The Topos Approach to Quantum Theory Nagoya Winter Workshop 21. February 2010

Andreas D¨ oring joint work with Chris Isham Oxford University Computing Laboratory [email protected]

Andreas D¨ oring (Oxford University)

The topos approach

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“A theory is something nobody believes, except the person who made it. An experiment is something everybody believes, except the person who made it.” (Unknown)

Andreas D¨ oring (Oxford University)

The topos approach

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Introduction

What’s the problem?

Why is quantum theory (QT) problematic conceptually? states do not assign values to all physical quantities, hence only probabilistic predictions interpretation needs classical structures: observers, preparations, measurements even with these classical structures: measurement problem quantum logic is too weak QT does not describe a system ‘in itself’, there is no picture of quantum reality arising ...

Andreas D¨ oring (Oxford University)

The topos approach

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Introduction

More problems... Why is quantum gravity (QG) even more problematic? we cannot write down QG, it is extremely hard technically space and time would become quantum objects, but where and when would a measurement on them take place? we don’t even have a good idea what QG is supposed to be (could be neither quantum nor gravity)

Andreas D¨ oring (Oxford University)

The topos approach

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Introduction

More problems... Why is quantum gravity (QG) even more problematic? we cannot write down QG, it is extremely hard technically space and time would become quantum objects, but where and when would a measurement on them take place? we don’t even have a good idea what QG is supposed to be (could be neither quantum nor gravity) What about quantum cosmology (QC)? can potentially arise from simplified QG, so simpler, but more generic clearly no external observer, no preparation, no measurements, no statistics not embedded into any classical structure, must be understood ‘out of itself’ Andreas D¨ oring (Oxford University)

The topos approach

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Introduction

A (tentative) goal One possible conclusion: it is not a good idea to search for a quantisation of GR using Hilbert space methods.

Andreas D¨ oring (Oxford University)

The topos approach

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Introduction

A (tentative) goal One possible conclusion: it is not a good idea to search for a quantisation of GR using Hilbert space methods. The odd one out seems to be quantum theory, not gravity. There are major conceptual issues with the former, not so much the latter. In particular with an eye to QG and QC, we need a mathematical formalism that allows to describe physical systems ‘in themselves’, without the need for external observers and other classical structures. In other words, we need a framework that allows us to formulate physical theories in a (more) realist way.

Andreas D¨ oring (Oxford University)

The topos approach

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Introduction

A (tentative) goal One possible conclusion: it is not a good idea to search for a quantisation of GR using Hilbert space methods. The odd one out seems to be quantum theory, not gravity. There are major conceptual issues with the former, not so much the latter. In particular with an eye to QG and QC, we need a mathematical formalism that allows to describe physical systems ‘in themselves’, without the need for external observers and other classical structures. In other words, we need a framework that allows us to formulate physical theories in a (more) realist way. This clearly is also interesting for QT, but can it be done? Has been tried many times, resulted in strong no-go theorems: Kochen-Specker, Bell, no-cloning, etc. Moreover, the Hilbert space formalism is very rigid. Andreas D¨ oring (Oxford University)

The topos approach

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Introduction

The topos approach The lack of success in finding a realist version of QT so far suggests that we have to be more radical from the outset. In particular, we may have to change the mathematical description of QT to obtain a new formulation that allows for a (more) realist interpretation.

Andreas D¨ oring (Oxford University)

The topos approach

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Introduction

The topos approach The lack of success in finding a realist version of QT so far suggests that we have to be more radical from the outset. In particular, we may have to change the mathematical description of QT to obtain a new formulation that allows for a (more) realist interpretation. Moreover, we want a framework in which potential theories ‘beyond quantum theory’ can be formulated.

Andreas D¨ oring (Oxford University)

The topos approach

6 / 32

Introduction

The topos approach The lack of success in finding a realist version of QT so far suggests that we have to be more radical from the outset. In particular, we may have to change the mathematical description of QT to obtain a new formulation that allows for a (more) realist interpretation. Moreover, we want a framework in which potential theories ‘beyond quantum theory’ can be formulated. The topos approach, initiated by Chris Isham and Jeremy Butterfield 12 years ago, aims to provide such a framework for the formulation of physical theories.

