Topos Formulation of History Quantum Theory

Cecilia Flori1

1 Perimeter Institute for Theoretical Physics

Birmingham, UK

21.09.2010

Cecilia Flori Topos Formulation of history Quantum Theory

1. INTRODUCTION

1.1 Why a History Theory?

The standard Copenhagen interpretation cannot describe closedsystems, since the existence of an external observer is required.

This causes problems in any theory of quantum cosmology.

One significant recent attempt to deal with closed systems inquantum mechanics is history theory:

– Omnes, Griffiths, Gell’mann & Hartle: Histories as products ofprojection operators (therefore not projectors).

–HPO formalism of consistent histories: History propositions areidentified with projection operators in bigger Hilbert space.

In this talk I will describe a new type of history quantum theory: atopos version of the temporal-logic part of the HPO formalism.

1.2 Why a topos version of the temporal logic part of the HPO formalism?

A topos reformulation of quantum mechanics was put forward byIsham and Döring.

An essential ingredient is a mapping of projection operators tocertain objects in a topos.

My aim is to extend this formulation to a history version ofquantum theory, therefore my starting point is the HPOformalism.

1.2 Why a topos version of the temporal logic part of the HPO formalism?

A topos reformulation of quantum mechanics was put forward byIsham and Döring.

An essential ingredient is a mapping of projection operators tocertain objects in a topos.

My aim is to extend this formulation to a history version ofquantum theory, therefore my starting point is the HPOformalism.

2.2 Why Topos Theory?

Kocken-Specher theorem =⇒ non-realist interpretation ofquantum theory

For quantum cosmology, need a reformulation of quantumtheory which is ‘more realist’. Isham, Butterfield & Döring: canbe done through topos theory.

Reformulate quantum theory to make it ‘look like’ classicalphysics:

– Classical physics uses Sets as its mathematical structure.A topos is a category which ‘looks like’ Sets.

– Logic of subsets in Sets is Boolean logic.Logic of subsets in a topos is a distributive logic

2.2 Why Topos Theory?

Kocken-Specher theorem =⇒ non-realist interpretation ofquantum theory

For quantum cosmology, need a reformulation of quantumtheory which is ‘more realist’. Isham, Butterfield & Döring: canbe done through topos theory.

Reformulate quantum theory to make it ‘look like’ classicalphysics:

– Classical physics uses Sets as its mathematical structure.A topos is a category which ‘looks like’ Sets.

– Logic of subsets in Sets is Boolean logic.Logic of subsets in a topos is a distributive logic

2.3 Which Topos?

Need for contexts comes from K-S theorem: only within abeliansubalgebras of B(H) can quantum theory ‘look like’ classicaltheory. Contexts form ‘classical snapshots’.

– The set of abelian subalgebras, V(H), forms a categoryunder subset inclusion: iV ′V : V

′ ⊆ V

i.e. consider all contexts at the same time!

– Example: V′′

= V ∩ V′ 6= ∅ then ∃ the inclusion maps iV ′′V

and iV ′′V ′ , therefore it is possible to ‘relate’ V and V′.

Topos of presheaves over the category of abelian subalgebras :SetsV(H)op

2.3 Which Topos?

Need for contexts comes from K-S theorem: only within abeliansubalgebras of B(H) can quantum theory ‘look like’ classicaltheory. Contexts form ‘classical snapshots’.

– The set of abelian subalgebras, V(H), forms a categoryunder subset inclusion: iV ′V : V

′ ⊆ V

i.e. consider all contexts at the same time!

– Example: V′′

= V ∩ V′ 6= ∅ then ∃ the inclusion maps iV ′′V

and iV ′′V ′ , therefore it is possible to ‘relate’ V and V′.

Topos of presheaves over the category of abelian subalgebras :SetsV(H)op

3. The Isham-Doering scheme

3.1 The State Object

State spaces in physics

1. Classical physics: Physical quantity A represented fA : S → R.

2. Quantum physics: Physical quantity A represented A : H → H.

3. Topos physics: Spectral presheaf Σ : V(H)→ Set such that

V 7→ ΣV := simultaneous eigenvalues of V; i.e., thepossible values of the physical quantities in V .

Given A, then fA : Σ→ R!

Σ is the ‘state object’ of the theory.

