topos-theoretic classiﬂcation of quantum events structures...

Topos-Theoretic Classification of Quantum Events

Structures In Terms of Boolean Reference Frames

ELIAS ZAFIRISUniversity of Athens

Institute of MathematicsPanepistimiopolis, 15784 Athens

Greece

Abstract

We construct a sheaf-theoretic representation of quantum eventsstructures, in terms of Boolean localization systems. These coveringsystems are constructed as ideals of structure-preserving morphismsof quantum events algebras from varying Boolean domains, identifiedwith physical contexts of measurement . The modeling sheaf-theoreticscheme is based on the existence of a categorical adjunction betweenpresheaves of Boolean events algebras and quantum events algebras.On the basis of this adjoint correspondence, we also prove the exis-tence of an object of truth values in the category of quantum logics,characterized as subobject classifier. This classifying object plays theequivalent role that the two-valued Boolean truth values object plays inclassical events structures. We construct the object of quantum truthvalues explicitly, and furthemore, demonstrate its functioning for thevaluation of propositions in a typical quantum measurement situation.

MSC : 18F05; 18F20; 18D30; 14F05; 53B50; 81P10.Keywords : Quantum Events Structures; Quantum Logic; Sheaves; Ad-

junction; Boolean Coverings; Topos Theory; Subobject Classifier; QuantumMeasurement.

0E-mail : [email protected]

1

1 Prologue

The notion of the logic of a physical theory has been introduced in 1936 byvon Neumann and G. Birkhoff in a paper entitled “The Logic of QuantumMechanics”. For classical theories the appropriate logic is a Boolean algebra;but for quantum theories a non-Boolean logical structure is necessary, whichcan be characterized as an orthocomplemented lattice, or a partial Booleanalgebra, or some other structure of a related form. The logic of a physicaltheory reflects the structure of the propositions describing the behavior ofa physical system in the domain of the corresponding theory.

Naturally, the typical mathematical structure associated with logic is anordered structure. The original quantum logical formulation of quantumtheory [1, 2] depends in an essential way on the identification of proposi-tions with projection operators on a complex Hilbert space. A non-classical,non-Boolean logical structure is effectively induced which has its originsin quantum theory. More accurately the Hilbert space quantum logic isaxiomatized as a complete, atomic, orthomodular lattice. Equivalently, itcan be isomorphic to the partial Boolean algebra of closed subspaces of theHilbert space associated with the quantum system, or alternatively, the par-tial Boolean algebra of projection operators of the system. On the contrary,the propositional logic of classical mechanics is Boolean logic, meaning thatthe class of models, over which validity and associated semantic notions aredefined for the propositions of classical mechanics, is the class of Booleanlogic structures.

In a previous work we have proposed a sheaf-theoretic scheme, that ac-commodates the formalization of quantum event and observable algebrasas structured interlocking families of Boolean event algebras [3]. In thepresent work our purpose is the study of the truth values structures suitedto represent accurately the quantum domain of discourse, according to thesheaf-theoretic representation established. We will argue that generalizedclassical logic structures, interconnected non-trivially, provide the buildingblocks of an appropriate conceptual environment by means of which it ispossible to comprehend the complexity of the structures of quantum propo-sitions. We hold the view that the logic of quantum propositions reflectsliteral ontological structures of the quantum domain of discourse, and theperspective offered by the proposed scheme, with respect to a logical truthvalues interpretation, reveals the relevant ontological aspects as well.

2

Traditionally, the vast majority of the attempts to explore the logicalstructures associated with quantum mechanical systems are based on a settheoretical language. We propose a transition in the syntax of the theoryinvolved, which as will see effects a transition in the semantics of quantumlogics. This transition hopefully clarifies the relationship between the on-tological structures associated with the classical and quantum domains ofdiscourse, as it is reflected on their logical intertransformability. The math-ematical language which is best suited to fulfill our objectives is provided bycategory and topos theory [4-10]. This is due to the fact that, these theoriesprovide the means to relate the form and meaning of non-Boolean quantumlogical structure with suitable interlocking local Boolean contexts, and mostimportantly, this can be done in a universal way.

The development of the conceptual and technical machinery of localiza-tion systems for generating non-trivial global events structures, as it hasbeen recently demonstrated in [11], effectuates a transition in the semanticsof events from a set-theoretic to a sheaf-theoretic one. This is a crucialsemantic difference that characterizes the present approach in comparisonto the vast literature on quantum measurement and quantum logic. Moreprecisely, quantum events algebras can be represented as sheaves for an ap-propriate covering system defined on the Boolean localizing category. Thisprocess is formalized categorically by the concept of localization systems,where, the specified maps from Boolean contexts induced by measurementsituations of observables, play the role of covers of a quantum structure ofevents. In more detail, the notion of local is characterized by a categori-cal Grothendieck topology, the axioms of which express closure conditionson the collection of covers. In this sense, the information available abouteach map of the specified covering system may be used to determine com-pletely a quantum events structure. In this paper, we will avoid to mentionGrothendieck topologies explicitly, although the definition of a Boolean lo-calization system, gives rise to a precise Grothendieck topology [11] .

The category of sheaves is a topos, and consequently, comes naturallyequipped with an object of generalized truth values, called subobject clas-sifier. This object of truth values, being remarkably a sheaf itself, namelyan object of the topos, is the appropriate conceptual tool for the organiza-tion of the logical dimension of the information included in the category ofquantum events algebras, as it is encoded in Boolean localization systems.

3

More concretely, we will show that the logic of propositions describing aquantum system can be comprehended via equivalence relations in the sheafof coefficients defined over the category of Boolean logical structures for anappropriate covering system of the latter, defined as a Boolean localizationsystem. We emphasize that the significance of the sheaf-theoretical concep-tion of a quantum logical structure, lies on the fact that it is supported bythe well defined underlying notion of multi-valued truth structure of a topos.

The fact that a quantum events algebra is actually a non-trivial globalobject is fully justified by Kochen-Specker theorem [12]. According to this,there are no two-valued homomorphisms on the algebra of quantum propo-sitions. Consequently, a quantum logical algebra cannot be embedded intoa Boolean one. We note that a two-valued homomorphism on a classicalevent algebra is a classical truth value assignment on the propositions ofthe physical theory, represented by the elements of the Boolean algebra,or a yes-no assignment on the corresponding properties represented by theelements of the algebra. In this work, we will show eventually that the cate-gorical environment specifying a quantum event algebra in terms of Booleanlocalization systems, contains an object of truth values, or classifying object,that constitutes the appropriate tool for the definition of a quantum truthvalues assignment, corresponding to valuations of propositions describingthe behavior of quantum systems.

Contextual topos theoretical approaches to quantum structures of truthvalues have been also considered, from a different viewpoint in [13, 14], anddiscussed in [15, 16]. Of particular relevance to the present work, regardingthe specification of a quantum truth values object, although not based oncategory theory methods, seems to be the approach to the foundations ofquantum logic by Takeuti and Davis [17, 18], according to whom, quanti-zation of a proposition of classical physics is equivalent to interpreting itin a Boolean extension of a set theoretical universe, where B is a completeBoolean algebra of projection operators on a Hilbert space.

