arXiv subjects

Alan Weinstein

Publications and source records attributed to Alan Weinstein.

At least 19 recordsLinked to original sources

arXiv2024

The Wehrheim-Woodward Category of Linear Canonical Relations between $G$-Spaces

We extend the work in a previous paper with David Li-Bland (arXiv:1401.7302) to construct the Wehrheim-Woodward category WW($G\mathbf{SLREL}$) of equivariant linear canonical relations between linear symplectic $G$-spaces for a compact group $G$. When $G$ is the trivial group, this reduces to the previous result that the morphisms in WW($\mathbf{SLREL}$) may be identified with pairs $(L,k)$ consisting of a linear canonical relation and a nonnegative integer.

math.SG↗

arXiv2024

Geometric States

We introduce a special family of distributional alpha-densities and give a transversality criterion stating when their product is defined, closely related to Hormander's criterion for general distributions. Moreover, we show that for the subspace of distributional half-densities in this family the distribution product naturally yields a pairing that extends the usual one on smooth half-densities.

math.FA↗

arXiv2023

Hamiltonian Lie algebroids over Poisson manifolds

We extend to Poisson manifolds the theory of hamiltonian Lie algebroids originally developed by two of the authors for presymplectic manifolds. As in the presymplectic case, our definition, involving a vector bundle connection on the Lie algebroid, reduces to the definition of hamiltonian action for an action Lie algebroid with the trivial connection. The clean zero locus of the momentum section of a hamiltonian Lie algebroid is an invariant coisotropic submanifold, the distribution being given by the image of the anchor. We study some basic examples: bundles of Lie algebras with zero anchor and cotangent and tangent Lie algebroids. Finally, we discuss a suggestion by Alejandro Cabrera that the conditions for a Lie algebroid $A$ to be hamiltonian may be expressed in terms of two bivector fields on $A^*$, the natural Poisson structure on the dual of a Lie algebroid and the horizontal lift by the connection of the given Poisson structure on the base.

math.SG↗

arXiv2022

A Lie-Rinehart algebra in general relativity

We construct a Lie-Rinehart algebra over an infinitesimal extension of the space of initial value fields for Einstein's equations. The bracket relations in this algebra are precisely those of the constraints for the initial value problem. The Lie-Rinehart algebra comes from a slight generalization of a Lie algebroid in which the algebra consists of sections of a sheaf rather than a vector bundle. (An actual Lie algebroid had been previously constructed by Blohmann, Fernandes, and Weinstein over a much larger extension.) The construction uses the BV-BFV (Batalin-Fradkin-Vilkovisky) approach to boundary value problems, starting with the Einstein equations themselves, to construct an $L_\infty$-algebroid over a graded manifold which extends the initial data. The Lie-Rinehart algebra is then constructed by a change of variables. One of the consequences of the BV-BFV approach is a proof that the coisotropic property of the constraint set follows from the invariance of the Einstein equations under space-time diffeomorphisms.

math-ph↗

arXiv2020

Symplectic Microgeometry IV: Quantization

We construct a special class of semiclassical Fourier integral operators whose wave fronts are symplectic micromorphisms. These operators have very good properties: they form a category on which the wave front map becomes a functor into the cotangent microbundle category, and they admit a total symbol calculus in terms of symplectic micromorphisms enhanced with half-density germs. This new operator category encompasses the semi-classical pseudo-differential calculus and offers a functorial framework for the semi-classical analysis of the Schr\"odinger equation. We also comment on applications to classical and quantum mechanics as well as to a functorial and geometrical approach to the quantization of Poisson manifolds.

math.SG↗

arXiv2019

Plasma in monopole background is not twisted Poisson

For a particle in the magnetic field of a cloud of monopoles, the naturally associated 2-form on phase space is not closed, and so the corresponding bracket operation on functions does not satisfy the Jacobi identity. Thus, it is not a Poisson bracket; however, it is twisted Poisson in the sense that the Jacobiator comes from a closed 3-form. The space $\mathcal D$ of densities on phase space is the state space of a plasma. The twisted Poisson bracket on phase-space functions gives rise to a bracket on functions on $\mathcal D$. In the absence of monopoles, this is again a Poisson bracket. It has recently been shown by Heninger and Morrison that this bracket is not Poisson when monopoles are present. In this note, we give an example where it is not even twisted Poisson.

