It was an old idea by Niels Bohr, one of the founding architects of quantum physics, that central features of quantum theory, such as complementarity, are also of pivotal significance beyond the domain of physics. Bohr became familiar with the notion of complementarity through the psychologist Edgar Rubin and, indirectly, William James (Holton Reference Holton1970). Although Bohr always insisted on the extraphysical relevance of complementarity, he never elaborated this idea in concrete detail, and for a long time no one else did so either. This situation has changed; there are now a number of research programs applying key notions of quantum theory beyond physics, in particular to psychology and cognitive science.
The first steps in this direction were made by Aerts and collaborators in the early 1990s (Aerts & Aerts Reference Aerts and Aerts1995) in the framework of non-Boolean logic of incompatible (complementary) propositions. Alternative ideas come from Khrennikov (Reference Khrennikov1999), focusing on non-classical probabilities, and Atmanspacher et al. (Reference Atmanspacher, Römer and Walach2002), proposing an algebraic framework with non-commuting operations. More recently, Bruza and colleagues as well as Busemeyer and colleagues have moved this novel field of research even more into the center of attention. The target article by Pothos & Busemeyer (P&B), and a novel monograph by Busemeyer and Bruza (Reference Busemeyer and Bruza2012), reflect these developments.
Intuitively, it is plausible that non-commuting operations or non-Boolean logic should be relevant, even inevitable, for mental systems. The non-commutativity of operations simply means that the sequence in which operations are applied matters for the final result. This is so if individual mental states are assumed to be dispersive (as individual quantum states are, as opposed to classical states). As a consequence, their observation amounts not only to registering a value, but entails a backreaction changing the observed state: something that seems evident for mental systems.
Non-Boolean logic refers to propositions that may have unsharp truth values beyond “yes” or “no.” However, this is not the result of subjective ignorance but must be understood as an intrinsic feature. The proper framework for a logic of incompatible propositions is a partial Boolean lattice (Primas Reference Primas2007), where locally Boolean sublattices are pasted together in a globally non-Boolean way – just like an algebra of generally non-commuting operations may contain a subset of commuting operations.
Although these formal extensions are essential for quantum theory, they have no dramatic effect on the way in which experiments are evaluated. The reason is that the measuring tools, even in quantum physics, are typically Boolean filters, and, therefore, virtually all textbooks of quantum physics get along with standard probability theory à la Kolmogorov. Only if incompatible experimental scenarios are to be comprehensively discussed in one single picture do the pecularities provided by non-classical thinking become evident and force us to leave outmoded classical reasoning.
In this sense, the authors use the notion of “quantum probability” for psychological and cognitive models and their predictions (cf. Gudder Reference Gudder1988; Redei & Summers Reference Redei and Summers2007). As Busemeyer points out, an experiment in psychology is defined as a collection of experimental conditions. Each one of them produces indivisible outcomes forming a complete set of mutually exclusive events. Whereas Kolmogorov probabilities refer to events for a single condition, quantum probabilities refer to the entire set of incompatible conditions, necessary for a comprehensive description of the experiment.
In such a description, events are represented as subspaces of a Hilbert space (as in quantum physics), and all subspaces correspond to orthogonal projectors. A state is defined as a vector, and the probability of an event equals the squared length of the projection of the state onto the subspace representing that event. As all events under each single experimental condition commute, they form a Boolean algebra and the probabilities assigned to them satisfy the axiomatics of Kolmogorov. However, all events of the entire experiment (i.e., the events of all experimental conditions) only form a partial Boolean algebra if some of them do not commute. And as Kolmogorov's axioms imply Bayes' rule, Bayesian reasoning, very influential in psychology, will generally fail to describe experiments with incompatible conditions properly.
Whereas the authors focus on decision theory, routes to be explored in more detail include uncertainty relations, in which order effects arise in variances in addition to mean shifts (Atmanspacher & Römer Reference Atmanspacher and Römer2012). A key feature of quantum theory, entanglement as tested by Bell-type inequalities, has been suggested by Atmanspacher and Filk (Reference Atmanspacher and Filk2010) for bistable perception and by Bruza et al. (Reference Bruza, Kitto, Ramm and Sitbon2012) for non-decomposable concept combinations.
Another possible move to incorporate complementarity and entanglement in psychology is based on a state space description of mental systems. If mental states are defined on the basis of cells of a state space partition, then this partition needs to be well tailored to lead to robustly defined states. Ad hoc chosen partitions will generally create incompatible descriptions (Atmanspacher & beim Graben Reference Atmanspacher and beim Graben2007) and states may become entangled (beim Graben et al. Reference beim Graben, Filk and Atmanspacher2013). This way it is possible to understand why mental activity may exhibit features of quantum behavior whereas the underlying neural dynamics are strictly classical.
A further important issue is the complexity or parsimony of Hilbert space models as compared with classical (Bayesian, Markov) models. Atmanspacher and Römer (Reference Atmanspacher and Römer2012) proposed an option to test limitations of Hilbert space modeling by outcomes of particular joint measurements. Such tests presuppose that the situation under study is framed well enough to enable well-defined research questions; a requirement that must be carefully observed to avoid superficial reasoning without sustainable substance.
With the necessary caution, I am optimistic that this novel field will grow from work in progress to an important subject area of psychology. A quantum theoretically inspired understanding of reality, including cognition, will force us to revise plugged-in cliches of thinking and resist overly naive world views. The Boolean “either-or” in logic and the law of commutativity in elementary calculations are special cases with their own significance, but it would be wrong to think that their generalization holds potential only for exotic particles and fields in physics. The opposite is the case.
