In the absence of an external frame of reference physical degrees of freedom must describe relations between systems. Using a simple model, we investigate how such a relational quantum theory naturally arises by promoting reference systems to the status of dynamical entities. Our goal is to demonstrate using elementary quantum theory how any quantum mechanical experiment admits a purely relational description at a fundamental level, from which the original "non-relational" theory emerges in a semi-classical limit. According to this thesis, the non-relational theory is therefore an approximation of the fundamental relational theory. We propose four simple rules that can be used to translate an "orthodox" quantum mechanical description into a relational description, independent of an external spacial reference frame or clock. The techniques used to construct these relational theories are motivated by a Bayesian approach to quantum mechanics, and rely on the noiseless subsystem method of quantum information science used to protect quantum states against undesired noise. The relational theory naturally predicts a fundamental decoherence mechanism, so an arrow of time emerges from a time-symmetric theory. Moreover, there is no need for a "collapse of the wave packet" in our model: the probability interpretation is only applied to diagonal density operators. Finally, the physical states of the relational theory can be described in terms of "spin networks" introduced by Penrose as a combinatorial description of geometry, and widely studied in the loop formulation of quantum gravity. Thus, our simple bottom-up approach (starting from the semi-classical limit to derive the fully relational quantum theory) may offer interesting insights on the low energy limit of quantum gravity.
Non-relativistic quantum mechanics is reformulated here based on the idea that relational properties among quantum systems, instead of the independent properties of a quantum system, are the most fundamental elements to construct quantum mechanics. This idea, combining with the emphasis that measurement of a quantum system is a bidirectional interaction process, leads to a new framework to calculate the probability of an outcome when measuring a quantum system. In this framework, the most basic variable is the relational probability amplitude. Probability is calculated as summation of weights …
Relational bundle geometric formulation of non-relativistic quantum mechanics
Masaryk University（马萨里克大学）；University of Graz（格拉茨大学）；Mons University（蒙斯大学）；Politecnico di Torino（都灵理工大学）；Istituto Nazionale di Fisica Nucleare, Section of Torino（意大利国家核物理研究所，都灵分部）
Relational formulation of quantum mechanics is based on the idea that relational properties among quantum systems, instead of the independent properties of a quantum system, are the most fundamental elements to construct quantum mechanics. In a recent paper (J. M. Yang, Sci. Rep. 8:13305, 2018), basic relational quantum mechanics framework is formulated to derive quantum probability, Born's Rule, Schrödinger Equations, and measurement theory. This paper further extends the reformulation effort in three aspects. First, it gives a clearer explanation of the key concepts behind the framework to c…
This article contains a local solution to the notorious Problem of Time in Quantum Gravity at the conceptual level and which is actually realizable for the relational triangle. The Problem of Time is that `time' in GR and `time' in ordinary quantum theory are mutually incompatible notions, which is problematic in trying to put these two theories together to form a theory of Quantum Gravity. Four frontiers to this resolution in full GR are identified, alongside three further directions not yet conquered even for the relational triangle. This article is also the definitive review on relational p…
In this comment we answer to the recent critique of our article [arXiv:2208.11793] about Relational Quantum Mechanics (RQM) by Aurélien Drezet [arXiv:2209.01237]. Here we point out that our critical analysis of RQM was precisely based on the most recent formulation of RQM, and that the theses found in the critique are based on neither RQM assumptions nor on our arguments.
