312 research outputs found
von Neumann algebras in JT gravity
We quantize JT gravity with matter on the spatial interval with two
asymptotically AdS boundaries. We consider the von Neumann algebra generated by
the right Hamiltonian and the gravitationally dressed matter operators on the
right boundary. We prove that the commutant of this algebra is the analogously
defined left boundary algebra and that both algebras are type II
factors. These algebras provide a precise notion of the entanglement wedge away
from the semiclassical limit. We comment on how the factorization problem
differs between pure JT gravity and JT gravity with matter.Comment: 35 pages + appendices. v2: typos fixed, appendix adde
Efficient parameterized algorithms on structured graphs
In der klassischen Komplexitätstheorie werden worst-case Laufzeiten von Algorithmen typischerweise einzig abhängig von der Eingabegröße angegeben. In dem Kontext der parametrisierten Komplexitätstheorie versucht man die Analyse der Laufzeit dahingehend zu verfeinern, dass man zusätzlich zu der Eingabengröße noch einen Parameter berücksichtigt, welcher angibt, wie strukturiert die Eingabe bezüglich einer gewissen Eigenschaft ist. Ein parametrisierter Algorithmus nutzt dann diese beschriebene Struktur aus und erreicht so eine Laufzeit, welche schneller ist als die eines besten unparametrisierten Algorithmus, falls der Parameter klein ist.
Der erste Hauptteil dieser Arbeit führt die Forschung in diese Richtung weiter aus und untersucht den Einfluss von verschieden Parametern auf die Laufzeit von bekannten effizient lösbaren Problemen. Einige vorgestellte Algorithmen sind dabei adaptive Algorithmen, was bedeutet, dass die Laufzeit von diesen Algorithmen mit der Laufzeit des besten unparametrisierten Algorithm für den größtmöglichen Parameterwert übereinstimmt und damit theoretisch niemals schlechter als die besten unparametrisierten Algorithmen und übertreffen diese bereits für leicht nichttriviale Parameterwerte.
Motiviert durch den allgemeinen Erfolg und der Vielzahl solcher parametrisierten Algorithmen, welche eine vielzahl verschiedener Strukturen ausnutzen, untersuchen wir im zweiten Hauptteil dieser Arbeit, wie man solche unterschiedliche homogene Strukturen zu mehr heterogenen Strukturen vereinen kann. Ausgehend von algebraischen Ausdrücken, welche benutzt werden können, um von Parametern beschriebene Strukturen zu definieren, charakterisieren wir klar und robust heterogene Strukturen und zeigen exemplarisch, wie sich die Parameter tree-depth und modular-width heterogen verbinden lassen. Wir beschreiben dazu effiziente Algorithmen auf heterogenen Strukturen mit Laufzeiten, welche im Spezialfall mit den homogenen Algorithmen übereinstimmen.In classical complexity theory, the worst-case running times of algorithms depend solely on the size of the input. In parameterized complexity the goal is to refine the analysis of the running time of an algorithm by additionally considering a parameter that measures some kind of structure in the input. A parameterized algorithm then utilizes the structure described by the parameter and achieves a running time that is faster than the best general (unparameterized) algorithm for instances of low parameter value.
In the first part of this thesis, we carry forward in this direction and investigate the influence of several parameters on the running times of well-known tractable problems.
Several presented algorithms are adaptive algorithms, meaning that they match the running time of a best unparameterized algorithm for worst-case parameter values. Thus, an adaptive parameterized algorithm is asymptotically never worse than the best unparameterized algorithm, while it outperforms the best general algorithm already for slightly non-trivial parameter values.
As illustrated in the first part of this thesis, for many problems there exist efficient parameterized algorithms regarding multiple parameters, each describing a different kind of structure.
In the second part of this thesis, we explore how to combine such homogeneous structures to more general and heterogeneous structures.
