Modeling observers as physical systems representing the world from within: Quantum theory as a physical and self-referential theory of inference John Realpe-G´omez1∗ Theoretical Physics Group, School of Physics and Astronomy, The University of Manchestery, Manchester M13 9PL, United Kingdom and Instituto de Matem´aticas Aplicadas, Universidad de Cartagena, Bol´ıvar130001, Colombia (Dated: June 13, 2019) In 1929 Szilard pointed out that the physics of the observer may play a role in the analysis of experiments. The same year, Bohr pointed out that complementarity appears to arise naturally in psychology where both the objects of perception and the perceiving subject belong to `our mental content'. Here we argue that the formalism of quantum theory can be derived from two related intu- itive principles: (i) inference is a classical physical process performed by classical physical systems, observers, which are part of the experimental setup|this implies non-commutativity and imaginary- time quantum mechanics; (ii) experiments must be described from a first-person perspective|this leads to self-reference, complementarity, and a quantum dynamics that is the iterative construction of the observer's subjective state. This approach suggests a natural explanation for the origin of Planck's constant as due to the physical interactions supporting the observer's information process- ing, and sheds new light on some conceptual issues associated to the foundations of quantum theory. It also suggests that fundamental equations in physics are typically of second order, instead of the more parsimonious first-order equations, due to the physical nature of the observer. It furthermore suggests some experimental conjectures: (i) the quantum of action could be understood as the re- sult of the additional energy required to transition from unconscious to conscious perception|this is consistent with available experimental data; (ii) humans can observe a single photon of visible light|this is related to (i) and is consistent with existing psychophysics experiments; (iii) the neural correlates of the self are composed of two complementary sub-processes that essentially model each other, much like the DNA molecule is composed of two strands that essentially produce a copy of each other|this may help explain why the brain is divided into hemispheres and suggests self-aware systems should have a similar architecture. Moreover, by explicitly and consistently incorporating us observers and our everyday first-person perspective into the foundations of physics, this approach may help bridge the gap between science and human experience. We discuss the potential implica- tions of these ideas for the modern research program on consciousness championed by Nobel laureate Francis Crick and the emerging field of contemplative science. As side results: (i) we show that message-passing algorithms and stochastic processes can be written in a quantum-like manner| this may suggest novel ways to simulate quantum systems with message-passing algorithms or to naturally implement these powerful distributed algorithms on quantum computers; (ii) we provide evidence that non-stoquasticity, a quantum computational resource, in some cases may be related to non-equilibrium phenomena|this suggests that some of the potential advantage of quantum computers associated to non-stoquasticity may be related to the type of computational advantages recently observed in non-equilibrium Monte Carlo methods where detailed balance is broken; (iii) we provide a different Hamiltonian function for a quantum particle in a classical electromagnetic field—this may suggest a probabilistic interpretation of electromagnetic phenomena. \Describe the real factual situation." A. Inference as a physical process7 B. First-person perspective and Albert Einstein self-reference9 arXiv:1705.04307v5 [quant-ph] 12 Jun 2019 IV. Quantum mechanics recasted 9 Contents A. Von Neumann equation as a pair of real matrix equations9 B. Some examples of quantum dynamics in I. Introduction 2 terms of non-negative real kernels 11 II. Overview and related work 6 1. Non-relativistic Schr¨odingerequation 11 2. From non-relativistic path integrals to III. The big picture 7 real convolutions 11 3. Particle in an electromagnetic field via asymmetric real kernels 12 V. Markov processes recasted 13 yA substantial part of this work was developed while being Research Associate at The University of Manchester. A. Principle of maximum caliber and factor ∗Electronic address: [email protected] graphs 13 2 B. Quantum-like formulation of stochastic 2. Can the scientific and contemplative processes via the cavity method 14 worldviews converge? 