Andreas D¨ oring (Oxford University)

The topos approach

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Introduction

The topos approach The lack of success in finding a realist version of QT so far suggests that we have to be more radical from the outset. In particular, we may have to change the mathematical description of QT to obtain a new formulation that allows for a (more) realist interpretation. Moreover, we want a framework in which potential theories ‘beyond quantum theory’ can be formulated. The topos approach, initiated by Chris Isham and Jeremy Butterfield 12 years ago, aims to provide such a framework for the formulation of physical theories. Most of the work so far is on QT, so we will focus on that.

Andreas D¨ oring (Oxford University)

The topos approach

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The main idea

The prototype The prototype of a realist theory is classical mechanics. Pure states form a state space S (symplectic manifold), physical quantities are real-valued functions fA on S. The evaluation (s, fA ) 7−→ fA (s) gives the value that the physical quantity A has in the state s. The subset fA−1 (∆) of S represents the proposition “A ε ∆”, that is, “the physical quantity A has a value in the (Borel) set ∆”. All states s ∈ S that lie in fA−1 (∆) make the proposition true, all other states make it false.

Andreas D¨ oring (Oxford University)

The topos approach

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The main idea

The prototype The prototype of a realist theory is classical mechanics. Pure states form a state space S (symplectic manifold), physical quantities are real-valued functions fA on S. The evaluation (s, fA ) 7−→ fA (s) gives the value that the physical quantity A has in the state s. The subset fA−1 (∆) of S represents the proposition “A ε ∆”, that is, “the physical quantity A has a value in the (Borel) set ∆”. All states s ∈ S that lie in fA−1 (∆) make the proposition true, all other states make it false. Conjunction (And) of propositions corresponds to intersection of Borel subsets of S, disjunction (Or) to union. The Borel subsets form a σ-complete Boolean algebra.

Andreas D¨ oring (Oxford University)

The topos approach

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The main idea

The prototype The prototype of a realist theory is classical mechanics. Pure states form a state space S (symplectic manifold), physical quantities are real-valued functions fA on S. The evaluation (s, fA ) 7−→ fA (s) gives the value that the physical quantity A has in the state s. The subset fA−1 (∆) of S represents the proposition “A ε ∆”, that is, “the physical quantity A has a value in the (Borel) set ∆”. All states s ∈ S that lie in fA−1 (∆) make the proposition true, all other states make it false. Conjunction (And) of propositions corresponds to intersection of Borel subsets of S, disjunction (Or) to union. The Borel subsets form a σ-complete Boolean algebra. We observe an interplay between the geometric and the logical structure of the theory. This is made precise by Stone’s theorem. Andreas D¨ oring (Oxford University)

The topos approach

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The main idea

Abstraction We want a more general scheme (for QT and beyond) with an analogue of the state space, an analogue of the real numbers and physical quantities as arrows between them.

Andreas D¨ oring (Oxford University)

The topos approach

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The main idea

Abstraction We want a more general scheme (for QT and beyond) with an analogue of the state space, an analogue of the real numbers and physical quantities as arrows between them. From the KS theorem, we know that there is no naive state space picture of quantum theory, and that we cannot assign real values to all physical quantities at once. The idea is to go beyond spaces, which are sets, to more general objects in a suitable category. This will also mean to go beyond Boolean logic. The collection of (representatives of) propositions of the form “A ε ∆” will not form a Boolean algebra anymore, but we still want some interpretable logical structure.

Andreas D¨ oring (Oxford University)

The topos approach

8 / 32

The main idea

Abstraction We want a more general scheme (for QT and beyond) with an analogue of the state space, an analogue of the real numbers and physical quantities as arrows between them. From the KS theorem, we know that there is no naive state space picture of quantum theory, and that we cannot assign real values to all physical quantities at once. The idea is to go beyond spaces, which are sets, to more general objects in a suitable category. This will also mean to go beyond Boolean logic. The collection of (representatives of) propositions of the form “A ε ∆” will not form a Boolean algebra anymore, but we still want some interpretable logical structure. All this suggests the use of topos theory. We will reformulate (large parts of) QT in a suitable topos. QG and QC will follow ;-) Andreas D¨ oring (Oxford University)

The topos approach

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Topoi and physics

What kind of beast is a topos? There are many facets of topos theory, and topoi show up in many mathematical situations. For us, the central idea is that an elementary topos E is a generalisation of the category Set of sets and functions (which is itself a topos, of course).