Using algebraic expressions, we define new combined graph classes
of heterogeneous structure in a clean and robust way, and we showcase this for the heterogeneous merge of the parameters tree-depth and modular-width, by presenting parameterized algorithms
on such heterogeneous graph classes and getting running times that match the homogeneous cases throughout
Discrete parametric graphical models with Dirichlet type priors
Typically, statistical graphical models are either continuous and parametric
(Gaussian, parameterized by the graph-dependent precision matrix) or discrete
and non-parametric (with graph-dependent probabilities of cells). Eventually,
the two types are mixed. We propose a way to break this dichotomy by
introducing two discrete parametric graphical models on finite decomposable
graphs: the graph negative multinomial and the graph multinomial distributions.
These models interpolate between the product of univariate negative multinomial
and negative multinomial distributions, and between the product of binomial and
multinomial distributions, respectively. We derive their Markov decomposition
and present probabilistic models leading to both. Additionally, we introduce
graphical versions of the Dirichlet distribution and inverted Dirichlet
distribution, which serve as conjugate priors for the two discrete graphical
Markov models. We derive explicit normalizing constants for both graphical
Dirichlet laws and demonstrate that their independence structure (a graphical
version of neutrality) yields a strong hyper Markov property for both Bayesian
models. We also provide characterization theorems for the generalized Dirichlet
distributions via strong hyper Markov property. Finally, we develop a Bayesian
model selection procedure for the graphical negative multinomial model with
respective Dirichlet-type priors.Comment: 36 page
Open Problems in (Hyper)Graph Decomposition
Large networks are useful in a wide range of applications. Sometimes problem
instances are composed of billions of entities. Decomposing and analyzing these
structures helps us gain new insights about our surroundings. Even if the final
application concerns a different problem (such as traversal, finding paths,
trees, and flows), decomposing large graphs is often an important subproblem
for complexity reduction or parallelization. This report is a summary of
discussions that happened at Dagstuhl seminar 23331 on "Recent Trends in Graph
Decomposition" and presents currently open problems and future directions in
the area of (hyper)graph decomposition
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Exclusive QCD Factorization and Single Transverse Polarization Phenomena at High-Energy Colliders
This Ph.D.~thesis is divided into two distinct parts. The first part focuses
on hard exclusive scattering processes in Quantum Chromodynamics (QCD) at high
energies, while the second part delves into spin phenomena at the Large Hadron
Collider (LHC).
Hard exclusive scattering processes play a crucial role in QCD at high
energies, providing unique insights into the confined partonic dynamics within
hadrons, complementing inclusive processes. Studying these processes within the
QCD factorization approach yields the generalized parton distribution (GPD), a
nonperturbative parton correlation function that offers a three-dimensional
tomographic parton image within a hadron. However, the experimental measurement
of these processes poses significant challenges. This thesis will review the
factorization formalism for related processes, examine the limitations of some
widely used processes, and introduce two novel processes that enhance the
sensitivity to GPD, particularly its dependence on the parton momentum fraction
.
