Categorial subsystem independence as morphism co-possibility
aa r X i v : . [ m a t h - ph ] F e b Categorial subsystem independence as morphism co-possibility ∗ Zal´an Gyenis † Mikl´os R´edei ‡ October 23, 2018
Abstract
This paper formulates a notion of independence of subobjects of an object in a general (i.e. notnecessarily concrete) category. Subobject independence is the categorial generalization of what isknown as subsystem independence in the context of algebraic relativistic quantum field theory. Thecontent of subobject independence formulated in this paper is morphism co-possibility: two subobjectsof an object will be defined to be independent if any two morphisms on the two subobjects of anobject are jointly implementable by a single morphism on the larger object. The paper investigatesfeatures of subobject independence in general, and subobject independence in the category of C ∗ -algebras with respect to operations (completely positive unit preserving linear maps on C ∗ -algebras)as morphisms is suggested as a natural subsystem independence axiom to express relativistic localityof the covariant functor in the categorial approach to quantum field theory. Keywords: Algebraic relativistic quantum field theory; Category theory; Subsystemindependence.
Subsystem independence is a crucial notion in the specific axiomatic approach to (relativistic) quantumfield theory known as “Local Quantum Physics” (also called “Algebraic Quantum Field Theory”). Thisapproach to quantum field theory was initiated by Haag and Kastler [19], and since its inception it hasdeveloped into a rich field. (For monographic summaries see [20], [16], [1]; for compact, more recentreviews we refer to [9], [10], [36]; the papers [13], [17], [18] recall some episodes in the history of thisapproach.) The key element in the approach is the implementation of locality, and “The locality conceptis abstractly encoded in a notion of independence of subsystems . . . ” [5]. It turns out that independence ofsubsystems of a larger system can be specified in a number of nonequivalent ways: Summers’ 1990 paper[34] gives a review of the rich hierarchy of independence notions; for a non-technical review of subsystemindependence concepts that include more recent developments as well see [35].Local Quantum Physics has recently been further developed into what can properly be called ‘Cate-gorial Local Quantum Physics’ [8]: in this new “paradigm” quantum field theory is a covariant functorfrom the category of certain spacetimes with isometric, smooth, causal embeddings of spacetimes as mor-phisms into the category of C ∗ -algebras with injective C ∗ -algebra homomorphisms as morphisms. (Fora self-contained review of this approach see [14]). This categorial approach is motivated by the desire to ∗ Forthcoming in the special issue of Communications in Mathematical Physics devoted to Rudolf Haag † Budapest University of Technology and Economics, Department of Algebra and Department of Logic, E¨otv¨os Lor´andUniversity; [email protected] ‡ Department of Philosophy, Logic and Scientific Method, London School of Economics and Political Science, HoughtonStreet, London WC2A 2AE, UK, [email protected] stablish a generally covariant quantum field theory on a general, non-flat spacetime that might not haveany non-trivial global symmetry. Lack of global spacetime symmetry makes it impossible to postulate co-variance of quantum fields in the usual way by requiring observables to transform covariantly with respectto representations of the global symmetry group of the spacetime. Instead, general covariance is imple-mented in Categorial Local Quantum Physics very naturally by postulating that the functor representingquantum field theory is a covariant functor. Locality also has to be implemented in categorial quantumfield theory. This is done by formulating axioms for the covariant functor that express independence ofsubsystems. In the original paper formulating Categorial Local Quantum Physics [8] Einstein Locality(local commutativity) is taken as the expression of locality as independence. In subsequent publications[6], [7] a categorial version of the split property is added to the Einstein Locality axiom. It is then shownin [7] that (under the further assumption of weak additivity) the categorial split property is equivalentto the functor being extendable to a tensor functor between the tensor category formed by spacetimeswith respect to disjoint union as tensor operation and the tensor category of C ∗ -algebras taken with theminimal tensor products of C ∗ -algebras.In what sense are Einstein Locality and Einstein Locality together with the categorial split property(i.e. the tensor property) of the functor subsystem independence conditions? This is a non-trivialquestion, which is shown by the remark of Buchholz and Summers on Einstein Locality:“This postulate, often called the condition of locality [16], has become one of the basic ingre-dients in both the construction and the analysis of relativistic theories [37]. Yet, in spite ofits central role in the theoretical framework, the question of whether locality can be deducedfrom physically meaningful properties of the physical states has been open for more than fourdecades.” [11]In particular, Buchholz and Summers point out [11] that some of the standard notions of subsystemindependence (such as C ∗ -independence) do not entail Einstein Locality. And conversely: EinsteinLocality alone does not entail C ∗ -independence. Nor does Einstein Locality, in and by itself, entailthe independence condition known as prohibition of superluminal signaling: local commutativity of C ∗ -algebras pertaining to spacelike separated spacetime regions only entails prohibition of superluminalsignaling with respect to measurements of local observables representable by the projection postulateand by the operations given by local Kraus operators – but not with respect to general operations thatdo not have a local Kraus representation (this was shown in [33]). Einstein Locality alone also doesnot ensure another subsystem independence called operational C ∗ -independence: That any two (non-selective) operations (completely positive, unit preserving linear maps on C ∗ -algebras) performed onspacelike separated subsystems of a larger system are jointly implementable as a single operation on thelarger system [32], [29]. Since the tensor product of two operations is again an operation ([4][p. 190], seealso Proposition 9. in [32]), the tensorial property of the functor does entail operational C ∗ -independenceof the components of the tensor product within the tensor product algebra; however, the tensor propertyentails more than this: it entails operational C ∗ -independence in the product sense [32], which is a strictlystronger condition than simple operational C ∗ -independence. But for the purposes of expressing localityin quantum field theory subsystem independence in the sense of operational C ∗ -independence does nothave to be implemented in the strong form of requiring existence of a joint product extension of operationson subsystems.Thus requiring only Einstein Locality of the functor seems too weak, imposing the categorial splitproperty (i.e. demanding the functor to be tensorial) seems to demand a bit more than needed toimplement locality interpreted as subsystem independence in Categorial Local Quantum Physics. What2s then the right concept of