Abstract Consciousness is a central issue in cognitive neuroscience. To explain the relationship between consciousness and its neural correlates, various theories have been proposed. We still lack a formal framework that can address the nature of the relationship between consciousness and its physical substrates though. Here, we provide a novel mathematical framework of Category Theory (CT), in which we can define and study the “sameness” between “different” domains of phenomena such as consciousness and its neural substrates. CT was designed and developed to deal with the “relationships” between various domains of phenomena. We introduce three concepts of CT including (i) category; (ii) inclusion functor and expansion functor; and (iii) natural transformation between the functors. Each of these mathematical concepts is related to specific features in the neural correlates of consciousness (NCC). In this novel framework, we will examine two of the major theories of consciousness: integrated information theory (IIT) of consciousness and temporo-spatial theory of consciousness (TTC). These theories concern the structural relationships among structures of physical substrates and subjective experiences. The three CT-based concepts, introduced in this paper, unravel some basic issues in our search for the NCC; while addressing the same questions, we show that IIT and TTC provide different albeit complementary answers. Importantly, our account suggests that we need to go beyond a traditional concept of NCC including both content-specific and full NCC. We need to shift our focus from the relationship between “one” neuronal and “one” phenomenal state to the relationship between a structure of neural states and a structure of phenomenal states. We conclude that CT unravels and highlights basic questions about the NCC in general which needs to be met and addressed by any future neuroscientific theory of consciousness. Author summary Neuroscience has made considerable progress in uncovering the neural correlates of consciousness (NCC). At the same time, recent studies demonstrated the complexity of the neuronal mechanisms underlying consciousness. To make further progress in the neuroscience of consciousness, we need proper mathematical formalization of the neuronal mechanisms potentially underlying consciousness. Providing a first tentative attempt, our paper addresses both by (i) pointing out the specific problems of and proposing a new approach to go beyond the traditional approach of the neural correlates of consciousness, and (ii) by recruiting a recently popular mathematical formalization, category theory (CT). With CT, we provide mathematical formalization of the broader neural correlates of consciousness by its application to two of the major theories, integrated information theory (IIT) and temporo-spatial theory of consciousness (TTC). Together, our CT-based mathematical formalization of the neural correlates of consciousness including its specification in the terms of IIT and TTC allows to go beyond the current concept of NCC in both mathematical and neural terms.
Interacting Fock space connects the study of quantum probability theory, classical random variables, and orthogonal polynomials. It is a pre-Hilbert space associated with creation, preservation, and annihilation processes. We prove that if three processes are asymptotically commutative, the arcsine law arises as the "large quantum number limits." As a corollary, it is shown that for many probability measures, asymptotic behavior of orthogonal polynomials is described by the arcsine function. A weaker form of asymptotic commutativity provides us a discretized arcsine law.
We define a new concept of local states in the framework of algebraic quantum field theory (AQFT). Local states are a natural generalization of states and give a clear vision of localization in the context of QFT. In terms of them, we can find a condition from which follows automatically the famous DHR selection criterion in DHR-DR theory. As a result, we can understand the condition as consequences of physically natural state preparations in vacuum backgrounds. Furthermore, a theory of orthogonal decomposition of completely positive (CP) maps is developed. It unifies a theory of orthogonal decomposition of states and order structure theory of CP maps. By using it, localized version of sectors is formulated, which gives sector theory for local states with respect to general reference representations.
Abstract We investigate operator-valued monotone independence, a noncommutative version of independence for conditional expectation. First we introduce operator-valued monotone cumulants to clarify the whole theory and show the moment-cumulant formula. As an application, one can obtain an easy proof of the central limit theorem for the operator-valued case. Moreover, we prove a generalization of Muraki’s formula for the sum of independent random variables and a relation between generating functions of moments and cumulants.
We prove that the Arcsine law as the time-averaged distribution for classical harmonic oscillators emerges from the distributions for quantum harmonic oscillators in terms of noncommutative algebraic probability. This is nothing but a simple and rigorous realization of "Quantum-Classical Correspondence" for harmonic oscillators.
Human consciousness is characterized by constant transitions in time. On the other hand, what is consciously experienced always possesses the temporal feature of “now.” In consciousness, “now” constantly holds different contents, yet it remains “now” no matter how far it goes. This duality is thematized in Husserlian phenomenology as “the standing-streaming now.” Although this phrase appears contradictory in everyday language, it has a structure that can be clearly understood and formalized. In this paper, we show that this structure can be described as a monoid in category theory. Furthermore, monoids can be transformed into the coslice category, which corresponds to the way of perceiving present moments as juxtaposed in succession. The seemingly contradictory nature of the “now” as both flowing and standing can be precisely structured and comprehended through the monoid, while the perspective of the “now” as discrete points on a timeline can be effectively formalized using the coslice category. This framework helps us more precisely understand the differences between ordinary consciousness and meditative consciousness, specifically the experience of the “eternal now.” We illustrate how the meditative states of consciousness presented in the early Buddhist scriptures (Pali Canon) and Dōgen’s Shōbōgenzō remarkably reflect a monoid structure.
Soft robotics is an emerging field of research where the robot body is composed of compliant and soft materials. It allows the body to bend, twist, and deform to move or to adapt its shape to the environment for grasping, all of which are difficult for traditional hard robots with rigid bodies. However, the theoretical basis and design principles for soft robotics are not well-founded despite their recognized importance. For example, the control of soft robots is outsourced to morphological attributes and natural processes; thus, the coupled relations between a robot and its environment are particularly crucial. In this paper, we propose a mathematical foundation for soft robotics based on category theory, which is a branch of abstract math where any notions can be described by objects and arrows. It allows for a rigorous description of the inherent characteristics of soft robots and their relation to the environment as well as the differences compared to conventional hard robots. We present a notion called the category of mobility that well describes the subject matter. The theory was applied to a model system and analysis to highlight the adaptation behavior observed in universal grippers, which are a typical example of soft robotics. This paper paves the way to developing a theoretical background and design principles for soft robotics.