Quantum measurement and quantum operation theory is developed here by taking the relational properties among quantum systems, instead of the independent properties of a quantum system, as the most fundamental elements. By studying how the relational probability amplitude matrix is transformed and how mutual information is exchanged during measurement, we derive the formulation that is mathematically equivalent to the traditional quantum measurement theory. More importantly, the formulation results in significant conceptual consequences. We show that for a given quantum system, it is possible t…
Every clock is a physical system and thereby ultimately quantum. A naturally arising question is how to describe time evolution relative to quantum clocks and, specifically, how the dynamics relative to different quantum clocks are related. This is a pressing issue in view of the multiple choice problem of time in quantum gravity, which posits that there is no distinguished choice of internal clock in generic general relativistic systems and that different choices lead to inequivalent quantum theories. Exploiting a recent approach to switching quantum reference systems (arXiv:1809.00556, arXiv…
The relational interpretation of quantum mechanics (RQM) has received a growing interest since its first formulation in 1996. Usually presented as an interpretational layer over the usual quantum mechanics formalism, it appears as a philosophical perspective without proper mathematical counterparts. This state of affairs has direct consequences on the scientific debate on RQM which still suffers from misunderstandings and imprecise statements. In an attempt to clarify those debates, the present paper proposes a radical reformulation of the mathematical framework of quantum mechanics which is r…
The Wigner's friend type of thought experiments manifest the conceptual challenge on how different observers can have consistent descriptions of a quantum measurement event. In this paper, we analyze the extended version of Wigner's friend thought experiment (Frauchiger and Renner, Nature Comm. 3711, 9 (2018)) in detail and show that the reasoning process from each agent that leads to the no-go theorem is inconsistent. The inconsistency is with respect to the requirement that an agent should make use of updated information instead of outdated information. We then apply the relational formulati…
The Quantum Measurement Problem is arguably one of the most debated issues in the philosophy of Quantum Mechanics, since it represents not only a technical difficulty for the standard formulation of the theory, but also a source of interpretational disputes concerning the meaning of the quantum postulates. Another conundrum intimately connected with the QMP is the Wigner friend paradox, a thought experiment underlining the incoherence between the two dynamical laws governing the behavior of quantum systems, i.e the Schrödinger equation and the projection rule. Thus, every alternative interpret…
Relational path integral, effective actions and quantum frame covariance in gravity
Okinawa Institute of Science and Technology Graduate University（冲绳科学技术大学院大学）；Friedrich-Alexander-Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡弗里德里希·亚历山大大学）；Kavli Institute for the Physics and Mathematics of the Universe (WPI), UTIAS, The University of Tokyo（东京大学宇宙线研究所（WPI） Kavli宇宙物理数学研究所）
Homomorphisms between relational structures play a central role in finite model theory, constraint satisfaction and database theory. A central theme in quantum computation is to show how quantum resources can be used to gain advantage in information processing tasks. In particular, non-local games have been used to exhibit quantum advantage in boolean constraint satisfaction, and to obtain quantum versions of graph invariants such as the chromatic number. We show how quantum strategies for homomorphism games between relational structures can be viewed as Kleisli morphisms for a quantum monad o…
Relational quantum mechanics (RQM) is an interpretation of quantum mechanics based on the idea that quantum states describe not an absolute property of a system but rather a relationship between systems. In this article, we observe that there is a tension between RQM's naturalistic emphasis on the physicality of information and the inaccessibility of certain sorts of information in current formulations of RQM. Therefore we propose a new postulate for RQM which requires that all of the information possessed by a certain observer is stored in physical variables of that observer and thus accessib…
The widely known but also somewhat esoteric Mach principle envisages a fully relational formulation of physical theories without any reference to a concept of `absolute space'. When applied to classical mechanics, under the guise of an extended symmetry group, this procedure is known to lead to an equation of motion with inertial-like forces that are sourced by the mass distribution of the system itself. In this paper we follow a similar procedure and reformulate the Schrödinger equation of non-relativistic quantum mechanics in a fully Machian way. Just like its classical counterpart, the resu…
We reappraise some of the hybrid classical-quantum models proposed in the literature with the goal of retrieving some of their common characteristics. In particular, first, we analyze in detail the Peres-Terno argument regarding the inconsistency of hybrid quantizations of the Sudarshan type. We show that to accept such hybrid formalism entails the necessity of dealing with additional degrees of freedom beyond those in the straight complete quantization of the system. Second, we recover a similar enlargement of degrees of freedom in the so-called statistical hybrid models. Finally, we use Wign…
We provide a reviewlike introduction into the quantum mechanical formalism related to non-Hermitian Hamiltonian systems with real eigenvalues. Starting with the time-independent framework we explain how to determine an appropriate domain of a non-Hermitian Hamiltonian and pay particular attention to the role played by PT-symmetry and pseudo-Hermiticity. We discuss the time-evolution of such systems having in particular the question in mind of how to couple consistently an electric field to pseudo-Hermitian Hamiltonians. We illustrate the general formalism with three explicit examples: i) the g…
