Contexts, Bi-Heyting Algebras and a New Logic for Quantum Systems Oberseminar Theoretische Informatik FAU Erlangen 4. November 2014

Andreas D¨oring

Theoretische Physik I, FAU Erlangen

[email protected]

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 1 / 49 “Never express yourself more clearly than you are able to think.” Niels Bohr

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 2 / 49 References

This talk is based on AD,“Topos-based Logic for Quantum Systems and Bi-Heyting Algebras”, to appear in Logic & Algebra in Quantum Computing, Lecture Notes in Logic, Association for Symbolic Logic in conjunction with Cambridge University Press (2012); [arXiv:1202.2750]

Some good references on standard quantum logic are: G. Birkhoff, J. von Neumann, “The Logic of Quantum Mechanics”, Annals of Mathematics 37, No. 4, 823–843 (1936). Varadarajan, Geometry of Quantum Theory, second ed., Springer (1985). M. Dalla Chiara, R. Giuntini, “Quantum Logics”, in Handbook of Philosophical Logic, Kluwer (2002); [arXiv:quant-ph/0101028v2]

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 3 / 49 Standard quantum logic

Standard quantum logic

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 4 / 49 Standard quantum logic Standard quantum logic

Some basics of standard quantum logic: Birkhoff and von Neumann suggested to use the complete orthomodular lattice P(H) of projections on a separable, complex Hilbert space H as representatives of propositions. (Fine point: modular vs. orthomodular lattices.) The link between (pre-mathematical) propositions “A ε ∆” and projections is provided by the spectral theorem. Pure quantum states |ψi (roughly) are models of this propositional logic, but only specific projections are assigned true or false: if Pˆ |ψi = |ψi, then the proposition represented by Pˆ is true in the state |ψi; if Pˆ |ψi = 0, then the proposition is false in the state |ψi. In general, only a probability between 0 and 1 for finding the outcome of a measurement of A to lie in a Borel set ∆ can be given. After measurement, the quantum state has changed to the eigenstate corresponding to the outcome.

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 5 / 49 Standard quantum logic Some (problematic) features of quantum logic

Propositions form a non-distributive orthomodular lattice. Quantum states do not assign true or false to all propositions “A ε ∆”; in general, one can only obtain probabilities. Lack of two-valued models; instrumentalism. A disjunction “A ε ∆ or B ε Γ” can be true in a state |ψi despite the fact that neither “A ε ∆” nor “B ε Γ” are true in the state. This is due to superposition. There a many meets in P(H) that are physically meaningless. This is due to the failure to take contextuality into account. The implication problem: there is no material implication, and the implicative rule does not hold. This is due to non-distributivity. There is no algebra of subsets representing propositions, but an algebra of subspaces. Most problematic features result from this (insufficient?) geometric underpinning of quantum logic.

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 6 / 49 A state space model for quantum systems

A state space model for quantum systems

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 7 / 49 A state space model for quantum systems The topos approach

The topos approach to the formulation of physical theories was initiated by Chris Isham ’97 and Isham/Butterfield ’98–’02, addresses certain structural and conceptual issues in the foundations of physics, uses topos theory to give new geometric and logical ideas for physics, aims to provide a new way of formulating physical theories in general, and quantum theory in particular – not based on Hilbert spaces, not ‘just another interpretation’, is motivated by questions on the way to quantum gravity (QG) and quantum cosmology (QC), most work so far is on standard, non-relativistic quantum theory – natural starting point, testing ground. Other researchers include: Landsman, Heunen, Spitters, Nakayama, Vickers, Fauser, Flori, ...

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 8 / 49 A state space model for quantum systems Where are we now? The topos approach is very much work in progress, so think of

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 9 / 49 A state space model for quantum systems Classical physics

In classical physics, a physical quantity A is described by a (measurable) real-valued function fA on the state space (phase space) P:

In a given state s ∈ P, a physical quantity A has the value fA(s) ∈ R.

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 10 / 49 A state space model for quantum systems Classical physics (2) Propositions such as “A ε ∆”, i.e., “the physical quantity A has a value in the Borel set ∆ ⊆ R”, are represented by subsets of the state space:

The Borel subsets of P form a σ-complete Boolean algebra. Pure states s ∈ P are models of this propositional logic. Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 11 / 49 A state space model for quantum systems The Kochen-Specker theorem and instrumentalism

But can quantum theory be cast in such a form? Obstacle: Theorem (Kochen-Specker 1967): If dim(H) ≥ 3, then there is no state space model of quantum theory (under mild and natural conditions). In logical terms, there is no way to assign true or false to all propositions “A ε ∆” at the same time.

