Paper I of this series introduced a nonlinear version of quantum mechanics that blocks cats, and paper II postulated a random part of the wavefunction to explain outcomes in experiments such as Stern-Gerlach or EPRB. However, an ad hoc extra parameter was assumed for the randomness. Here I provide some analytic and simulation evidence that the nonlinear theory exhibits sensitive dependence on initial conditions in measurement scenarios, perhaps implying that the magnitude of randomness required is determined by structural features of the model, and does not require a free parameter.
Paper I of this series introduced a nonlinear version of quantum mechanics that blocks cats, and paper II postulated a random part of the wavefunction to explain outcomes in experiments such as Stern-Gerlach or EPRB. However, an ad hoc extra parameter was assumed for the randomness. Here I provide some analytic and simulation evidence that the nonlinear theory exhibits sensitive dependence on initial conditions in measurement scenarios, perhaps implying that the magnitude of randomness required is determined by structural features of the model, and does not require a free parameter.
In the first paper of this series, I introduced a non-linear, Hamiltonian, generalization of Schroedinger's theory that blocks formation of macroscopic dispersion ("cats"). But that theory was entirely deterministic, and so the origin of random outcomes in experiments such as Stern-Gerlach or EPRB was left open. Here I propose that Schroedinger's wavefunction has a random component and demonstrate that such an improvised stochastic theory can violate Bell's inequality. Repeated measurements and the back-reaction on the microsystem are discussed in a toy example. Experiments that might falsify the theory are described.
In a prior paper, the author described an instability in a nonlinear wavefunction model. Proposed in connection with the Measurement Problem, the model contained an external potential creating a ``classical'' instability. However, it is interesting to ask whether such models possess an intrinsic randomness -- even ``chaos" -- independent of external potentials. In this work, I investigate the criterion analytically and simulate from a small (``3 qubit") model, demonstrating that the Lyapunov exponent -- a standard measure of ``chaos" -- is positive. I also extend the instability criterion to models in the continuum. These results suggest that the boundary between classical and wavefunction physics may also constitute the threshold of chaos, and present an alternative to Max Born's ad hoc probability hypothesis: random outcomes in experiments result not from ``wave-particle duality" or ``the existence of the quantum," but from sensitive dependence on initial conditions, as is common in the other sciences.
Working entirely within the Schroedinger paradigm, meaning wavefunction only, I present a modification of his theory that prevents formation of macroscopic dispersion (MD; "cats"). The proposal is to modify the Hamiltonian based on a method introduced by Steven Weinberg in 1989, as part of a program to test quantum mechanics at the atomic or nuclear level. By contrast, the intent here is to eliminate MD without affecting the predictions of quantum mechanics at the microscopic scale. This restores classical physics at the macro level. Possible experimental tests are indicated and the differences from previous theories discussed. In a second paper, I will address the other difficulty of wavefunction physics without the statistical (Copenhagen) interpretation: how to explain random outcomes in experiments such as Stern-Gerlach, and whether a Schroedingerist theory with a random component can violate Bell's inequality.
Quantum non-integrable systems have pseudochaos in the phase of the eigenfunctions. In particular, correlation function of wave functions at two different points disappears, when observation process, which requires space and/or time average over a small range, is taken into account. This gives rise to the realization of decoherence in measuring processes. By virtue of this property of quantum chaos, various problems and paradoxes are explained in the framework of conventional quantum mechanics. The subjects treated here are the duality of wave and particle, the wave function collapse in measurement, Stern–Gerlach experiments, Schrödinger's cat paradox, the Einstein–Podolsky–Rosen paradox and quantum Zeno effect.
In recent years we have suggested that many of the so-called paradoxes resulting from the Copenhagen interpretation of quantum mechanics could well have more logical parallels based in nonlinear dynamics and chaos theory. Perhaps quantum mechanics might not be strictly linear as has been commonly postulated, and indeed, during the past year experimentalists have discovered signatures of chaos in a definitely quantum system. As an illustration of what can go wrong when quantum effects are forced into a linear interpretation, I examine Bell-type inequalities. In conventional derivations of such inequalities, classical systems are found to impose upper limits on the statistical correlations between, say, the properties of a pair of separated but entangled particles, whereas quantum systems allow greater correlations. Numerous experiments have upheld the quantum predictions (greater statistical correlations than allowed classically), which has led to inferences such as the instantaneous transmission of information between effectively infinitely separated particles — Einstein's "spooky action-at-a-distance," incompatible with relativity. I argue that there is nothing wrong with the quantum mechanical side of such derivations (the usual point of attack by those attempting to debunk Bell-type arguments), but implicit in the derivations on the classical side is the assumption of independent, uncorrelated particles. As a result, one is comparing uncorrelated probabilities versus conditional probabilities rather than comparing classical versus quantum mechanics, making moot the experimental inferences. Further, nonlinear classical systems are known to exhibit correlations that can easily be as great as and overlap with quantum correlations — so-called nonextensive thermodynamics with its nonadditive entropy has verified this with numerous examples. Perhaps quantum mechanics does contain fundamental nonlinear elements. Nonlinear dynamics and chaos theory could well provide a bridge between the determinism so dear to Einstein and the statisical interpretation of the Copenhagen school. Einstein and Bohr both could have been right in their debates.
Collapse models including some external noise of unknown origin are routinely used to describe phenomena on the quantum-classical border; in particular, quantum measurement. Although containing nonlinear dynamics and thereby exposed to the possibility of superluminal signaling in individual events, such models are widely accepted on the basis of fully reproducing the non-signaling statistical predictions of quantum mechanics. Here we present a deterministic nonlinear model without any external noise, with randomness emerging from deterministic irregular dynamics of the detectors, shown to be just as non-signaling as noise-based collapse models are: it fully reproduces quantum statistical predictions, while being still vulnerable to signaling in individual events. The model is based on a minimally nonlinear von Neumann equation for a Stern-Gerlach or Bell-type measuring setup, containing coordinate and momentum operators in a skew-symmetric, split scalar product structure over the configuration space. The microscopic states of the detectors act as a nonlocal set of hidden parameters, controlling individual outcomes. Following Pearle's gambler's ruin scheme, the model is shown to display pumping of weights between setup-defined basis states, with a single winner randomly selected and the rest collapsing to zero. Outcome probabilities obey Born's rule under a no-drift or fair-game condition, implying that collapse must run to conclusion during the metastable pre-firing period of the detector system. Hidden signaling in individual events remains to be handled by future research; what is reached here is to put deterministic nonlinear models and noise-based collapse models on equal footing.