Mathematical structures of the universe / edited by Michał Eckstein, Michael Heller, Sebastian J. Szybka.

Format:

Book

Contributor:

Szybka, Sebastian J., editor.

Eckstein, Michał, editor.

Heller, Michał, editor.

Language:

English

Subjects (All):

Cosmology--Mathematics.

Cosmology.

Physical Description:

1 online resource (459 pages)

Edition:

1st ed.

Place of Publication:

Kraków, Poland : Copernicus Center Press, [2014]

Summary:

This book contains a collection of essays on mathematical structures that serve to model the Universe. The contributions discuss such topics as the interplay between mathematics and physics, geometrical structures in physical models, and observational and conceptual aspects of cosmology. The reader can also contemplate the scientific method on the verge of its limits. [Subject: Physics, Mathematics, Cosmology, Natural Philosophy]

Contents:

Intro

Mathematical Structures of the Universe

Table of Contents

Introduction

Part I. General Relativity and Cosmology

Observer dependent geometries

1. Geometry for observers and observables

2. Geometry of the clock postulate: Finsler spacetimes

2.1. De˝nition of Finsler spacetimes

2.2. Causal structure and observers

2.3. Dynamics for point masses

2.4. Observers and observations

2.5. Field theory

2.6. Gravity

3. The local perspective: Cartan geometry of observer space

3.1. Definition of observer space

3.2. Introduction to Cartan geometry

3.3. Cartan geometry of observer space

3.4. Observers and observations

3.5. Gravity

3.6. The role of spacetime

Acknowledgments

References

Classification of classical singularities: a differential spaces approach

1. Motivation

2. Fundamental concepts

3. Spectral properties

4. B-boundary

5. Singularities

Acknowledgment

The smooth beginning of the Universe

1. Introduction

2. Sikorski's di˙erential spaces and GR

3. A differential space for the flat FRW d-manifold

4. Time orientability

5. A smooth evolution with respect to cosmological time

6. The simplest smoothly evolving models

7. Interpretation

8. Smoothly evolved models in a neighbourhood of singularity

9. Summary

Are singularities the limits of cosmology?

1. What are singularities?

2. Big-Bang and non-Big-Bang singularities in cosmology

2.1. The strength of singularities

2.2. Geodesics and geodesic deviation

2.3. Spacetime averaging

2.4. Energy conditions

3. Properties and classification of singularities

4. Varying constants removing or changing singularities

4.1. Removing a Big-Bang singularity - VG

4.2. Removing SFS or FSF - VSL

4.3. Removing SFS or FSF - VG.

4.4. A hybrid case - VG

5. Singularities and the limits of cosmology

Acknowledgements

Simplicity in cosmology: add virialisation, remove Λ, keep classical GR

Computer algebra tests physical theories: the case of relativistic astrophysics

The meaning of conformal curvature

Riemann curvature invariants: Schwarzschild space-time

Oppenheimer-Volkoff problem

Oppenheimer paradigm

Counterexample

Speculations or reality?

Acknowledgement

On gravitational interactions between two bodies

Historical introduction

From Newton to Einstein

Two-body problem in General Relativity

Summary

Part II. Quantum Geometries

Geometry of quantum correlations

2. The geometry of quantum mechanics

3. A symplectic setting for classical mechanics

4. Classical Hamiltonian systems with symmetry

5. Symplectic structures in spaces of quantum states

6. Composite quantum systems

separable and entangled states

7. Quantum correlations and symplectic geometry

7.1. Symplectic measures of entanglement

7.2. Local unitary equivalence of states

7.3. Stratification of multiqubit states by reduced one-particle density matrices

7.4. Generalizations. Indistinguishable particles

mixed states

7.5. Summary and conclusions

Gauge field theories: various mathematical approaches

2. Ordinary di˙erential geometry

2.1. Ehresmann connections and Yang-Mills theory

2.2. Linear connections and Einstein theory of gravitation

2.3. Cartan geometry and gravitation theories

3. Noncommutative geometry

3.1. Basic structures in noncommutative geometry

3.2. Spectral triples

3.3. Derivation-based noncommutative geometry.

4. Transitive Lie algebroids

4.1. Generalities on transitive Lie algebroids

4.2. Differential structures

4.3. Gauge field theories

5. Conclusion

Towards a construction of a quantum field theory in four dimensions

1.1. Rigorous quantum field theories

1.2. Regularisation and renormalisation

1.3. TheMoyal space as symmetry-enhancing regulator

2. Exact solution of the quartic matrix model

2.1. Ward identity and topological expansion

2.2. Schwinger-Dyson equations

3. Renormalisation and integral representation

4. Schwinger functions and re˛ection positivity

4.1. Schwinger functions

4.2. Osterwalder-Schrader axioms

5. Computer simulations

6. Summary and outlook

Unweaving the fabric of the Universe: the interplay between mathematics and physics

1. The plot

2. The plot thickens

3. The plot unfolds

3.1. String/M-theory

3.2. Loop Quantum Gravity

3.3. Noncommutative Geometry

4. Epilogue

Quantum gravity models - a brief conceptual summary

2. Problems with Einstein quantum gravity (QG) as aquantum field-theoretic model and dynamical noncommutative quantum space-time

3. Basic QG models

3.1. General remarks

3.2. Loop quantum gravity (LQG)

3.3. QG as QFT on noncommutative space-time

3.4. Functional formulation of QG and lattice approach

4. Final remarks

Pointless geometry

2. The illusion of points

2.1. Points from an algebraic point of view

2.2. What are points in mathematics?

2.3. Points in physics

3. What is Noncommutative Geometry?

3.1. The first dictionary

3.2. How to differentiate in algebras?

3.3. How to measure in a geometry without points?

4. The Spectral Triples.

4.1. The Dirac operator

4.2. Dirac operators for noncommutative algebras

5. On distances between representations

5.1. Open problems

6. Conclusions

Noncommutative geometry, Lorentzian structures and causality

2. Noncommutative geometry as a new tool for physical models

3. Lorentzian structures in noncommutative geometry

4. Causality in noncommutative geometry

5. An almost commutative toy model

6. Conclusions and outlook

Ontology and noncommutative geometry

2. The origin of noncommutative geometry

3. Limits of classical ontology

4. Ontology and pointless spaces

5. Ontological questions related to nonlocality and nonseparability

6. Logic for noncommutative mathematics?

7. Pathways towards a new ontology?

Part III. Overviews

The self-representing Universe

2. Relative Realism and the nature of Reality

3. Relative Realism and morality

4. From Relative Realism to self-duality

5. Self-duality in quantum gravity

6. Self-duality v Hodge duality

7. Self-duality v deMorgan duality

8. Relative realism and the origin of Riemannian geometry

Reflections on the geometrization of physics

1. From tangible to intangible?

2. Special relativity and Minkowski geometry

3. General relativity and differential geometry

4. Quantum theory and spinors

5. Gauge theories and fibre bundles

6. Unifications and yet more geometries

7. The intangible becomes tangible?

Metacosmology and the limits of science

2. The macro-micro connection

3. The anthropic principle

4. The multiverse

5. Metacosmology and the lessons of history

6. The changing nature of science

Acknowledgment.

References

The price for mathematics

The field of rationality and category theory

1. Field of rationality

2. Beginnings of the theory

3. The concept of category

4. Functor categories

5. Category of categories

6. n-categories

7. Category theory and logic

8. Some methodological caveats

9. The 'field of categories'

References.

Notes:

Includes bibliographical references.

Description based on print version record.

ISBN:

83-7886-471-5

OCLC:

1125224976