Andreas D¨ oring (Oxford University)

The topos approach

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Topoi and physics

What kind of beast is a topos? There are many facets of topos theory, and topoi show up in many mathematical situations. For us, the central idea is that an elementary topos E is a generalisation of the category Set of sets and functions (which is itself a topos, of course). The objects in a topos can be seen as generalised sets, the arrows are generalised functions. Given two sets, we can form their cartesian product, disjoint union and the set of all functions from one set to the other. In a topos, there are analogous operations defined on all objects.

Andreas D¨ oring (Oxford University)

The topos approach

9 / 32

Topoi and physics

What kind of beast is a topos? There are many facets of topos theory, and topoi show up in many mathematical situations. For us, the central idea is that an elementary topos E is a generalisation of the category Set of sets and functions (which is itself a topos, of course). The objects in a topos can be seen as generalised sets, the arrows are generalised functions. Given two sets, we can form their cartesian product, disjoint union and the set of all functions from one set to the other. In a topos, there are analogous operations defined on all objects. Just as the topos Set comes equipped with Boolean logic, each topos E has an internal logic, which is of intuitionistic type, i.e., the law of excluded middle does not hold in general. But, interestingly, topoi allow for a well-defined notion of partial truth. Andreas D¨ oring (Oxford University)

The topos approach

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Topoi and physics

The genesis of topos ideas in physics In the beginning, there was Lawvere, who decided to reinvent mathematics in order to do physics properly (1960s). Generalising Grothendieck, he invented elementary topoi (together with Miles Tierney). Regarding physics, Lawvere was and mostly is interested in continuum mechanics.

Andreas D¨ oring (Oxford University)

The topos approach

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Topoi and physics

The genesis of topos ideas in physics In the beginning, there was Lawvere, who decided to reinvent mathematics in order to do physics properly (1960s). Generalising Grothendieck, he invented elementary topoi (together with Miles Tierney). Regarding physics, Lawvere was and mostly is interested in continuum mechanics. In 1996, Chris Isham first suggested to use topos theory in foundations of quantum theory (in the consistent histories approach). IJTP 36 (1997), 785–814 A central idea was already there: Consider the commutative/distributive parts of a non-commutative/ non-distributive structure and their relations and build presheaves over them. The latter form a topos, whose internal logic is then employed in the interpretation. The commutative/distributive parts serve as stages of truth. Andreas D¨ oring (Oxford University)

The topos approach

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Topoi and physics

The genesis of topos ideas in physics (2) The idea to look at a quantum system from the collection of classical perspectives was then developed considerably in four papers (1998-2001) with Jeremy Butterfield, concerned with the KS theorem (and how to get around it). IJTP 37, 2669–2733 (1998), IJTP 38, 827–859 (1999), IJTP 39, 1413–1436 (2000), IJTP 41, 613–639 (2002)

Andreas D¨ oring (Oxford University)

The topos approach

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Topoi and physics

The genesis of topos ideas in physics (2) The idea to look at a quantum system from the collection of classical perspectives was then developed considerably in four papers (1998-2001) with Jeremy Butterfield, concerned with the KS theorem (and how to get around it). IJTP 37, 2669–2733 (1998), IJTP 38, 827–859 (1999), IJTP 39, 1413–1436 (2000), IJTP 41, 613–639 (2002)

In particular, they considered the set V(N ) of abelian subalgebras of the non-abelian von Neumann algebra N of physical quantities, partially ordered under inclusion. Each abelian subalgebra or context V ∈ V(N ) gives a classical perspective on the quantum system. Importantly, each context V has a spectrum ΣV such that V ' C (ΣV ). Each ΣV is a local state space. Butterfield and Isham introduced the spectral presheaf Σ, which collects all these local state spaces into one large structure.

Andreas D¨ oring (Oxford University)

The topos approach

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The spectral presheaf

The spectral presheaf The spectral presheaf Σ is the functor assigning to each V ∈ V(N ) its Gel’fand spectrum, and to each inclusion V 0 ⊂ V of contexts the restriction function Σ(iV 0 V ) : ΣV −→ ΣV 0 λ 7−→ λ|V 0 .

Andreas D¨ oring (Oxford University)

The topos approach

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The spectral presheaf

The spectral presheaf The spectral presheaf Σ is the functor assigning to each V ∈ V(N ) its Gel’fand spectrum, and to each inclusion V 0 ⊂ V of contexts the restriction function Σ(iV 0 V ) : ΣV −→ ΣV 0 λ 7−→ λ|V 0 . This presheaf can be seen as a kind of spectrum of the non-abelian algebra N . As such, it is a non-commutative space. Importantly, it has ho points (global elements) – a fact that is equivalent to the Kochen-Specker theorem.