math-ph↗

arXiv2019

(Co)isotropic triples and poset representations

We study triples of coisotropic or isotropic subspaces in symplectic vector spaces; in particular, we classify indecomposable structures of this kind. The classification depends on the ground field, which we only assume to be perfect and not of characteristic 2. Our work uses the theory of representations of partially ordered sets with (order reversing) involution; for (co)isotropic triples, the relevant poset is "$2 + 2 + 2$" consisting of three independent ordered pairs, with the involution exchanging the members of each pair. A key feature of the classification is that any indecomposable (co)isotropic triple is either "split" or "non-split". The latter is the case when the poset representation underlying an indecomposable (co)isotropic triple is itself indecomposable. Otherwise, in the "split" case, the underlying representation is decomposable and necessarily the direct sum of a dual pair of indecomposable poset representations; the (co)isotropic triple is a "symplectification". In the course of the paper we develop the framework of "symplectic poset representations", which can be applied to a range of problems of symplectic linear algebra. The classification of linear Hamiltonian vector fields, up to conjugation, is an example; we briefly explain the connection between these and (co)isotropic triples. The framework lends itself equally well to studying poset representations on spaces carrying a non-degenerate symmetric bilinear form; we mainly keep our focus, however, on the symplectic side.

math.SG↗

arXiv2019

A self-consistent method to estimate the rate of compact binary coalescences with a Poisson mixture model

The recently published GWTC-1 - a journal article summarizing the search for gravitational waves (GWs) from coalescing compact binaries in data produced by the LIGO-Virgo network of ground-based detectors during their first and second observing runs - quoted estimates for the rates of binary neutron star, neutron star black hole binary, and binary black hole mergers, as well as assigned probabilities of astrophysical origin for various significant and marginal GW candidate events. In this paper, we delineate the formalism used to compute these rates and probabilities, which assumes that triggers above a low ranking statistic threshold, whether of terrestrial or astrophysical origin, occur as independent Poisson processes. In particular, we include an arbitrary number of astrophysical categories by redistributing, via mass-based template weighting, the foreground probabilities of candidate events, across source classes. We evaluate this formalism on synthetic GW data, and demonstrate that this method works well for the kind of GW signals observed during the first and second observing runs.

astro-ph.HE↗

arXiv2019

The GstLAL Search Analysis Methods for Compact Binary Mergers in Advanced LIGO's Second and Advanced Virgo's First Observing Runs

After their successful first observing run (September 12, 2015 - January 12, 2016), the Advanced LIGO detectors were upgraded to increase their sensitivity for the second observing run (November 30, 2016 - August 26, 2017). The Advanced Virgo detector joined the second observing run on August 1, 2017. We discuss the updates that happened during this period in the GstLAL-based inspiral pipeline, which is used to detect gravitational waves from the coalescence of compact binaries both in low latency and an offline configuration. These updates include deployment of a zero-latency whitening filter to reduce the over-all latency of the pipeline by up to 32 seconds, incorporation of the Virgo data stream in the analysis, introduction of a single-detector search to analyze data from the periods when only one of the detectors is running, addition of new parameters to the likelihood ratio ranking statistic, increase in the parameter space of the search, and introduction of a template mass-dependent glitch-excision thresholding method.

gr-qc↗

arXiv2018

Template bank for spinning compact binary mergers in the second observation run of Advanced LIGO and the first observation run of Advanced Virgo