In Section 2, we recapitulate the categorical framework that leads to thesheaf-theoretic representation of quantum events algebras, by formulatingthe existence of a categorical adjunction between the categories of Booleanpresheaves and quantum events algebras. Moreover we explain the notions ofBoolean systems of localizations for quantum events algebras and formulatea representation theorem in terms of the counit of the Boolean-quantum

4

adjunction. In Section 3, we introduce the notion of a subobject functor andspecify the necessary and sufficient conditions for being representable by anobject in the category of quantum logics, to be identified as a quantum truthvalues object. In Section 4, we construct the representation of the quantumtruth values object in tensor product form, and moreover, we prove thatit plays the role of subobject classifier in the category of quantum eventsalgebras. Furthermore, we formulate explicitly, the relevant criterion oftruth for a complete description of reality. In Section 5, we propose the useof quantum truth values as the proper range for valuations of propositionsassociated with the behavior of quantum systems and demonstrate theirfunctioning. Finally we conclude in Section 6.

2 The Categorical Framework of Representation

2.1 Sheaf-Theoretic Modeling of Quantum Events Algebras

Definition: A Quantum events structure is a small cocomplete cate-gory, denoted by L, which is called the category of quantum events algebras.The objects of L are quantum events algebras and the arrows are quantumalgebraic homomorphisms.

Definition: A quantum events algebra L in L, is defined as an or-thomodular σ-orthoposet, that is, as a partially ordered set of quantumevents, endowed with a maximal element 1, and with an operation of ortho-complementation [−]∗ : L qqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqq L, which satisfy, for all l ∈ L, the following

conditions: [a] l ≤ 1, [b] l∗∗ = l, [c] l ∨ l∗ = 1, [d] l ≤ l ⇒ l∗ ≤ l∗, [e]l⊥l ⇒ l ∨ l ∈ L, [f] for l, l ∈ L, l ≤ l implies that l and l are compatible,where 0 := 1∗, l⊥l := l ≤ l∗, and the operations of meet ∧ and join ∨ aredefined as usually.

Remark: We recall that l, l ∈ L are compatible if the sublattice generatedby l, l∗, l, l∗ is a Boolean algebra, namely if it is a Boolean sublattice.The σ-completeness condition, namely that the join of countable families ofpairwise orthogonal events must exist, is also required in order to have awell defined theory of observables over L.

5

Definition: A quantum algebraic homomorphism in L is a morphismK H qqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqq L, which satisfies, for all k ∈ K, the following conditions: [a] H(1) =

1, [b] H(k∗) = [H(k)]∗, [c] k ≤ k ⇒ H(k) ≤ H(k), [d] k⊥k ⇒ H(k ∨ k) ≤H(k) ∨ H(k), [e] H(

∨nkn) =

∨nH(kn) , where k1, k2, . . . countable family

of mutually orthogonal events.

Definition: A Classical events structure is a small category, denotedby B, which is called the category of Boolean events algebras. The objectsof B are σ-Boolean algebras of events and the arrows are the correspondingBoolean algebraic homomorphisms.

Definition: A functor of local Boolean coefficients, A : B qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq L,assigns to Boolean events algebras in B, that instantiates a model coor-dinatizing category, the underlying quantum events algebras from L, andto Boolean homomorphisms the underlying quantum algebraic homomor-phisms.

Remark: The functor M acts as a forgetful functor, forgetting the extraBoolean structure of B.

Definition: The functor category of presheaves on Boolean eventsalgebras, denoted by SetsB

op, has objects all functors P : Bop qqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqq Sets,

and morphisms all natural transformations between such functors, whereBop is the opposite category of B.

Definition: Each object P in SetsBop

is a contravariant set-valued func-tor on B, called a presheaf on B.

Remark: For each Boolean algebra B of B, P(B) is a set, and for eacharrow f : C qqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqq B, P(f) : P(B) qqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqq P(C) is a set function. If P is a

presheaf on B and p ∈ P(B), the value P(f)(p) for an arrow f : C qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq B inB is called the restriction of x along f and is denoted by P(f)(p) = p · f .

Remark: Each object B of B gives rise to a contravariant Hom-functory[B] := HomB(−, B). This functor defines a presheaf on B. Its action onan object C of B is given by

y[B](C) := HomB(C, B)

6

whereas its action on a morphism D x qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq C, for v : C qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq B is given by

y[B](x) : HomB(C, B) qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq HomB(D, B)

y[B](x)(v) = v x

Furthermore, y can be made into a functor from B to the contravariantfunctors on B

y : B qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq SetsBop

such that B 7→HomB(−, B). This is called the Yoneda embedding and it isa full and faithful functor.

Remark: The functor category of presheaves on Boolean events algebrasSetsB

op, provides an instantiation of a structure known as topos. A topos

exemplifies a well defined notion of a universe of variable sets. It can beconceived as a local mathematical framework corresponding to a generalizedmodel of set theory or as a generalized space.

Definition : The category of elements of the presheaf P, denotedby

∫(P,B) has objects all pairs (B, p), and morphisms (B, p)→(B, p) are

those morphisms u : B→B of B for which p · u = p.

Definition: Projection on the second coordinate of∫(P,B), defines a func-

tor∫

P :∫(P,B)→B.

∫(P,B) together with the projection functor

∫P is

defined as the split discrete fibration induced by P, where B is the basecategory of the fibration as in the diagram below.

Remark: We note that the fibers are categories in which the only arrowsare identity arrows. If B is an object of B, the inverse image under

∫P of B

is simply the set P(B), although its elements are written as pairs so as toform a disjoint union. The construction of the fibration induced by P, is aninstance of the general Grothendieck construction [8].

∫(P,B)

∫P

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

B P qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq Sets

7

Remark: The construction of the split discrete fibration induced by P,where B is the base category of the fibration, incorporates the physicallyimportant requirement of uniformity [11]. The notion of uniformity, requiresthat for any two events observed over the same domain of measurement, thestructure of all Boolean contexts that relate to the first cannot be distin-guished in any possible way from the structure of Boolean contexts relatingto the second. In this sense, all the observed events within any particularBoolean context should be uniformly equivalent to each other. It is easy tonotice that the composition law in the category of elements of the presheafP, expresses precisely the above uniformity condition.

