Interpreting the Modal Kochen Specker Theorem: On Quantum Possibility and Potentiality

CHRISTIAN DE RONDE

Munich

Encuentro; Foundations of Physics ? The 17th UK and European Meeting; 2013

The idea that a preexistent ??pre? making reference to the consideration of existence independent of observation or measurement? set of definite prop- erties constitute or describe reality is one of the basic ideas which remains the fundament of all classical physical theories and determines the possi- bility to discuss about an independent objective world, a world which does not depend on our choices or consciousness. Physical reality can be then conceived and analyzed in terms of a theory ?which describes a preexis- tent world? independently of actual observation. But, as it is well known, this description of physical reality faces several inconveniences when presup- posed in the interpretation of the quantum formalism. In formal terms, this is demonstrated by the Kochen-Specker (KS) theorem, which states that if we consider three physical magnitudes represented by operators A, B and C, with A commuting with B and C but B non-commuting with C, the value of A depends on the choice of the context of inquiry; i.e. whether A is considered together with B or together with C [2]. From an instrumentalist point of view, this is bypassed by considering the context (in KS sense) as the experimental arrangement. However, if we attempt to go beyond the discourse regarding measurement results and provide some kind of realist representation of what is going on according to quantum mechanics (QM), we need to make sense of the indeterminateness of definite valued properties. Contextuality can be directly related to the impossibility to represent a piece of the world as constituted by a set of definite valued properties inde- pendently of the choice of the context. This definition makes reference only 1 to the actual realm. But as we know, QM makes probabilistic assertions about measurement results. Therefore, it seems natural to assume that QM does not only deal with actualities but also with possibilities. Then the question arises whether the space of possibilities is subject to the same re- strictions as the space of actualities. Formally, on the one hand, the set of actualities is structured as the orthomodular lattice of subspaces of the Hilbert space of the states of the system and, as Michael Dickson remarks in [1], the KS theorem (i.e., the absence of a family of compatible valua- tions from subalgebras of the orthomodular lattice to the Boolean algebra of two elements 2) can be understood as a consequence of the failure of the distributive law in the lattice. On the other hand, the set of possibilities is the center of the enlarged structure that might be constructed with an adequate definition of the possibility operator Q. This construction was de- veloped elsewhere. Since the elements of the center of a structure are those which commute with all other elements, one might think that the possible propositions defined in this way escape from the constrains arising from the non-commutative character of the algebra of operators. Thus, at first sight one might conclude that possibilities behave in a classical manner. In order to explicitly verify whether modal propositions escape from KS- type contradictions, in previous works we have developed a mathematical scheme which allowed us to deal with both actual and possible propositions in the same structure. Within this frame we were able to prove a theorem which describes the algebraic relations between both kinds of propositions. The theorem shows explicitly the formal limits of possible actualizations, in short, that no enrichment of the orthomodular lattice with modal proposi- tions allows to circumvent the contextual character of the quantum language. For obvious reasons, it was called the Modal Kochen-Specker (MKS) theo- rem. It is important to remark that our formalism also provides a formal meaning in an algebraic frame to the Born rule. In this paper we attempt to physically interpret the Modal Kochen- Specker theorem. In order to do so, we analyze the features of the possible properties about quantum systems arising from the elements in an ortho- modular lattice and distinguish the use of ?possibility? in the classical and quantum realms as related to their particular formalisms. 2 References [1] Dickson, W. M., 2001, ?Quantum logic is alive ∧ (It is true ∨ It is false)?, Proceedings of the Philosophy of Science Association 2001, 3, S274-S287. [2] Kochen, S. and Specker, E., 1967, ?On the problem of Hidden Variables in Quantum Mechanics?, Journal of Mathematics and Mechanics, 17, 59-87. Reprinted in Hooker, 1975, 293-328.