Categorical quantum mechanics II: Classical-quantum interaction

A wave dynamics of fields φ(p, q; t) ∈ L2(Γ) over the phase space Γ(p, q) of a classical system 𝒮 is derived from the Liouville theorem. We define the energy contained in a given field φ(p, q; t). We show that for a special class of fields, selected on physical grounds, the energy spectrum is given by a time-independent Schrödinger equation. This allows us to associate with 𝒮 an ordinary quantum system Q such that the values of the quantized energy coincide for the fields in the phase space of 𝒮 and for Q. Then we make use of Wiener's stochastic integral based on the theory of Brownian motion to derive probabilities which are the same as those one would obtain through Born's statistical postulate of quantum theory. From this it follows that we can regard normalized fields φ(p, q; t) as ``probability amplitudes'' leading to a probability density function ρ(p, q; t) = φφ* in the sense of Gibbs' statistical mechanics. Our work therefore appears as a bridge between a statistical theory (in the sense of Gibbs) of a mechanical system 𝒮 and the usual quantum theory of the related quantum system Q.

Recent analyses of classical integrable structures in quantum integrable models solved by various versions of the Bethe ansatz are reviewed. Similarities between elements of quantum and classical theories of integrable systems are discussed. Some key ideas in quantum theory, now standard in the quantum inverse scattering method, are identified with typical constructions in classical soliton theory. Functional relations for quantum transfer matrices become the classical Hirota bilinear difference equation; solving this classical equation gives all the basic results for the spectral properties of quantum systems. Vice versa, typical Bethe ansatz formulas under certain boundary conditions yield solutions of this classical equation. The Baxter T-Q relation and its generalizations arise as auxiliary linear problems for the Hirota equation.

Feynman insisted 'no one understands quantum theory'. Yet, experimentalists tell us quantum theory is the most successful theory in history. Quantum theory cannot be understood as a classical mechanical theory since it arose through the 'interpolation' of two highly successful but complementary classical mechanics: Newtonian particle mechanics and Maxwellian wave mechanics. The two-slit experiment illustrates that what is experienced depends on choice of experimental set-up. Quantum theory is properly understood within the more general framework of engineering thermodynamics. In Part One, I point to four essential characteristics of quantum theory that cannot be understood in any framework defined by the classical mechanical presuppositions of symmetry and conservation. These four characteristics are the participatory, the complementary, the indeterminate and the new non-commutative geometry. In Part Two, articulating engineering thermodynamics, I note there are two histories and two formulations of thermodynamics: Carnot's engineering thermodynamics and the 'rational mechanical' tradition of Clausius-Boltzmann. These four essential characteristics of quantum theory are also characteristics of engineering thermodynamics. In Part Three, I trace the precursors of Lazare Carnot's engineering thermodynamics to earlier insights of Huygens, d'Alembert, Leibniz and the Bernoullis. Leibniz brought these forth in his meta-paradigm shift from Statics to Dynamics. This article is part of the theme issue 'Thermodynamics 2.0: Bridging the natural and social sciences (Part 2)'.

Non-Hermitian (NH) extension of quantum-mechanical Hamiltonians represents one of the most significant advancements in physics. During the past two decades, numerous captivating NH phenomena have been revealed and demonstrated, but all of which can appear in both quantum and classical systems. This leads to the fundamental question: what NH signature presents a radical departure from classical physics? The solution of this problem is indispensable for exploring genuine NH quantum mechanics, but remains experimentally untouched so far. Here, we resolve this basic issue by unveiling distinct exceptional entanglement phenomena, exemplified by an entanglement transition, occurring at the exceptional point of NH interacting quantum systems. We illustrate and demonstrate such purely quantum-mechanical NH effects with a naturally dissipative light-matter system, engineered in a circuit quantum electrodynamics architecture. Our results lay the foundation for studies of genuinely quantum-mechanical NH physics, signified by exceptional-point-enabled entanglement behaviors.