Definition : The functor of generalized elements of a quantumevents algebra L in the environment of the category of presheaves onBoolean events algebras, SetsB

op, or functor of Boolean frames of a

quantum event algebra L, is defined by:

R : L → SetsBop

from L to the category of presheaves of Boolean events algebras SetsBop

,where, the action on an object B in B is given by

R(L)(B) := RL(B) = HomL(M(B), L)

whereas, the action on a morphism D x qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq B in B, for v : B qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq L is givenby

R(L)(x) : HomL(M(B), L) qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq HomB(M(D), L)

R(L)(x)(v) = v x

Theorem: There exists a pair of adjoint functors L a R as follows (for aproof see [3]):

L : SetsBop

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

q

L : R

The Boolean-quantum adjunction consists of the functors L and R, calledleft and right adjoints with respect to each other respectively, as well as thenatural bijection:

Nat(P,R(L)) ∼= HomL(LP, L)

8

Remark: The established bijective correspondence, interpreted functo-rially, says that the functor R from L to presheaves given by

R(L) : B 7→HomL(M(B), L)

has a left adjoint L : SetsBop → L, which is defined for each presheaf of

Boolean algebras P in SetsBop

as the colimit

L(P) = Colim∫

(P,B)∫

P qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq B M qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq L

Corollary: The Boolean coefficients functor M(B) is characterized as thecolimit of the representable presheaf on the category of Boolean algebras (fora proof see [3]), as follows:

Ly[B](B) ∼= M ∫

y[B](B, 1B) = M(B)

Remark: The following diagram (with the Yoneda embedding y) com-mutes.

B

y

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

@@

@@

@@qqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqq

M

SetsBop Lp p p p p p p p p p p p p p p pqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqq L

Corollary: The colimit in the category of elements of the functor ofBoolean frames L(R(L)) is a quantum event algebra (for a proof see [3],[11]).

Definition: A system of Boolean prelocalizations for a quantumevents algebra L in L is a subfunctor of the Hom-functor R(L) of the formS : Bop → Sets, that is for all B in B, it satisfies S(B) ⊆ [R(L)](B).

Remark: A system of Boolean prelocalizations for a quantum eventsalgebra L in L is equivalent to a right ideal S . R(L), defined by the re-quirement that, for each B in B, S(B) is a set of quantum algebraic homo-morphisms of the form ψB : M(B) qqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqq L, satisfying the following property:

9

〈 If ψB : M(B) qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq L ∈ S(B), and M(v) : M(B) qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq M(B) in L, forv : B qqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqq B in B, then ψB M(v) : M(B) qqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqq L ∈ S(B) 〉.

Definition: A family of Boolean covers ψB : M(B) qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq L, B in B, is thegenerator of a system of Boolean prelocalizations S, if and only if,this system is the smallest among all that contains that family.

Remark: The systems of Boolean prelocalizations constitute a partiallyordered set under inclusion. The minimal system is the empty one, namelyS(B) = ∅ for all B in B, whereas the maximal system is the Hom-functorR(L) itself.

Definition: The pullback of the Boolean covers: ψB : M(B) qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq L,B in B, and ψB : M(B) qqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqq L, B in B, with common codomain the quantum

events algebra L, consists of the object M(B)×LM(B) and two arrows ψBB

and ψBB, called projections, as shown in the following diagram. The squarecommutes and for any object T and arrows h and g that make the outersquare commute, there is a unique u : T qqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqq M(B)×LM(B) that makes the

whole diagram commute.

T

@@

@@

@@qqqqqqqqqqq


u

HHHHHHHHHHHHHHqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

h

AAAAAAAAAAAAAAqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqq

g M(B)×LM(B)ψB,B

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq M(B)

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

ψB,B

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

ψB

M(B)ψB qqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqq L

Remark: If ψB and ψB are injective, then their pullback is isomorphicwith the intersection M(B) ∩M(B).

10

Definition: The pasting isomorphism of Boolean covers, is defined asfollows:

ΩB,B : ψBB(M(B)⋂

LM(B)) qqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqq ψBB(M(B)

⋂LM(B))

ΩB,B = ψBB ψBB−1

Theorem: The Boolean coordinatizing maps ψBB(M(B)×LM(B)) andψBB(M(B)×LM(B)) cover the same part of a quantum events algebra in acompatible way.

Proof: An immediate consequence of the previous definition is the satis-faction of the following Boolean coordinates cocycle conditions:

ΩB,B = 1B 1B : identity of B

ΩB,B ΩB,

´B

= ΩB,

´B

if M(B) ∩M(B) ∩M( ´B) 6= 0

ΩB,B = Ω−1B,B if M(B) ∩M(B) 6= 0

Thus, the pasting morhism assures that ψBB(M(B)×LM(B)) and ψBB(M(B)×LM(B))cover the same part of a quantum events algebra in a compatible way.

Definition: Given a system of prelocalizations for a quantum events alge-bra L ∈ L, we call it a system of Boolean localizations, or equivalently,a structure sheaf of Boolean coefficients, iff the Boolean coordinatescocycle conditions are satisfied, and moreover, the quantum algebraic eventsstructure is preserved.

Definition: For any presheaf P ∈ SetsBop, the unit is defined as

δP : P qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq RLP

On the other side, for each quantum event algebra L in L the counit isdefined as

εL : LR(L) qqqqqqqqqqqqqqqqqq


Boolean Representation Theorem: The representation of a quantumevents algebra L in L, in terms of a coordinatization system of localiza-tions, consisting of Boolean coefficients, is full and faithful, if and only if thecounit of the Boolean-quantum adjunction, restricted to that system, is anisomorphism, that is, structure-preserving, injective and surjective.

11

Remark: The counit of the adjunction, restricted to a system of local-izations is a quantum algebraic isomorphism, if and only if the right ad-joint functor is full and faithful, or equivalently, if and only if the coconefrom the functor M ∫

R(L) to the quantum event algebra L is universal foreach L in L. In the latter case we characterize the coordinatization functorM : B qqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqq L, a proper modeling functor.

2.2 Physical Semantics

The physical significance of this representation lies on the fact that the wholeinformation content in a quantum events algebra is preserved by every cov-ering Boolean system, qualified as a system of measurement localizations.The preservation property is established by the counit isomorphism. It isremarkable that the categorical notion of adjunction provides the appropri-ate formal tool for the formulation of invariant properties, giving rise topreservation principles of a physical character.

If we return to the intended representation, we realize that the surjectiveproperty of the counit guarantees that the Boolean domain covers, beingthemselves objects in the category of elements

∫(R(L), B), cover entirely

the quantum event algebra L, whereas its injective property guarantees thatany two covers are compatible in a system of measurement localizations.Moreover, since the counit is also a homomorphism, it preserves the algebraicstructure.

In the physical state of affairs, each cover corresponds to a set of Booleanevents actualized locally in a measurement situation. The equivalence classesof Boolean domain covers represent quantum events in L through compatiblecoordinatizations by Boolean coefficients. Consequently, the structure ofa quantum event algebra is being generated by the information that itsstructure preserving maps, encoded as Boolean covers in localization systemscarry, as well as their compatibility relations. Most significantly, the samecompatibility conditions provide the necessary relations for understandinga system of localizations for a quantum event algebra as a structure sheafof Boolean coefficients associated with local contexts of measurement ofobservables.

