11,285 research outputs found

### 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

### 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

### Projective Representations of the Inhomogeneous Hamilton Group: Noninertial Symmetry in Quantum Mechanics

Symmetries in quantum mechanics are realized by the projective
representations of the Lie group as physical states are defined only up to a
phase. A cornerstone theorem shows that these representations are equivalent to
the unitary representations of the central extension of the group. The
formulation of the inertial states of special relativistic quantum mechanics as
the projective representations of the inhomogeneous Lorentz group, and its
nonrelativistic limit in terms of the Galilei group, are fundamental examples.
Interestingly, neither of these symmetries includes the Weyl-Heisenberg group;
the hermitian representations of its algebra are the Heisenberg commutation
relations that are a foundation of quantum mechanics. The Weyl-Heisenberg group
is a one dimensional central extension of the abelian group and its unitary
representations are therefore a particular projective representation of the
abelian group of translations on phase space. A theorem involving the
automorphism group shows that the maximal symmetry that leaves invariant the
Heisenberg commutation relations are essentially projective representations of
the inhomogeneous symplectic group. In the nonrelativistic domain, we must also
have invariance of Newtonian time. This reduces the symmetry group to the
inhomogeneous Hamilton group that is a local noninertial symmetry of Hamilton's
equations. The projective representations of these groups are calculated using
the Mackey theorems for the general case of a nonabelian normal subgroup

### 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

### Constraint quantisation of a worldline system invariant under reciprocal relativity. II

We consider the world-line quantisation of a system invariant under the
symmetries of reciprocal relativity. 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(3,1) ~ U(3,1) x H(4),
the semi-direct product of the pseudo-unitary group with the Weyl-Heisenberg
group. In our previous paper, J Phys A 40 (2007) 12095--12111, the `spin'
degrees of freedom were handled as covariant oscillators, leading to a unique
choice of cosmological constant, required for projecting out negative-norm
states from the physical gauge-invariant states. In the present paper the spin
degrees of freedom are treated as standard oscillators with positive norm
states (wherein Lorentz boosts are not number-conserving in the auxiliary
space; reciprocal transformations are of course not spin-conserving in
general). As in the covariant approach, the spectrum of the square of the
energy-momentum vector is continuous over the entire real line, and thus
includes tachyonic (spacelike) and null branches. Adopting standard frames, the
Wigner method on each branch is implemented, to decompose the auxiliary space
into unitary irreducible representations of the respective little algebras and
additional degeneracy algebras. The physical state space is vastly enriched as
compared with the covariant approach, and contains towers of integer spin
massive states, as well as unconventional massless representations, with
continuous Euclidean momentum and arbitrary integer helicity.Comment: 21 pages, LaTe

### A redshift survey of IRAS galaxies

Results are presented from a redshift survey of all 72 galaxies detected by IRAS in Band 3 at flux levels equal to or greater then 2 Jy. The luminosity function at the high luminosity end is proportional to L sup -2, however, a flattening was observed at the low luminosity end indicating that a single power law is not a good description of the entire luminosity function. Only three galaxies in the sample have emission line spectra indicative of AGN's, suggesting that, at least in nearby galaxies, unobscured nuclear activity is not a strong contributor to the far infrared flux. Comparisons between the selected IRAS galaxies and an optically complete sample taken from the CfA redshift survey show that they are more narrowly distributed than those optically selected, in the sence that the IRAS sample includes few galaxies of low absolute blue luminosity. It was also found that the space distributions of the two samples differ: the density enhancement or IRAS galaxies is only approx. 1/3 that of the optically selected galaxies in the core of the Coma cluster

### Ballistic-Ohmic quantum Hall plateau transition in graphene pn junction

Recent quantum Hall experiments conducted on disordered graphene pn junction
provide evidence that the junction resistance could be described by a simple
Ohmic sum of the n and p mediums' resistances. However in the ballistic limit,
theory predicts the existence of chirality-dependent quantum Hall plateaus in a
pn junction. We show that two distinctively separate processes are required for
this ballistic-Ohmic plateau transition, namely (i) hole/electron Landau states
equilibration and (ii) valley iso-spin dilution of the incident Landau edge
state. These conclusions are obtained by a simple scattering theory argument,
and confirmed numerically by performing ensembles of quantum magneto-transport
calculations on a 0.1um-wide disordered graphene pn junction within the
tight-binding model. The former process is achieved by pn interface roughness,
where a pn interface disorder with a root-mean-square roughness of 10nm was
found to suffice under typical experimental conditions. The latter process is
mediated by extrinsic edge roughness for an armchair edge ribbon and by
intrinsic localized intervalley scattering centers at the edge of the pn
interface for a zigzag ribbon. In light of these results, we also examine why
higher Ohmic type plateaus are less likely to be observable in experiments.Comment: 9 pages, 6 figure

### 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.

### Nonlocal reflection by photonic barriers

The time behaviour of microwaves undergoing partial reflection by photonic
barriers was measured in the time and in the frequency domain. It was observed
that unlike the duration of partial reflection by dielectric layers, the
measured reflection duration of barriers is independent of their length. The
experimental results point to a nonlocal behaviour of evanescent modes at least
over a distance of some ten wavelengths. Evanescent modes correspond to
photonic tunnelling in quantum mechanics.Comment: 8 pages, 5 figure

### Light Sheets and the Covariant Entropy Conjecture

We examine the holography bound suggested by Bousso in his covariant entropy
conjecture, and argue that it is violated because his notion of light sheet is
too generous. We suggest its replacement by a weaker bound.Comment: 5 pages, to appear in Classical and Quantum Gravit

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