We establish the explicit correspondence between the theory of soliton gases in classical integrable dispersive hydrodynamics, and generalized hydrodynamics (GHD), the hydrodynamic theory for many-body quantum and classical integrable systems. This is done by constructing the GHD description of the soliton gas for the Korteweg–de Vries equation. We further predict the exact form of the free energy density and flux, and of the static correlation matrices of conserved charges and currents, for the soliton gas. For this purpose, we identify the solitons’ statistics with that of classical particles, and confirm the resulting GHD static correlation matrices by numerical simulations of the soliton gas. Finally, we express conjectured dynamical correlation functions for the soliton gas by simply borrowing the GHD results. In principle, other conjectures are also immediately available, such as diffusion and large-deviation functions for fluctuations of soliton transport.

Quantum processes on phase space

Geometry and non-adiabatic response in quantum and classical systems

Symmetry principles play an important part in theoretical physics, both as a means of understanding conservation laws and as computational devices providing quick and elegant solutions to problems that would otherwise be tedious to solve. In quantum theory, their role is especially significant since they not only clarify basic concepts and simplify the formal development of the subject, but they also provide detailed information about physical systems for which no explicit dynamical models exist. However, the methods used to investigate symmetry in quantum systems are often based on abstract group-theoretical techniques, which bypass the canonical formalism. They include no general rules for going over to the classical limit and for directly relating the symmetries of the quantum system to those of a corresponding classical system.

PurposeAs any attempts at explaining quantum theory in terms of simple, local “cause‐and‐effect” models have remained unsatisfactory, approaches from the perspectives of systems theory seem called for, which is rich in a variety of more complex understandings of causality.Design/methodology/approachThis paper presents one option for such approaches, which the author has introduced previously as “quantum cybernetics”: considering waves (but not “wave functions”!) and “particles” as mutually dependent system components, and thus defining “organizationally closed systems” characterized by a fundamental circular causality. Using such an approach, a new look can be achieved on both classical and quantum physics.FindingsIt was found that quantum theory's most fundamental equation, the Schrödinger equation, can actually be derived from classical physics, once the latter is considered anew, i.e. under said approach involving both particles and (Huygens) waves. In fact, the only difference to existing views is that Huygens waves are here considered to be real, physically effective waves in some hypothesized sub‐quantum medium, rather than mere formal tools.Originality/valueWhat is particularly new in the present paper is that quantum systems can be described by what Heinz von Foerster has called “nontrivial machines”, whereas the corresponding classical counterparts turn out to behave as “trivial machines”. This should provide enough stimulus for discussing system theoretical issues also in the context of the foundations of quantum theory.

Exact predictions for most quantum systems are computationally inaccessible. This is the so-called many body problem, which is present in most common interpretations of quantum mechanics. Therefore, predictions of natural quantum phenomena have to rely on some approximations (assumptions or simplifications). In the literature, there are different types of approximations, ranging from those whose justification is basically based on theoretical developments to those whose justification lies on the agreement with experiments. This last type of approximations can convert a quantum theory into an “unfalsifiable” quantum theory, true by construction. On the practical side, converting some part of a quantum theory into an “unfalsifiable” one ensures a successful modeling (i.e. compatible with experiments) for quantum engineering applications. An example of including irreversibility and dissipation in the Bohmian modeling of open systems is presented. On the ontological level, however, the present-day foundational problems related to controversial quantum phenomena have to avoid (if possible) being contaminated by the unfalsifiability originated from the many body problem. An original attempt to show how the Bohmian theory itself (minimizing the role of many body approximations) explains the transitions from a microscopic quantum system towards a macroscopic classical one is presented.

The first three of these axioms describe quantum theory and classical mechanics as statistical theories from the very beginning. With these, it can be shown in which sense a more general than the conventional measure theoretic probability theory is used in quantum theory. One gets this generalization defining transition probabilities on pairs of events (not sets of pairs) as a fundamental, not derived, concept. A comparison with standard theories of stochastic processes gives a very general formulation of the non existence of quantum theories with hidden variables. The Cartesian product of probability spaces can be given a natural algebraic structure, the structure of an orthocomplemented, orthomodular, quasi-modular, not modular, not distributive lattice, which can be compared with the quantum logic (lattice of all closed subspaces of an infinite dimensional Hubert space). It is shown how our given system of axioms suggests generalized quantum theories, especially Schrodinger equations, for phase space amplitudes.

