Ebook: A First Course in Topos Quantum Theory

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