Finally, the operational substantiation of the sheaf theoretical schemeof representation of quantum event algebras, is naturally provided by theapplication of Stone’s representation theorem for Boolean algebras. Accord-

12

ing to this theorem, it is legitimate to replace Boolean algebras by fields ofsubsets of a space, playing the equivalent role of a local context for measure-ment. We note that in an equivalent topological interpretation, we couldconsider a local measurement space as a compact Hausdorff space, the com-pact open subsets of which are the maximal filters or the prime ideals of theunderlying Boolean algebra. If we replace each Boolean algebra B in B byits set-theoretical representation [Σ, BΣ], consisting of a local measurementspace Σ and its local field of subsets BΣ, it is possible to define local mea-surement space covers (BΣ, ψBΣ

: A(BΣ) qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq L) and corresponding spacelocalization systems for a quantum event algebra L in L. Again from lo-cal measurement space covers (BΣ, ψBΣ

: M(BΣ) qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq L) we may form theirequivalence classes by using the colimits construction in the category of ele-ments of R(L). Then by taking into account the conditions for compatibilityon overlaps we can establish a full and faithful representation of quantumevents in L by equivalence classes of local measurement space covers. Un-der these circumstances we may interpret these equivalence classes as thestatistical experimental actualizations of the quantum events in L. Thepullback compatibility condition, which is in bijective correspondence withthe one in L since it holds in a localization system, may be interpreted inthe operational context as denoting that two local space representations ofquantum events satisfy the compatibility condition on overlaps if and onlyif they support measurements of observables sharing the same experimentalarrangement.

The above set-up could be the ideal starting point for the developmentof quantum probability, as a contextual probability theory on a structuresheaf of Boolean coefficients associated with local contexts of measurementof observables. Following this line of thought we may obtain important in-sights regarding probabilistic notions in quantum theory. In the prologuewe have expressed the thesis that the logic of quantum propositions reflectsliteral ontological structures of the quantum domain of discourse. Of coursethe substantiation of this claim necessitates a thorough investigation of thetruth values structures suited to express the quantum domain of discourse.In classical theories it is well known that the logic of events, or equivalently,propositions referring to the behavior of a classical system is character-ized by valuations into the trivial Boolean two-valued truth values object2 := 0, 1 stating that a proposition is true or false. Moreover the notion

13

of probability has been designed as a superstructure on the truth valuesobject 2, expressing an ignorance of all the relevant details permitting asharp true/false value assignment on the propositions of the theory. In thissense classical probabilities are not objective, but constitute a measure ofignorance. On the other side, in quantum theories a true/false value assign-ment is possible under the specification of a Boolean preparatory context ofmeasurement and only after a measurement device provides a response as aresult of its interaction with a quantum system. This state of affairs is atthe heart of the problem of quantum measurement and makes necessary acareful re-examination of all the relevant assumptions concerning valuationsof propositions that belong in quantum event structures. In this manner,if the truth values structures suited for valuations of quantum propositionsprove to be different from the trivial two-valued classical ones, the notion ofquantum probability acquires an objective meaning and its interpretationcannot be based on ignorance. Rather, it can be conceived as a measureof indistinguishability in the generalized topological sense of covering sys-tems on categories, being in agreement at the same time, with the physicalsemantics of a sheaf theoretical interpretation. In the sequel, our objec-tive will be exactly the investigation of the truth values structures suited toexpress valuations in quantum event algebras. Fortunately the categoricalframework provides all the necessary means for this purpose.

3 The Subobject Functor

3.1 Existence of the Subobject Functor in LWe have seen previously that the counit of the fundamental adjunction, re-stricted to localization systems of a quantum event algebra is a quantumalgebraic isomorphism, iff the right adjoint functor is full and faithful. Thisfact is important, because it permits us to consider the category of quantumevents algebras as a reflection of the category of presheaves of Boolean eventalgebras SetsBop

. It is methodologically appropriate to remind that the co-ordinatization functor, M : B qqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqq L, is called a proper modeling functor iff

the right adjoint functor of the established adjunction is full and faithful.In this sense, a proper modeling functor guarantees a full and faithful cor-responding representation of quantum event algebras in terms of Boolean

14

localization systems, such that the whole information content contained ina quantum structure of events is totally preserved by its covering systemsof Boolean domain coordinatizations. Furthermore, the fact that L can beconceived as reflection of SetsBop

, secures that L is a complete category,as well as that, monic arrows are preserved by the right adjoint functor R.Since L is a complete category, there is a terminal object for insertion ofinformation related with the structure of events it represents, and also, thereexist pullbacks securing the satisfaction of compatibility relations. In par-ticular, since pullbacks of monic arrows also exist, there exists a subobjectfunctor.

Definition: The subobject functor of the category of quantum eventalgebras is defined as:

Sub : Lop → Sets

This is, remarkably, a contravariant functor by pulling back. Compositionof this functor with a proper modeling functor provides a presheaf functorin SetsBop

as follows:

Sub M : Bop → Lop → Sets

In a compact notation we obtain:

Θ(M(−)) : Bop → Sets

such that:Bop 3 B 7→ [Dom(m)→mM(B)] ∈ Sets

where the range denotes an equivalence class of monic algebraic homomor-phisms to M(B).

Definition: We say that Θ(M(B)), is the set of all subobjects of M(B)in the category of quantum events algebras L.

Furthermore it is easy to verify that, Θ(M(B)) is a partially ordered setunder inclusion of subobjects.

3.2 Representation of the Subobject Functor in LA natural question that arises in this context, is if it could be possible torepresent the subobject functor by means of a quantum events algebra, Ω,

15

that is an object of L, which would play the role of a classifying object inL. The representation of the subobject functor in L, is significant from aphysical perspective, since it would allow to associate the concrete classify-ing object Ω, with the functioning of a truth values object, in a sense similarto the role played by the two-valued Boolean object 2 := 0, 1, in charac-terization of the logic of propositions referring to the behavior of classicalsystems. In that case, subobjects of a quantum events algebra should becharacterized in terms of characteristic functions, that take values, not in2, but precisely, in the truth values object Ω in L. Most importantly, inthat case the category of quantum events algebras L, is endowed with asubobject classifier, defined categorically as follows:

Definition: The subobject classifier of the category of quantum eventsalgebras is a universal monic quantum homomorphism,

T := True : 1 → Ω

such that, to every monic arrow, m : K → L in L, there is a uniquecharacteristic arrow φm, which, with the given monic arrow m, forms apullback diagram

K! qqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqq 1

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

m

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

T

Lφm qqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqq Ω

This is equivalent to saying that every subobject of L in L, is uniquely apullback of the universal monic T.

