Between Classical and Quantum

Abstract

The relationship between classical and quantum theory is of central importance to the philosophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this relationship is partly historical and conceptual, but mostly technical and mathematically rigorous, including over 500 references. For example, we sketch how certain intuitive ideas of the founders of quantum theory have fared in the light of current mathematical knowledge. One such idea that has certainly stood the test of time is Heisenberg's 'quantum-theoretical Umdeutung (reinterpretation) of classical observables', which lies at the basis of quantization theory. Similarly, Bohr's correspondence principle (in somewhat revised form) and Schrödinger's wave packets (or coherent states) continue to be of great importance in understanding classical behaviour from quantum mechanics. On the other hand, no consensus has been reached on the Copenhagen Interpretation, but in view of the parodies of it one typically finds in the literature we describe it in detail.On the assumption that quantum mechanics is universal and complete, we discuss three ways in which classical physics has so far been believed to emerge from quantum physics, namely in the limit → 0 of small Planck's constant (in a finite system), in the limit N → ∞ of a large system with N degrees of freedom (at fixed ), and through decoherence and consistent histories. The first limit is closely related to modern quantization theory and microlocal analysis, whereas the second involves methods of C * -algebras and the concepts of superselection sectors and macroscopic observables. In these limits, the classical world does not emerge as a sharply defined objective reality, but rather as an approximate appearance relative to certain "classical" states and observables. Decoherence subsequently clarifies the role of such states, in that they are "einselected", i.e. robust against coupling to the environment. Furthermore, the nature of classical observables is elucidated by the fact that they typically define (approximately) consistent sets of histories.This combination of ideas and techniques does not quite resolve the measurement problem, but it does make the point that classicality results from the elimination of certain states and observables from quantum theory. Thus the classical world is not created by observation (as Heisenberg once claimed), but rather by the lack of it.

Unsharp Quantum Reality

“…The classical limit is commonly thought to be associated with the formal limit of Planck's constant approaching zero,h → 0. However, it takes considerable effort to carry out this limit in a mathematically satisfactory way [22]. Moreover it is not clear whether this simple process is capable of capturing the emergence of classical behavior in operational terms.…”

When champions meet: Rethinking the Bohr–Einstein debate

“…At the same time, this approach (together with the Many Worlds Interpretation) provides the strongest indications that the classical world is an appearance! It seems to follow that whereas Bohr's doctrine stands, its original motivation as being a requirement for objective science is questionable (Landsman, 2005). De Muynck (2004) has drawn attention to a different reason why classical physics cannot be objective in the sense envisaged by Bohr and Heisenberg.…”

“…Indeed, most experts on the foundations of quantum theory would nowadays agree that the classical world is an appearance relative to the perspective of a certain class of observers, whereas the quantum world is real (though its peculiar reality is "veiled"). See Landsman (2005) for a recent overview of this issue. Provided that Nature is quantum-mechanical, classical physics is therefore deprived of its objective status altogether, undermining at least Bohr's reasoning.…”

“…as the system becomes infinitely large or as → 0) form a commutative algebra (Landsman, 2005). The 31 Here we take leave from Howard's (1994) analysis; it is frightening that Bohr seems to leave room for such differing implementations of his doctrine of classical concepts as Heisenberg's (which we follow, siding e.g.…”

“…Howard's arguments against Heisenberg's implementation are that it 'introduces a new dualism into our ontology' and that 'one would like to think that the classical/quantum distinction corresponds to an objective feature of the world'. The first has just been dealt with; the second takes a 100 page article to answer (Landsman, 2005). 32 In the writings of Bohr terms like 'observer', 'subject', 'apparatus', 'measuring device', 'experimental conditions', etc.…”