3.2 Propositions

Propositions

1. Classical physics:

“A ∈ ∆”→ f−1A (∆) = s ∈ S|fA(s) ∈ ∆ ⊆ S

2. Quantum theory:

P = E [A ∈ ∆] ∈ P(H)

3. Topos physics: Need to identify P with sub-object of Σ; i.e., foreach V need subset of ΣV ; i.e., a projection operator in V .

3.3 Daseinisation

‘Daseinisation’:δ : P(H)→ P(V )

P 7→ δ(P)V

where δ(P)V :=∧α ∈ P(V )|α ≥ P: the ‘best’ approximation of

P (from above) by projectors in V .

Relation to Sub(Σ):

Any projector in V gives a subset of ΣV . Therefore get map, foreach V , P 7→ δ(P)

Can show that this corresponds to δ : P(H)→ Sub(Σ)!

Thus δ maps propositions about a quantum system to adistributive lattice in a contextual manner.

3.3 Daseinisation

‘Daseinisation’:δ : P(H)→ P(V )

P 7→ δ(P)V

where δ(P)V :=∧α ∈ P(V )|α ≥ P: the ‘best’ approximation of

P (from above) by projectors in V .

Relation to Sub(Σ):

Any projector in V gives a subset of ΣV . Therefore get map, foreach V , P 7→ δ(P)

Can show that this corresponds to δ : P(H)→ Sub(Σ)!

Thus δ maps propositions about a quantum system to adistributive lattice in a contextual manner.

3.4 States

States in classical physics

In classical physics a microstate is a point in the state space.

States in the topos formulation of quantum theory

– Σ has no ‘points’ =⇒ Equivalent to the K-S theorem

– Topos analogues of a state is a (non-point!) sub-object ofthe state object Σ:

Pseudo-state: w|ψ〉 := δ(|ψ〉〈ψ|) ⊆ Σ ,

δ( |ψ〉〈ψ|)V :=∧α ∈ P(V ) | |ψ〉〈ψ| ≤ α

w|ψ〉 is the ‘closest’ one can get to defining a point in Σ.

3.4 States

States in classical physics

In classical physics a microstate is a point in the state space.

States in the topos formulation of quantum theory

– Σ has no ‘points’ =⇒ Equivalent to the K-S theorem

– Topos analogues of a state is a (non-point!) sub-object ofthe state object Σ:

Pseudo-state: w|ψ〉 := δ(|ψ〉〈ψ|) ⊆ Σ ,

δ( |ψ〉〈ψ|)V :=∧α ∈ P(V ) | |ψ〉〈ψ| ≤ α

w|ψ〉 is the ‘closest’ one can get to defining a point in Σ.

3.5 Truth Values

Truth values in classical physics

A proposition Q ⊆ S is true in a state s iff s ∈ Q.This is equivalent to s ⊆ Q.

Truth values in the topos formulation of quantum theory

– In topos quantum theory, we define the proposition δ(P) tobe ‘totally true’ given the pseudo state w |ψ〉 iff w |ψ〉 ⊆ δ(P).

This means that, for all V , w |ψ〉V ≤ δ(P)V .

– However, in a topos a proposition can be ‘partially true’using ‘contextual truth values’. At stage V we define

v(w |ψ〉 ⊆ δ(P))V := V′⊆ V | w |ψ〉V ′ ⊆ δ(P)

= V′⊆ V | 〈ψ|δ(P)V ′ |ψ〉 = 1

3.5 Truth Values

Truth values in classical physics

A proposition Q ⊆ S is true in a state s iff s ∈ Q.This is equivalent to s ⊆ Q.

Truth values in the topos formulation of quantum theory

– In topos quantum theory, we define the proposition δ(P) tobe ‘totally true’ given the pseudo state w |ψ〉 iff w |ψ〉 ⊆ δ(P).

This means that, for all V , w |ψ〉V ≤ δ(P)V .

– However, in a topos a proposition can be ‘partially true’using ‘contextual truth values’. At stage V we define

v(w |ψ〉 ⊆ δ(P))V := V′⊆ V | w |ψ〉V ′ ⊆ δ(P)

= V′⊆ V | 〈ψ|δ(P)V ′ |ψ〉 = 1

3.5 Truth values

– This is called a sieve on V .

The collection, ΩV , of all sieves on V forms a Heyting algebra!

– For varying V such truth values form a global section ΓΩ of thesub-object classifier Ω.

The set ΓΩ is itself a Heyting algebra!

– Thus we have a Heyting algebra of propositions and a Heytingalgebra of ‘generalised’ truth values!