39 1. Cavity messages as imaginary-time 3. Not just bare rationality: A call for wave functions 14 action 41 2. Belief propagation as imaginary-time quantum dynamics 15 Acknowledgments 43 C. Euclidean quantum mechanics: From linear chains to cycles 16 A. We as physical systems 43 VI. Third-person perspective 17 B. The modern and well-respected A. Observations as internal representations 17 approach to consciousness 45 B. Experiments as circular interactions 18 1. Third-person perspective and the `easy' 1. Simulation interpretation 18 problem of consciousness 45 2. Ideomotor interpretation 18 2. First-person perspective and the `hard' C. Imaginary-time von Neumann equation 20 problem of consciousness 46 1. Circularity entails non-commutativity 20 C. Self-reference and the recursion 2. Pure states and causality 21 theorem 47 VII. First-person perspective 22 D. Derivation of real kernel A. Self-reference and complementarity 22 representaions in Sec.IVB 48 B. First-person observers and quantum 1. Non-relativistic Schr¨odingerequation 48 dynamics 23 2. From non-relativistic path integrals to 1. General considerations 23 real convolutions 49 2. Equilibrium environment and 3. Particle in an electromagnetic field via symmetric kernels 25 asymmetric real kernels 51 3. Nonequilibrium environment and a. Prelude: Hermitian kernels via asymmetric kernels 25 complex Hamiltonian functions 51 b. Postlude: real asymmetric kernels via VIII. Occam's razor favors the reverse real Hamiltonian functions 54 paradigm 26 A. Some relevant assumptions in the E. Effective kernels with negative entries 55 mainstream paradigm 26 B. Corresponding interpretation in the F. Phase-less quantum-like formulation reverse paradigm 27 of Markov processes 56 C. Quantum foundations in reverse mode 27 1. Time-symmetric evolution equations 56 2. Time-symmetric kernels are not IX. Quantumness and consciousness 29 normalized 57 A. Consciousness as a rigorous scientific 3. Future-past symmetry and quantum-like subject 29 Markov processes 57 B. The message of the quantum: observers are physical 29 References 59 C. Planck constant from psychophysics experiments 30 1. General considerations 30 I. INTRODUCTION 2. `Classical' pshycophysics experiments 31 3. `Quantum' psychophysics Perhaps some of the most difficult transitions in the experiments 32 evolution of our understanding of the universe have D. Self-reference and the global architecture been those that removed our special status in some of self-aware systems and the self 33 way|like the resistance against the concept that our planet is not the center of the universe, attributed to X. Discusion 35 Copernicus, or against the concept that we are not A. Summary of main points 35 as different as we thought from other animals, at- B. Quantum computing, artificial general tributed to Darwin. Yet history has taught us again intelligence, and quantum cognition 36 and again that once we surrender and accept the C. On science, our worldview, and how we new status a previously hidden simplicity suddenly live 36 emerges. 1. Enlarging our toolbox with In part because our subjective biases are often mis- first-person methods 37 leading, we have usually made an effort to keep the 3 subjective, ourselves, out of our picture of the uni- Although substantial progress has been done (see e.g. verse in search of an objective reality. Even studies Refs. [2{10] and Sec. II), no general consensus has of the human brain have mostly focused on a third- been reached [11, 12]. This elusive character of quan- person perspective (see Fig.2a), i.e. scientist usually tum theory contrasts with its outstanding success. study others' brains, not their own. This has granted Here we argue that the resistance we have developed us the special status of being able to understand the against human experience as a key aspect for the sci- world as if we were not part of it, independently of our entific understanding of nature has prevented us from everyday human experience. However, at the same better grasping the essential message of quantum the- time that we gained the special status of doing sci- ory [8, 13]. Indeed, there has usually been an under- ence without the scientist, we also created a deep standable skepticism of any suggestion that observers tension between science and human experience (see or consciousness might play a special role in quantum below). theory. However, we are witnessing today a radical This work is a kind invitation to reconsider the re- shift in our understanding and control of aspects that sistance that mainstream physics has understandably we previously thought were intrinsically human, per- developped against the role that human experience haps even unreachable to the powerful methods of might play on the foundations of science. Such type science (see AppendixA). of invitation is not new, of course. Indeed, a simi-