Can decoherence make quantum theories unfalsifiable? Understanding the quantum-to-classical transition without it

Approaches for improving students' understanding of quantum mechanics

Semi-classical theories are approximations to quantum theory that treat some degrees of freedom classically and others quantum mechanically. In the usual approach, the quantum degrees of freedom are described by a wave function which evolves according to some Schrödinger equation with a Hamiltonian that depends on the classical degrees of freedom. The classical degrees of freedom satisfy classical equations that depend on the expectation values of quantum operators. In this paper, we study an alternative approach based on Bohmian mechanics. In Bohmian mechanics the quantum system is not only described by the wave function, but also with additional variables such as particle positions or fields. By letting the classical equations of motion depend on these variables, rather than the quantum expectation values, a semi-classical approximation is obtained that is closer to the exact quantum results than the usual approach. We discuss the Bohmian semi-classical approximation in various contexts, such as nonrelativistic quantum mechanics, quantum electrodynamics and quantum gravity. The main motivation comes from quantum gravity. The quest for a quantum theory for gravity is still going on. Therefore a semi-classical approach where gravity is treated classically may be an approximation that already captures some quantum gravitational aspects. The Bohmian semi-classical theories will be derived from the full Bohmian theories. In the case there are gauge symmetries, like in quantum electrodynamics or quantum gravity, special care is required. In order to derive a consistent semi-classical theory it will be necessary to isolate gauge-independent dependent degrees of freedom from gauge degrees of freedom and consider the approximation where some of the former are considered classical.

This and the following chapter consider Bohr’s new ways (plural) of thinking about quantum phenomena and quantum theory in terms of complementarity that emerged in the wake of Heisenberg’s discovery of quantum mechanics and then of the uncertainty relations. Bohr’s thought on these subjects underwent several changes even at the earlier stages to be discussed in these two chapters, and then was further refined in the wake of EPR’s argument. The first change is the shift from Bohr’s pre-complementarity view of quantum theory in the wake of Heisenberg’s 1925 discovery of quantum mechanics to his view following Schrodinger’s wave mechanics, Dirac’s and Jordan’s transformations theory, and Heisenberg’s uncertainty relations—developments that were instrumental to the invention of complementarity, introduced in the Como lecture in 1927. This change will be discussed in this chapter. The second change, discussed in Chapter 7, was marked by Bohr’s rethinking of the question of causality in quantum theory and occurred under the impact of his exchanges with Einstein in 1927. Section 6.1 gives an introductory discussion of the developments of Bohr’s thought and of the concept of complementarity. Sections 6.2 and 6.3 offer an analysis of the Como lecture. Section 6.3 also critically examines some of the problematic implications of the Como argument, in particular those concerning quantum causality. Section 6.4 considers, from the same critical perspective, the arguments concerning quantum causality offered—in part following Bohr’s Como argument—by Dirac, Heisenberg, and von Neumann.

Ladyman and Ross (LR) argue that quantum objects are not individuals (or are at most weakly discernible individuals) and use this idea to ground their metaphysical view, ontic structural realism, according to which relational structures are primary to things. LR acknowledge that there is a version of quantum theory, namely the Bohm theory (BT), according to which particles do have definite trajectories at all times. However, LR interpret the research by Brown {\em et al.} as implying that "raw stuff" or {\em haecceities} are needed for the individuality of particles of BT, and LR dismiss this as idle metaphysics. In this paper we note that Brown {\em et al.}'s research does not imply that {\em haecceities} are needed. Thus BT remains as a genuine option for those who seek to understand quantum particles as individuals. However, we go on to discuss some problems with BT which led Bohm and Hiley to modify it. This modified version underlines that, due to features such as context-dependence and non-locality, Bohmian particles have a very limited autonomy in situations where quantum effects are non-negligible. So while BT restores the possibility of quantum individuals, it also underlines the primacy of the whole over the autonomy of the parts. The later sections of the paper also examine the Bohm theory in the general mathematical context of symplectic geometry. This provides yet another way of understanding the subtle, holistic and dynamic nature of Bohmian individuals. We finally briefly consider Bohm's other main line of research, the "implicate order", which is in some ways similar to LR's structural realism.

For more than eighty years, quantum theory (QT) has dominated physical science. This domination remains unchallenged to this day. Some physicists celebrate this remarkable stability. Others lament this fact and argue that QT inhibits our understanding of physical reality. They feel that numerous problems that have accumulated in physical science require stepping beyond the horizon outlined by QT. The article offers a critical examination of the foundational assumptions of QT that shape its practice. It uses two interpretations of QT as case studies—Copenhagen interpretation and the theoretical perspective advanced by American physicist David Bohm—to explain one major problem that continues to plague our study of physics. The article sketches a path to a possible solution and outlines a new science practice that this solution will require.

