Category-Theoretic Interpretative Framework of the Complementarity Principle in Quantum Mechanics

Abstract

This study aims to provide an analysis of the complementarity principle in quantum theory through the establishment of partial structural congruence relations between the quantum and Boolean kinds of event structure. Specifically, on the basis of the existence of a categorical adjunction between the category of quantum event algebras and the category of presheaves of variable Boolean event algebras, we establish a twofold complementarity scheme consisting of a generalized/global and a restricted/local conceptual dimension, where the latter conception is subordinate to and constrained by the former. In this respect, complementarity is not only understood as a relation between mutually exclusive experimental arrangements or contexts of comeasurable observables, as envisaged by the original conception, but it is primarily comprehended as a reciprocal relation concerning information transfer between two hierarchically different structural kinds of event structure that can be brought into partial congruence by means of the established adjunction. It is further argued that the proposed category-theoretic framework of complementarity naturally advances a contextual realist conceptual stance towards our deeper understanding of the microphysical nature of reality.

Appendix A

1.1 A.1 Related Work

The major source of inspiration in what is presently called categorical or topos-theoretic approaches to the foundations of quantum physics emanates from the 1998-1999 work of Isham and Butterfield (IB) [28, 29], who provided a concrete topos reformulation of the Kochen-Specker theorem, albeit proposals targeting the utility and relevance of these new mathematical methods appeared earlier (e.g., [42]). For this purpose, they considered the partially ordered set (poset) of commutative von Neumann subalgebras of the non-commutative algebra of all bounded operators on a quantum Hilbert space as a “category of contexts” where the only arrows are inclusions. The latter “category of contexts” served as a base category for defining the topos of presheaves of sets over the poset of commutative subalgebras. This topos-theoretic initiative has been extended, elaborated and developed further around ten years later by Döring and Isham (DI) (e.g., [20] and references therein). The central principle of their work is that the construction of a theory of physics, like quantum mechanics, is equivalent to finding a representation in “a topos of a certain formal language” that is attached to the system. Conceptually, the (DI) scheme is based on Lawvere’s notion of an “elementary topos”, i.e., the logical embodiment of the topos concept, serving as a generalized model of set theory being equipped with a subobject classifier, specified by a distributive Heyting algebra forcing an intuitionistic type of semantics [30].

Shortly afterwards, in relation to the (IB) approach, appeared an alternative topos-theoretic approach to the foundations of quantum physics by the first author [44], which has been motivated, not by the notion of an “elementary topos”, as in the (IB) case, but by the notion of a “Grothendieck topos’” [39]. Since 2000, this approach has been developed in a series of works targeting the observable [45, 46], measure-theoretic [47], topological [48], geometric [49], logical [50], semantical [33] and conceptual-philosophical [21, 33, 34, 50] aspects of quantum theory. The notion of a “Grothendieck topos” interprets a topos as a generalized geometric environment, which makes it possible to constitute the structural information content of certain global objects, like quantum event algebras, from the non-trivial and non-classical localization properties of observables, which are utilized to “cover” locally these global objects in terms of interconnected multitudes of measurement contexts at various overlapping scales. This form of synthetic amalgamation of a global quantum object from the joint, pairwise compatible, and colimiting gluing of locally congruent with it Boolean objects culminates in the concept of a sheaf defined over a base category of Boolean frames. It should be emphasized that in contradistinction to the (IB) and (DI) approach who consider only the partial ordering of “contexts” as their base category, the full category of complete Boolean algebras with Boolean homomorphisms is used as a category of possible frames (by virtue of the spectral theorem) without any restriction to inclusions or injective morphisms. Most important, in our categorical approach to quantum mechanics, the “Grothendieck topos” interpretation is founded on the existence of a categorical adjunction between the category of presheaves of variable Boolean probing frames and the category of quantum event algebras. As systematically developed in the present paper, it is precisely the semantics of the resulting pair of adjoint functors (L ⊣R) that gives rise to the proposed re-conceptualization of the complementarity principle in quantum meshanics. The basic underlying idea formalized by this adjunctive pair of functors concerns the partial or local structural congruence between the Boolean and quantum levels of event structure as well as the encoding and decoding bridges from one to the other. The congruence relations are induced by the joint action of Boolean covering families of local frames on a global quantum event algebra at various layers of “observable resolution”. The left and right adjoint functors in this Boolean frames–quantum adjunction subsume the role of natural bridges between the Boolean and quantum structural levels shedding new light on the complementarity principle via a local and a global conceptual dimension interpreted sheaf-theoretically.

