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A First Course in Topos Quantum Theory

دسته بندی: نظریه نسبیت و گرانش
ویرایش: 2013 
نویسندگان: Cecilia Flori  
سری: Lecture Notes in Physics 
ISBN (شابک) : 3642357121, 9783642357121 
ناشر: Springer 
سال نشر: 2013 
تعداد صفحات: 464 
زبان: English 
فرمت فایل : PDF (درصورت درخواست کاربر به PDF، EPUB یا AZW3 تبدیل می شود) 
حجم فایل: 4 مگابایت 

کلمات کلیدی مربوط به کتاب اولین دوره در تئوری کوانتومی Topos: فیزیک، نسبیت و نظریه های جایگزین گرانش، گرانش کوانتومی

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توضیحاتی درمورد کتاب به خارجی

In the last five decades various attempts to formulate theories of quantum gravity have been made, but none has fully succeeded in becoming the quantum theory of gravity. One possible explanation for this failure might be the unresolved fundamental issues in quantum theory as it stands now. Indeed, most approaches to quantum gravity adopt standard quantum theory as their starting point, with the hope that the theory’s unresolved issues will get solved along the way. However, these fundamental issues may need to be solved before attempting to define a quantum theory of gravity. The present text adopts this point of view, addressing the following basic questions: What are the main conceptual issues in quantum theory? How can these issues be solved within a new theoretical framework of quantum theory? A possible way to overcome critical issues in present-day quantum physics – such as a priori assumptions about space and time that are not compatible with a theory of quantum gravity, and the impossibility of talking about systems without reference to an external observer – is through a reformulation of quantum theory in terms of a different mathematical framework called topos theory. This course-tested primer sets out to explain to graduate students and newcomers to the field alike, the reasons for choosing topos theory to resolve the above-mentioned issues and how it brings quantum physics back to looking more like a “neo-realist” classical physics theory again. Table of Contents Cover A First Course in Topos Quantum Theory ISBN 9783642357121 ISBN 9783642357138 Acknowledgement Contents Chapter 1 Introduction Chapter 2 Philosophical Motivations 2.1 What Is a Theory of Physics and What Is It Trying to Achieve? 2.2 Philosophical Position of Classical Theory 2.3 Philosophy Behind Quantum Theory 2.4 Conceptual Problems of Quantum Theory Chapter 3 Kochen-Specker Theorem 3.1 Valuation Functions in Classical Theory 3.2 Valuation Functions in Quantum Theory 3.2.1 Deriving the FUNC Condition 3.2.2 Implications of the FUNC Condition 3.3 Kochen Specker Theorem 3.4 Proof of the Kochen-Specker Theorem 3.5 Consequences of the Kochen-Specker Theorem Chapter 4 Introducing Category Theory 4.1 Change of Perspective 4.2 Axiomatic Definitio of a Category 4.2.1 Examples of Categories 4.3 The Duality Principle 4.4 Arrows in a Category 4.4.1 Monic Arrows 4.4.2 Epic Arrows 4.4.3 Iso Arrows 4.5 Elements and Their Relations in a Category 4.5.1 Initial Objects 4.5.2 Terminal Objects 4.5.3 Products 4.5.4 Coproducts 4.5.5 Equalisers 4.5.6 Coequalisers 4.5.7 Limits and Colimits 4.6 Categories in Quantum Mechanics 4.6.1 The Category of Bounded Self Adjoint Operators 4.6.2 Category of Boolean Sub-algebras Chapter 5 Functors 5.1 Functors and Natural Transformations 5.1.1 Covariant Functors 5.1.2 Contravariant Functor 5.2 Characterising Functors 5.3 Natural Transformations 5.3.1 Equivalence of Categories Chapter 6 The Category of Functors 6.1 The Functor Category 6.2 Category of Presheaves 6.3 Basic Categorical Constructs for the Category of Presheaves 6.4 Spectral Presheaf on the Category of Self-adjoint Operators with Discrete Spectra Chapter 7 Topos 7.1 Exponentials 7.2 Pullback 7.3 Pushouts 7.4 Sub-objects 7.5 Sub-object Classifie (Truth Object) 7.6 Elements of the Sub-object Classifier Sieves 7.7 Heyting Algebras 7.8 Understanding the Sub-object Classifie in a General Topos 7.9 Axiomatic Definitio of a Topos Chapter 8 Topos of Presheaves 8.1 Pullbacks 8.2 Pushouts 8.3 Sub-objects 8.4 Sub-object Classifie in the Topos of Presheaves 