Quantum interference, manifest in the two slit experiment, lies at the heart of several quantum computational speed-ups and provides a striking example of a quantum phenomenon with no classical counterpart. An intriguing feature of quantum interference arises in a variant of the standard two slit experiment, in which there are three, rather than two, slits. The interference pattern in this set-up can be written in terms of the two and one slit patterns obtained by blocking one, or more, of the slits. This is in stark contrast with the standard two slit experiment, where the interference pattern cannot be written as a sum of the one slit patterns. This was first noted by Rafael Sorkin, who raised the question of why quantum theory only exhibits irreducible interference in the two slit experiment. One approach to this problem is to compare the predictions of quantum theory to those of operationally-defined ‘foil’ theories, in the hope of determining whether theories that do exhibit higher-order interference suffer from pathological—or at least undesirable—features. In this paper two proposed extensions of quantum theory are considered: the theory of Density Cubes proposed by Dakić, Paterek and Brukner, which has been shown to exhibit irreducible interference in the three slit set-up, and the Quartic Quantum Theory of Życzkowski. The theory of Density Cubes will be shown to provide an advantage over quantum theory in a certain computational task and to posses a well-defined mechanism which leads to the emergence of quantum theory—analogous to the emergence of classical physics from quantum theory via decoherence. Despite this, the axioms used to define Density Cubes will be shown to be insufficient to uniquely characterise the theory. In comparison, Quartic Quantum Theory is a well-defined theory and we demonstrate that it exhibits irreducible interference to all orders. This feature of Życzkowski’s theory is argued not to be a genuine phenomenon, but to arise from an ambiguity in the current definition of higher-order interference in operationally-defined theories. Thus, to begin to understand why quantum theory is limited to a certain kind of interference, a new definition of higher-order interference is needed that is applicable to, and makes good operational sense in, arbitrary operationally-defined theories.

Interpretations of quantum theory have traditionally assumed a “Galilean” observer, a bare “point of view” implemented physically by a quantum system. This paper investigates the consequences of replacing such an informationally-impoverished observer with an observer that satisfies the requirements of classical automata theory, i.e., an observer that encodes sufficient prior information to identify the system being observed and recognize its acceptable states. It shows that with reasonable assumptions about the physical dynamics of information channels, the observations recorded by such an observer will display the typical characteristics predicted by quantum theory, without requiring any specific assumptions about the observer’s physical implementation.

As earlier conjectured by several authors and much later established by Solèr, from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex Hilbert space framework. This complex description is unique and more precise than the real one as, for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group. The complex picture fulfils the thesis of Solér’s theorem and permits the standard formulation of the quantum Noether’s theorem. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic Hilbert space. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. In the initial part of this work, we extend some results of group representation theory and von Neumann algebra theory from the real and complex cases to the quaternionic Hilbert space case. We prove the double commutant theorem also for quaternionic von Neumann algebras (whose proof requires a different procedure with respect to the real and complex cases) and we extend to the quaternionic case a result established in the previous paper concerning the classification of irreducible von Neumann algebras into three categories. In the second part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the quaternionic one, all self-adjoint operators represent observables in agreement with Solèr’s thesis, the standard quantum version of Noether theorem may be formulated and the notion of composite system may be given in terms of tensor product of elementary systems. In the third part of the paper, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. The overall conclusion is that relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulation and this complex description is uniquely fixed by physics.

This is the second part of a three-part overview, in which we derive the category-theoretic backbone of quantum theory from a process ontology, treating quantum theory as a theory of systems, processes and their interactions. In this part, we focus on classical–quantum interaction. Classical and quantum systems are treated as distinct types, of which the respective behavioral properties are specified in terms of processes and their compositions. In particular, classicality is witnessed by ‘spiders’ which fuse together whenever they connect. We define mixedness and show that pure processes are extremal in the space of all processes, and we define entanglement and show that quantum theory indeed exhibits entanglement. We discuss the classification of tripartite qubit entanglement and show that both the GHZ-state and the W-state come from spider-like families of processes, which differ only in how they behave when they are connected by two or more wires. We define measurements and provide fully comprehensive descriptions of several quantum protocols involving classical data flow. Finally, we give a notion of ‘genuine quantumness’, from which special processes called ‘phase spiders’ arise, and get a first glimpse of quantum nonlocality.

Time reversal is in a particular position with respect to quantum theory, because of the intervention of antilinear operators. A discussion of this point should explain the role played by complex numbers in quantum theory. It is suggested to look upon that as an effect of an operational irreversibility of time, this being a consequence of the interaction between quantum system and measuring apparatus, when the irreversibility inherent to this is taken into account.

Sum uncertainty relation in quantum theory

Quantum theory has been criticized for not being deterministic and therefore not universal: it cannot completely describe a measurement aimed at verifying its predictions (although any given apparatus can be considered as a quantum system). We investigate the possible alternatives. Theories where the observed world is deterministic but the observer is not (whatever the reason for that) lead to Bell’s nonseparability theorem. If, on the other hand, the observer too is deterministic, the theory is not verifiable. It follows that quantum theory must be the logically preferred option. Its inability to completely describe the measurement process appears to be not a flaw of the theory but a logical necessity which is analogous to Gödel’s undecidability theorem.

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.

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.

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.