Andreas D¨ oring (Oxford University)

The topos approach

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The spectral presheaf

The spectral presheaf The spectral presheaf Σ is the functor assigning to each V ∈ V(N ) its Gel’fand spectrum, and to each inclusion V 0 ⊂ V of contexts the restriction function Σ(iV 0 V ) : ΣV −→ ΣV 0 λ 7−→ λ|V 0 . This presheaf can be seen as a kind of spectrum of the non-abelian algebra N . As such, it is a non-commutative space. Importantly, it has ho points (global elements) – a fact that is equivalent to the Kochen-Specker theorem. Physically, the spectral presheaf Σ is the analogue of the state space S of a classical system. op

Σ is an object in the topos SetV(N ) of presheaves over the context category V(N ). This is the topos associated to a quantum system (with N as algebra of physical quantities). Andreas D¨ oring (Oxford University)

The topos approach

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The spectral presheaf

Contravariance

Some remarks: The fact that we are using contravariant functors incorporates the important concept of coarse-graining; the idea of coarse-graining goes back to Kochen-Specker pairs of physical quantities.

Andreas D¨ oring (Oxford University)

The topos approach

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Daseinisation

Daseinisation of projections

The spectral presheaf Σ is the analogue of state space, so propositions should be represented by subobjects of Σ. The relevant mapping is called daseinisation of projections. It maps every proposition “A ε ∆” to a (clopen) subobject of Σ, the analogue of a Borel subset fA−1 (∆) of state space. Daseinisation takes coarse-graining into account. The subobjects form a Heyting algebra.

Andreas D¨ oring (Oxford University)

The topos approach

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Daseinisation

Daseinisation of projections

The spectral presheaf Σ is the analogue of state space, so propositions should be represented by subobjects of Σ. The relevant mapping is called daseinisation of projections. It maps every proposition “A ε ∆” to a (clopen) subobject of Σ, the analogue of a Borel subset fA−1 (∆) of state space. Daseinisation takes coarse-graining into account. The subobjects form a Heyting algebra. The resulting new form of quantum logic (Butterfield, Isham, D) is intuitionistic. This topos quantum logic has many sensible properties (different from standard quantum logic). It is distributive, multi-valued, contextual and intuitionistic.

Andreas D¨ oring (Oxford University)

The topos approach

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Daseinisation

Daseinisation of self-adjoint operators

There is another mapping, called daseinisation of self-adjoint operators. Though mathematically related to the daseinisation of projections, it does something quite different: it ‘translates’ physical quantities A into arrows op ˘ δ(A) in our topos SetV(N ) .

Andreas D¨ oring (Oxford University)

The topos approach

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Daseinisation

Daseinisation of self-adjoint operators

There is another mapping, called daseinisation of self-adjoint operators. Though mathematically related to the daseinisation of projections, it does something quite different: it ‘translates’ physical quantities A into arrows op ˘ δ(A) in our topos SetV(N ) . ˘ Specifically, δ(A) is an arrow from the spectral presheaf to a certain presheaf of ‘values’, basically given by intervals of real numbers. The ˘ arrow δ(A) : Σ → R↔ is the analogue of the function fA : S → R in classical physics.

Andreas D¨ oring (Oxford University)

The topos approach

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States

Pure states In the topos formulation, a pure state ψ of a quantum system corresponds to a certain subobject wψ of the spectral presheaf Σ, called a pseudostate. Since Σ has no points, wψ must be ‘bigger than a point’, but it is minimal with respect to a certain natural condition.

Andreas D¨ oring (Oxford University)

The topos approach

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States

Pure states In the topos formulation, a pure state ψ of a quantum system corresponds to a certain subobject wψ of the spectral presheaf Σ, called a pseudostate. Since Σ has no points, wψ must be ‘bigger than a point’, but it is minimal with respect to a certain natural condition. Using the internal logic of our topos of presheaves, we can assign a truth-value to every proposition “A ε ∆” in any given state. In fact, the truth-value assignment is a direct generalisation of the truth-value assignment in classical physics.