subsystem independence that expresses locality in Categorial Local QuantumPhysics?The subsystem independence hierarchy in Local Quantum Physics suggests a general concept ofsubsystem independence that has a natural formulation in terms of categories objects of which are setswith morphisms as maps: independence as morphism co-possibility. According to this independenceconcept two objects are independent in a larger object with respect to a class of morphisms if any twomorphisms on the two smaller objects have a joint extension to a morphism on the larger object. Thisindependence notion appeared in [30] and was suggested in [31] as a possible axiom to require in CategorialLocal Quantum Physics. Taking specific subclasses of the operations as the class of morphisms, one canrecover the standard concepts of subsystem independence in the independence hierarchy as special casesof morphism co-possibility (see [30]).The way subsystem independence as morphism co-possibility was formulated above and in the papers[30] and [31] is not entirely satisfactory however because it is not purely categorial: in a general categoryobjects are not necessarily sets and morphisms are not necessarily functions – not every category isa concrete category (e.g. the real numbers R regarded as a poset category) [3]. In a general categorysubsystem independence as morphism co-possibility should be formulated as subobject independence withrespect to some class of morphisms. The aim of the present paper is to define subobject independencein this way, as morphism co-possibility in a general category, and to investigate the basic properties ofsuch an independence notion. This notion is of interest in its own right and, after defining it in section2, we give several examples of this sort of independence in different categories in section 3. Section 4proves some propositions on the relation of subobject independence and tensor structure in a category. Insection 5 subobject independence is specified in the context of the category of C ∗ -algebras taken with theclass of operations between C ∗ -algebras. The resulting notion of operational independence is suggestedthen in section 6 as a possible axiom to express relativistic locality of the covariant functor describing agenerally covariant quantum field theory. In this section C = ( Ob , Mor ) denotes a general category, and
Hom is a subclass of
Mor such that( Ob , Hom ) also is a category. It is not assumed that C is a concrete category; i.e. that it is categoricallyequivalent to a category objects of which are sets and monomorphisms are functions. Morphisms in Hom will be referred to as
Hom -morphisms, morphisms in
Mor will be called
Mor -morphisms. We wish todefine a notion of independence of subobjects
A, B of an object C , where the concept of subobject is un-derstood with respect to Hom -morphisms, and the independence expresses that any two
Mor -morphismson the
Hom -subobjects A and B are jointly implementable by a single Mor -morphism on C . The twomorphism classes Hom and
Mor should be considered as variables in this categorial concept of indepen-dence: Choosing different morphism classes one obtains independence notions contents of which can varyconsiderably.Recall that a
Mor -morphism f : A → B is a monomorphism (“mono”, for short) if for any object C ∈ Ob and any Mor -morphisms g , g : C → A it holds that g f = g f implies g = g . Monomorphismsare the categorial equivalents of injective functions.The notion of Hom -subobject is formulated in terms of
Hom -monomorphisms: a
Hom -subobject of anobject X is an equivalence class of Hom -monomorphisms
Hom ∋ i A : A → X where i A is defined to beequivalent to i B : B → X if there is an Hom -isomorphism h : A → B such that hi B = i A and h − i A = i B .In what follows, | i A | Hom denotes the equivalence class of
Hom -morphisms equivalent to i A .3 efinition 2.1 ( Mor -independence of
Mor -morphisms) . Mor -morphisms
A X B f A f B are called Mor -independent if for any two
Mor -morphisms
A A α A , B B α B there is Mor -morphism
X X α such that the diagram below commutes. A X BA X B f A α A α f B α B f A f B (cid:3) We are now in the position to give the definition of the central concept of this paper:
Definition 2.2 ( Mor -independence of
Hom -subobjects) . Two
Hom -subobjects of object X represented bythe two equivalence classes | f | Hom and | f | Hom are called
Mor -independent if any two
Hom -monomorphisms D X D g g g ∈ | f | Hom and g ∈ | f | Hom are
Mor -independent. (cid:3)
The content of
Mor -independence of
Hom -subobjects is that two
Hom -subobjects of object C are Mor -independent if and only if any two
Mor -morphisms on any representations of the
Hom -subobjectsare jointly implementable by a single
Mor -morphism on C . This independence concept expresses theindependence of the of the substructures that are invariant with respect to Hom from the perspective ofthe structural properties embodied in the
Mor -morphisms. (See examples in section 3.)The next proposition is useful when it comes to determine whether two subobjects are independent.By definition, independence of
Hom -subobjects implies
Mor -independence of any of their representatives.The following proposition states the converse: if one pair of representatives of two
Hom -subobjects are
Mor -independent, then the two
Hom -subobjects are
Mor -independent.
Proposition 2.3.
If for
Hom -monomorphisms C X C f f and D X D g g we have | f | Hom = | g | Hom and | f | Hom = | g | Hom (i.e. the
Hom -monomorpisms f i and g i ( i = 1 , Hom -subobject), then f and f are Mor -independent if and only if g and g are Mor -independent.
Proof.
Suppose C X C f f are Mor -independent and consider the diagram below. D D C X C C X C D D i g α f β j f g γ f β f j g i α g Since | f | Hom = | g | Hom and | f | Hom = | g | Hom there are
Hom -isomorphisms i , j and i , j as figured.Take and arbitrary Mor -morphism α : D → D . Let β = i − α j − : C → C (1) β = i − α j − : C → C (2)4y assumption C X C f f are Mor -independent, therefore there is a suitable
Mor -morphism γ : X → X . Then we obtain g γ = i f γ = i β f = i β j g = α g and similarly g γ = i f γ = i β f = i β j g = α g This completes the proof.Our next proposition formulates a very natural necessary condition for independence. The content ofthe necessary condition can be illustrated on the example of the category C of structures. Let A and B substructures of C . If we take two morphisms α A : A → A , α B : B → B , then a morphism γ : C → C that extends both α A and α B can exist only in the case when α A and α B act on Y = A ∩ B exactly thesame way, i.e. if one has α A ↾ Y = α B ↾ Y (3)The next proposition we wish to establish expresses this condition in the case of every category. To statethe proposition, first we formulate the condition (3) in general categorial terms. Since the intersection Y in the category of structures is the pullback A × C B , for the next definition it is assumed that pullbacksexist in C . Definition 2.4 ( Mor -compatibility) . We say that
Mor -morphisms
A C B f A f B are Mor -com-patible if the diagram A × C BA C BA C Bp A p B f A f B α A α B f A f B commutes for all Mor -morphisms α A , α B . Here A × C B is the pullback. (cid:3) The next proposition states the sought-after necessary condition for independence in a general cate-gory:
Proposition 2.5. If A C B f A f B are Mor -independent, then they are also
Mor -compatible.
Proof.