The problem of emergence in physical theories makes necessary to build a\ngeneral theory of the relationships between the observed system and the\nobserving system. It can be shown that there exists a correspondence between\nclassical systems and computational dynamics according to the Shannon-Turing\nmodel. A classical system is an informational closed system with respect to the\nobserver; this characterizes the emergent processes in classical physics as\nphenomenological emergence. In quantum systems, the analysis based on the\ncomputation theory fails. It is here shown that a quantum system is an\ninformational open system with respect to the observer and able to exhibit\nprocesses of observational, radical emergence. Finally, we take into\nconsideration the role of computation in describing the physical world.\n

A quantum system at equilibrium is represented by a corresponding classical system, chosen to reproduce the thermodynamic and structural properties. The objective is to develop a means for exploiting strong coupling classical methods (e.g., MD, integral equations, DFT) to describe quantum systems. The classical system has an effective temperature, local chemical potential, and pair interaction that are defined by requiring equivalence of the grand potential and its functional derivatives with respect to the external and pair potentials for the classical and quantum systems. Practical inversion of this mapping for the classical properties is effected via the hypernetted chain approximation, leading to representations as functionals of the quantum pair correlation function. As an illustration, the parameters of the classical system are determined approximately such that ideal gas and weak coupling RPA limits are preserved (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

We develop an operational approach for reconstructing the quantum theory of qubit systems from elementary rules on information acquisition. The focus lies on an observerOinterrogating a systemSwith binary questions andS's state is taken asO's `catalogue of knowledge' aboutS. The mathematical tools of the framework are simple and we attempt to highlight all underlying assumptions. Four rules are imposed, asserting (1) a limit on the amount of information available toO; (2) the mere existence of complementary information; (3)O's total amount of information to be preserved in-between interrogations; and, (4)O's `catalogue of knowledge' to change continuously in time in-between interrogations and every consistent such evolution to be possible. This approach permits a {\it constructive} derivation of quantum theory, elucidating how the ensuing independence, complementarity and compatibility structure ofO's questions matches that of projective measurements in quantum theory, how entanglement and monogamy of entanglement, non-locality and, more generally, how the correlation structure of arbitrarily many qubits and rebits arises. The rules yield a reversible time evolution and a quadratic measure, quantifyingO's information aboutS. Finally, it is shown that the four rules admit two solutions for the simplest case of a single elementary system: the Bloch ball and disc as state spaces for a qubit and rebit, respectively, together with their symmetries as time evolution groups. The reconstruction for arbitrarily many qubits is completed in a companion paper [P. A. Höhn and C. S. P. Wever, Phys. Rev. A 95 (2017) 012102] where an additional rule eliminates the rebit case. This approach is inspired by (but does not rely on) the relational interpretation and yields a novel formulation of quantum theory in terms of questions.

Quantum theory was discovered in an adventurous way, under the urge to solve puzzles—like the spectrum of the blackbody radiation—that haunted the physics community at the beginning of the 20th century. It soon became clear, though, that quantum theory was not just a theory of specific physical systems, but rather a new language of universal applicability. Can this language be reconstructed from first principles? Can we arrive at it from logical reasoning, instead of ad hoc guesswork? A positive answer was provided in Phys Rev A, 81:062348, 2010 [34], Phys Rev A, 84:012311, 2011 [26], where we put forward six principles that identify quantum theory uniquely in a broad class of theories. We first defined a class of “theories of information”, constructed as extensions of probability theory in which events can be connected into networks. In this framework, we formulated the six principles as rules governing the control and the accessibility of information. Directly from these rules, we reconstructed a number of quantum information features, and eventually, the whole Hilbert space framework. In short, our principles characterize quantum theory as the theory of information that allows for maximal control of randomness.