8.4.1 Elements of the Sub-object Classifie 8.5 Global Sections 8.6 Local Sections 8.7 Exponential Chapter 9 Topos Analogue of the State Space 9.1 The Notion of Contextuality in the Topos Approach 9.1.1 Category of Abelian von Neumann Sub-algebras 9.1.2 Example 9.1.3 Topology on V(H) 9.2 Topos Analogue of the State Space 9.2.1 Example 9.3 The Spectral Presheaf and the Kochen-Specker Theorem Chapter 10 Topos Analogue of Propositions 10.1 Propositions 10.1.1 Physical Interpretation of Daseinisation 10.2 Properties of the Daseinisation Map 10.3 Example Chapter 11 Topos Analogues of States 11.1 Outer Daseinisation Presheaf 11.2 Properties of the Outer-Daseinisation Presheaf 11.3 Truth Object Option 11.3.1 Example of Truth Object in Classical Physics 11.3.2 Truth Object in Quantum Theory 11.3.3 Example 11.4 Pseudo-state Option 11.4.1 Example 11.5 Relation Between Pseudo-state Object and Truth Object Chapter 12 Truth Values 12.1 Representation of Sub-object Classifie 12.1.1 Example 12.2 Truth Values Using the Pseudo-state Object 12.3 Example 12.4 Truth Values Using the Truth-Object 12.4.1 Example 12.5 Relation Between the Truth Values Chapter 13 Quantity Value Object and Physical Quantities 13.1 Topos Representation of the Quantity Value Object 13.2 Inner Daseinisation 13.3 Spectral Decomposition 13.3.1 Example of Spectral Decomposition 13.4 Daseinisation of Self-adjoint Operators 13.4.1 Example 13.5 Topos Representation of Physical Quantities 13.6 Interpreting the Map Representing Physical Quantities 13.7 Computing Values of Quantities Given a State 13.7.1 Examples Chapter 14 Sheaves 14.1 Sheaves 14.1.1 Simple Example 14.2 Connection Between Sheaves and �tale Bundles 14.3 Sheaves on a Partially Ordered Set 14.4 Adjunctions 14.4.1 Example 14.5 Geometric Morphisms 14.6 Group Action and Twisted Presheaves 14.6.1 Spectral Presheaf 14.6.2 Quantity Value Object 14.6.3 Daseinisation 14.6.4 Truth Values Chapter 15 Probabilities in Topos Quantum Theory 15.1 General Definitio of Probabilities in the Language of Topos Theory 15.2 Example for Classical Probability Theory 15.3 Quantum Probabilities 15.4 Measure on the Topos State Space 15.5 Deriving a State from a Measure 15.6 New Truth Object 15.6.1 Pure State Truth Object 15.6.2 Density Matrix Truth Object 15.7 Generalised Truth Values Chapter 16 Group Action in Topos Quantum Theory 16.1 The Sheaf of Faithful Representations 16.2 Changing Base Category 16.3 From Sheaves on the Old Base Category to Sheaves on the New Base Category 16.4 The Adjoint Pair 16.5 From Sheaves over V(H) to Sheaves over V(Hf ) 16.5.1 Spectral Sheaf 16.5.2 Quantity Value Object 16.5.3 Truth Values 16.6 Group Action on the New Sheaves 16.6.1 Spectral Sheaf 16.6.2 Sub-object Classifie 16.6.3 Quantity Value Object 16.6.4 Truth Object 16.7 New Representation of Physical Quantities Chapter 17 Topos History Quantum Theory 17.1 A Brief Introduction to Consistent Histories 17.2 The HPO Formulation of Consistent Histories 17.3 The Temporal Logic of Heyting Algebras of Sub-objects 17.4 Realising the Tensor Product in a Topos 17.5 Entangled Stages 17.6 Direct Product of Truth Values 17.7 The Representation of HPO Histories Chapter 18 Normal Operators 18.1 Spectral Ordering of Normal Operators 18.1.1 Example 18.2 Normal Operators in a Topos 18.2.1 Example 18.3 Complex Number Object in a Topos 18.3.1 Domain-Theoretic Structure Chapter 19 KMS States 19.1 Brief Review of the KMS State 19.2 External KMS State 19.3 Deriving the Canonical KMS State from the Topos KMS State 19.4 The Automorphisms Group 19.5 Internal KMS Condition Chapter 20 One-Parameter Group of Transformations and Stone's Theorem 20.1 Topos Notion of a One Parameter Group 20.1.1 One Parameter Group Taking Values in the Real Valued Object 20.1.2 One Parameter Group Taking Values in Complex Number Object 20.2 Stone's Theorem in the Language of Topos Theory Chapter 21 Future Research 21.1 Quantisation 21.2 Internal Approach 21.3 Configuratio Space 21.4 Composite Systems 21.5 Differentiable Structure Appendix A Topoi and Logic Appendix B Worked out Examples References Index

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