Subobject Representation Theorem: The subobject functor can berepresented in the category of quantum event algebras, L, iff there exists aclassifying object Ω in L, that is, iff there exists an isomorphism for eachBoolean domain object of the model category, as follows:

Θ(M(−)) ' R(Ω) := HomL(M(−), Ω)

16

Proof: We have seen previously, that the counit of the adjunction, foreach quantum event algebra object L of L, is

εL : LR(L) qqqqqqqqqqqqqqqqqq


εL, being a a quantum algebraic isomorphism, guarantees a full and faithfulrepresentation of a quantum event algebra in terms of a covering or localiza-tion system consisting of Boolean domain coordinatizations via the actionof a proper modeling functor. From the other side, we have seen that forany presheaf P ∈ SetsBop

, the unit is defined as

δP : P qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq RLP

It is easy to see that if we consider as P ∈ SetsBop, the subobject functor

Θ(M(−)) we obtain the following arrow:

δΘ(M(−)) : Θ(M(−)) qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq RLΘ(M(−))

or equivalently:

δΘ(M(−)) : Θ(M(−)) qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq HomL(M(−),LΘ(M(−))

Hence, by inspecting the unit of the adjunction arrow δΘ(M(−)), we concludethat the subobject functor becomes representable in the category of quantumevents algebras if we, equivalently, prove the following:

Subobject Unit Theorem: Given that the counit of the Boolean-quantumadjunction is an isomorphism, if the unit δΘ(M(−)) is also an isomorphism,then the subobject functor becomes representable in L, by the quantumevents algebra classifying object Ω, characterized explicitly as, Ω := LΘ(M(−)),and thus, the category of quantum events algebras L is endowed with a sub-object classifier, that functions as a quantum truth values object in L. Theinverse of the theorem also obviously holds.

Proof: Firstly, it is easy to notice that, in case, the unit δΘ(M(−)) is anisomorphism, then the classifying quantum event algebra Ω, is constructedby application of the left adjoint functor, that is, as the colimit taken in thecategory of elements of the modeled subobject functor, according to:

Ω := LΘ(M(−))

17

We may verify the above immediately, by realizing that if the unit δΘ(M(−))

is an isomorphism, then;

Ω := LΘ(M(−)) ' L[RLΘ(M(−))] ' LRΩ

is precisely an expression of the counit isomorphism for the quantum eventalgebra Ω.

Next, we will show explicitly that, if the unit δΘ(M(−)) is an isomorphism,the category of quantum events algebras L is endowed with a subobjectclassifier, that functions as a quantum truth values object in L. For thispurpose, we consider a monic quantum homomorphism l : K → L, denotinga subobject of L, in L, and subsequently, we define a natural transformationin SetsBop

:Φl : R(L) → Θ(M(−))

specified for each Boolean event algebra B, in B by:

[Φl]B : R(L)(B) → Θ(M(B))

such that for an element e in R(L)(B), we have:

[Φl]B(e) := l ∗ e

where the monic arrow l ∗ e, denotes the pullback of l along e in L, as in thefollowing diagram:

Dom(l ∗ e) qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq K

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

l∗ e

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

l

M(B) e qqqqqqqqqqqqqqqqqq


Furthermore, if we take into account the subobjects of the terminal object 1in L, denoted by the uniquely defined monic quantum algebraic homomor-phisms κ : K → 1, we may define a natural transformation in SetsBop

:

Υ : R(1) → Θ(M(−))


[Υ]B : R(1)(B) → Θ(M(B))

18

such that for the unique element α(B) in R(1)(B), we have:

[Υ]B(α(B)) := idM(B)

At a next stage, we may combine the natural transformations, defined previ-ously, in order to obtain, for each monic quantum algebraic homomorphisml : K → L, the following commutative diagram in SetsBop

, that by con-struction is a pullback as it can be easily seen.

R(K) qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq R(1)

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

R(l)

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

Υ

R(L) Φl qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq Θ(M(−))

Moreover we consider the arrows obtained by composing, the arrows [Φl]and [Υ], with the unit isomorphism δΘ(M(−)) as follows:

δΘ(M(−)) Φl : R(L) → Θ(M(−)) → R(Ω)

δΘ(M(−)) Υ : R(1) → Θ(M(−)) → R(Ω)

Concerning the latter composite arrow, we may define:

R(T) := δΘ(M(−)) Υ : R(1) → R(Ω)

and using the fact that the right adjoint functor is full and faithful, by thecounit isomorphism, we obtain a uniquely defined monic quantum homo-morphism

T := true : 1 → Ω

The previous pullback diagram, together with the composite arrowsδΘ(M(−)) Φl, δΘ(M(−))Υ, facilitate the immediate verification of the claim,as follows: We wish to show that, if the unit of the adjunction δΘ(M(−)) isan isomorphism, then

Sub(L) ' HomL(L,Ω)

19

such that that the category of quantum event algebras L is endowed with asubobject classifier. So we define a map

$L : Sub(L) → HomL(L,Ω)

such that the element e of the range, defined by:

Sub(L) 3 l 7→ [e : L → Ω]

is specified by the requirement:

R(e) = δΘ(M(−)) Φl : R(L) → R(Ω)

Hence, for the subobject l of L, in L, and the element e of HomL(L,Ω),with e = $L(l), we obtain the following pullback diagram in SetsBop

,

R(K) qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq R(1)

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

R(l)

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

R(T )

R(L) R(e)qqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqq R(Ω)

Using again the argument of the counit isomorphism, that specifies the rightadjoint as a full and faithful functor, we obtain a pullback diagram in L:

K! qqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqq 1

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

l

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

T

Le qqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqq Ω

Moreover, it is straightforward to show that, $L : Sub(L) → HomL(L,Ω)is 1-1 and epi. Thus, we have verified that, if the unit δΘ(M(−)) is an isomor-phism, then the category of quantum events algebras, L, is endowed witha subobject classifier, according to the above pullback diagram. Preciselystated, the subobject classifier in L, is specified by the monic quantum alge-braic homomorphism T := True : 1 → Ω, such that, R(T) := δΘ(M(−)) Υ.It is easy to notice that the inverse obviously holds. As a consequence weconclude that the diagram below;

20

Dom(λ) ! qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq LR(1)

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

λ

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

LΥ = T

M(B)$M(B)(λ) = ζ

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq LΘ(M(−))

is a pullback square for each quantum algebraic homomorphism

$M(B)(λ) = ζ : M(B) → LΘ(M(−))

from a Boolean domain modelled object, such that λ is a subobject of M(B).It is important to notice, that in this case:

$M(B)(λ) = δΘ(M(B))(λ)

and since this holds for arbitrary subobject λ of M(B), we have

$M(B) = δΘ(M(B))

The above completes the proof of the theorem.It is instructive to remark the significance of the subobject representa-

tion theorem, in relationship with the notion of quantum sets. This notioncan acquire a precise meaning in the present framework, if we remind theanalogy with classical sets. We notice that classical sets are specified by therule which states that the subsets of any set are represented as characteristicfunctions into 2. By analogy, we may say that quantum sets admit a spec-ification by the rule according to which, the subsets of a quantum set arerepresented as characteristic arrows in the quantum truth values object Ω.We may easily associate a quantum set, specified as above, by the colimit inthe category of elements of a presheaf of local spaces, where each local spaceis the representation of a Boolean event algebra using Stone’s representationtheorem, as has already been explained in Section 2.2.