Usual answer: interpret propositions in an instrumentalist way, and use probabilities, because repeatedly measuring the same physical quantity A in the same state |ψi can give different outcomes – states do not assign truth values to all propositions.

This is not what we want to do – we want to provide a quantum state space in spite of the KS theorem.

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 12 / 49 A state space model for quantum systems C ∗-algebras and von Neumann algebras As a reminder: Definition A C ∗-algebra A is a complex Banach algebra with an involution (−)∗ : A → A such that

∀a ∈ A : |a∗a| = |a|2.

A W ∗-algebra is a C ∗-algebra that is the dual of a Banach space M.

Every C ∗-algebra can be faithfully represented as a norm-closed subalgebra of B(H), the algebra of bounded linear operators on some Hilbert space H. Every W ∗-algebra can be represented as a weakly closed subalgebra of some B(H). Weakly closed subalgebras of B(H) are called von Neumann algebras. We will often use ‘hats’ (as in Aˆ, Bˆ, ...) when denoting elements of operator algebras. Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 13 / 49 A state space model for quantum systems The basic idea

With a quantum system (described by a noncommutative von Neumann algebra N , e.g. N = B(H)), we associate a topos and define a spectral object Σ in the topos. The spectral presheaf Σ is a generalised set, playing the role of a state space for the quantum system (notwithstanding the Kochen-Specker theorem!). Structure of state space determines logical structure of a theory: Classical physics: Borel subsets of state space S represent propositions “A ε ∆”, form Boolean algebra. Quantum physics: clopen subobjects of the state space Σ represent propositions “A ε ∆”, form bi-Heyting algebra.

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 14 / 49 A state space model for quantum systems The context category

Let S be a quantum system, described by a noncommutative von Neumann algebra N ⊆ B(H). Let V be a non-trivial commutative von Neumann subalgebra of N which has the same unit element as N . This gives and is given by a set of commuting self-adjoint operators in N . We call V a context.

Each context V is a partial classical perspective on the quantum system. The main idea: take all classical perspectives/contexts together to obtain a complete picture of the quantum system.

Concretely, we consider the set V(N ) of all contexts, partially ordered under inclusion. This poset is called the context category.

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 15 / 49 A state space model for quantum systems Gelfand spectra

Each commutative von Neumann algebra V has a Gelfand spectrum ΣV , consisting of the algebra morphisms

λ : V −→ C.

These maps are also pure states of V . The Gelfand topology is the ∗ (relative) weak -topology on ΣV . With respect to this topology, ΣV is an extremely disconnected compact Hausdorff space. For a commutative von Neumann algebra V , the projection lattice P(V ) is a complete Boolean algebra. Let λ ∈ ΣV , and let Pˆ ∈ P(V ), representing some proposition about an Aˆ ∈ Vsa. Note that

λ(Pˆ) = λ(Pˆ 2) = λ(Pˆ)2 ∈ {0, 1}'{false, true},

so the elements of ΣV are the models of the classical logic described by P(V ).

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 16 / 49 A state space model for quantum systems The spectral presheaf

If V 0 ⊂ V is a von Neumann subalgebra, then there is a canonical restriction map

rV ,V 0 :ΣV −→ ΣV 0

λ 7−→ λ|V 0 .

This map is continuous, closed, open and surjective.

Definition Let N be a von Neumann algebra, and let V(N ) be its context category. The spectral presheaf Σ of N over V(N ) is defined

(a) on objects: for all V ∈ V(N ), let ΣV := ΣV , 0 (b) on arrows: for all inclusions iV 0V : V ,→ V , let Σ(iV 0V ) = rV ,V 0 .

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 17 / 49 The bi-Heyting algebra of clopen subobjects

The bi-Heyting algebra of clopen subobjects

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 18 / 49 The bi-Heyting algebra of clopen subobjects Bi-Heyting algebras – history

Rauszer: bi-Heyting algebras in superintuitionistic logic (’73–’77) Lawvere: co-Heyting and bi-Heyting algebras in category and topos theory, in particular in connection with continuum physics (’86, ’91) Reyes/Makkai (’95) and Reyes/Zolfaghari (’96): bi-Heyting algebras and modal logic Bezhanishvili et al. (’10): new duality theorems for bi-Heyting algebras based on bitopological spaces Majid (’95, ’08): Heyting and co-Heyting algebras within a tentative representation-theoretic approach to the formulation of quantum gravity As far as I am are aware, nobody has connected quantum systems and their logic with bi-Heyting algebras before.