4. TOPOS THEORY AND THE HPO FORMALISM

4.1 HPO Formulation of Quantum Temporal Logic

Identify the set of all history propositions with projectionoperators in a new Hilbert space Ht1 ⊗Ht2 ⊗ · · · ⊗ Htn

– ‘Homogeneous histories’: Tensor products of projectionoperators

α := αt1 ⊗ αt2 ⊗ · · · ⊗ αtn

– ‘Inhomogeneous histories’:

¬(αt1 ⊗ αt2 ) := 1Ht1⊗ 1Ht2

− αt1 ⊗ αt2

= (¬αt1 ⊗ αt2 ) ∨ (αt1 ⊗ ¬αt2 ) ∨ (¬αt1 ⊗ ¬αt2 )

– ‘Type III’ propositions

Example of entangled inhomogeneous proposition

Entangled state:| ↑〉| ↓〉 − | ↓〉| ↑〉

Associated projection operator

Pentangled = (| ↑〉| ↓〉 − | ↓〉| ↑〉)(〈↑ |〈↓ | − 〈↓ |〈↑ |)

Pentangled 6= Pud ∨ Pdu = (| ↑〉| ↓〉)(〈↓ |〈↑ |) ∨ (| ↓〉| ↑〉)(〈↑ |〈↓ |)

Logical entanglement given by inhomogeneous propositions isnot sufficient to account for the full quantum entanglement.

4. TOPOS THEORY AND THE HPO FORMALISM

4.1 HPO Formulation of Quantum Temporal Logic

Identify the set of all history propositions with projectionoperators in a new Hilbert space Ht1 ⊗Ht2 ⊗ · · · ⊗ Htn

– ‘Homogeneous histories’: Tensor products of projectionoperators

α := αt1 ⊗ αt2 ⊗ · · · ⊗ αtn

– ‘Inhomogeneous histories’:

¬(αt1 ⊗ αt2 ) := 1Ht1⊗ 1Ht2

− αt1 ⊗ αt2

= (¬αt1 ⊗ αt2 ) ∨ (αt1 ⊗ ¬αt2 ) ∨ (¬αt1 ⊗ ¬αt2 )

– ‘Type III’ propositions

I.e., the tensor product implements the ‘and then’ temporalconnective in quantum temporal logic.

4.2 Temporal Structure in a Heyting algebra

Aim: Find a topos representation of the homogeneous historyα = (A1 ∈ ∆1)t1 u (A2 ∈ ∆2)t2 · · · u (An ∈ ∆n)tn

– Individual-time propositions are identified with sub-objectsof the spectral presheaf ΣV(Hti ). Collection of all suchsub-objects, Sub(ΣV(Hti )), forms a Heyting algebra.

– Temporal structure of Heyting algebras requires tensorproduct of Heyting algebras? Yes!

I will identify homogeneous histories with elements inSub(ΣV(Ht1 ))⊗ Sub(ΣV(Ht2 ))⊗ · · · ⊗ Sub(ΣV(Htn ))

The definition of the tensor product of Heyting algebras allowsfor both homogeneous and inhomogeneous propositions.

Aim: Find a topos representation of the homogeneous historyα = (A1 ∈ ∆1)t1 u (A2 ∈ ∆2)t2 · · · u (An ∈ ∆n)tn

– Individual-time propositions are identified with sub-objectsof the spectral presheaf ΣV(Hti ). Collection of all suchsub-objects, Sub(ΣV(Hti )), forms a Heyting algebra.

– Temporal structure of Heyting algebras requires tensorproduct of Heyting algebras? Yes!

I will identify homogeneous histories with elements inSub(ΣV(Ht1 ))⊗ Sub(ΣV(Ht2 ))⊗ · · · ⊗ Sub(ΣV(Htn ))

The definition of the tensor product of Heyting algebras allowsfor both homogeneous and inhomogeneous propositions.