21

4 Tensor Product Representation of the Quantum

Subobject Classifier

4.1 Quantum Truth Values Representation

The quantum truth values object Ω, has been characterized as Ω = LΘ(M(−)),that is, as the colimit taken in the category of elements of the modeled sub-object functor. In what follows, we are going to exploit the categoricalconstruction of the colimit defined above, as a coequalizer of a coproduct.For this purpose, it is necessary to consider the category of elements ofΘ(M(−)), denoted by

∫(Θ(M(−)),B). Its objects are all pairs (B, ϕM(B)),

where ϕM(B) is a subobject of M(B), that is, a monic quantum homomor-phism in M(B). The morphisms of the category of elements of Θ(M(−))are given by the arrows (B, φM(B)) qqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqq (B,ϕM(B)), namely they are those

morphisms u : B qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq B of B for which ϕM(B) ∗ u = φM(B), where ϕM(B) ∗ u

denotes the pullback of the subobject of M(B), ϕM(B), along u.

Tensor Product Representation Theorem: The quantum truth val-ues object Ω, given by the colimit in the category of elements of the modeledsubobject functor, Ω = LΘ(M(−)), admits the following coequalizer repre-sentation in tensor product form:

∐v:B→BM(B) ζ

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqη

∐(B,ϕM(B)))

M(B) χqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqq Θ(M(−))⊗BM

In the diagram above the second coproduct is over all the objects (B, ϕM(B))with ϕM(B) ∈ Θ(M(B)) of the category of elements, while the first coprod-uct is over all the maps v : (B, φM(B)) qqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqq (B,ϕM(B)) of that category, so

that v : B qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq B and the condition ϕM(B) ∗ u = φM(B) is satisfied.The proof of the theorem above, makes use of standard category theo-

retical arguments, regarding the representation of a colimit as a coequalizerof coproduct [5]. For physical purposes, it is essential to describe the truthvalues of Ω = Θ(M(−))⊗BM explicitly, and then, demonstrate how theycan be used for valuations of propositions in typical quantum measurementsituations.

22

Quantum Truth Values Theorem: The quantum truth values in Ωadmit a tensor product representation in the following form:

[δΘ(M(B))]ϕM(B)(b) = ϕM(B) ⊗ b

where

[ϕM(B)∗v]⊗b = ϕM(B)⊗v(b), ϕM(B) ∈ Θ(M(B), b ∈ M(B), v : B → B, v(b) = b

and a Boolean cover of the truth values object in a localization system, usingthe unit of the adjunction, is expressed as:

δΘ(M(B))(ϕM(B)) = [δΘ(M(B))]ϕM(B)

Proof: First of all, it is essential to prove that, the set Θ(M(−))⊗BMendowed with the relevant structure, is actually a quantum event algebra,for every B in B, so that, its elements can be interpreted as quantum truthvalues.

According to the coequalizer representation of the colimit establishedpreviously, if we consider its interpretation in the category of Sets, thecoproduct qϕM(B))M(B) is a coproduct of sets, which is equivalent to theproduct Θ(M(B))×M(B) for B ∈ B. The coequalizer is thus the definitionof the tensor product Θ(M(−))⊗A of the set valued functors:

Θ(M(−)) : Bop qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq Sets, M : B qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq Sets

∐B,BΘ(M(B))×Hom(B, B)×M(B) ζ

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq


ζqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq


∐BΘ(M(B))×M(B) χ

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq Θ(M(−))⊗M(B)

According to the preceding diagram for elements ϕM(B) ∈ Θ(M(B)), v :B → B and q ∈ M(B) the following equations hold:

ζ(ϕM(B), v, q) = (ϕM(B) ∗ v, q), η(ϕM(B), v, q) = (ϕM(B), v(q))

symmetric in Θ(M(B)) and M. Hence the elements of the set Θ(M(B))⊗BMare all of the form χ(ϕM(B)), q). This element can be written as

χ(ϕM(B)), q) = ϕM(B) ⊗ q, ϕM(B) ∈ Θ(M(B)), q ∈ M(B)

23

Thus if we take into account the definitions of ζ and η above, we obtain

[ϕM(B)∗v]⊗q = ϕM(B)⊗v(q), ϕM(B) ∈ Θ(M(B), q ∈ M(B), v : B qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq B

We conclude that the set Θ(M(B))⊗BM is actually the quotient of the setqBΘ(M(B)) × M(B) by the equivalence relation generated by the aboveequations. Furthermore, if we define:

[ϕM(B) ∗ v] = φM(B)

v(q) = q

where φM(B) is a subobject of M(B) and q ∈ M(B), we obtain the equations:

φM(B) ⊗ q = ϕM(B) ⊗ q

At a next stage, since pullbacks exist in L , we may consider the arrowsh : M(D) → M(B) and h : M(D) → M(B) and the following pullbackdiagram in L:

M(D) h qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq M(B)

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

h

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

M(B) qqqqqqqqqqqqqqqqqq


such that the relations that follow are satisfied: h(d) = q, h(d) = q andϕM(B) ⊗ h = φM(B) ⊗ h. Then we obtain:

ϕM(B)⊗q = ϕM(B)⊗h(d) = [ϕM(B)∗h]⊗d = [φM(B)∗h]⊗d = φM(B)⊗h(d) = φM(B)⊗q

We may further define:

ϕM(B) ∗ h = φM(B) ∗ h = εM(D)

Then, it is obvious that:

ϕM(B) ⊗ q = εM(D) ⊗ d

φM(B) ⊗ q = εM(D) ⊗ d

24

It is then evident that we may define a partial order on the set Θ(M(B))⊗BMas follows:

ϕM(B) ⊗ b ≤ %M(C) ⊗ c

iff there exist quantum algebraic homomorphisms β : M(D) → M(B) andγ : M(D) → M(C), and some d1, d2 in M(D), such that: β(d1) = b,γ(d2) = c, and ϕM(B) ∗ β = %M(C) ∗ γ = εM(D). Thus we obtain:

ϕM(B) ⊗ b = εM(D) ⊗ d1

%M(C) ⊗ c = εM(D) ⊗ d2

We conclude that:ϕM(B) ⊗ b ≤ %M(C) ⊗ c

iffεM(D) ⊗ d1 ≤ εM(D) ⊗ d2 ⇐⇒ d1 ≤ d2

The set Θ(M(B))⊗BM may be further endowed with a maximal elementwhich admits the following presentations:

1 = εM(Z) ⊗ 1 ∀εM(Z) ∈ Θ(M(Z))

1 = idM(B) ⊗ b ∀b ∈ M(B)

and an orthocomplementation operator:

[εM(Z) ⊗ z]? = εM(Z) ⊗ z?

Then it is easy to verify that the set Θ(M(−))⊗BM endowed with theprescribed operations is actually a quantum event algebra, for every B inB. Consequently, the truth values in Ω are represented in the form


where




25

4.2 Criterion of Truth in the Quantum Regime

In order to understand the functioning of the quantum truth values object Ωin the category of quantum event algebras, we need to establish a criterionof truth, that can be used for valuations of propositions describing quantumevents.