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 19 / 49 The bi-Heyting algebra of clopen subobjects Definition

A bi-Heyting algebra K is a lattice which is a Heyting algebra and a co-Heyting algebra. For each A ∈ K, the functor A ∧ : K → K has a right adjoint A ⇒ : K → K, and the functor A ∨ : K → K has a left adjoint A ⇐ : K → K. We write ¬ for the Heyting negation and ∼ for the co-Heyting negation. A bi-Heyting algebra K is called complete if it is complete as a Heyting algebra and complete as a co-Heyting algebra. Canonical example: Boolean algebra B (where ¬ = ∼).

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 20 / 49 The bi-Heyting algebra of clopen subobjects Projections and clopen subsets

Let V be a commutative von Neumann algebra, and let Cl(ΣV ) denote the clopen subsets of the Gelfand spectrum of V . There is an isomorphism of complete Boolean algebras

αV : P(V ) −→ Cl(ΣV )

Pˆ 7−→ {λ ∈ ΣV | λ(Pˆ) = 1}.

Hence, within each context, i.e., each commutative subalgebra V ⊂ N , we can freely switch between clopen subsets of ΣV and projections in V .

ˆ ˆ −1 We will write SPˆ := αV (P) and PS := αV (S). Note that for a commutative von Neumann algebra V , the Gelfand spectrum ΣV is homeomorphic to the Stone space of the complete Boolean algebra P(V ).

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 21 / 49 The bi-Heyting algebra of clopen subobjects Clopen subobjects of the spectral presheaf

A subobject of the spectral presheaf Σ is simply a subfunctor S. It is determined by specifying a subset SV of ΣV for each context V such that

0 0 ∀V , V ∈ V(N ): V ⊂ V implies Σ(iV 0V )(SV ) ⊆ SV 0 .

A subobject S of Σ is called clopen if SV ⊆ ΣV is a clopen subset for all V ∈ V(N ). Equivalently, we can consider the family (Pˆ ) of corresponding SV V ∈V(N ) projections. The subobject condition becomes

∀V , V 0 ∈ V(N ): V 0 ⊂ V implies Pˆ ≤ Pˆ . SV SV 0

The set of clopen subobjects of Σ is denoted by Subcl(Σ).

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 22 / 49 The bi-Heyting algebra of clopen subobjects Contextuality and coarse-graining

The concept of contextuality is implemented by this construction: Σ is a presheaf over the context category V(N ). Within each context, we have classical Boolean logic; local propositions about the value of some physical quantity in V are represented by elements of the complete Boolean algebra P(V ). Moreover, the concept of coarse-graining is implemented by the fact that we use subobjects of Σ: if V 0 ⊂ V , then Pˆ ≥ Pˆ (since S is a SV 0 SV subobject), so SV 0 represents a local proposition at the smaller context V 0 ⊂ V that is coarser than (i.e., it is weaker than, a consequence of) the local proposition represented by SV .

Clopen subobjects S ∈ Subcl(Σ) hence are interpreted as contextualised families of local propositions, compatible w.r.t. coarse-graining.

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 23 / 49 The bi-Heyting algebra of clopen subobjects

Subcl(Σ) as a lattice

It is obvious that Subcl(Σ) is a partially ordered set if we set

∀S, T ∈ Subcl(Σ): S ≤ T iff (∀V ∈ V(N ): SV ⊆ T V ). Meets and joins with respect to this order are defined as follows: for all families (Si )i∈I ⊆ Subcl(Σ) and all V ∈ V(N ), ^ \ ( Si )V := int Si;V , i∈I i∈I _ [ ( Si )V := cl Si;V . i∈I i∈I Since the lattice operations are defined locally, i.e., at each stage V ∈ V(N ) separately, we obtain a distributive lattice by the fact that for all V ∈ V(N ), Cl(ΣV ) 'P(V ) is distributive. Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 24 / 49 The bi-Heyting algebra of clopen subobjects

Subcl(Σ) as a complete Heyting algebra

In fact, we can say more: each Cl(ΣV ) is a complete Boolean algebra, so for each S ∈ Subcl(Σ) the functor

∧ S : Subcl(Σ) −→ Subcl(Σ) R 7−→ R ∧ S

preserves all joins (note that meets and joins are defined stagewise) and hence has a right adjoint

S ⇒ : Subcl(Σ) −→ Subcl(Σ).