We describe the methods used to construct the aligned-spin template bank of gravitational waveforms used by the GstLAL-based inspiral pipeline to analyze data from the second observing run of Advanced LIGO and the first observing run of advanced Virgo. The bank expands upon the parameter space covered during the first observing run, including coverage for merging compact binary systems with total mass between 2 $\mathrm{M}_{\odot}$ and 400 $\mathrm{M}_{\odot}$ and mass ratios between 1 and 97.988. Thus the systems targeted include merging neutron star-neutron star systems, neutron star-black hole binaries, and black hole-black hole binaries expanding into the intermediate-mass range. Component masses less than 2 $\mathrm{M}_{\odot}$ have allowed (anti-)aligned spins between $\pm0.05$ while component masses greater than 2 $\mathrm{M}_{\odot}$ have allowed (anti-)aligned between $\pm0.999$. The bank placement technique combines a stochastic method with a new grid-bank method to better isolate noisy templates, resulting in a total of 677,000 templates.

astro-ph.IM↗

arXiv2018

Hamiltonian Lie algebroids

In previous work with M.C. Fernandes, we found a Lie algebroid symmetry for the Einstein evolution equations of general relativity. The present work was motivated by the effort to explain the coisotropic structure of the constraint subset for the initial value problem by extending the notion of hamiltonian structure from Lie algebra actions to general Lie algebroids over presymplectic manifolds. After comparing possible compatibility conditions between the anchor $A\to TM$ and the presymplectic structure on the base $M$, we choose the most natural of them, given by a suitably chosen connection on $A$. We define a notion of momentum section of $A^*$ and a condition for compatibility with the Lie bracket. A Lie algebroid over a presymplectic manifold with compatible anchor and momentum section is then called hamiltonian. For an action Lie algebroid, we retrieve the conditions of a hamiltonian action. The clean zero locus of the momentum section of a hamiltonian Lie algebroid is a coisotropic submanifold. We show that a bracket-compatible momentum map is equivalent to a closed basic extension of the presymplectic form, within the generalization of the BRST model of equivariant cohomology to Lie algebroids. We construct groupoids by reduction of an action Lie groupoid $G\times M$ by a subgroup $H$ of $G$ which is not necessarily normal, and we find conditions which imply that a hamiltonian structure descends to their Lie algebroids. We consider many examples and, in particular, find that the tangent Lie algebroid over a symplectic manifold is hamiltonian with respect to some connection if and only if the symplectic structure has a nowhere vanishing primitive. Recent results of Stratmann and Tang show that this is the case whenever the symplectic structure is exact.

math.SG↗

arXiv2016

Analysis Framework for the Prompt Discovery of Compact Binary Mergers in Gravitational-wave Data

We describe a stream-based analysis pipeline to detect gravitational waves from the merger of binary neutron stars, binary black holes, and neutron-star-black-hole binaries within ~ 1 minute of the arrival of the merger signal at Earth. Such low-latency detection is crucial for the prompt response by electromagnetic facilities in order to observe any fading electromagnetic counterparts that might be produced by mergers involving at least one neutron star. Even for systems expected not to produce counterparts, low-latency analysis of the data is useful for deciding when not to point telescopes, and as feedback to observatory operations. Analysts using this pipeline were the first to identify GW151226, the second gravitational-wave event ever detected. The pipeline also operates in an offline mode, in which it incorporates more refined information about data quality and employs acausal methods that are inapplicable to the online mode. The pipeline's offline mode was used in the detection of the first two gravitational-wave events, GW150914 and GW151226, as well as the identification of a third candidate, LVT151012.

astro-ph.IM↗

arXiv2016

Gravitational-Wave Constraints on the Progenitors of Fast Radio Bursts

The nature of fast radio bursts (FRBs) remains enigmatic. Highly energetic radio pulses of millisecond duration, FRBs are observed with dispersion measures consistent with an extragalactic source. A variety of models have been proposed to explain their origin. One popular class of theorized FRB progenitor is the coalescence of compact binaries composed of neutron stars and/or black holes. Such coalescence events are strong gravitational-wave emitters. We demonstrate that measurements made by the LIGO and Virgo gravitational-wave observatories can be leveraged to severely constrain the validity of FRB binary coalescence models. Existing measurements constrain the binary black hole rate to approximately $5\%$ of the FRB rate, and results from Advanced LIGO's O1 and O2 observing runs may place similarly strong constraints on the fraction of FRBs due to binary neutron star and neutron star--black hole progenitors.