Consider the diagram, where α A , α B are arbitrary Mor -morphisms and A × C B is the pullback. A × C BA C BA C Bi A i B f A f B α A α B f A f B γ
5e need to show that the diagram without the dashed arrow commutes. By
Mor -independence, for
Mor -morphisms α A , α B there exists a suitable Mor -morphism γ . Then i A α A f A = i A f A γ = i B f B γ = i B α B f B which we had to show.For completeness we note that the existence of the pullback A × C B in the definition of Mor -compatibility could be relaxed by replacing the pullback A × C B in the diagram by any Y that canbe mapped into A and B (i.e. there are Mor -arrows Y → A and Y → B ) and universally quantifyingover Y .In the next proposition let ⊕ be a coproduct in the category ( Ob , Mor ), i.e. X ⊕ X be an elementsuch that there exist Mor -morphisms (called the coproduct injections) i : X → X ⊕ X and i : X → X ⊕ X having the universal property. Proposition 2.6.
Coproduct injections X X ⊕ X X i i are Mor -independent.
Proof.
Let X X ⊕ X X i i be a coproduct with coproduct injections i and i . Fromthe diagram below on the left-hand side, by composing arrows, one gets the diagram on the right-handside which is a coproduct diagram, therefore a suitable m with the dotted arrow (which is the copair[ m i , m i ]) exists and completes the proof. X X ⊕ X X X X ⊕ X X i m i i m i X X ⊕ X X X ⊕ X i m i m i m i We remark that coproduct injections in general are not necessarily monic, however, in certain cate-gories (such as extensive or distributive categories) coproduct injections are automatically monic.
Set is the category of sets as objects with functions as
Mor -morphisms. Let
Hom = Mor and consider
A C B f A f B . Speaking about subobjects we may assume A, B ⊆ C , that is, f A and f B are theinclusion mappings. The pullback A × C B is just the intersection A ∩ B . A and B are Hom -compatibleif and only if A ∩ B = ∅ since otherwise one could take permutations of A and B that act differently onthe intersection. It is straightforward to check that A and B are Mor -independent if and only if they aredisjoint. 6 .2 Vector spaces
Let
Vect F be the category of vector spaces over the field F with linear mappings as Mor -morphisms. Take
Hom = Mor . If C is a vector space and A , B are subspaces then the pullback A × C B is the subspace A ∩ B . Recall that two subspaces A , B are linearly independent if and only if A ∩ B = { } . We claimthat Mor -independence and linear independence coincide. Take two
Mor -morphisms α A : A → A and α B : B → B . Then α A and α B act on the bases h a i : i ∈ I i = A and h b j : j ∈ J i = B . Any functiondefined on bases can be extended to a linear mapping, therefore α A and α B have a common extension γ : h A ∪ B i → h A ∪ B i if and only if they act on A ∩ B the same way. As α A , α B were arbitrary, the lattercondition is equivalent to A ∩ B = { } . Finally, one can extend the set { a i , b j : i ∈ I, j ∈ J } to a basisof C and extend γ to be defined on the entire C . A moment of thought shows that Mor -compatibility isalso equivalent to A ∩ B = { } . Pregeometries (or matroids in the combinatorial terminology) are defined in order to capture the notionof independence in a very general framework. Formally, a pregeometry is a tuple ( X, cl) where X isa set and cl : ℘ ( X ) → ℘ ( X ) is a closure operator having a finite character satisfying the Steinitzexchange principle. Independence and basis can be defined as in vector spaces. A morphism betweentwo pregeometries f : ( X, cl X ) → ( Y, cl Y ) is a function f : X → Y that preserves closed sets, that is, itsatisfies cl X ( f − [ Z ]) = f − [cl Y ( Z )] for all Z ⊆ Y . If the two closure operators are topological closure,then morphisms are just the continuous functions. If ( X, cl) is a pregeometry, then a sub-pregeometryis a tuple ( Y, cl ↾ Y ) where Y ⊆ X is closed: Y = cl( Y ). We denote sub-pregeometries by Y ≤ X . Let Pregeom be the category of pregeometries with
Mor = Hom as described above. Then
Mor -independenceof
A, B ≤ C coincides with independence of A and B in the pregeometry sense. The proof is similar tothat of the vector space case. A, B ≤ C are independent if and only if A ∩ B = cl( {∅} ). This holds onlyif neither A nor B has basis elements in A ∩ B . In this case any basis of A and B can be concatenatedand extended to a basis of C in the same way as in the case of vector spaces. The result follows thenfrom the observation that any Mor -morphism can be identified with an action on the elements of a basis.
Let
Bool be the category of Boolean algebras as objects with injective homomorphisms as
Mor -morphisms.As before, we set
Hom = Mor . Two subalgebras
A, B ≤ C are called Boole-independent if for all a ∈ A , b ∈ B we have a ∧ b = 0 provided a = 0 = b . Boole independence is logical independence if the Booleanalgebras are viewed as the Tarski-Lindenbaum algebra of a classical propositional logic: a ∧ b = 0entails that there is an interpretation on C that makes a ∧ b hence both a and b true; i.e. any twopropositions that are not contradictions can be jointly true in some interpretation. How is this Boolean(logical) independence related to Mor -independence? The connection between Boole-independence and
Mor -independence is a bit more subtle than in the previous examples.(1)
Mor -independence does not imply Boole-independence. Consider the case when C is finite, { c , . . . , c n } is the set of atoms of C and the subalgebras A and B are generated by distinct set ofatoms A = h c , . . . , c k i , B = h c k +1 , . . . , c n i . Clearly A and B are not Boole-independent. However,any Mor -morphisms (i.e. automorphism, because in the finite case every injective homomorphisminto itself is an automorphism) of A (resp. B ) comes from a permutation of atoms generating A .Conversely any permutation of atoms extend to an automorphism. Given automorphisms α A and7 B of A and B , respectively, give rise to a permutation of all the atoms of C which extends to anautomorphism of C . Consequently, A and B are Mor -independent.(2) Boole-independence of
A, B ≤ C implies Mor -independence if A ∪ B generates C . A Boolean algebra C is the internal sum of the subalgebras A and B just in case the union A ∪ B generates C andwhenever a ∈ A , b ∈ B are non-zero elements, then a ∧ b = 0. (Internal) sum of Boolean algebras isjust the coproduct of the algebras (up to isomorphism) whence by Proposition 2.6 Mor -independenceof A and B follows.If A, B ≤ C are Boole-independent but A ∪ B does not generate C (i.e. the internal sum A ⊕ B of A and B is a proper subalgebra of C ), then a similar argument shows that any Mor -morphism α A : A → A and α B : B → B can be jointly extended to a Mor -morphism γ : A ⊕ B → A ⊕ B .The question whether γ can be further extended to an Mor -morphism C → C is non-trivial and isrelated to the injectivity of C . Injective Boolean algebras are essentially the complete ones in thecategory Bool [25][p.117, 16(c,e)]. Consequently Boole-independence implies
Mor -independence inany complete Boolean algebra.