Given two Heyting algebras A and B the tensor product A⊗ B isdefined as the Heyting algebra generated by elements a⊗ b for alla ∈ A and b ∈ B subject to

– (a1 ⊗ b1) ∧ (a2 ⊗ b2) := (a1∧

a2)⊗ (b1∧

– (a1 ⊗ b) ∨ (a2 ⊗ b) := (a1∨

a2)⊗ b

– (a⊗ b1) ∨ (a⊗ b2) := a⊗ (b1 ∨ b2)

Therefore

(a1∨b1)⊗(a2∨b2) = (a1⊗a2)∨(a1⊗b2)∨(b1⊗a2)∨(b1⊗b2) ≥ a1⊗a2∨b1⊗b2

4.3 Tensor Product in a Topos

We need to relate the Heyting algebraSub(ΣV(Ht1 ))⊗ Sub(ΣV(Ht2 )) to sub-objects of some ‘stateobject’ in some topos related to quantum theory.

αt1 → δ(αt1 ) ∈ Sub(ΣV(Ht1 )) ∈ SetsV(Ht1 )op

To what topos does the history propositionδ(αt1 )⊗ δ(αt2 ) ∈ Sub(ΣV(Ht1 ))⊗ Sub(ΣV(Ht2 )) belong?

Need to find a common topos to which both the topoi SetsV(Ht1 )op

and SetsV(Ht2 )op

can be related.

We need to relate the Heyting algebraSub(ΣV(Ht1 ))⊗ Sub(ΣV(Ht2 )) to sub-objects of some ‘stateobject’ in some topos related to quantum theory.

To what topos does the history propositionδ(αt1 )⊗ δ(αt2 ) ∈ Sub(ΣV(Ht1 ))⊗ Sub(ΣV(Ht2 )) belong?

Need to find a common topos to which both the topoi SetsV(Ht1 )op

and SetsV(Ht2 )op

can be related.

Intermediate topos: SetsV(Ht1 )op×V(Ht2 )

p1 : V(Ht1 )× V(Ht2 )→ V(Ht1 )

〈V1,V2〉 7→ V1

p2 : V(Ht1 )× V(Ht2 )→ V(Ht2 )

〈V1,V2〉 7→ V2

from which we can obtain

p∗1 : SetsV(Ht1 )op→ SetsV(Ht1 )

op×V(Ht2 )op

p∗2 : SetsV(Ht2 )op→ SetsV(Ht1 )

op×V(Ht2 )op

which gives the well-defined product in SetsV(Ht1 )op×V(Ht2 )

(ΣV(Ht1 ) × ΣV(Ht2 ))〈V1,V2〉 := (p∗1(ΣV(Ht1 ))× p∗2(ΣV(Ht2 )))〈V1,V2〉

= ΣV(Ht1 )

V1× Σ

V(Ht2 )

4.4 Topos Formulation of HPO

Theorem: Sub(ΣV(Ht1 ))⊗ Sub(ΣV(Ht2 )) ∼= Sub(ΣV(Ht1 ) × ΣV(Ht2 )

)Conjecture: Proposition α1 u α2 should be represented by

δ(α1)⊗ δ(α2) ∈ Sub(ΣV(Ht1 ) × ΣV(Ht2 )

)But, the HPO-representative, α1 ⊗ α2, of the history propositionα1 u α2 belongs to Ht1 ⊗Ht2 and is daseinised by

δ(α1 ⊗ α2) ∈ ΣV(Ht1⊗Ht2 ) ∈ SetsV(Ht1⊗Ht2 )op

V(Ht1 ⊗Ht2 ) contains entangled contexts W = V1⊗V2 + V3⊗V4

and so SetsV(Ht1⊗Ht2 )op

6= SetsV(Ht1 )op×V(Ht2 )

Want to find a relation betweenδ(α1)⊗ δ(α2) ∈ p∗1(ΣV(Ht1 ))× p∗2(ΣV(Ht2 )) andδ(α1 ⊗ α2) ∈ ΣV(Ht1⊗Ht2 ).

Must find a relation between SetsV(Ht1 )op×V(Ht2 )

and SetsV(Ht1⊗Ht2 )op

– Relation between context categories:

θ : V(Ht1 )× V(Ht2 ) → V(Ht1 ⊗Ht2 )

〈V1,V2〉 7→ V1 ⊗ V2

– Induced relation between topoi:

θ∗ : SetsV(Ht1⊗Ht2 )op

→ SetsV(Ht1 )op×V(Ht2 )

(θ∗ΣV(Ht1⊗Ht2 ))〈V1,V2〉 := (ΣV(Ht1⊗Ht2 ))θ(〈V1,V2〉) = ΣV(Ht1⊗Ht2 )

V1⊗V2

∼= ΣV(Ht1 )

V1× Σ

V(Ht2 )

using the fact that, for contexts of the form V1 ⊗ V2,

ΣV(Ht1⊗Ht2 )

V1⊗V2∼= Σ

V(Ht1 )

V1× Σ

V(Ht2 )

Conclusion: To account for both homogeneous and inhomogeneous(‘logically entangled’) propositions the intermediate toposSetsV(Ht1 )

op×V(Ht2 )op

suffices.

