Interpreting the Modal Kochen–Specker theorem: Possibility and many worlds in quantum mechanics

Abstract

In this paper we attempt to physically interpret the Modal Kochen–Specker (MKS) theorem. In order to do so, we analyze the features of the possible properties of quantum systems arising from the elements in an orthomodular lattice and distinguish the use of “possibility” in the classical and quantum formalisms. Taking into account the modal and many worlds non-collapse interpretation of the projection postulate, we discuss how the MKS theorem rules the constraints to actualization, and thus, the relation between actual and possible realms.

Introduction

In classical physics, every physical system may be described exclusively by means of its actual properties, taking ‘actuality’ as expressing the preexistent mode of being of the properties themselves, independently of observation—the ‘pre’ referring to its existence previous to measurement. The evolution of the system may be described by the change of its actual properties. Mathematically, the state is represented by a point $(p,q)$ in the corresponding phase space $\Gamma$ and, given the initial conditions, the equation of motion tells us how this point evolves in $\Gamma$.² Physical magnitudes are represented by real functions over $\Gamma$. These functions commute with each other and can be all interpreted as possessing definite values at any time, independently of physical observation. In this scheme, speaking about potential or possible properties usually refers to functions defined on points in $\Gamma$ to which the state of the system will arrive at a future instant of time; these points, in turn are completely determined by the equations of motion and the initial conditions.

In the orthodox formulation of quantum mechanics (QM), the representation of the state of a system is given by a ray in Hilbert space $\mathcal{H}$. Contrary to the classical scheme, physical magnitudes are represented by operators on $\mathcal{H}$ that, in general, do not commute. This mathematical fact has extremely problematic interpretational consequences for it is then difficult to affirm that these quantum magnitudes are simultaneously preexistent. In order to restrict the discourse to sets of commuting magnitudes, different Complete Sets of Commuting Operators (CSCOs) have to be chosen. This choice has not found until today a clear justification and remains problematic. In the literature this feature is called quantum contextuality—it will be discussed in Section 2. Another fundamental feature of QM is due to the linearity of the Schrödinger equation which implies the existence of entangled states involving the measuring device. The path from such an entangled state, i.e. a superposition of eigenstates of the measured observable to the eigenstate corresponding to the measured eigenvalue is given, formally, by an axiom added to the formalism: the projection postulate. In Section 3 we will discuss the different physical interpretations of this postulate which is, either thought in terms of a “collapse” of the wave function (i.e., as a real physical interaction) or in terms of non-collapse proposals, such as the modal and many worlds interpretations. After having introduced and discussed these two main features of QM we will present, in Section 4, our formal analysis regarding possibility in orthomodular structures. In Section 5, we shall discuss and analyze the distinction between mathematical formalism and physical interpretation, a distinction which can raise many pseudo-problems if not carefully taken into account. As a consequence of this distinction we will also put forward the difference between ‘classical possibility’ and ‘quantum possibility’. In Section 6, we are ready to advance towards a physical interpretation of both quantum possibility and the MKS theorem—taking into account the specific formal constraints to modality implied by it. In Section 7 we will discuss the consequences of the MKS theorem regarding the many worlds interpretation. Finally, in Section 8, we provide the conclusions of our work.