Logical independence is meaningful in categories of lattices that are not distributive. The relevantexamples for physics are the von Neumann lattices (in particular Hilbert lattices) that are interpreted asquantum logic: If A and B are two von Neumann subalgebras of von Neumann algebra C , and P ( A ) , P ( B )and P ( C ) denote the corresponding orthomodular lattices of projections, then P ( A ) and P ( B ) can bedefined to be logically independent if a ∧ b = 0 whenever P ( A ) ∋ a = 0 and P ( B ) ∋ b = 0 ([26], [27][28][Section 11]). Taking the category of von Neumann lattices with orthomodular lattice homomorphismsas morphisms the notion of subobject independence becomes meaningful and the problem of relation oflogical independence of von Neumann lattices and the subobject independence emerges in this categoryjust like in the category of Boolean algebras. We clarify here the relation of logical independence tosubobject independence in the context of general orthomodular lattices [21].Let OML be the category of orthomodular lattices as objects with injective ortho-homomorphismsas
Mor -morphisms. Take
Hom = Mor . Logical independence of orthomodular sublattices A and B of theorthomodular lattice C is defined as in case of Boolean algebras: a ∧ b = 0 whenever A ∋ a = 0 and B ∋ b = 0. The connection between logical independence and Mor -independence in this general contextis similar to the one in the category of Boolean algebras. To describe the relation, recall first the notionof internal direct sum for lattices (see e.g. [23]): If L is a lattice (not necessarily orthomodular) and x, y ∈ L , then write x ▽ y if for all z ∈ L we have ( x ∨ z ) ∧ y = z ∧ y . Clearly x ∧ y = 0 implies x ▽ y . Let S and Q be subsets of L . We say that L is the internal direct sum of S and Q (and we write L = S ⊕ Q ) if(1) each x ∈ L can be written as x = s ∨ q with s ∈ S and q ∈ Q ;(2) s ∈ S , q ∈ Q entails s ▽ q .If S and Q are (orthomodular) lattices, then their direct product is an (orthomodular) lattice, and thereis a natural (ortho)-isomorphism between their direct product and their internal direct sum given by( s, q ) s ∨ q (see [23]). It follows that any homomorphisms given on the direct summands S and Q extends to a homomorphism on their internal direct sum. We have then the following characterization ofthe relation of logical independence and subobject independence in the category of orthomodular lattices:(1) Mor -independence does not imply logical independence of sub-orthomodular lattices. This followsfrom what was said about Boolean algebras in section 3.4 because every Boolean algebra is an8rthomodular lattice and we saw that
Mor -independence does not imply Boole-independence. (
Bool is a complete subcategory of
OML ).(2) Logical independence of
A, B ≤ C implies Mor -independence if A ∪ B generates C . In this case C = A ⊕ B is the internal sum of A and B since logical independence ensures a ▽ b for each a ∈ A , b ∈ B . On the other hand, each x ∈ C can be written as x = a ∨ b with a ∈ A and b ∈ B as A ∪ B generates C .If A, B ≤ C are logically independent but A ∪ B does not generate C (i.e. the internal sum A ⊕ B of A and B is a proper subalgebra of C ), then a similar argument shows that any Mor -morphism α A : A → A and α B : B → B can be jointly extended to a Mor -morphism γ : A ⊕ B → A ⊕ B .The question whether γ can be further extended to an Mor -morphism C → C is non-trivial and isrelated to the injectivity of C . We are not aware of any useful characterization of injective objectsin OML . Components of tensor products are typically regarded “independent” within the tensor product. Theparadigm example is the standard product of probability measure spaces with the product measure onthe product of the component measurable spaces. In this section we investigate the relation of categorialsubobject independence and the tensor product structure in a category. We will see that categorialsubobject independence of the components of the tensor product is not automatic. We will howeverisolate conditions on the tensor category that entail subobject independence of the components in thetensor product (Proposition 4.4).Recall first the definition of a tensor product in a category (cf. Section 7.8 in [3])
Definition 4.1.
A bifunctor ⊗ : C × C → C is a tensor product if it is associative up to a naturalisomorphism and there is an element I that acts as a left and right identity (up to isomorphism). (cid:3) A category with a tensor product ( C , ⊗ ) is a tensorial category (monoidal category) if ⊗ satisfies thepentagon and triangle axioms. If a category has products or coproducts for all finite sets of objects, thenthe category can be turned into a tensor category by adding the product or coproduct as a bifunctor (dueto the universal property of products and coproducts).For the next definition suppose that ( C , ⊗ ) is such that for any two objects A , B there are Mor -morphisms
A A ⊗ B B i A i B Definition 4.2 ( ⊗ -independence of Mor -morphisms) . Mor -morphisms
A C B f A f B are called ⊗ -independent if there exists a Mor -morphism h : A ⊗ B → C such that the following diagram commutes. A A ⊗ B BCf A f B i A i B h (cid:3) ⊗ is the coproduct, then the universal property of coproducts implies the existence of such a h inthe definition. Mor -independence of components of tensor products is not automatic. As a counterexample considerthe category of sets with the tensor product being the union operation. Then (
Set , ∪ ) is a monoidalcategory; yet if A and B are non-disjoint sets, then A A ∪ B B ⊆ ⊆ are not
Mor -independent(see subsection 3.1). Also note that there are tensorial categories where components of a tensor productcannot even be mapped into the tensor product hence they are not subobjects (an example is the categoryof rings with homomorphisms). This motivates Definition 4.3 below. Note that ⊗ being a bifunctor meansthat it acts on Mor × Mor too; that is to say: if f : A → A ′ and g : B → B ′ are two morphisms, thenthere is a morphism f ⊗ g : A ⊗ B → A ′ ⊗ B ′ . Definition 4.3.
The tensorial category ( Ob , Mor , ⊗ ) is called Hom -regular if (i) and (ii) below hold.(i) For all objects
A, B there are
Hom -monomorphisms
A A ⊗ B B i A i B We call these
Hom -monomorphisms canonical injections .(ii) For any pairs of
Mor -morphisms m A : A → A ′ and m B : B → B ′ the tensor product arrow m A ⊗ m B makes the following diagram commute. A A ⊗ B BA ′ A ′ ⊗ B ′ B ′ i A m A i A ′ i B m B i B ′ m A ⊗ m B (cid:3) We then have as an immediate consequence of regularity:
Corollary 4.4.
If ( Ob , Mor , ⊗ ) is a Hom -regular tensorial category, then the canonical injections are
Mor -independent.
Proof.