But, in full topos SetsV(Ht1⊗Ht2 )op

there also arise (i) entangledcontexts W = V1 ⊗ V2 + V3 ⊗ V4; and (ii) a third type that cannot beexpressed in this way. The physical significance of these needs to beexplored further.

5. Truth Values

Claim: because of the absence of state-vector reduction in the toposapproach, it is meaningful to define the truth value of a(homogeneous) history proposition in terms of the truth values of theindividual time components:

v((A1 ∈ ∆1)t1u(A2 ∈ ∆2)t2 ; |ψ〉t1

(A1 ∈ ∆1; |ψ〉t1

)⊗v(A2 ∈ ∆2; |ψ〉t2

)where |ψ〉t2 = U(t2, t1) |ψ〉t1 .

Want to find a ‘topos interpretation’ of the above equation.

Truth values

Problem: truth values belong to different topoi

v(A1 ∈ ∆1; |ψ〉t1

):= v(w |ψ〉t1 ⊆ δ(P1)) ∈ ΓΩV(Ht1 )

v(A2 ∈ ∆2; |ψ〉t2

):= v(w |ψ〉t2 ⊆ δ(P2)) ∈ ΓΩV(Ht2 )

Solution: pull back to the ‘intermediate topos’ SetsV(Ht1 )op×V(Ht2 )

.In SetsV(Ht1 )

op×V(Ht2 )op

we have:

v(w |ψ〉t1 ⊆ δ(Pt1 )

)⊗v(w |ψ〉t2 ⊆ δ(Pt2 )

) ∼= v(w |ψ〉t1⊗w |ψ〉t2 ⊆ δ(Pt1 )⊗δ(Pt2 )

)using the fact that (theorem)

ΓΩV(Ht1 )⊗ΓΩV(Ht2 ) ' ΓΩV(Ht1 )×V(Ht2 )

Tensor products of Truth values

Therefore, so long as entangled contexts (plus type III) are notconsidered it is possible to define truth values of history propositionsas tensor products of truth values of individual-time propositions.

In particular, to obtain a topos formulation of quantum history theoriesthe intermediate topos suffices.

5. Summary

The aim was to obtain a topos formulation of the temporalquantum logic in the HPO formalism.The strategy adopted was to extend the topos formulation ofquantum mechanics put forward by Isham and Döring so as toinclude temporal propositions.

I have been able to represent both homogeneous andinhomogeneous history propositions as elements of the Heytingalgebra Sub(ΣV(Ht1 ))⊗ Sub(ΣV(Ht2 ))

Sub(ΣV(Ht1 ))⊗ Sub(ΣV(Ht2 )) ∈ SetsV(Ht1 )op×V(Ht2 )

; i.e., noentangled contexts

The topos that represents the full HPO formalism is the toposSets(V(Ht1⊗Ht2 ))

which includes entangled and type III contexts.

For non-entangled contexts the two topoi coincide.

5. Summary

Future Work

1. Is it possible to represent, with a novel mathematical structure,type III propositions in topos-theoretical terms?

– Logical entanglement is captured by the notion of tensorproduct of Heyting algebra.

– Quantum entanglement might be captured by the notion(yet to be defined) of ‘quantum’ tensor product.

2. Extend the topos reformulation of history theory to the case ofcontinuous time.

Future Work

1. Is it possible to represent, with a novel mathematical structure,type III propositions in topos-theoretical terms?

– Logical entanglement is captured by the notion of tensorproduct of Heyting algebra.

– Quantum entanglement might be captured by the notion(yet to be defined) of ‘quantum’ tensor product.

2. Extend the topos reformulation of history theory to the case ofcontinuous time.

Every frame A defines a complete Heyting algebra (cHa) in such away that the operations ∧ and ∨ are preserved, and the implicationrelation→ is defined as follows

a→ b =∨c : c ∧ a ≤ b

Frame distributivity implies that (a→ b) ∧ a ≤ b, from which it follows

c ≤ a→ b iff c ∧ a ≤ b

Topos Formulation of History Quantum Theory

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