Extending Quantum Probability from Real Axis to Complex Plane
arXiv:2103.05518 · doi:10.3390/e23020210
Abstract
Probability is an important question in the ontological interpretation of quantum mechanics. It has been discussed in some trajectory interpretations such as Bohmian mechanics and stochastic mechanics. New questions arise when the probability domain extends to the complex space, including the generation of complex trajectory, the definition of the complex probability, and the relation of the complex probability to the quantum probability. The complex treatment proposed in this article applies the optimal quantum guidance law to derive the stochastic differential equation governing a particle random motion in the complex plane. The probability distribution of the particle position over the complex plane is formed by an ensemble of the complex quantum random trajectories, which are solved from the complex stochastic differential equation. Meanwhile, this probability distribution is verified by the solution of the complex Fokker Planck equation. It is shown that quantum probability and classical probability can be integrated under the framework of complex probability, such that they can both be derived from the same probability distribution by different statistical ways of collecting spatial points.
References in corpus (18)
- Observing single quantum trajectories of a superconducting qubit
- Quantum Averages of Weak Values
- Observing and Verifying the Quantum Trajectory of a Mechanical Resonator
- Modified de Broglie-Bohm approach to quantum mechanics
- DeDonder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory
- Bohmian trajectories for photons
- Quantum Tomography twenty years later
- Probability and complex quantum trajectories
- Weak measurement of photon polarization by back-action induced path interference
- Experimental observation of anomalous trajectories of single photons
- Interplay of causticity and vorticality within the complex quantum Hamilton-Jacobi formalism
- Experimental parity-time symmetry quantum walks on a directed graph
- Probability Density in the Complex Plane
- Interfering trajectories in experimental quantum-enhanced stochastic simulation
- Quantum Trajectories based on the Weak Value
- Probability representation of quantum states as a renaissance of hidden variables -- God plays coins
- Hypercomplex quantum mechanics
- Trajectory Interpretation of Correspondence Principle: Solution of Nodal Issue