This map, the Heyting implication from S, makes Subcl(Σ) into a complete Heyting algebra. The Heyting implication is given by the adjunction R ∧ S ≤ T if and only if R ≤ (S ⇒ T ). Note that this is the implicative rule, and ⇒ is a material implication. Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 25 / 49 The bi-Heyting algebra of clopen subobjects

Subcl(Σ) as a complete co-Heyting algebra

There is more structure: again since each Cl(ΣV ) is a complete Boolean algebra, for each S ∈ Subcl(Σ) the functor

S ∨ : Subcl(Σ) −→ Subcl(Σ)

preserves all meets, so it has a left adjoint

S ⇐ : Subcl(Σ) −→ Subcl(Σ)

which we call co-Heyting implication from S. This map makes Subcl(Σ) into a complete co-Heyting algebra. It is characterised by the adjunction

(S ⇐ T ) ≤ R iff S ≤ T ∨ R,

which means that ^ (S ⇐ T ) = {R ∈ Subcl(Σ) | S ≤ T ∨ R}.

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 26 / 49 The bi-Heyting algebra of clopen subobjects

Subcl(Σ) as a complete bi-Heyting algebra The Heyting implication ⇒ induces the Heyting negation op ¬ : Subcl(Σ) −→ Subcl(Σ) _ S 7−→ ¬S := (S ⇒ ∅) = {T ∈ Subcl(Σ) | S ∧ T = ∅}. Analogously, the co-Heyting implication ⇐ induces the co-Heyting negation op ∼: Subcl(Σ) −→ Subcl(Σ) ^ S 7−→∼ S := (Σ ⇐ S) = {T ∈ Subcl(Σ) | S ∨ T = Σ}. Summing up, we have shown Proposition

(Subcl(Σ), ∧, ∨, ∅, Σ, ⇒, ¬, ⇐, ∼) is a complete bi-Heyting algebra.

It is easy to see that Subcl(Σ) is not a Boolean algebra. Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 27 / 49 The bi-Heyting algebra of clopen subobjects Two small results on negation Lemma

For all S ∈ Subcl(Σ), we have ¬S ≤ ∼ S.

Proof.

For all V ∈ V(N ), it holds that (¬S)V ⊆ ΣV \SV , since (¬S ∧ S)V = (¬S)V ∩ SV = ∅, while (∼ S)V ⊇ ΣV \SV since (∼ S ∨ S)V = (∼ S)V ∪ SV = ΣV . The above lemma and the fact that ¬S is the largest subobject such that ¬S ∧ S = ∅ imply Corollary In general, ∼ S ∧ S ≥ ∅.

This means that the co-Heyting negation does not give a system in which a central axiom of most logical systems, viz. freedom from contradiction, holds. We have a paraconsistent logic for quantum systems. Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 28 / 49 Daseinisation

Daseinisation

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 29 / 49 Daseinisation Daseinisation

Standard quantum theory: proposition “A ε ∆” represented by projection Pˆ = Eˆ[A ε ∆]. There is a way of ‘translating’ from standard quantum logic to the new logic based on clopen subobjects of the quantum state space Σ: first consider a single commutative subalgebra V ⊂ N . There is an inclusion

P(V ) ,→ P(N )

that is a morphism of complete orthomodular lattices, so it preserves all meets in particular. Hence, it has a left adjoint

o δN ,V : P(N ) −→ P(V ) ˆ o ˆ ^ ˆ ˆ ˆ P 7−→ δN ,V (P) = {Q ∈ P(V ) | Q ≥ P}.

Note that commutativity of V plays no role here.

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 30 / 49 Daseinisation Daseinisation (2)

Then consider this map for all contexts V ∈ V(N ) at once: Definition Let N be a von Neumann algebra, and let P(N ) be its lattice of projections. The map

o δ : P(N ) −→ Subcl(Σ) ˆ o ˆ o ˆ P 7−→ δ (P) := (αV (δN ,V (P)))V ∈V(N )

is called outer daseinisation of projections.