The second part of the thesis centers on spin phenomena, specifically single
spin production, at the LHC. Noting that a single transverse polarization can
be generated even in an unpolarized collision, this research proposes two new
jet substructure observables: one for boosted top quark jets and another for
high-energy gluon jets. The observation of these phenomena paves the way for
innovative tools in LHC phenomenology, enabling both precision measurements and
the search for new physics.Comment: Ph.D. thesis. 417 page
Spherical and Hyperbolic Toric Topology-Based Codes On Graph Embedding for Ising MRF Models: Classical and Quantum Topology Machine Learning
The paper introduces the application of information geometry to describe the
ground states of Ising models by utilizing parity-check matrices of cyclic and
quasi-cyclic codes on toric and spherical topologies. The approach establishes
a connection between machine learning and error-correcting coding. This
proposed approach has implications for the development of new embedding methods
based on trapping sets. Statistical physics and number geometry applied for
optimize error-correcting codes, leading to these embedding and sparse
factorization methods. The paper establishes a direct connection between DNN
architecture and error-correcting coding by demonstrating how state-of-the-art
architectures (ChordMixer, Mega, Mega-chunk, CDIL, ...) from the long-range
arena can be equivalent to of block and convolutional LDPC codes (Cage-graph,
Repeat Accumulate). QC codes correspond to certain types of chemical elements,
with the carbon element being represented by the mixed automorphism
Shu-Lin-Fossorier QC-LDPC code. The connections between Belief Propagation and
the Permanent, Bethe-Permanent, Nishimori Temperature, and Bethe-Hessian Matrix
are elaborated upon in detail. The Quantum Approximate Optimization Algorithm
(QAOA) used in the Sherrington-Kirkpatrick Ising model can be seen as analogous
to the back-propagation loss function landscape in training DNNs. This
similarity creates a comparable problem with TS pseudo-codeword, resembling the
belief propagation method. Additionally, the layer depth in QAOA correlates to
the number of decoding belief propagation iterations in the Wiberg decoding
tree. Overall, this work has the potential to advance multiple fields, from
Information Theory, DNN architecture design (sparse and structured prior graph
topology), efficient hardware design for Quantum and Classical DPU/TPU (graph,
quantize and shift register architect.) to Materials Science and beyond.Comment: 71 pages, 42 Figures, 1 Table, 1 Appendix. arXiv admin note: text
overlap with arXiv:2109.08184 by other author
Energy inequalities in integrable quantum field theory
Negative energy densities are an abundant and necessary feature of quantum field
theory (QFT) and may lead to surprising measurable effects. Some of these stand
in contrast to classical physics, so that the accumulation of negative energy, also
in quantum field theory, must be subject to some constraints. One class of such
constraints is commonly referred to as quantum energy inequalities (QEI). These
are lower bounds on the averaged stress-energy tensor which have been established
quite generically in quantum field theory, however, mostly excluding models with
self-interaction.
A rich but mathematically tractable class of interacting models are those subject
to integrability. In this thesis, we give an overview of the construction of integrable
models via the inverse scattering approach, extending previous results on the char-
acterization of local observables to models with more than one particle species and
inner degrees of freedom.
We apply these results to the stress-energy tensor, leading to a characterization
of the stress-energy tensor at one-particle level. In models with simple interaction,
where the S-matrix is independent of the particles’ momenta, this suffices to con-
struct the full stress-energy tensor and provide a state-independent QEI. In models
with generic interaction, we obtain QEIs at the one-particle level and find that they
substantially constrain the choice of reasonable stress-energy tensors, in some cases
fixing it uniquely.:Acknowledgements 4
Contents 5
1 Introduction 7
2 Constructive aspects of integrable quantum field theories 13
2.1 General notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Particle spectrum and one-particle space . . . . . . . . . . . . . . . . 15
2.3 The scattering function . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Full state space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5 Asymptotic completeness; closing the circle . . . . . . . . . . . . . . . 29
2.6 Connection to algebraic quantum field theory . . . . . . . . . . . . . 33
3 Locality and the form factor series 37
3.1 Locality and the form factor series . . . . . . . . . . . . . . . . . . . 38
3.2 Local commutativity theorem for one- and two-particle form factors . 44
3.3 Transformation properties of the form factors . . . . . . . . . . . . . 58
3.3.1 Form factors of invariant operators and derivatives . . . . . . 62
4 Structure of form factors and the minimal solution 64
4.1 Classification of two-particle form factors . . . . . . . . . . . . . . . . 64