First of all, it is necessary to provide a definition of the value true. Forthis purpose we remind the following:

True : LR(1) → LΘ(M(−))

Υ : R(1) → Θ(M(−))


[Υ]B : R(1)(B) → Θ(M(B))

such that for the unique element α(B) in R(1)(B), we have:

[Υ]B(α(B)) := idM(B)

Then by the commutativity of the diagram below

M(B)

¡¡

¡¡

¡¡qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

[δR(1)]α(B)

@@

@@

@@qqqqqqqqqqq


[δΘ(M(B))]idM(B)

LR(1) True qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq LΘ(M(−))

we may easily conclude that

1 = idM(B) ⊗ b = True(([δR(1)]α(B))(b)) := true

Having specified the value true of the quantum truth values object Ω, wedefine the notion of truth with respect to the category of quantum eventalgebras as follows:

26

Criterion of Truth: The notion of truth with respect to the category ofquantum events algebras is specified as follows:

[δΘ(M(B))]ϕM(B)(b) = ϕM(B) ⊗ b = true iff b ∈ Dom(ϕM(B))

where




Furthermore according to the pullback diagram below, ϕA(B) = l ? e, for asubobject of a quantum event algebra l : K → L, and a Boolean domaincover e : A(B) → L.

Dom(l ∗ e) qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq K

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

l∗ e

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

l

A(B) e qqqqqqqqqqqqqqqqqq


We conclude that, the characteristic function of a subobject of a quantumevent algebra l : K → L, is specified as an equivalence class of pullbacks ofthe subobject along its restrictions on a localization system of compatibleBoolean domain covers. In particular, if the Boolean covers are monic mor-phisms, each pullback is expressed as the intersection of the subobject withthe corresponding cover in the Boolean localization system. Moreover, thevalue 1 = true in Ω is assigned to all those b that belong in Dom(ϕM(B))according to the pullback diagram above, or equivalently, to all those b, thatbelong to the restrictions of a subobject of a quantum event algebra alongthe covers of a localization system of the latter. We emphasize that thespecification of the classifying object Ω, characterized by the truth values[δΘ(M(B))]ϕM(B)(b) = ϕM(B) ⊗ b, implies a localization of truth in the quan-tum regime, with respect to Boolean points belonging to a covering systemof a quantum events algebra.

27

Furthermore, we may identify the value 1 = true in Ω, with the maximalelement of Ω, as follows:

1 = εM(Z) ⊗ 1 ∀εM(Z) ∈ Θ(M(Z))

The identity above implies that 1 belongs in εM(Z), for all εM(Z) ∈ Θ(M(Z)).Thus, the quantum truth values can be characterized as equivalence classesof filters of covers in a Boolean localization system, and the maximal valuetrue corresponds to an equivalence class of ultrafilters. Using this obser-vation, it is straightforward to state the truth value criterion, in case ofmonic covers in a Boolean localization system of a quantum events algebraas follows:

Criterion of Truth for Monic Boolean Covers:

[δΘ(M(C))]ϕM(C)(c) = ϕM(C) ⊗ c = true iff c = ΩB,C(1)

that is, if and only if, c is in the image of the maximal element in M(B),via the isomorphism pasting map ΩB,C , where;

ΩB,B : ρM(B)(M(B)⋂

M(B)) qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq %M(B)(M(B)⋂

M(B))

according to the pullback diagram below:

M(B)⋂M(B)

%M(B)qqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqq M(B)

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

ρM(B)

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

ψB

M(B)ψB qqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqq L

is defined by;ΩB,B = ρM(B) %M(B)

−1

5 Application in Quantum Measurement

The use of the quantum truth values object Ω, in conjunction with thelanguage of Boolean reference frames, for valuations of propositions related

28

with the behavior of quantum systems, provide a powerful formal mecha-nism capable of resolving the problems associated with the quantum regimeof description of reality. The provocative claim that follows from the topos-theoretic logical framework developed in the previous Sections, has to dowith the realization that the quantum measurement problem, as well as itsassociated problem of quantum state reduction, are not related with anyactual physical mechanism, but on the contrary, are logical consequencesof the use of an inappropriate classifying object in the category of quan-tum event algebras. More concretely, the two elements Boolean algebra 2,used for valuations of propositions related with the behavior of a classicalsystem, cannot be also used for valuations of propositions of quantum sys-tems, because it cannot play the role of a subobject classifier in the categoryof quantum event algebras, as in the classical case. Thus, the conceptualessence of existence of a quantum truth values object Ω in the category ofquantum event algebras, as specified concretely in the previous Section, isassociated with the fact that Ω constitutes the appropriate quantum algebraor quantum logic for valuations of propositions describing the behavior of aquantum system, in correspondence with the classical case, where the twoelements Boolean algebra 2 is properly used.

In this sense, propositions associated with the description of the behaviorof a quantum system in various contexts of observation, identified by Booleancovers in localization systems of a quantum event algebra, are being properlyassigned truth values in Ω, by means of:


where


and furthermore b may be though as representing the element (for instanceprojection operator) that identifies a proposition p in the context of M(B).It is instructive to notice that the description of reality in the quantumregime, by means of the quantum truth values in Ω, is relativized and local-ized with respect to Boolean reference frames.

More specifically, a complete description of reality is characterizedby the requirement that:

true = 1 = εM(Z) ⊗ 1 ∀εM(Z) ∈ Θ(M(Z))

29

true = 1 = idM(B) ⊗ b ∀b ∈ M(B)

Now, we are in a position to interpret from our perspective, a typical mea-surement situation referring to a quantum system prepared to pass througha slit, where a counter has been put to record by clicking, the passagethrough the slit. If we denote a Boolean domain preparation context, thatcontains both the measuring apparatus as well as the system observed, byM(B), then we may form the propositions: 〈p〉 :=counter clicks, 〈q〉 :=sys-tem passes through the slit, as well as, the composite proposition 〈 Counterclicks ⇒ system passes through the slit 〉:=〈 p → q 〉

The proposition 〈 p → q 〉 is assigned the value true in Ω, expressinga complete description of the state of affairs. Moreover, in every Booleancover of a localization system, the maximal element corresponds to 〈p →q〉 = 〈¬p

∨q〉. We notice that, the above is not enough to infer that 〈q〉 is

true. In order to infer the above, we need to use the Boolean reference framethat contains only the measuring apparatus, being obviously a subobject ofthe preparatory Boolean frame M(B). If we denote by εM(B), the monicthat corresponds to the specified subobject, we easily deduce that

εM(B) ⊗ p = true

since obviously p is contained in Dom(εM(B)), and for notational conveniencewe have identified the proposition 〈p〉 with its corresponding element p inM(B). Now, it is evident that, with respect to the Boolean frame contain-ing only the apparatus, we can say that the proposition 〈q〉 :=system passesthrough the slit is true. In this perspective, the existence of a measuringapparatus plays the role of an ultrafilter in the preparatory context M(B),transforming truth with respect to Ω, into two-valued truth with respect to2. This is effectuated by the fact that the monic subobject of M(B), con-taining only the measuring apparatus, is equivalent to a classical valuationmap A(B) → 2, as can be easily seen from the ultrafilter characterization.

