15,352 research outputs found
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 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
A complement to Le Cam's theorem
This paper examines asymptotic equivalence in the sense of Le Cam between
density estimation experiments and the accompanying Poisson experiments. The
significance of asymptotic equivalence is that all asymptotically optimal
statistical procedures can be carried over from one experiment to the other.
The equivalence given here is established under a weak assumption on the
parameter space . In particular, a sharp Besov smoothness
condition is given on which is sufficient for Poissonization,
namely, if is in a Besov ball with . Examples show Poissonization is not possible whenever .
In addition, asymptotic equivalence of the density estimation model and the
accompanying Poisson experiment is established for all compact subsets of
, a condition which includes all H\"{o}lder balls with smoothness
.Comment: Published at http://dx.doi.org/10.1214/009053607000000091 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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