Andreas D¨ oring (Oxford University)

The topos approach

16 / 32

States

Pure states In the topos formulation, a pure state ψ of a quantum system corresponds to a certain subobject wψ of the spectral presheaf Σ, called a pseudostate. Since Σ has no points, wψ must be ‘bigger than a point’, but it is minimal with respect to a certain natural condition. Using the internal logic of our topos of presheaves, we can assign a truth-value to every proposition “A ε ∆” in any given state. In fact, the truth-value assignment is a direct generalisation of the truth-value assignment in classical physics. In every pure state, every proposition has a truth-value, independent of measurements, observers or any other instrumentalist concepts. We have a neo-realist formulation of quantum theory.

Andreas D¨ oring (Oxford University)

The topos approach

16 / 32

States and measures

Measures from states

States of a classical system

The most general kind of state of a classical system is a probability measure on the state space S. In the case of a Dirac measure (or point measure) δs , concentrated at some point s ∈ S, we can identify the point with the measure itself.

Andreas D¨ oring (Oxford University)

The topos approach

17 / 32

States and measures

Measures from states

States of a classical system

The most general kind of state of a classical system is a probability measure on the state space S. In the case of a Dirac measure (or point measure) δs , concentrated at some point s ∈ S, we can identify the point with the measure itself. The physical quantities of a classical system are represented by measurable, real-valued functions on the state space S. These functions form a commutative algebra A.

Andreas D¨ oring (Oxford University)

The topos approach

17 / 32

States and measures

Measures from states

States of a classical system

The most general kind of state of a classical system is a probability measure on the state space S. In the case of a Dirac measure (or point measure) δs , concentrated at some point s ∈ S, we can identify the point with the measure itself. The physical quantities of a classical system are represented by measurable, real-valued functions on the state space S. These functions form a commutative algebra A. A probability measure on the state space S also defines an integral on the algebra A by the Riesz representation theorem.

Andreas D¨ oring (Oxford University)

The topos approach

17 / 32

States and measures

Measures from states

States of a quantum system The states of a quantum system are positive linear functionals of norm 1, i.e., integrals on the non-commutative von Neumann algebra N of physical quantities.

Andreas D¨ oring (Oxford University)

The topos approach

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States and measures

Measures from states

States of a quantum system The states of a quantum system are positive linear functionals of norm 1, i.e., integrals on the non-commutative von Neumann algebra N of physical quantities. Kadison/Ringrose, Fundamentals of the Theory of Operator Algebras, Vol. 2, p485/6: ... In this view, a state of a non-commutative C ∗ -algebra may be thought of as a “non-commutative integral” and corresponding, in an implicit sense, to a “non-commutative measure”.

Andreas D¨ oring (Oxford University)

The topos approach

18 / 32

States and measures

Measures from states

States of a quantum system The states of a quantum system are positive linear functionals of norm 1, i.e., integrals on the non-commutative von Neumann algebra N of physical quantities. Kadison/Ringrose, Fundamentals of the Theory of Operator Algebras, Vol. 2, p485/6: ... In this view, a state of a non-commutative C ∗ -algebra may be thought of as a “non-commutative integral” and corresponding, in an implicit sense, to a “non-commutative measure”. What is lacking is a suitable state space on which these measures could live. In fact, the Kochen-Specker theorem shows that such a state space cannot exist.

Andreas D¨ oring (Oxford University)

The topos approach

18 / 32

States and measures

Measures from states

States and measures

Classical state

Quantum state

Integral on A

Integral on N

Prob. measure on S

???

Andreas D¨ oring (Oxford University)

The topos approach

19 / 32

States and measures

Measures from states

States and measures

Classical state

Quantum state

Integral on A

Integral on N

Prob. measure on S

???

In the following, I will show how states of the von Neumann algebra N correspond bijectively to measures on its spectral presheaf Σ, thus filling the gap above. This gives a mathematically and conceptually new formulation of quantum states.

Andreas D¨ oring (Oxford University)

The topos approach

19 / 32

States and measures

Measurable subobjects

Measurable subobjects In the classical case, we considered Borel (i.e., measurable) subsets of the state space. Analogously, we have to find measurable subobjects of our ˆV )V ∈V(N ) of ‘quantum state space’ Σ. These will be collections (P projections such that ˆV ∈ ΣV for all V ∈ V(N ); P ˆV 0 ≥ P ˆV (coarse-graining). if V 0 ⊂ V , then P

Andreas D¨ oring (Oxford University)

