A Real Ensemble Interpretation of Quantum Mechanics

Dipankar Home, M. A. B. Whitaker

Thomas A. Brody

Glenn F. Webb

Ciann‐Dong Yang, Chia-Hung Wei

The current trajectory interpretation of quantum mechanics is based on an ensemble viewpoint that the evolution of an ensemble of Bohmian trajectories guided by the same wavefunction Ψ converges asymptotically to the quantum probability . Instead of the Bohm's ensemble-trajectory interpretation, the present paper gives a single-trajectory interpretation of quantum mechanics by showing that the distribution of a single chaotic complex-valued trajectory is enough to synthesize the quantum probability. A chaotic complex-valued trajectory manifests both space-filling (ergodic) and ensemble features. The space-filling feature endows a chaotic trajectory with an invariant statistical distribution, while the ensemble feature enables a complex-valued trajectory to envelop the motion of an ensemble of real trajectories. The comparison between complex-valued and real-valued Bohmian trajectories shows that without the participation of its imaginary part, a single real-valued trajectory loses the ensemble information contained in the wavefunction Ψ, and this explains the reason why we have to employ an ensemble of real-valued Bohmian trajectories to recover the quantum probability . © 2015 Wiley Periodicals, Inc.

Pedro, Leonardo

One attractive interpretation of quantum mechanics is the ensemble interpretation, where Quantum Mechanics merely describes a statistical ensemble of objects and not individual objects. But this interpretation does not address why the wave-function plays a central role in the calculations of probabilities, unlike most other interpretations of quantum mechanics. We prove: 1) the wave-function is a parametrization of any probability distribution of a statistical ensemble: there is a surjective map from an hypersphere to the set of all probability distributions; 2) for a quantum system defined in a 2-dimensional real Hilbert space, the role of the (2-dimensional real) wave-function is identical to the role of the Euler's formula in engineering, while the collapse of the wave-function is identical to selecting the real part of a complex number; 3) the collapse of the wave-function of any quantum system is a recursion of collapses of 2-dimensional real wave-functions. The wave-function plays a central role because it is a good parametrization that allows us to represent a group of transformations using linear transformations of the hypersphere. It is precisely the fact that the hypersphere is not the phase-space of the theory that implies the collapse of the wave-function. Without collapse, the wave-function parametrization would be inconsistent.