Take A ′ = A and B ′ = B in the definition of regularity. Definition 4.5 (( Hom , ⊗ )-independence of Hom -morphisms) . Hom -morphisms
A C B f A f B are called ( Hom , ⊗ )-independent if there exists a Hom -morphism h : A ⊗ B → C such that the followingdiagram commutes. A A ⊗ B BCf A f B i A i B h (cid:3) Definition 4.6 ( Hom -injectivity) . An object Q is Hom -injective if for all A and arrows in the diagrambelow we have 10 QQ ∈ Hom ∈ Mor ∃ ∈
Mor (cid:3)
Proposition 4.7.
In a
Hom -regular tensorial category ( Ob , Mor , ⊗ ) we have that ( Hom , ⊗ )-independenceof Hom -subobjects in a
Hom -injective object implies
Mor -independence of the
Hom -subobjects.
Proof.
Suppose
A Q B f A f B are Hom -monomorphisms representing two
Hom -subobjects whichare (
Hom , ⊗ )-independent. Let Q be Hom -injective and consider the diagram below.
A A ⊗ B BQQA A ⊗ B Bi A i B f A f B ∃ uα A α B i A i B f A f B ∃ vh j By (
Hom , ⊗ )-independence of A Q B f A f B there is u, v : A ⊗ B → Q with u, v ∈ Hom andby regularity of the tensorial category there is h : A ⊗ B → A ⊗ B , h = α A ⊗ α B making the diagramcommute. Applying Hom -injectivity of Q for u and hv we get a suitable j : Q → Q with uj = hv . Then f A j = i A uj = i A hv = α A i A v = α A f A and similarly f B j = i B uj = i B hv = α B i B v = α B f B The intuitive content of Proposition 4.7 is as follows. Suppose A and B are Hom -subobjects of an
Hom -injective object Q . The subobject relations are witnessed by the Hom -arrows f A and f B . ( Hom , ⊗ )-independence tells us that A and B , as subobjects, lie in Q in a similar manner as they lie in the tensorproduct A ⊗ B , i.e. the tensor product can be mapped into Q via some Hom -arrow u in such a way thatthe canonical injections (that witness that A and B are Hom -subobjects of the tensor product) commutewith f A , f B and u . Take any two Mor -morphisms α A : A → A and α B : B → B . By Hom -regularity ofthe tensor product this two mappings are jointly implementable by a single morphism h on the tensorproduct . The question is whether this mapping h can be extended to a mapping defined on the entire Q . Hom -injectivity of Q does this favour to us: Hom -injectivity guarantees that any
Mor -morphism definedon a
Hom -subobject can be extended as a
Mor -morphism acting on Q . ⊗ -independence of Mor -morphisms (Definition 4.2) and the notion of a regular category (Definition4.3) was introduced and studied in [12] under different names. In [12] the notion of a tensor product with rojections or with inclusions has been defined (essentially, this is our Definition 4.3). It was shown in [12]that the definition of stochastic independence relies on such a structure and that independence can bedefined in an arbitrary category with a tensor product with inclusions or projections in a manner similarto Definition 4.2. It turns out that the standard notion of stochastic independence of classical randomvariables is equivalent to ⊗ -independence of objects in the category of random variables (for more detailsee [12]). Moreover, the classifications of quantum stochastic independence by Muraki, Ben Ghorbal, andSch¨urmann has been shown to be classifications of the tensor products with inclusions for the categoriesof algebraic probability spaces and non-unital algebraic probability spaces. Thus ⊗ -independence of Mor -morphisms is directly relevant for stochastic independence in the context of quantum probabilityspaces. C ∗ -algebras with respect to operations as mor-phisms In this section (
Alg , Op
Alg ) denotes the category of C ∗ -algebras, where the elements in the class ofmorphisms Op Alg are the non-selective operations: completely positive, unit preserving linear maps on C ∗ -algebras. Operations represent physical operations performed on quantum physical systems whose algebraof observables are represented by the (selfadjoint) part of the C ∗ -algebra the operation is defined on.Examples of operations include states, conditional expectations (in particular the projection postulate),operations that are given by Kraus operators, and more (see [22] for the elementary theory and physicalinterpretation of operations, [2] for some basic properties, and [24] for a systematic treatment of operationsfrom the perspective of operator spaces.) Specifically, C ∗ -algebra homomorphisms are completely positive;hence the class hom Alg of injective C ∗ -algebra homomorphism is a subclass of Op Alg . Thus it is meaningfulto talk about Op Alg -independence in the sense of the following definition:
Definition 5.1. C ∗ -subalgebras A , B of C ∗ -algebra C are called Op Alg -independent in C if A and B are Op Alg -independent as hom
Alg -subobjects of object C in the category ( Alg , Op
Alg ) of C ∗ -algebras in thesense of Definition 2.2. (cid:3) The notion of Op Alg -independence of C ∗ -subalgebras was first formulated in categorial terms in [30]but its content, expressed in a non-categorial terminology and called “operational C ∗ -independence”appeared already in [32]. The content of Op Alg -independence of C ∗ -subalgebras A , B of C ∗ -algebra C is that operations on the C ∗ -subalgebras A , B have a joint extension to the C ∗ -algebra C . This kind ofindependence has a direct physical interpretation: The physical content of Op Alg -independence is thatany two physical operations (for instance measurement interaction) performed on the two subsystemsobservables of which are represented by A and B , respectively, can be performed as a single physicaloperation on the larger system observables of which are represented by C .Note that Op Alg -independence of A , B in C has two components: (i) that operations on A and B can be extended to C ; and (ii) that there exists a joint extension. Already (i) is a non-trivial demand becauseoperations on C ∗ -subalgebras are not always extendable to the larger algebra [2]. Formulated differently:Not all C ∗ -algebras are injective. This fact complicates the implementation of subsystem independenceas Op Alg -independence in the categorial formulation of quantum field theory (see the end of the finalsection of the paper). Also note that Op Alg -independence does not require that the extension of theoperations on A and B factorize across A and B ; i.e. the extension need not be a product extension. One12an strengthen the notion of Op Alg -independence by requiring the existence of a product extension; wecall the resulting concept of independence Op Alg -independence in the product sense .( Alg , Op
Alg ) is a tensor category with respect to the minimal C ∗ -tensor product A ⊗ B of C ∗ -algebras A and B . Since algebras A and B have units I A , I B , they can be injected into the tensor product by the hom Alg -morphisms
A ∋ A A ⊗ I B and B ∋ B I A ⊗ B . Thus the canonical hom Alg -injections inDefinition 4.3 (i) exist and item (ii) in Definition 4.3 is also fulfilled. Thus (
Alg , Op
Alg , ⊗ ) is a hom Alg -regular category in the sense of Definition 4.3. It follows that Proposition 4.7 applies and we obtain
Proposition 5.2. C ∗ -algebras A ≈ A ⊗ I B and B ≈ I A ⊗ B are Op Alg -independent in
A ⊗ B .As a corollary:
Corollary 5.3. If C is an injective C ∗ -algebra and A ⊗ B is a C ∗ -subalgebra of C , then A ≈ A ⊗ I B and B ≈ I A ⊗ B are Op Alg -independent in C .The joint extension to A⊗ B of operations on A and B guaranteed by Proposition 5.2 is just the tensorproduct of the two operations, which is again an operation [4][p. 190], (see also Proposition 9. in [32]).Note that C ∗ -algebras A , B are not just Op Alg -independent in
A ⊗ B , they are Op Alg -independent in
A ⊗ B in the product sense : the tensor product of two operations factorizes over the components. Op Alg -independence in the product sense is a very strong independence property. It is known to be strictlystronger than Op Alg -independence simpliciter: Op Alg -independence in C of commuting C ∗ -subalgebras A , B of C in the product sense is equivalent to C ∗ -independence of A , B in the product sense (Proposition10, [32]) but C ∗ -independence of A , B is strictly weaker than C ∗ -independence of A , B in the productsense [34] (cf. Proposition 1. in [32]).The difference between Op Alg -independence and Op Alg -independence in the product sense, and thefact that the latter concept relies on the morphisms in Op Alg being functions, lead to the question ofwhether there is a purely categorial version of subobject independence as morphism co-possibility “in theproduct sense”. We do not have such a concept and leave it is a problem for further investigation.Note that the definition of
Mor -independence of
Hom -subobjects (Definition 2.2) remains meaningfuleven if the class
Hom is not a subclass of
Mor : As long as morphisms in
Hom and
Mor can be composed,one can meaningfully talk about
Mor -independence of
Hom -subobjects. This enables one to recover themajor subsystem independence concepts that occur in algebraic quantum (field) theory by choosing specialsubclasses of the class of all non-selective operations Op Alg . For instance, taking states as a subclass ofoperations, one obtains C ∗ -independence; if algebras A , B and C are von Neumann algebras, taking normal states as the subset of operations one obtains W ∗ -independence; taking normal operations as subclass ofoperations, one obtains operational W ∗ -independence (cf. [29]). One also can define the product versionsof these specific independence concepts by considering Op Alg -independence in the product sense withrespect to the respective subclasses of operations. One has then the notions of C ∗ -and W ∗ -independencein the product sense, and operational C ∗ -and W ∗ -independence in the product sense. Specificationsof further sub-types of independence obtains by considering particular operations such as conditionalexpectations, or Kraus operations (see [29]). The logical relation of these independence concepts emergesthen as a non-trivial problem, some of which are still open [29]. Viewed from the perspective of theresulting hierarchy of independence notions, Op Alg -independence serves as a general, categorial frame inwhich independence can be formulated and analyzed.Given the concept of Op Alg -independence, it is natural to consider it as a possible condition to imposeit on the covariant functor F representing quantum field theory in order to express causal locality in termsof it. To do so we recall first the definition of the functor F describing quantum field theory.13 Op Alg -independence as locality condition in categorial quan-tum field theory
The functor F representing a general covariant quantum field theory is between two categories: (i)( Man , hom
Man ), the category of spacetimes with isometric embeddings of spacetimes as morphisms; and(ii) (
Alg , hom
Alg ), the category of C ∗ -algebras with injective C ∗ -algebra homomorphisms as morphisms.The category ( Man , hom
Man ) is specified by the following stipulations (see [8] for more details):(i) The objects in
Obj ( Man ) are 4 dimensional C ∞ spacetimes ( M, g ) with a Lorentzian metric g andsuch that ( M, g ) is Hausdorff, connected, time oriented and globally hyperbolic.(ii) The morphisms in hom
Man are isometric smooth embeddings ψ : ( M , g ) → ( M , g ) that preservethe time orientation and are causal in the following sense: if the endpoints γ ( a ) , γ ( b ) of a timelikecurve γ : [ a, b ] → M are in the image ψ ( M ), then the whole curve is in the image: γ ( t ) ∈ ψ ( M ) for all t ∈ [ a, b ]. The composition of morphisms is the usual composition of maps. Definition 6.1.
A locally covariant quantum field theory is a functor F between the categories ( Man , hom Man ) and (
Alg , hom
Alg ): For any object (
M, g ) in
Man the F ( M, g ) is a C ∗ -algebra in Alg ; for anyhomomorphism ψ in hom Man the F ( ψ ) is an injective C ∗ -algebra homomorphism in hom Alg . The functor F is required to have the properties 1.-4. below:1. Covariance : F ( ψ ◦ ψ ) = F ( ψ ) ◦ F ( ψ ) F ( id Man ) = id Alg Einstein Causality : Whenever the embeddings ψ : ( M , g ) → ( M, g ) and ψ : ( M , g ) → ( M, g )are such that ψ ( M ) and ψ ( M ) are spacelike in M , then h F ( ψ ) (cid:16) F ( M , g ) (cid:17) , F ( ψ ) (cid:16) F ( M , g ) (cid:17)i F ( M,g ) − = { } (4)where [ , ] F ( M,g ) − in (4) denotes the commutator in the C ∗ -algebra F ( M, g ).3.
Time slice axiom : If (
M, g ) and ( M ′ , g ′ ) and the embedding ψ : ( M, g ) → ( M ′ , g ′ ) are such that ψ ( M, g ) contains a Cauchy surface for ( M ′ , g ′ ) then F ( ψ ) F ( M, g ) = F ( M ′ , g ′ )4. Op Alg - independence : Whenever the embeddings ψ : ( M , g ) → ( M, g ) and ψ : ( M , g ) → ( M, g ) are such that ψ ( M ) and ψ ( M ) are spacelike in M , then the objects F ( M , g ) and F ( M , g ) are Op Alg -independent in F ( M, g ) in the sense of Definition 5.1. (cid:3)
The axiom system specified by Definition 6.1 differs from the one originally proposed in [8] by theaddition of the Op Alg -independence condition. Following the terminology introduced in [31], we call theoriginal axiom system in [8]
BASIC , to distinguish it from the one given by Definition 6.1, which wecall
OPIND . One also can strengthen
OPIND by requiring in 4. in Definition 6.1 that the objects F ( M , g ) and F ( M , g ) are Op Alg -independent in F ( M, g ) in the product sense . We call the resultingaxiom system
OPIND × . 14ther stipulations on the functor are also possible and have been formulated: The axiom system BASIC was amended by Brunetti and Fredenhagen by replacing the Einstein Causality condition byan axiom that requires a tensorial property of F (Axiom 4 in [6]; also see [15]). To formulate thisaxiom one first extends ( Man , hom
Man ) to a tensor category (
Man ⊗ , hom ⊗ Man ). This tensor categoryhas, by definition, as its objects finite disjoint unions of objects from
Man , and the empty set as unitobject. The morphisms h ⊗ in hom ⊗ Man are embeddings of unions of disjoint spacetimes that are hom
Man -homomorphisms when restricted to the (connected) elements of the disjoint union of spacetimes and havethe feature that the images under h ⊗ of disjoint spacetimes are spacelike. The functor F is then requiredto be extendable to a tensor functor F ⊗ between ( Man ⊗ , hom ⊗ Man ) in a natural way. We call the resultingaxiom system
TENSOR .One obtains yet another axiom system if one requires a categorial version of the split property. Thiscondition was formulated in [7] – together with the categorial version of weak additivity. The definitionsare:
Definition 6.2.