4.2 Existence of the minimal solutions and asymptotic growth . . . . . . 68
4.3 Computing a characteristic function . . . . . . . . . . . . . . . . . . . 74
5 The stress-energy tensor 77
5.1 The stress-energy tensor from first principles . . . . . . . . . . . . . . 77
5.2 The stress-energy tensor at one-particle level . . . . . . . . . . . . . . 83
5.3 Characterization at one-particle level . . . . . . . . . . . . . . . . . . 88
6 State-independent QEI for constant scattering functions 94
6.1 Candidate for the stress-energy tensor . . . . . . . . . . . . . . . . . 94
6.2 A generic estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.3 Derivation of the QEI . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.4 Discussion of the QEI . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.5 Supplementary computations . . . . . . . . . . . . . . . . . . . . . . 108
7 QEIs at one-particle level for generic scattering functions 110
7.1 Derivation of the QEI at one-particle level . . . . . . . . . . . . . . . 111
7.2 Extending the scope of the QEI result . . . . . . . . . . . . . . . . . 117
7.3 A general recipe to obtain QEIs at one-particle level . . . . . . . . . 119
8 Examples 123
8.1 Models with one scalar particle type without bound states . . . . . . 123
8.2 Generalized Bullough-Dodd model . . . . . . . . . . . . . . . . . . . 125
8.3 Federbush model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8.4 O(n)-nonlinear sigma model . . . . . . . . . . . . . . . . . . . . . . . 130
9 Conclusion, discussion, and outlook 134
A Constructive aspects of integrable quantum field theory 137
A.1 Representation theory of the Poincaré group in 1+1d . . . . . . . . . 137
A.2 Discrete symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
A.3 S-function and ZF operators in a basis . . . . . . . . . . . . . . . . . 143
A.4 Improper rapidity eigenstates . . . . . . . . . . . . . . . . . . . . . . 145
A.5 Bound states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
A.6 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
B Literature survey: Form factor conventions 158
C Stress-energy tensor 159
C.1 Stress-energy tensors for the free scalar field . . . . . . . . . . . . . . 159
C.2 A weaker notion for the density property . . . . . . . . . . . . . . . . 163
C.3 Stress-energy tensor at one-particle level generating the boosts . . . . 164
Bibliography 16
Resolving Prime Modules: The Structure of Pseudo-cographs and Galled-Tree Explainable Graphs
The modular decomposition of a graph is a natural construction to capture
key features of in terms of a labeled tree whose vertices are
labeled as "series" (), "parallel" () or "prime". However, full
information of is provided by its modular decomposition tree only,
if is a cograph, i.e., does not contain prime modules. In this case,
explains , i.e., if and only if the lowest common
ancestor of and has label "". Pseudo-cographs,
or, more general, GaTEx graphs are graphs that can be explained by labeled
galled-trees, i.e., labeled networks that are obtained from the modular
decomposition tree of by replacing the prime vertices in by
simple labeled cycles. GaTEx graphs can be recognized and labeled galled-trees
that explain these graphs can be constructed in linear time.
In this contribution, we provide a novel characterization of GaTEx graphs in
terms of a set of 25 forbidden induced subgraphs.
This characterization, in turn, allows us to show that GaTEx graphs are closely
related to many other well-known graph classes such as -sparse and
-reducible graphs, weakly-chordal graphs, perfect graphs with perfect
order, comparability and permutation graphs, murky graphs as well as interval
graphs, Meyniel graphs or very strongly-perfect and brittle graphs. Moreover,
we show that every GaTEx graph as twin-width at most 1.Comment: 18 pages, 3 figure
Computing Well-Covered Vector Spaces of Graphs using Modular Decomposition
A graph is well-covered if all its maximal independent sets have the same
cardinality. This well studied concept was introduced by Plummer in 1970 and
naturally generalizes to the weighted case. Given a graph , a real-valued
vertex weight function is said to be a well-covered weighting of if all
its maximal independent sets are of the same weight. The set of all
well-covered weightings of a graph forms a vector space over the field of
real numbers, called the well-covered vector space of . Since the problem of
recognizing well-covered graphs is --complete, the
problem of computing the well-covered vector space of a given graph is
--hard. Levit and Tankus showed in 2015 that the
problem admits a polynomial-time algorithm in the class of claw-free graph. In
this paper, we give two general reductions for the problem, one based on
anti-neighborhoods and one based on modular decomposition, combined with
Gaussian elimination. Building on these results, we develop a polynomial-time
algorithm for computing the well-covered vector space of a given fork-free
graph, generalizing the result of Levit and Tankus. Our approach implies that
well-covered fork-free graphs can be recognized in polynomial time and also
generalizes some known results on cographs.Comment: 25 page
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