The topos approach

20 / 32

States and measures

Measurable subobjects

Measurable subobjects In the classical case, we considered Borel (i.e., measurable) subsets of the state space. Analogously, we have to find measurable subobjects of our ˆV )V ∈V(N ) of ‘quantum state space’ Σ. These will be collections (P projections such that ˆV ∈ ΣV for all V ∈ V(N ); P ˆV 0 ≥ P ˆV (coarse-graining). if V 0 ⊂ V , then P For each V ∈ V(N ), we obtain a clopen subset S PˆV of the Gel’fand spectrum ΣV by ˆV ) = 1}. S PˆV := {λ ∈ ΣV | λ(P A collection (S PˆV )V ∈V(N ) is called a clopen subobject of Σ. Notation: S = (S PˆV )V ∈V(N ) .

Andreas D¨ oring (Oxford University)

The topos approach

20 / 32

States and measures

Measurable subobjects

Measurable subobjects In the classical case, we considered Borel (i.e., measurable) subsets of the state space. Analogously, we have to find measurable subobjects of our ˆV )V ∈V(N ) of ‘quantum state space’ Σ. These will be collections (P projections such that ˆV ∈ ΣV for all V ∈ V(N ); P ˆV 0 ≥ P ˆV (coarse-graining). if V 0 ⊂ V , then P For each V ∈ V(N ), we obtain a clopen subset S PˆV of the Gel’fand spectrum ΣV by ˆV ) = 1}. S PˆV := {λ ∈ ΣV | λ(P A collection (S PˆV )V ∈V(N ) is called a clopen subobject of Σ. Notation: S = (S PˆV )V ∈V(N ) . Theorem The clopen subobjects form a complete Heyting algebra Subcl (Σ). Andreas D¨ oring (Oxford University)

The topos approach

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States and measures

Definition of the measure

Definition of the measure Definition Let ρ be a state on the von Neumann algebra N . The probability measure on clopen subobjects of Σ associated to ρ is the mapping µρ : Subcl (Σ) −→ A(V(N )) ˆV )V ∈V(N ) 7−→ (ρ(P ˆV ))V ∈V(N ) . S = (P Here, A(V(N )) is the set of antitone (order-reversing) functions from V(N ) to the unit interval [0, 1].

Andreas D¨ oring (Oxford University)

The topos approach

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States and measures

Definition of the measure

Definition of the measure Definition Let ρ be a state on the von Neumann algebra N . The probability measure on clopen subobjects of Σ associated to ρ is the mapping µρ : Subcl (Σ) −→ A(V(N )) ˆV )V ∈V(N ) 7−→ (ρ(P ˆV ))V ∈V(N ) . S = (P Here, A(V(N )) is the set of antitone (order-reversing) functions from V(N ) to the unit interval [0, 1]. Some properties are immediate: for all states ρ, µρ (Σ) = 1V(N ) , where 1V(N ) denotes the function that is constantly 1 on V(N ); µρ (0) = 0V(N ) ; for all S ∈ Subcl (Σ), if V 0 ⊂ V , then µρ (S)(V 0 ) ≥ µρ (S)(V ). Andreas D¨ oring (Oxford University)

The topos approach

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States and measures

Definition of the measure

A property of the measure Let S 1 , S 2 be two arbitrary clopen subobjects of Σ. Then, for all V ∈ V(N ), µρ (S 1 ∨ S 2 )(V ) + µρ (S 1 ∧ S 2 )(V ) ˆ1;V ∨ P ˆ2;V ) + ρ(P ˆ1;V ∧ P ˆ2;V ) = ρ(P ˆ1;V ∨ P ˆ2;V + P ˆ1;V ∧ P ˆ2;V ) = ρ(P ˆ1;V + P ˆ2;V ) = ρ(P ˆ1;V ) + ρ(P ˆ2;V ) = ρ(P = µρ (S 1 )(V ) + µρ (S 2 )(V ), which gives globally µρ (S 1 ∨ S 2 ) + µρ (S 1 ∧ S 2 ) = µρ (S 1 ) + µρ (S 2 ).

Andreas D¨ oring (Oxford University)

The topos approach

22 / 32

States and measures

Axiomatisation

Axiomatisation of measures We now axiomatise measures: a mapping µ : Subcl (Σ) −→ A(V(N )) ˆV )V ∈V(N ) 7−→ (µ(P ˆV ))V ∈V(N ) S = (P is called a probability measure on the clopen subobjects of Σ if the following two conditions are fulfilled: 1

µ(Σ) = 1V(N ) ;

2

for all S 1 , S 2 ∈ Subcl (Σ), it holds that µ(S 1 ∨ S 2 ) + µ(S 1 ∧ S 2 ) = µ(S 1 ) + µ(S 2 ).