It was an old idea by Niels Bohr, one of the founding architects of quantum physics, that central features of quantum theory, such as complementarity, are also of pivotal significance beyond the domain of physics. Bohr became familiar with the notion of complementarity through the psychologist Edgar Rubin and, indirectly, William James (Holton Reference Holton1970). Although Bohr always insisted on the extraphysical relevance of complementarity, he never elaborated this idea in concrete detail, and for a long time no one else did so either. This situation has changed; there are now a number of research programs applying key notions of quantum theory beyond physics, in particular to psychology and cognitive science.
The first steps in this direction were made by Aerts and collaborators in the early 1990s (Aerts & Aerts Reference Aerts and Aerts1995) in the framework of non-Boolean logic of incompatible (complementary) propositions. Alternative ideas come from Khrennikov (Reference Khrennikov1999), focusing on non-classical probabilities, and Atmanspacher et al. (Reference Atmanspacher, Römer and Walach2002), proposing an algebraic framework with non-commuting operations. More recently, Bruza and colleagues as well as Busemeyer and colleagues have moved this novel field of research even more into the center of attention. The target article by Pothos & Busemeyer (P&B), and a novel monograph by Busemeyer and Bruza (Reference Busemeyer and Bruza2012), reflect these developments.
Intuitively, it is plausible that non-commuting operations or non-Boolean logic should be relevant, even inevitable, for mental systems. The non-commutativity of operations simply means that the sequence in which operations are applied matters for the final result. This is so if individual mental states are assumed to be dispersive (as individual quantum states are, as opposed to classical states). As a consequence, their observation amounts not only to registering a value, but entails a backreaction changing the observed state: something that seems evident for mental systems.
Non-Boolean logic refers to propositions that may have unsharp truth values beyond “yes” or “no.” However, this is not the result of subjective ignorance but must be understood as an intrinsic feature. The proper framework for a logic of incompatible propositions is a partial Boolean lattice (Primas Reference Primas2007), where locally Boolean sublattices are pasted together in a globally non-Boolean way – just like an algebra of generally non-commuting operations may contain a subset of commuting operations.
Although these formal extensions are essential for quantum theory, they have no dramatic effect on the way in which experiments are evaluated. The reason is that the measuring tools, even in quantum physics, are typically Boolean filters, and, therefore, virtually all textbooks of quantum physics get along with standard probability theory à la Kolmogorov. Only if incompatible experimental scenarios are to be comprehensively discussed in one single picture do the pecularities provided by non-classical thinking become evident and force us to leave outmoded classical reasoning.
In this sense, the authors use the notion of “quantum probability” for psychological and cognitive models and their predictions (cf. Gudder Reference Gudder1988; Redei & Summers Reference Redei and Summers2007). As Busemeyer points out, an experiment in psychology is defined as a collection of experimental conditions. Each one of them produces indivisible outcomes forming a complete set of mutually exclusive events. Whereas Kolmogorov probabilities refer to events for a single condition, quantum probabilities refer to the entire set of incompatible conditions, necessary for a comprehensive description of the experiment.
In such a description, events are represented as subspaces of a Hilbert space (as in quantum physics), and all subspaces correspond to orthogonal projectors. A state is defined as a vector, and the probability of an event equals the squared length of the projection of the state onto the subspace representing that event. As all events under each single experimental condition commute, they form a Boolean algebra and the probabilities assigned to them satisfy the axiomatics of Kolmogorov. However, all events of the entire experiment (i.e., the events of all experimental conditions) only form a partial Boolean algebra if some of them do not commute. And as Kolmogorov's axioms imply Bayes' rule, Bayesian reasoning, very influential in psychology, will generally fail to describe experiments with incompatible conditions properly.
Whereas the authors focus on decision theory, routes to be explored in more detail include uncertainty relations, in which order effects arise in variances in addition to mean shifts (Atmanspacher & Römer Reference Atmanspacher and Römer2012). A key feature of quantum theory, entanglement as tested by Bell-type inequalities, has been suggested by Atmanspacher and Filk (Reference Atmanspacher and Filk2010) for bistable perception and by Bruza et al. (Reference Bruza, Kitto, Ramm and Sitbon2012) for non-decomposable concept combinations.
Another possible move to incorporate complementarity and entanglement in psychology is based on a state space description of mental systems. If mental states are defined on the basis of cells of a state space partition, then this partition needs to be well tailored to lead to robustly defined states. Ad hoc chosen partitions will generally create incompatible descriptions (Atmanspacher & beim Graben Reference Atmanspacher and beim Graben2007) and states may become entangled (beim Graben et al. Reference beim Graben, Filk and Atmanspacher2013). This way it is possible to understand why mental activity may exhibit features of quantum behavior whereas the underlying neural dynamics are strictly classical.
A further important issue is the complexity or parsimony of Hilbert space models as compared with classical (Bayesian, Markov) models. Atmanspacher and Römer (Reference Atmanspacher and Römer2012) proposed an option to test limitations of Hilbert space modeling by outcomes of particular joint measurements. Such tests presuppose that the situation under study is framed well enough to enable well-defined research questions; a requirement that must be carefully observed to avoid superficial reasoning without sustainable substance.
With the necessary caution, I am optimistic that this novel field will grow from work in progress to an important subject area of psychology. A quantum theoretically inspired understanding of reality, including cognition, will force us to revise plugged-in cliches of thinking and resist overly naive world views. The Boolean “either-or” in logic and the law of commutativity in elementary calculations are special cases with their own significance, but it would be wrong to think that their generalization holds potential only for exotic particles and fields in physics. The opposite is the case.