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arXiv:1210.2353

A Flea on Schroedinger's Cat

Klaas Landsman, Robin Reuvers

Published Oct 8, 2012• 5 citations

Abstract

We propose a technical reformulation of the measurement problem of quantum mechanics, which is based on the postulate that the final state of a measurement is classical. Unlike the usual formulation (in which the post-measurement state is a unit vector in Hilbert space), our version actually opens the possibility of admitting a purely technical solution within the confines of conventional quantum theory (as opposed to solutions that either modify this theory, or introduce unusual and controversial interpretative rules and/or ontologies). To that effect, we recall a remarkable phenomenon in the theory of Schroedinger operators (discovered in 1981 by Jona-Lasinio et al), according to which the ground state of a symmetric double-well Hamiltonian becomes exponentially sensitive to tiny perturbations of the potential as h -> 0. We show that this instability emerges also from the textbook WKB approximation, extend it to time-dependent perturbations, and study the dynamical transition from the ground state of the double well to the perturbed ground state (in which the cat is typically either dead or alive, depending on the details of the perturbation). Numerical simulations show that adiabatically arising perturbations may (quite literally) cause the collapse of the wave-function in the classical limit. Thus, at least in the context of a simple mathematical model, we combine the technical and conceptual virtues of decoherence (which fails to solve the measurement problem but launches the key idea that perturbations may come from the environment) with those of dynamical collapse models a la GRW (which do solve the measurement problem but are ad hoc), without sharing their drawbacks: single measurement outcomes are obtained (instead of merely diagonal reduced density matrices), and no modification of quantum mechanics is needed.

DOI: 10.48550/arxiv.1210.2353

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The measurement problem of quantum theory has troubled the minds of physicists and philosophers for decades. This paper will discuss a relatively new attempt at resolving the issue, known as the mechanism of the “flea”, as put forward by Landsman and Reuvers in [13] and by Landsman in [12]. This paper first reviews the formalism of algebraic quantum theory. Chapter 3 then contains the greatest conceptual work, given by a novel understanding of the measurement problem. In this context, a small, asymmetric perturbation that arises from the environment is postulated, as described in chapter 4. This “flea” perturbation provides the symmetry breaking required for a classical state to emerge from a quantum system, thereby bringing us one step closer to a possible solution of the measurement problem. Chapter 5 hopes to offer an evaluation of the flea mechanism. These comments focus on the limitations of the models considered, as well as the nature of the flea perturbation. In particular, the challenges in attempting to recover the Born rule are discussed. This discussion is informed by the work of Van Heugten and Wolters in [22]. Finally, the conclusion contains some general reflections on the flea proposal, portraying it as the start of a new way of making the idea of a wave function collapse mathematically precise.

The quantum state of Schroedinger's cat is usually incorrectly described as a superposition of "dead" and "alive," despite an argument by Rinner and Werner that, locally, the cat should be considered to be in a mixture of non-superposed states. Here, it is rigorously proven that the cat is not in a superposition. This is central to the measurement problem. Nonlocal two-photon interferometry experiments throw further light on the measurement state by probing the effect of a variable phase factor inserted between its superposed terms. These experiments demonstrate that both subsystems really are in locally mixed states rather than superpositions, and they tell us what the measurement state superposition actually superposes. They show that measurement transfers the coherence in Schroedinger's nuclear superposition neither to the cat nor to the nucleus, but only to the correlations between them. This explains the collapse process--but not its subsequent irreversible dissipation--within the context of unitary dynamics with no need for external entities such as the environment, a human mind, other worlds, or collapse mechanisms.

The basic idea of quantum mechanics is that the property of any system can be in a state of superposition of various possibilities (or eigen states).This state of superposition is also known as wave function and it evolves linearly with time in a deterministic way in accordance with the Schrodinger equation.However, when a measurement is carried out on the system to determine the value of that property (say position), the system instantaneously transforms to one of the eigen states and thus we get only a single value as an outcome of the measurement.Quantum measurement problem seeks to find the cause and exact mechanism governing this transformation.In an attempt to solve the above problem, in this paper, we will first define what the wave function represents in real-world and will identify the root cause behind the stochastic nature of events.Then, we will develop a model to explain the mechanism of collapse of the quantum mechanical wave function in response to a measurement.In the process of development of model, we will explain Schrodinger cat paradox and will show how Born's rule for probability becomes a natural consequence of the measurement process.

Schrödinger's cat paradox has persisted for nearly a century as a demonstration of the apparent absurdity of quantum superposition when extended to macroscopic scales. This technical note demonstrates that the paradox dissolves entirely within the CAT'S Theory framework through the triadic invariant Reality = Pattern × Intent × Presence (P × I × Pr). We argue that the cat is neither "both alive and dead" nor the victim of mysterious collapse. Instead, superposition represents a state where Presence (Pr) has not yet instantiated, yielding P × I × 0 = 0 (potential but not actual). Measurement does not "cause" collapse through mysterious observer effects; it provides the Presence component required for actualization. This resolution eliminates the need for Multiple Worlds or hidden variables while maintaining consistency with quantum mechanical predictions.

The interaction between Schrodinger’s cat and a radioactive atom is the sort of thing that happens in a quantum measurement. The “measurement problem” of quantum mechanics is to explain how the cat can be considered to end up either definitely alive or definitely dead if the final state is an entangled state of the radioactive atom and the cat. This chapter discusses Bohm’s theory and the Everett interpretation as solutions to the measurement problem. The Everett interpretation is fundamentally a proposal to solve what Pitowsky calls the “big” measurement problem: how a quantum measurement can produce a definite outcome. On the information-theoretic interpretation proposed here, the “big” measurement problem is a pseudo-problem. The chapter concludes with some remarks on the “small” measurement problem: how a classical probability distribution over macroscopic measurement outcomes emerges in a measurement process – how probabilities of “what you’ll obtain if you measure” become transformed to probabilities of “what’s there.”

The Schrödinger cat paradox is often presented as evidence that quantum mechanics fundamentally conflicts with macroscopic reality. This short conceptual paper proposes a resolution based on the TUO–NS (Theory of the Universe as an Operator – Navier–Stokes–like) framework. In this approach, physical states are interpreted as elastic deformations of a coercive vacuum field. The linear superposition principle is shown to be an effective property, valid only in the regime of small, linearized perturbations around a stable background configuration:

\mathcal{L}[\delta\psi_1 + \delta\psi_2] \approx \mathcal{L}[\delta\psi_1] + \mathcal{L}[\delta\psi_2]

At macroscopic scales, the vacuum energy functional

\mathcal{E}[\Psi]

is coercive and admits only isolated local minima. Superpositions of distinct macroscopic configurations correspond to energetically unstable states:

\mathcal{E}[\Psi_{dead} + \Psi_{alive}] \gg \mathcal{E}[\Psi_{dead}] + \mathcal{E}[\Psi_{alive}]

These states relax dynamically toward one of the admissible minima. The apparent “collapse of the wave function” is thus reinterpreted as a physical relaxation process, rather than an observer-dependent or probabilistic axiom. This framework provides a physically grounded explanation for the absence of macroscopic superpositions, without invoking external observers, measurement postulates, or modifications of quantum dynamics. This document is intended as a conceptual clarification complementary to the main TUO–NS technical papers, rather than as a replacement for standard quantum mechanics.