Andreas D¨ oring (Oxford University)

The topos approach

23 / 32

States and measures

Axiomatisation

Axiomatisation of measures We now axiomatise measures: a mapping µ : Subcl (Σ) −→ A(V(N )) ˆV )V ∈V(N ) 7−→ (µ(P ˆV ))V ∈V(N ) S = (P is called a probability measure on the clopen subobjects of Σ if the following two conditions are fulfilled: 1

µ(Σ) = 1V(N ) ;

2

for all S 1 , S 2 ∈ Subcl (Σ), it holds that µ(S 1 ∨ S 2 ) + µ(S 1 ∧ S 2 ) = µ(S 1 ) + µ(S 2 ).

There is no condition of countable or complete additivity. We want a formalism that is able to accommodate non-normal states also. Andreas D¨ oring (Oxford University)

The topos approach

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States and measures

Axiomatisation

Axiomatisation of measures (2)

Some remarks: Note that the definition of a measure µ does not refer in any direct way to the non-commutativity of the vNa N from which Σ is constructed.

Andreas D¨ oring (Oxford University)

The topos approach

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States and measures

Axiomatisation

Axiomatisation of measures (2)

Some remarks: Note that the definition of a measure µ does not refer in any direct way to the non-commutativity of the vNa N from which Σ is constructed. Conditions (1.) and (2.) are local, since meet and join of subobjects are defined locally, i.e., component-wise at each V , as is the addition in A(V(N )).

Andreas D¨ oring (Oxford University)

The topos approach

24 / 32

States and measures

Axiomatisation

Axiomatisation of measures (2)

Some remarks: Note that the definition of a measure µ does not refer in any direct way to the non-commutativity of the vNa N from which Σ is constructed. Conditions (1.) and (2.) are local, since meet and join of subobjects are defined locally, i.e., component-wise at each V , as is the addition in A(V(N )). The only piece of information relating different contexts is the fact that the codomain of µ is A(V(N )), i.e., smaller contexts are assigned larger numbers.

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States and measures

The bijection between states and measures

The bijection between states and measures Somewhat surprisingly, one can show: Theorem Let N be a vNa without direct summand of type I2 , and let µ be a measure on Σ. Then there exists a unique state ρµ of the vNa N such that µ = µρµ .

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States and measures

The bijection between states and measures

The bijection between states and measures Somewhat surprisingly, one can show: Theorem Let N be a vNa without direct summand of type I2 , and let µ be a measure on Σ. Then there exists a unique state ρµ of the vNa N such that µ = µρµ . Since conversely every state ρ gives a measure µρ as we saw, we have Theorem For every vNa N without type I2 summand, there is a bijection between the space of states S(N ) and the set M(Subcl (Σ)) of probability measures on the clopen subobjects of the spectral presheaf Σ of N .

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States and measures

The bijection between states and measures

The bijection between states and measures (2)

Idea of proof: Given a probability measure µ, define a (unique) finitely additive probability measure m : P(N ) → [0, 1] on the projections in N . Then, by the generalised version of Gleason’s theorem, there is a unique state ρµ ∈ S(N ) extending m.

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The topos approach

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States and measures

The bijection between states and measures

The bijection between states and measures (2)

Idea of proof: Given a probability measure µ, define a (unique) finitely additive probability measure m : P(N ) → [0, 1] on the projections in N . Then, by the generalised version of Gleason’s theorem, there is a unique state ρµ ∈ S(N ) extending m. While in the classical case, the Riesz representation theorem is the link between integrals and measures, in the quantum case it is Gleason’s theorem. The measures corresponding to quantum states can be used to calculate the usual expectation values.

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States and measures

Normal states

Characterisation of normal states

Normality is a certain continuity condition that a state ρ may or may not fulfil.

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States and measures

Normal states

Characterisation of normal states

Normality is a certain continuity condition that a state ρ may or may not fulfil. Proposition A measure µ : Subcl (Σ) → A(V(N )) corresponds to a normal state ρ if and only if µ preserves joins (where the join in A(V(N )) is the componentwise supremum). Of course, if the Hilbert space is separable, preservation of countable joins is sufficient.