astro-ph.HE↗

arXiv2015

Decomposition of (co)isotropic relations

We identify thirteen isomorphism classes of indecomposable coisotropic relations between Poisson vector spaces and show that every coisotropic relation between finite-dimensional Poisson vector spaces may be decomposed as a direct sum of multiples of these indecomposables. We also find a list of thirteen invariants, each of which is the dimension of a space constructed from the relation, such that the 13-vector of multiplicities and the 13-vector of invariants are related by an invertible matrix over $\mathbb Z$. It turns out to be simpler to do the analysis above for isotropic relations between presymplectic vector spaces. The coisotropic/Poisson case then follows by a simple duality argument.

math.SG↗

arXiv2015

Categories of (co)isotropic linear relations

In categories of linear relations between finite dimensional vector spaces, composition is well-behaved only at pairs of relations satisfying transversality and monicity conditions. A construction of Wehrheim and Woodward makes it possible to impose these conditions while retaining the structure of a category. We analyze the resulting category in the case of all linear relations, as well as for (co)isotropic relations between symplectic vector spaces. In each case, the Wehrheim-Woodward category is a central extension of the original category of relations by the endomorphisms of the unit object, which is a free submonoid with two generators in the additive monoid of pairs of nonnegative integers.

math.SG↗

arXiv2015

(Co)isotropic Pairs in Poisson and Presymplectic Vector Spaces

We give two equivalent sets of invariants which classify pairs of coisotropic subspaces of finite-dimensional Poisson vector spaces. For this it is convenient to dualize; we work with pairs of isotropic subspaces of presymplectic vector spaces. We identify ten elementary types which are the building blocks of such pairs, and we write down a matrix, invertible over $\mathbb{Z}$, which takes one 10-tuple of invariants to the other.

math.SG↗

arXiv2015

The Needle in the 100 deg2 Haystack: Uncovering Afterglows of Fermi GRBs with the Palomar Transient Factory

The Fermi Gamma-ray Space Telescope has greatly expanded the number and energy window of observations of gamma-ray bursts (GRBs). However, the coarse localizations of tens to a hundred square degrees provided by the Fermi GRB Monitor instrument have posed a formidable obstacle to locating the bursts' host galaxies, measuring their redshifts, and tracking their panchromatic afterglows. We have built a target-of-opportunity mode for the intermediate Palomar Transient Factory in order to perform targeted searches for Fermi afterglows. Here, we present the results of one year of this program: 8 afterglow discoveries out of 35 searches. Two of the bursts with detected afterglows (GRBs 130702A and 140606B) were at low redshift (z=0.145 and 0.384 respectively) and had spectroscopically confirmed broad-line Type Ic supernovae. We present our broadband follow-up including spectroscopy as well as X-ray, UV, optical, millimeter, and radio observations. We study possible selection effects in the context of the total Fermi and Swift GRB samples. We identify one new outlier on the Amati relation. We find that two bursts are consistent with a mildly relativistic shock breaking out from the progenitor star, rather than the ultra-relativistic internal shock mechanism that powers standard cosmological bursts. Finally, in the context of the Zwicky Transient Facility, we discuss how we will continue to expand this effort to find optical counterparts of binary neutron star mergers that may soon be detected by Advanced LIGO and Virgo.

astro-ph.HE↗

arXiv2014

Selective Categories and Linear Canonical Relations

A construction of Wehrheim and Woodward circumvents the problem that compositions of smooth canonical relations are not always smooth, building a category suitable for functorial quantization. To apply their construction to more examples, we introduce a notion of highly selective category, in which only certain morphisms and certain pairs of these morphisms are "good". We then apply this notion to the category $\mathbf{SLREL}$ of linear canonical relations and the result ${\rm WW}(\mathbf{SLREL})$ of our version of the WW construction, identifying the morphisms in the latter with pairs $(L,k)$ consisting of a linear canonical relation and a nonnegative integer. We put a topology on this category of indexed linear canonical relations for which composition is continuous, unlike the composition in $\mathbf{SLREL}$ itself. Subsequent papers will consider this category from the viewpoint of derived geometry and will concern quantum counterparts.

math.SG↗