The functor F has the categorial split property if the following two conditions hold:1. For spacetimes ( M, g M ) , ( N, g N ) in Man and morphism ψ : ( M, g M ) → ( N, g N ) such that the closureof ψ ( M, g M ) is compact, connected and in the interior of M , there exists a type I von Neumannfactor R such that F ( ψ )( F ( M, g M )) ⊂ R ⊂ F ( N, g N ) (5)2. σ -continuity of the F ( ψ ′ ) with respect to the inclusion R ⊂ R ′ , where ψ ′ : ( M, g M ) → ( L, g L ) and( F ( ψ ′ ) ◦ F ( ψ ))( F ( M, g M )) ⊂ F ( ψ ′ )( R ) (6) ⊂ F ( ψ ′ )( F ( N, g N )) ⊂ R ′ ⊂ F ( L, g L ) (7) (cid:3) Definition 6.3 (weak additivity of the functor F ) . The functor F satisfies weak additivity if for anyspacetime ( M, g ) and any family of spacetimes ( M i , g i ) with morphisms ψ i : ( M i , g i ) → ( M, g ) such that M ⊆ ∪ i ψ i ( M i ) (8)we have F ( M, g ) = ∪ i F ( ψ i )( F ( M i , g i ))) norm (9) (cid:3) We call
BASIC+SPLIT the axiom system that requires of the covariant functor F to have weakadditivity and the categorial split property, in addition to Einstein Locality and Time Slice axiom.As these different conditions imposed on the functor F show, one can articulate the concept ofphysical locality understood as independence of the algebras of observables of spatio-temporaly localphysical systems localized in causally disjoint spacetime regions in more than one way. Thus the questionor relation of the different axiom systems arise, and one also can ask: which one of the axiom systems isthe most adequate.The problem of the relation of the axiom systems was raised in [31], where it was argued that theimplications in the following diagram depicting the logical relations hold. Here we comment on the reverseof the indicated implications below. 15 ENSOR ⇐ OPIND × ⇒ OPIND ⇒ BASIC m BASIC + SPLIT
We have seen in Section 1 that
BASIC does not entail
OPIND . The technical obstacle prohibiting thereverse of the implication
OPIND × ⇒ TENSOR to hold trivially is that operations on C ∗ -subalgebrasof a C ∗ -algebra C need not be extendable to C . Hence, although C ∗ -subalgebras A , B are Op Alg -independent in the tensor product
A ⊗ B , this does not entail without further conditions that A , B are Op Alg -independent in a C ∗ -algebra C containing A ⊗ B as a C ∗ -subalgebra. Injectivity of C would entailthis; however, it is not clear to us whether the C ∗ -algebras F ( M, g ) are injective in general – or at leastfor some specific, typical spacetime regions such as double cones.The reverse of the implication
OPIND × ⇒ OPIND is unlikely to hold, given that operational C ∗ -independence in the product sense is a strictly stronger independence condition than operational C ∗ -independence – but we do not have a rigorous proof of OPIND OPIND × in terms of a model of theaxioms displaying the non-implication.In view of the logical (in)dependencies of the axiom systems depicted in the chart, the conclusion wepropose is that the most natural independence condition to stipulate to hold for the functor F in order toexpress physical locality is Op Alg -independence. This condition has a very natural physical interpretationand it does not require more than what is contained in the notion of subsystem independence as co-possibility. So, if some physically relevant models existed which violate
TENSOR but satisfy
OPIND ,that model would still be entirely acceptable from the perspective of a causal behavior of the quantumfiled theory represented by the functor satisfying
OPIND .Our final remark concerns a possible characterization of spacelike separatedness of spacetime regions assubobject independence with respect to some embeddings of spacetimes as morphisms. Specifically, onewould like to know if the causal embeddings defining the homomorphisms in the category (
Man , hom
Man )have this feature. If indeed hom
Man -independence of spacetimes in the category (
Man , hom
Man ) (in thesense of Definition 2.2) entails spacelike separetedness, then causal locality of the functor F could bedefined in a nice, compact manner as independence-faithfulness of the functor, where independence bothin the domain and in the range of F is captured completely by categorial subobject independence withrespect to natural classes of morphisms. Acknowledgement
Research supported in part by the Hungarian Scientific Research Found (OTKA). Contract numbers: K115593 and K 100715. Zal´an Gyenis was partially supported by the Premium Postdoctoral Grant of theHungarian Academy of Sciences.
References [1] H. Araki.
Mathematical Theory of Quantum Fields , volume 101 of
International Series of Monograps inPhysics . Oxford University Press, Oxford, 1999. Originally published in Japanese by Iwanami ShotenPublishers, Tokyo, 1993.[2] W. Arveson. Subalgebras of C ∗ -algebras. Acta Mathematica , 123:141–224, 1969. This problem was raised by K. Fredenhagen in the discussion after a talk based on this paper was delivered at the“Local Quantum Physics and Beyond – in Memoriam Rudolf Haag”, September 26-27, Hamburg, Germany.
3] S. Awodey.
Category Theory . Oxford University Press, 2010. Second edition.[4] B. Blackadar.