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 31 / 49 Daseinisation Daseinisation (3)

Some of the main properties of daseinisation are: o δ : P(N ) → Subcl(Σ) is monotone. δo(0)ˆ = ∅ and δo(1)ˆ = Σ. δo is injective, but not surjective. δo preserves all joins (disjunctions). This means that the part of standard quantum logic that comes from superposition is preserved, despite the fact that Σ is not a vector space. For meets (conjunctions), we have

∀Pˆ, Qˆ ∈ P(N ): δo(Pˆ ∧ Qˆ ) ≤ δo(Pˆ) ∧ δo(Qˆ ).

All conjunctions exist, but coarse-graining and contextuality guarantee that they all have a good physical interpretation.

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 32 / 49 Negations and regular elements

Negations and regular elements

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 33 / 49 Negations and regular elements The Heyting negation explicitly

Recall that ¬S is the largest element of Subcl(Σ) such that S ∧ ¬S = 0.

The stagewise expression for ¬S is

0 (¬S)V = {λ ∈ ΣV | ∀V ⊆ V : λ|V 0 ∈/ SV 0 }. One can show: Proposition

Let S ∈ Subcl(Σ), and let V ∈ V(N ). Then _ Pˆ = 1ˆ − Pˆ , (¬S)V SV 0 0 V ∈mV

0 0 where mV = {V ⊆ V | V minimal}.

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 34 / 49 Negations and regular elements Heyting-regular elements

With this, it is easy to check that double negation is given (stagewise) by ^ Pˆ = Pˆ ≥ Pˆ , (¬¬S)V SV 0 SV 0 V ∈mV so ¬¬S ≥ S as expected. We obtain: Proposition

An element S of Subcl(Σ) is Heyting-regular, i.e., ¬¬S = S, if and only if for all V ∈ V(N ), it holds that ^ Pˆ = Pˆ , SV SV 0 0 V ∈mV

0 0 where mV = {V ⊆ V | V minimal}.

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 35 / 49 Negations and regular elements The co-Heyting negation explicitly

Dually, ∼ S is the smallest element of Subcl(Σ) such that

S∨ ∼ S = Σ.

One can show: Proposition

Let S ∈ Subcl(Σ), and let V ∈ V(N ). Then _ Pˆ = (δo (1ˆ − Pˆ )), (∼S)V V˜ ,V SV˜ ˜ V ∈MV

where MV = {V˜ ⊇ V | V˜ maximal}.

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 36 / 49 Negations and regular elements Co-Heyting regular elements

One checks that double co-Heyting negation is given (stagewise) by _ Pˆ = δo (Pˆ ), (∼∼S)V V˜ ,V SV˜ ˜ V ∈MV which implies ∼∼ S ≤ S. We obtain: Proposition

An element S of Subcl(Σ) is co-Heyting-regular, i.e., ∼∼ S = S, if and only if for all V ∈ V(N ) it holds that _ Pˆ = δo (Pˆ ), SV V˜ ,V SV˜ ˜ V ∈MV

where MV = {V˜ ⊇ V | V˜ maximal}.

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 37 / 49 Negations and regular elements Daseinisation and regularity

Definition

A clopen subobject S ∈ Subcl(Σ) is called tight if

Σ(iV 0V )(SV ) = SV 0

for all V 0, V ∈ V(N ) such that V 0 ⊆ V .

Proposition Tight subobjects are both Heyting-regular and co-Heyting regular.

Theorem A subobject δo(Pˆ) obtained from daseinisation is tight and hence both regular and co-Heyting regular.

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 38 / 49 States and truth values

States and truth values

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 39 / 49 States and truth values Pure states

In classical physics, a (pure) state is given by an element s of the state space S.A proposition “A ε ∆” is represented by a Borel subset −1 fA (∆) ⊂ S. The truth value of the proposition in the given state is

v(B; s) = (s ∈ B),

which is a Boolean formula that is either false or true. For quantum theory, we need an analogue of the pure state s ∈ S. But: Theorem (Isham, Butterfield ’00, D ’05): The spectral presheaf Σ of a von Neumann algebra N has no global elements if N has no summand of type I2. This is equivalent to the Kochen-Specker theorem.

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 40 / 49 States and truth values Pseudo-states

In standard quantum theory (for N = B(H)), one uses vector states: let ψ ∈ H be a unit vector, then

ψ w : B(H) −→ C Aˆ 7−→ hψ, Aˆ(ψ)i.

We simply ‘daseinise’ such a vector state: let Pˆψ the projection onto the ψ line Cψ, then the pseudo-state w is given as

ψ o w := δ (Pˆψ).

This is a ‘small subobject’, (one of) the smallest non-empty subobjects one can obtain from daseinisation.

