1,885 research outputs found
Uncertainty Relations for Angular Momentum
In this work we study various notions of uncertainty for angular momentum in
the spin-s representation of SU(2). We characterize the "uncertainty regions''
given by all vectors, whose components are specified by the variances of the
three angular momentum components. A basic feature of this set is a lower bound
for the sum of the three variances. We give a method for obtaining optimal
lower bounds for uncertainty regions for general operator triples, and evaluate
these for small s. Further lower bounds are derived by generalizing the
technique by which Robertson obtained his state-dependent lower bound. These
are optimal for large s, since they are saturated by states taken from the
Holstein-Primakoff approximation. We show that, for all s, all variances are
consistent with the so-called vector model, i.e., they can also be realized by
a classical probability measure on a sphere of radius sqrt(s(s+1)). Entropic
uncertainty relations can be discussed similarly, but are minimized by
different states than those minimizing the variances for small s. For large s
the Maassen-Uffink bound becomes sharp and we explicitly describe the
extremalizing states. Measurement uncertainty, as recently discussed by Busch,
Lahti and Werner for position and momentum, is introduced and a generalized
observable (POVM) which minimizes the worst case measurement uncertainty of all
angular momentum components is explicitly determined, along with the minimal
uncertainty. The output vectors for the optimal measurement all have the same
length r(s), where r(s)/s goes to 1 as s tends to infinity.Comment: 30 pages, 22 figures, 1 cut-out paper model, video abstract available
on https://youtu.be/h01pHekcwF
Observational Signatures of Quantum Gravity in Interferometers
We consider the uncertainty in the arm length of an interferometer due to
metric fluctuations from the quantum nature of gravity, proposing a concrete
microscopic model of energy fluctuations in holographic degrees of freedom on
the surface bounding a causally connected region of spacetime. In our model,
fluctuations longitudinal to the beam direction accumulate in the infrared and
feature strong long distance correlation in the transverse direction. This
leads to a signal that could be observed in a gravitational wave
interferometer. We connect the positional uncertainty principle arising from
our calculations to the 't Hooft gravitational S-matrix.Comment: 6 pages, 1 figur
Uncertainty from Heisenberg to Today
We explore the different meanings of "quantum uncertainty" contained in
Heisenberg's seminal paper from 1927, and also some of the precise definitions
that were explored later. We recount the controversy about "Anschaulichkeit",
visualizability of the theory, which Heisenberg claims to resolve. Moreover, we
consider Heisenberg's programme of operational analysis of concepts, in which
he sees himself as following Einstein. Heisenberg's work is marked by the
tensions between semiclassical arguments and the emerging modern quantum
theory, between intuition and rigour, and between shaky arguments and
overarching claims. Nevertheless, the main message can be taken into the new
quantum theory, and can be brought into the form of general theorems. They come
in two kinds, not distinguished by Heisenberg. These are, on one hand,
constraints on preparations, like the usual textbook uncertainty relation, and,
on the other, constraints on joint measurability, including trade-offs between
accuracy and disturbance.Comment: 36 pages, 1 figur
High Spectral Resolution Measurement of the Sunyaev–Zel'dovich Effect Null with Z-Spec
The Sunyaev-Zel'dovich (SZ) effect spectrum crosses through a null where ΔT_CMB = 0 near ν_0 = 217 GHz. In a cluster of galaxies, ν0 can be shifted from the canonical thermal SZ effect value by corrections to the SZ effect scattering due to the properties of the inter-cluster medium. We have measured the SZ effect in the hot galaxy cluster RX J 1347.5 – 1145 with Z-Spec, an R ~ 300 grating spectrometer sensitive between 185 and 305 GHz. These data comprise a high spectral resolution measurement around the null of the SZ effect and clearly exhibit the transition from negative to positive ΔT_CMB over the Z-Spec band. The SZ null position is measured to be ν_0 = 225.8 ± 2.5(stat.) ± 1.2(sys.) GHz, which differs from the canonical null frequency by 3.0σ and is evidence for modifications to the canonical thermal SZ effect shape. Assuming the measured shift in ν0 is due only to relativistic corrections to the SZ spectrum, we place the limit kT_e = 17.1 ± 5.3 keV from the zero-point measurement alone. By simulating the response of the instrument to the sky, we are able to generate likelihood functions in {y_0, T_e, v_pec} space. For v_pec = 0 km s^(–1), we measure the best-fitting SZ model to be y_0 = 4.6^(+0.6)_(–0.9) × 10^(–4), T_e, 0 = 15.2^(+12)_(–7.4) keV. When v pec is allowed to vary, a most probable value of v_pec = + 450 ± 810 km s^(–1) is found