Operator Algebras: Theory of C*-Algebras and von Neumann Algebras . Encyclopaedia ofMathematical Sciences. Springer, 1. edition, 2005.[5] R. Brunetti and K. Fredenhagen. Algebraic approach to quantum field theory. In Jean-Pierre Francoise,Gregory L. Naber, and Tsou Sheung Tsun, editors,
Elsevier Encyclopedia of Mathematical Physics , pages198–204. Academic Press, Amsterdam, 2006. arXiv:math-ph/0411072.[6] R. Brunetti and K. Fredenhagen. Quantum field theory on curved backgrounds. In C. B¨ar and K. Fre-denhagen, editors,
Quantum Field Theory on Curved Spacetimes , volume 786 of
Lecture Notes in Physics ,chapter 5, pages 129–155. Springer, Dordrecht, Heidelberg, London, New York, 2009.[7] R. Brunetti, K. Fredenhagen, I. Paniz, and K. Rejzner. The locality axiom in quantum field theory and tensorproducts of C ∗ -algebras. Reviews of Mathematical Physics , 26:1450010, 2014. arXiv:1206.5484 [math-ph].[8] R. Brunetti, K. Fredenhagen, and R. Verch. The generally covariant locality principle – a new paradigmfor local quantum field theory.
Communications in Mathematical Physics , 237:31–68, 2003. arXiv:math-ph/0112041.[9] D. Buchholz. Algebraic quantum field theory: A status report. In A. Grigoryan, A. Fokas, T. Kibble, andB. Zegarlinski, editors,
XIIIth International Congress on Mathematical Physics, Imperial College, London,UK , pages 31–46. International Press of Boston, Sommervile, MA U.S.A., 2001. arXiv:math-ph/0011044.[10] D. Buchholz and R. Haag. The quest for understanding in relativistic quantum physics.
Journal of Mathe-matical Physics , 41:3674–3697, 2000. arXiv:hep-th/9910243.[11] D. Buchholz and S.J. Summers. Quantum statistics and locality.
Physics Letters A , 337:17–21, 2005.[12] U. Franz. What is stochastic independence? In N. Obata, T. Matsui, A. Hora, and S¯uri Kaiseki Kenky¯ujoKy¯oto Daigaku, editors,
Non-commutativity, Infinite-dimensionality and Probability at the Crossroads: Pro-ceedings of the RIMS Workshop on Infinite-Dimensional Analysis and Quantum Probability: Kyoto, Japan,20-22 November, 2001 , QP-PQ Quantum Probability and White Noise Analysis, pages 254–274. World Sci-entific, 2002. arxiv.org/abs/math/0206017.[13] K. Fredenhagen. Lille 1957: The birth of the concept of local algebras of observables.
The European PhysicalJournal H , 35:239–241, 2010.[14] K. Fredenhagen and K. Reijzner. Quantum field theory on curved spacetimes: Axiomatic framework andexamples.
Journal of Mathematical Physics , 57:031101, 2016.[15] K. Fredenhagen and K. Rejzner. Local covariance and background independence. In F. Finster, O. M¨uller,M. Nardmann, J. Tolksdorf, and E. Zeidler, editors,
Quantum Field Theory and Gravity. Conceptual andMathematical Advances in the Search for a Unified Framework , pages 15–24. Birkh¨auser Springer Basel,Basel, 2012. arXiv:1102.2376 [math-ph].[16] R. Haag.
Local Quantum Physics: Fields, Particles, Algebras . Springer Verlag, Berlin and New York, 1992.[17] R. Haag. Discussion of the ‘axioms’ and the asymptotic properties of a local field theory with compositeparticles.
The European Physical Journal H , 35:243–253, 2010. English translation and re-publication of atalk given at the international conference on mathematical problems of the quantum theory of fields, Lille,June 1957.[18] R. Haag. Local algebras. A look back at the early years and at some achievements and missed opportunities.
The European Physical Journal H , 35:255–261, 2010.[19] R. Haag and D. Kastler. An algebraic approach to quantum field theory.
Journal of Mathematical Physics ,5:848–861, 1964.[20] S.S. Horuzhy.
Introduction to Algebraic Quantum Field Theory . Kluwer Academic Publishers, Dordrecht,1990.[21] G. Kalmbach.
Orthomodular Lattices . Academic Press, London, 1983.
22] K. Kraus.
States, Effects and Operations , volume 190 of
Lecture Notes in Physics . Springer, New York, 1983.[23] F. Maeda. Direct sums and normal ideals of lattices.
Journal of Science of the Hiroshima University. SeriesA , 14:85–92, 1949.[24] V. Paulsen.
Completely Bounded Maps and Operator Algebras , volume 78 of
Cambridge Studies in AdvancedMathematics . Cambridge University Press, Cambridge, 2003.[25] Richard S. Pierce.
Introduction to the Theory of Abstract Algebras . Dover Publications, Mineola, New York,2014. Originally published by Holt, Rinehart and Winston, Inc., New York, 1968.[26] M. R´edei. Logical independence in quantum logic.
Foundations of Physics , 25:411–422, 1995.[27] M. R´edei. Logically independent von Neumann lattices.
International Journal of Theoretical Physics ,34:1711–1718, 1995.[28] M. R´edei.
Quantum Logic in Algebraic Approach , volume 91 of
Fundamental Theories of Physics . KluwerAcademic Publisher, 1998.[29] M. R´edei. Operational independence and operational separability in algebraic quantum mechanics.
Founda-tions of Physics , 40:1439–1449, 2010.[30] M. R´edei. A categorial approach to relativistic locality.
Studies in History and Philosophy of Modern Physics ,48:137–146, 2014.[31] M. R´edei. Categorial local quantum physics. In J. Butterfield, H. Halvorson, M. R´edei, J. Kitajima,F. Buscemi, and M. Ozawa, editors,
Reality and Measurement in Algebraic Quantum Theory , Proceedings inMathematics & Statistics (PROMS). Springer, 2016. under review.[32] M. R´edei and S.J. Summers. When are quantum systems operationally independent?
International Journalof Theoretical Physics , 49:3250–3261, 2010.[33] M. R´edei and G. Valente. How local are local operations in local quantum field theory?
Studies in Historyand Philosophy of Modern Physics , 41:346–353, 2010.[34] S.J. Summers. On the independence of local algebras in quantum field theory.
Reviews in MathematicalPhysics , 2:201–247, 1990.[35] S.J. Summers. Subsystems and independence in relativistic microphysics.
Studies in History and Philosophyof Modern Physics , 40:133–141, 2009. arXiv:0812.1517 [quant-ph].[36] S.J. Summers. A perspective on constructive quantum field theory. arXiv:1203.3991 [math-ph], 2012. Thisis an expanded version of an article commissioned for UNESCO’s Encyclopedia of Life Support Systems(EOLSS).[37] S. Weinberg.
The Quantum Theory of Fields. Vol. 1: Foundations . Cambridge University Press, Cambridge,1995.. Cambridge University Press, Cambridge,1995.