Thus, the role of the measuring apparatus in a typical measurement sit-uation as the above, provides precisely the means for the transformation ofthe quantum truth values object Ω, into the classical object 2. In this sense,we conclude that the physics of the apparatus specifies a Boolean frame, inwhich a unique decomposition of the proposition p → q is possible, such thatthe proposition q is legitimately assigned the value true, only with respect to

30

that frame, namely an ultrafilter in M(B). From that perspective, Kochen-Specker theorem is an expression of the fact that a unique apparatus cannotreduce all propositions in a quantum event algebra to classical two-valuedtruth, or equivalently, that truth/false assignments cannot be performed ina global quantum logic with respect to a unique Boolean cover. The latterrealization justifies again, a posteriori, the use of variable Boolean contextsinterlocking non-trivially in localization systems of a quantum algebra ofevents, interpreted as Boolean reference frames in a sheaf-theoretic environ-ment. We may further argue that, the variation of the base Boolean eventsalgebra is actually arising from any operational procedure aiming to fix thestate of a quantum system, and corresponds in this sense, to the variation ofall possible Boolean preparatory contexts for measurement. In this setting,the notion of truth is adjacent to equivalent classes of compatible filters,instantiating subobjects of preparatory contexts for measurement, whereasthe value true, that provides a complete description of reality, is prescribedby the rule true = 1 = εM(B) ⊗ 1 ∀εM(B) ∈ Θ(M(B)).

6 Epilogue

In this paper, we have proposed a category-theoretic framework for the in-terpretation of quantum events structures and their logical semantics. Thescheme of interpretation is based on the existence of the Boolean-quantumadjunction. From that adjunction, characterized by means of the counit andunit natural transformations, we have constructed a sheaf-theoretic repre-sentation of quantum events algebras in terms of Boolean localization sys-tems, as well as, a quantum subobject classifier, that plays the role of aclassifying object in the quantum universe of discourse. In this sense, theBoolean-quantum adjunction incorporates both, the semantics of represen-tation of quantum logics as sheaves of local Boolean coefficients, and, thesemantics of truth values encoded in the specification of a classifying objectin the category of quantum logics.

Thus, from a physical viewpoint, the Boolean-quantum adjunction standsas a theoretical platform for decoding the global structural information con-tained in quantum algebras of events via processes of localization in Booleanreference frames, realized as physical contexts for measurement of observ-ables, and subsequent processes of information classification in terms of

31

truth values. The functioning of this platform is based on the establishmentof a bidirectional dependence between the Boolean and quantum structurallevels of events in local or partial congruence. Most significantly, the de-pendence takes place through the topos-theoretic universe of sheaves of setsover the Boolean points of quantum events algebras, where these generalizedpoints play the role of local Boolean covers, effectuating in this sense theidea of partial congruence mentioned above. Of course, the sheaf-theoreticrequirements secure the compatibility of the Booleanized information in theoverlapping regions of physical measurement contexts.

Additionally the global closure of this bidirectional dependence, is nec-essary to be constrained to obey certain conditions, such that its total con-stitutive information content, unfolded in the multitude of local Booleanreference frames, is both, preserved and, coherently organized in a logicalmanner. Remarkably, the necessary and sufficient conditions for both ofthese requirements, that is:

[i] preservation of the quantum information content in Boolean localiza-tion systems, and,

[ii] logical classification of quantum information by means of truth values,is supplied by the Boolean-quantum adjunctive correspondence itself, via thecounit and unit natural transformations respectively.

More concretely, regarding the first condition, we conclude that it issatisfied if the counit of the adjunction is an isomorphism for each quan-tum events algebra. In this case, there exists a full and faithful sheaf-theoretic representation of quantum events algebras in the descriptive termsof Boolean covering systems, characterized as Boolean localization systemsof measurement. Regarding the second condition, we conclude that it issatisfied, given the validity of the first condition, if the unit of the adjunc-tion is an isomorphism for the subobject functor. In this case respectively,the subobject functor is representable in the category of quantum eventsalgebras by a classifying object in that category, called subobject classifier.The classifying object plays the role of a quantum truth values object andmay be legitimately used for valuations of quantum propositions, in exactcorrespondence with the use of the two-valued Boolean object, used for val-uations of classical propositions. Moreover, the quantum subobject classifierprovides the key logical device for the analysis of typical quantum measure-ment situations, providing a criterion of truth for a complete description of

32

reality in the quantum regime.

Acknowledgments: I would gratefully like to acknowledge support ofthis work by a Hellenic State research grant in Mathematical Physics.

References

[1] G. Birkhoff and J. von Neumann, The logic of Quantum Mechanics,Annals of Mathematics 37, 823 (1936).

[2] V. S. Varadarajan, Geometry of Quantum Mechanics, Vol. 1, Van Nos-trand, Princeton, New Jersey, 1968.

[3] E. Zafiris, Boolean Coverings of Quantum Observable Structure: A Set-ting for an Abstract Differential Geometric Mechanism , Journal of Ge-ometry and Physics 50, 99 (2004).

[4] F. W. Lawvere and S. H. Schanuel, Conceptual Mathematics, Cam-bridge University Press, Cambridge, 1997.

[5] S. MacLane, Categories for the Working Mathematician, Springer-Verlag, New York, 1971.

[6] S. MacLane and I. Moerdijk, Sheaves in Geometry and Logic, Springer-Verlag, New York, 1992.

[7] J. L. Bell, Toposes and Local Set Theories, Oxford University Press,Oxford, 1988.

[8] M. Artin, A. Grothendieck, and J. L. Verdier, Theorie de topos et co-homologie etale des schemas, (Springer LNM 269 and 270, Springer-Verlag, Berlin, 1972).

[9] Bell J. L., From Absolute to Local Mathematics, Synthese 69, 1986.

[10] Bell J. L., Categories, Toposes and Sets, Synthese, 51, No.3, 1982.

33

[11] E. Zafiris, Generalized Topological Covering Systems on QuantumEvents Structures, Journal of Physics A: Mathematical and General39, (2006).

[12] S. Kochen and E. Specker, The problem of hidden variables in QuantumMechanics, J. Math. and Mech. 17, 59 (1967).

[13] J. Butterfield and C. J. Isham, A Topos Perspective on the Kochen-Specker Theorem: I. Quantum States as Generalized Valuations, Inter-national Journal of Theoretical Physics 37, 2669 (1998).

[14] J. Butterfield and C. J. Isham, A Topos Perspective on the Kochen-Specker Theorem: II. Conceptual Aspects and Classical Analogues, In-ternational Journal of Theoretical Physics 38, 827 (1999).

[15] J. P. Rawling and S. A. Selesnick, Orthologic and Quantum Logic.Models and Computational Elements, Journal of the Association forComputing Machinery 47, 721 (2000).

[16] J. Butterfield and C. J. Isham, Some Possible Roles for Topos Theory inQuantum Theory and Quantum Gravity, Foundations of Physics 30,1707 (2000).

[17] G. Takeuti, Two Applications of Logic to Mathematics, MathematicalSociety of Japan 13, Kano Memorial Lectures 3 (1978).

[18] M. Davis, A Relativity Principle in Quantum Mechanics, InternationalJournal of Theoretical Physics 16, 867 (1977).

34

topos-theoretic classiﬂcation of quantum events structures...

Documents