In this connection, there seems to exist an apparent affinity with another topos-theoretic approach, called “Bohrification”, by Heunen, Landsman and Spitters (HLS) circa 2010 ([24] and references therein). This approach has been an outgrowth of the (DI) program motivated by the adoption of a dual covariant perspective. They construct an internal commutative C^∗-algebra (or an internal Boolean algebra) within the topos of co-presheaves over the poset of “classical contexts”, so their topos comes equipped with an “internal commutative algebraic object”, which is interpreted, to some extent, as a formal expression of Bohr’s doctrine of classical concepts. In relation to both (DI) and (HLS) frameworks, it has to be observed that the partial order relation of “classical contexts” is not adequate to capture the global structural information of a quantum event structure, and at least, the inclusion morphisms of the “Boolean contexts” to the quantum lattice should be also taken into account. In comparison, the present topos-approach, not based on the partial order relation of “Boolean contexts”, specifies globally a quantum event structure via the left-adjoint functor of the established adjunction.

Our particular interest to the “Bohrification program” here stems from the fact that it has been applied recently by Heunen [25] as a means to explicate the complementarity principle in quantum mechanics. So it is quite elucidating to point out a fundamental difference underlying our respective perspectives and interpretation of this principle. According to Heunen, “complete complementarity” refers to taking all pairwise partially compatible classical contexts together. Specifically, given a certain C^∗-algebra A, representing the observables of a physical system, Heunen’s notion of “complete complementarity” is formalized by collecting all commutative C^∗-subalgebras C of A into a single mathematical object, a partially ordered set ${\mathcal C}(A)$, whose partial ordering is given by set-theoretic inclusion. Each C in ${\mathcal C}(A)$ represents a classical or experimental context, which has been disconnected from the others, except for the inclusion relations which relate compatible experiments. Then, “complete complementarity” means that it suffices to examine all commutative subalgebras (or “classical contexts”) C to determine the empirical content of the system modeled by the initial algebra A. It is instructive to note, however, that this construction applies in an identical manner even when A itself is commutative. Otherwise, should A be non-commutative, then the best one can do is to approximate it with ever larger commutative subalgebras C. Hence, it is clear that ${\mathcal C}(A)$ in itself is not adequate to reconstruct A. In a purely mathematical setting, ${\mathcal C}(A)$ may be seen as a useful tool for classifying C^∗-algebras through the operation of set-theoretic inclusion, but it cannot be appropriately regarded as the mathematical space of complementarity, the latter requiring the existence of pairs of incompatible observables standing in a complementary relation in each other’s determination. In our view, the preceding notion of “complete complementarity” can be rather considered as an elaboration of contextuality in relation to a quantum structure; it does not qualify properly the meaning of quantum complementarity. No doubt, complementarity embraces the notion of contextuality, but the principle of complementarity is neither reduced to contextuality nor exhausted by it. The reason is that partial compatibility among “classical contexts” within an overall quantum structure can be properly described in local sheaf-theoretic terms. In the absence of the crucial local-global distinction, the complementarity principle can be wrongly conflated with the notion of contextuality, since there is no functorial bridge of transferring information from the level of local contexts to the hierarchically different level of quantum structures and inversely. For this reason, we distinguish between a local and a global conceptual dimension of the complementarity principle, which only if considered together, they convey the essential meaning of this principle, formalized by means of a pair of adjoint functors.

It is worth mentioning, in passage, that there have been also developed recently other sheaf-theoretic and categorical approaches for studying, for instance, the issue of non-locality in relation to contextuality (e.g., [1]), providing context-dependent operations in logic (e.g., [19]), and representing various physical processes (e.g., [16]). Finally, we note that the interested reader may find particularly helpful the Appendix A.1. in [50], where it is presented a detailed comparison of the technical part of our category-theoretic approach with all other major categorical approaches to the foundations of quantum physics, including those mentioned above.