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 41 / 49 States and truth values Truth values

ψ Let w be a pseudo-state, and let S ∈ Subcl(Σ) be a proposition (e.g., S = δo(Eˆ[A ε ∆])). We can interpret the expression

v(S; wψ) := (wψ ∈ S)

op in the Mitchell-Benabou language of the topos SetV(N ) , which gives a truth value in the multi-valued, intuitionistic logic of the topos. Concretely, such a truth value is a global element of the subobject classifier of the topos, which is the presheaf of sieves Ω. Since the base category V(N ) of our topos is a poset, this becomes particularly simple: the global elements of Ω correspond bijectively to the lower sets in V(N ), Γ(Ω) = L(V(N )).

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 42 / 49 Generalisation to OMLs Spectral presheaves of OMLs

What about generalisations to orthomodular lattices (OMLs)? Given an OML L, one can consider the poset B(L) of its Boolean subalgebras. Each Boolean subalgebra B has a Stone space Ω(B), so one can define a spectral presheaf Ω(L) of L. Moreover, there is a suitable category in which these spectral presheaves are objects. Morphisms φ : L1 → L2 of OMLs induce morphisms Φ:Ω(L2) → Ω(L1).

Theorem

(Sarah Cannon, AD ’13; unpublished) Let L1, L2 be two OMLs. There is an isomorphism Φ:Ω(L2) → Ω(L1) if and only if there is an isomorphism φ : L1 → L2.

Hence, the spectral presheaf of an OML L is a complete invariant. The clopen subobjects form a bi-Heyting algebra Subcl(Ω(L)). Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 43 / 49 Summary

Summary

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 44 / 49 Summary Some features of topos-based logic for quantum systems

Propositions form a distributive, complete bi-Heyting algebra Subcl(Σ). Each pseudo-state wψ assigns a truth-value to every proposition “A ε ∆”, represented by δo(Eˆ[A ε ∆]). Multi-valued, contextual logic. o ‘Translation’ map δ : P(N ) → Subcl(Σ) preserves all joins – ‘superposition without linearity’.

All meets in Subcl(Σ) can be interpreted due to coarse-graining and contextuality. There is a material implication, the Heyting implication, and the implicative rule holds. Additional paraconsistent logic. All this relates to the fact that this new form of logic for quantum systems has a geometric underpinning in the form of generalised sets (i.e., objects in a topos).

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 45 / 49 Summary Two ways of generalising classical logic

Birkhoff and von Neumann explicitly emphasised in their 1936 article that they find it most plausible to give up distributivity, but to keep negation intact when generalising from classical Boolean logic. This can be seen as a claim that intuitionistic logic, which was much debated at the time, is not appropriate for quantum systems. In our new form of topos-based logic for quantum systems, we depart from classical logic in a different way, by keeping distributivity, but ‘splitting’ negation into two concepts. This leads to a much better-behaved logic for quantum systems. Bi-Heyting algebras are a comparatively mild generalisation of Boolean algebras. Moreover, many other features of the spectral presheaf Σ have good physical meaning in the topos approach to quantum theory.

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 46 / 49 Summary Some open questions

Dynamics: Time evolution? (See [arXiv:1212.4882].) State change by measurements? (More) physical interpretation of intuitionistic and paraconsistent parts of this logic? Universal property of outer daseinisation δo? Higher-order aspects, making more use of the internal logic of the topos? Behaviour of logic under morphisms between topoi? → current work with Rui Soares Barbosa, Jonathon Funk, Pedro Resende Treatment of composite systems? ...

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 47 / 49 Summary References

Selection of references on the new topos-based form of quantum logic:

AD, C.J. Isham, “Topos Theory in the Foundations of Physics 1–4”, JMP 49, Issue 5, 053515–18 (2008). [arXiv:quant-ph/0703060,62,64 and 66] AD, “Topos theory and ‘neo-realist’ quantum theory”, in Quantum Field Theory, Competitive Models, eds. B. Fauser et al., Birkh¨auser (2009). [arXiv:0712.4003] AD, “Topos-Based Logic for Quantum Systems and Bi-Heyting Algebras”, to appear in Logic & Algebra in Quantum Computing, Lecture Notes in Logic, published by the ASL/CUP. [arxiv:1202.2750]

(There’s more stuff in the ArXiv.)

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 48 / 49 Summary

Thanks for listening!

Andreas D¨oring (Erlangen) Bi-Heyting algebras and quantum systems 49 / 49