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States and measures

Open questions

Some open problems There are a number of open questions:

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The topos approach

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States and measures

Open questions

Some open problems There are a number of open questions: We know that every state of a quantum system corresponds to a measure. How can entanglement be described in terms of these measures?

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The topos approach

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States and measures

Open questions

Some open problems There are a number of open questions: We know that every state of a quantum system corresponds to a measure. How can entanglement be described in terms of these measures? Can we use methods from measure theory to understand and classify entanglement?

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The topos approach

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States and measures

Open questions

Some open problems There are a number of open questions: We know that every state of a quantum system corresponds to a measure. How can entanglement be described in terms of these measures? Can we use methods from measure theory to understand and classify entanglement? Dynamics is described by linear automorphisms of the state space S(N ), and hence M(Subcl (Σ)) – what does this imply for the geometry of Σ?

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The topos approach

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States and measures

Open questions

Some open problems There are a number of open questions: We know that every state of a quantum system corresponds to a measure. How can entanglement be described in terms of these measures? Can we use methods from measure theory to understand and classify entanglement? Dynamics is described by linear automorphisms of the state space S(N ), and hence M(Subcl (Σ)) – what does this imply for the geometry of Σ? How does the difference between classical and quantum physics manifest itself with respect to the measures describing states?

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The topos approach

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Summary

The table Abstract

Classical phys.

Standard QT

Topos QT

state space propos. “A ∈ ∆” latt. of props. topos logic deductive sys. truth values interpretation quant.-val. obj. phys. quant. A pure state general state

set S subset of S Boolean algebra Set Boolean yes true, false realist R fA : S → R Dirac meas. δs prob. m. µ on S

Hilb. sp. H? ˆ projection P P(N ) (non-distrib.) Set (quantum log.) no ? (probabilities) instrumentalist R? ˆ :H→H A vector st. wψ state ρ of vNa

presheaf Σ subobj. of Σ Heyting algebra op SetV(N ) intuitionistic yes elements of Ω neo-realist R↔ ˘ A) ˆ : Σ → R↔ δ( pseudo-st. wψ prob. m. µ on Σ

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Summary

Quantum theory is...

The fact that quantum theory can be reformulated in a suitable topos casts some doubt on old dogmas like there is no state space for a quantum system; quantum theory is fundamentally probabilistic; there is no realist formulation of quantum theory.

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Summary

Quantum theory is...

The fact that quantum theory can be reformulated in a suitable topos casts some doubt on old dogmas like there is no state space for a quantum system; quantum theory is fundamentally probabilistic; there is no realist formulation of quantum theory. All these aspects are interrelated. Topos theory delivers tools to generalise both Boolean logic and set-based mathematics/topology/geometry.

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The topos approach

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Summary

Related work Recently, Caspers, Heunen, Landsman and Spitters used covariant functors over the context category to define a topos-internal commutative algebra N from the external non-commutative algebra N . The commutative algebra N has a Gel’fand spectrum, which is a certain locale in the topos SetV(N ) of functors. CMP in print (2009), FOP in print (2009), arXiv:0905.2275 They suggest to use opens in this locale as representatives of propositions. The resulting formalism, and in particular the intuitionistic quantum logic, are closely related to the earlier work in the contravariant approach. It will be interesting to see the interpretational differences. Flori has developed a topos-based version of the consistent histories formalism. arXiv:0812.1290

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The topos approach

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Summary

References A. D¨oring, C. J. Isham, “A Topos Foundation for Theories of Physics I-IV”, J. Math. Phys. 49, 5, 053515–18 (2008), arXiv:quant-ph/0703060, 62, 64 and 66 A. D¨oring, “Topos theory and ‘neo-realist’ quantum theory”, in Quantum Field Theory: Competitive models, eds. B. Fauser, J. Tolksdorf, E. Zeidler (Birkh¨auser 2009), arXiv:0712.4003 A. D¨oring, C. J. Isham, “‘What is a thing?’: Topos Theory in the Foundations of Physics”, to appear in New Structures in Physics, ed. Bob Coecke (Springer 2010), arXiv:0803.0417 A. D¨oring, “Quantum States and Measures on the Spectral Presheaf”, Adv. Sci. Lett. 2, Number 2 (Special Issue on “Quantum Gravity, Cosmology and Black Holes”, ed. M. Bojowald), 291–301 (2009), arXiv:0809.4847 Andreas D¨ oring (Oxford University)

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