Hohenberg, “Consistent quantum theory,” or Copenhagen done right!

Consistent quantum theory (CQT) was introduced over 25 years ago by Robert Griffiths (1984), and further elaborated by Gell-Mann and Hartle (1993). This theory is a version of orthodox quantum mechanics based in the principle of complementarity. Complementarity is the essence of quantum mechanics, at the center of the Heisenberg indeterminacy principle, according to which two observables represented by noncommuting operators not only cannot be measured simultaneously with arbitrary precision but also cannot be defined simultaneously with arbitrary precision. Questions of measurement, entanglement, nonlocality, collapse of the wave function, and the myriad quantum paradoxes which have plagued the “Copenhagen interpretation” of the theory for over 80 years become add-ons that are relatively easily incorporated once one has faced the fundamental issue of complementarity which already arises for the simplest system. Let us follow P. C. Hohenberg (Department of Physics, New York University, NY, USA), “Colloquium: An introduction to consistent quantum theory,” REVIEWS OF MODERN PHYSICS 82: 2835-2844, OCTOBER–DECEMBER 2010 [free in ArXiv].

“CQT formulation of quantum mechanics is based on the following assumptions: (A) Physical objects and their properties are represented by states in an appropriate Hilbert space. (B) The predictions of quantum mechanics are not deterministic but intrinsically probabilistic. The quantum-mechanical wave function evolving by the deterministic Schrödinger equation yields only probabilistic information about physical systems. (C) The principle of unicity does not hold: there is not a unique exhaustive description of a physical system or process.

CQT takes a realistic point of view in that it deals with real, i.e., intrinsic properties of physical systems, independent of observers or measurement. However, reality can be be described in various alternative incompatible ways using descriptions, called “frameworks,” which cannot be compared or combined. This is the principle of complementarity or incompatibility, expressed in terms of the “single-framework rule,” to be elucidated in what follows. The essential feature of the theory is that its “quantum realism” posits multiple incompatible but coexisting layers of reality, expressed in the theory as different frameworks. Any quantum-mechanical statement must refer to one and only one framework (the single framework rule).

Let us define a framework as any basis in which we choose to represent a given state of the system. An arbitrary quantum state can be expanded in an infinite number of ways by choosing an appropriate basis. There are an infinite number of different frameworks in which to represent a given state, but any quantum-mechanical statement only makes sense, and thus can only be correct, relative to a particular framework. The full mystery and weirdness of the theory is captured by the existence of multiple frameworks, each one of which provides an account of quantum-mechanical “truth,” but different frameworks are mutually incompatible. Note that although each framework is associated with one and only one basis in the Hilbert space, a framework and a basis are not identical. Different bases are in some sense equivalent, but a  framework  identifies the properties (eigenstates) associated with the corresponding basis but only retains the real probability for those properties and discards the remaining information. Thus, different frameworks capture different real aspects of a quantum system and a full description of that system requires the set of all frameworks.

CQT modifies the Einstein-Podolsky-Rosen (1935) argument by substituting the “classical realism” assumption by quantum realism, whereby physical properties of a system are real relative to a well-defined framework, but properties belonging to incompatible frameworks are not simultaneously real. Two quantum mechanically entangled particles have correlations in each and every framework, but different correlations that cannot be incorporated into a single framework are not simultaneously meaningful. Einstein would have embraced the quantum realism of CQT, as the best way to reconcile quantum mechanics whose predictions Einstein did not doubt, with the requirements of locality. According to CQT reality is relative to a framework just as simultaneity is relative to a reference frame in special relativity.

Schrödinger cat paradox is the most important problem with orthodox quantum mechanics. The Schrödinger cat paradox can be thought to arise because one is implicitly thinking in terms of the unitary framework and at the same time thinking of the system as containing a superposition of live and dead cats. According to CQT, however, there are no cats at all in the unitary framework, whereas in the cat framework there are no superpositions. In CQT, in neither case (micro or macro) is the superposition a valid quantum-mechanical description. Entanglement is a description of how properties contained in a framework appear in the language of another framework.

The wave function encodes all the information about a system, but could not provide a realistic representation of a quantum system, isolated from observation, i.e., from some measurement apparatus. In CQT the wave function encodes all the dynamical information about a system, but it does not provide a direct universal representation of quantum reality.” Maybe, Einstein would have been happy if he would have kown CQT, at least this is the opinion of Hohenberg.

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One Response to Hohenberg, “Consistent quantum theory,” or Copenhagen done right!

  1. Pingback: La mecánica cuántica que le hubiera gustado conocer a Einstein | Ciencia | La Ciencia de la Mula Francis

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