains SU(3) and the latter is Os(2) which contains SU(2)*U(1).Comment: 18 pages, PDF, to be published in J. Math. Phys., Mathematica 3.0 computation files available from author at [email protected]

Relativity group for noninertial frames in Hamilton's mechanics

The group E(3)=SO(3) *s T(3), that is the homogeneous subgroup of the Galilei group parameterized by rotation angles and velocities, defines the continuous group of transformations between the frames of inertial particles in Newtonian mechanics. We show in this paper that the continuous group of transformations between the frames of noninertial particles following trajectories that satisfy Hamilton's equations is given by the Hamilton group Ha(3)=SO(3) *s H(3) where H(3) is the Weyl-Heisenberg group that is parameterized by rates of change of position, momentum and energy, i.e. velocity, force and power. The group E(3) is the inertial special case of the Hamilton group.Comment: Final versio

Reciprocal relativity of noninertial frames: quantum mechanics

Noninertial transformations on time-position-momentum-energy space {t,q,p,e} with invariant Born-Green metric ds^2=-dt^2+dq^2/c^2+(1/b^2)(dp^2-de^2/c^2) and the symplectic metric -de/\dt+dp/\dq are studied. This U(1,3) group of transformations contains the Lorentz group as the inertial special case. In the limit of small forces and velocities, it reduces to the expected Hamilton transformations leaving invariant the symplectic metric and the nonrelativistic line element ds^2=dt^2. The U(1,3) transformations bound relative velocities by c and relative forces by b. Spacetime is no longer an invariant subspace but is relative to noninertial observer frames. Born was lead to the metric by a concept of reciprocity between position and momentum degrees of freedom and for this reason we call this reciprocal relativity. For large b, such effects will almost certainly only manifest in a quantum regime. Wigner showed that special relativistic quantum mechanics follows from the projective representations of the inhomogeneous Lorentz group. Projective representations of a Lie group are equivalent to the unitary reprentations of its central extension. The same method of projective representations of the inhomogeneous U(1,3) group is used to define the quantum theory in the noninertial case. The central extension of the inhomogeneous U(1,3) group is the cover of the quaplectic group Q(1,3)=U(1,3)*s H(4). H(4) is the Weyl-Heisenberg group. A set of second order wave equations results from the representations of the Casimir operators

Technology transfer and cultural exchange: Western scientists and engineers encounter late Tokugawa and Meiji Japan

[FIRST PARAGRAPH] During the last decade of the nineteenth century, the Engineer was only one of many British and American publications that took an avid interest in the rapid rise of Japan to the status of a fully industrialized imperial power on a par with major European nations. In December 1897 this journal published a photographic montage of "Pioneers of Modem Engineering Education in Japan" (Figure I), showing a selection of the Japanese and Western teachers who had worked to bring about this singular transformation.' The predominance of Japanese figures in this representation is highly significant: it is an acknowledgment by British observers that the industrialization of Japan-the "Britain of the East"-was not a feat accomplished solely by Western experts who transferred their science and technology to passive Japanese recipients. Yet in focusing primarily on native teachers active in Japan after 1880, this image excludes several of the very foreigners who had trained this indigenous workforce in the preceding decade. Rather than attempting to assess the careers of each of the many international experts involved in Western encounters with Japan before and after the Meiji restoration in 1868, we will focus on disaggregating the highly individualized responses of just some of the Englishspeaking characters. In documenting their diverse encounters with Japanese people and technologies, we will look at the complex phenomena of cultural exchange in which they participated, not always without chauvinism or resistance

Discussion of "Frequentist coverage of adaptive nonparametric Bayesian credible sets"

Discussion of "Frequentist coverage of adaptive nonparametric Bayesian credible sets" by Szab\'o, van der Vaart and van Zanten [arXiv:1310.4489v5].Comment: Published at http://dx.doi.org/10.1214/15-AOS1270D in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Adaptive confidence balls

Adaptive confidence balls are constructed for individual resolution levels as well as the entire mean vector in a multiresolution framework. Finite sample lower bounds are given for the minimum expected squared radius for confidence balls with a prespecified confidence level. The confidence balls are centered on adaptive estimators based on special local block thresholding rules. The radius is derived from an analysis of the loss of this adaptive estimator. In addition adaptive honest confidence balls are constructed which have guaranteed coverage probability over all of $\mathbb{R}^N$ and expected squared radius adapting over a maximum range of Besov bodies.Comment: Published at http://dx.doi.org/10.1214/009053606000000146 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Nonparametric estimation over shrinking neighborhoods: Superefficiency and adaptation

A theory of superefficiency and adaptation is developed under flexible performance measures which give a multiresolution view of risk and bridge the gap between pointwise and global estimation. This theory provides a useful benchmark for the evaluation of spatially adaptive estimators and shows that the possible degree of superefficiency for minimax rate optimal estimators critically depends on the size of the neighborhood over which the risk is measured. Wavelet procedures are given which adapt rate optimally for given shrinking neighborhoods including the extreme cases of mean squared error at a point and mean integrated squared error over the whole interval. These adaptive procedures are based on a new wavelet block thresholding scheme which combines both the commonly used horizontal blocking of wavelet coefficients (at the same resolution level) and vertical blocking of coefficients (across different resolution levels).Comment: Published at http://dx.doi.org/10.1214/009053604000000832 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Virtual copies of semisimple Lie algebras in enveloping algebras of semidirect products and Casimir operators

Given a semidirect product $\frak{g}=\frak{s}\uplus\frak{r}$ of semisimple Lie algebras $\frak{s}$ and solvable algebras $\frak{r}$, we construct polynomial operators in the enveloping algebra $\mathcal{U}(\frak{g})$ of $\frak{g}$ that commute with $\frak{r}$ and transform like the generators of $\frak{s}$, up to a functional factor that turns out to be a Casimir operator of $\frak{r}$. Such operators are said to generate a virtual copy of $\frak{s}$ in $\mathcal{U}(\frak{g})$, and allow to compute the Casimir operators of $\frak{g}$ in closed form, using the classical formulae for the invariants of $\frak{s}$. The behavior of virtual copies with respect to contractions of Lie algebras is analyzed. Applications to the class of Hamilton algebras and their inhomogeneous extensions are given.Comment: 20 pages, 2 Appendice