Sourcing semiclassical gravity from spontaneously localized quantum matter
The possibility that a classical space-time and quantum matter cohabit at the
deepest level, i.e. the possibility of having a fundamental and not
phenomenological semiclassical gravity, is often disregarded for lack of a good
candidate theory. The standard semiclassical theory suffers from fundamental
inconsistencies (e.g.: Schr\"odinger cat sources, faster-than-light
communication and violation of the Born rule) which can only be ignored in
simple typical situations. We harness the power of spontaneous localization
models, historically constructed to solve the measurement problem in quantum
mechanics, to build a consistent theory of (stochastic) semiclassical gravity
in the Newtonian limit. Our model makes quantitative and potentially testable
predictions: we recover the Newtonian pair potential up to a short distance
cut-off (hence we predict no 1 particle self-interaction) and uncover an
additional gravitational decoherence term which depends on the specifics of the
underlying spontaneous localization model considered. We hint at a possible
program to go past the Newtonian limit, towards a consistent general
relativistic semiclassical gravity.Comment: 9 pages + refs, 1 figure, typos corrected and minor modification
Efficient Subgraph Isomorphism using Graph Topology
Subgraph isomorphism or subgraph matching is generally considered as an
NP-complete problem, made more complex in practical applications where the edge
weights take real values and are subject to measurement noise and possible
anomalies. To the best of our knowledge, almost all subgraph matching methods
utilize node labels to perform node-node matching. In the absence of such
labels (in applications such as image matching and map matching among others),
these subgraph matching methods do not work. We propose a method for
identifying the node correspondence between a subgraph and a full graph in the
inexact case without node labels in two steps - (a) extract the minimal unique
topology preserving subset from the subgraph and find its feasible matching in
the full graph, and (b) implement a consensus-based algorithm to expand the
matched node set by pairing unique paths based on boundary commutativity. Going
beyond the existing subgraph matching approaches, the proposed method is shown
to have realistically sub-linear computational efficiency, robustness to random
measurement noise, and good statistical properties. Our method is also readily
applicable to the exact matching case without loss of generality. To
demonstrate the effectiveness of the proposed method, a simulation and a case
study is performed on the Erdos-Renyi random graphs and the image-based affine
covariant features dataset respectively.Comment: Authors contributed equally. Names listed in alphabetical orde
The Emergence of Gravitational Wave Science: 100 Years of Development of Mathematical Theory, Detectors, Numerical Algorithms, and Data Analysis Tools
On September 14, 2015, the newly upgraded Laser Interferometer
Gravitational-wave Observatory (LIGO) recorded a loud gravitational-wave (GW)
signal, emitted a billion light-years away by a coalescing binary of two
stellar-mass black holes. The detection was announced in February 2016, in time
for the hundredth anniversary of Einstein's prediction of GWs within the theory
of general relativity (GR). The signal represents the first direct detection of
GWs, the first observation of a black-hole binary, and the first test of GR in
its strong-field, high-velocity, nonlinear regime. In the remainder of its
first observing run, LIGO observed two more signals from black-hole binaries,
one moderately loud, another at the boundary of statistical significance. The
detections mark the end of a decades-long quest, and the beginning of GW
astronomy: finally, we are able to probe the unseen, electromagnetically dark
Universe by listening to it. In this article, we present a short historical
overview of GW science: this young discipline combines GR, arguably the
crowning achievement of classical physics, with record-setting, ultra-low-noise
laser interferometry, and with some of the most powerful developments in the
theory of differential geometry, partial differential equations,
high-performance computation, numerical analysis, signal processing,
statistical inference, and data science. Our emphasis is on the synergy between
these disciplines, and how mathematics, broadly understood, has historically
played, and continues to play, a crucial role in the development of GW science.
We focus on black holes, which are very pure mathematical solutions of
Einstein's gravitational-field equations that are nevertheless realized in
Nature, and that provided the first observed signals.Comment: 41 pages, 5 figures. To appear in Bulletin of the American
Mathematical Societ
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