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Extending Quantum Probability from Real Axis to Complex Plane.

Ciann-Dong Yang1, Shiang-Yi Han2

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Summary
This summary is machine-generated.

This study introduces complex probability to quantum mechanics, unifying quantum and classical probabilities within a complex framework. It enables deriving both from a single complex probability distribution.

Keywords:
complex Fokker–Planck equationcomplex probabilitycomplex stochastic differential equationoptimal quantum guidance lawquantum trajectory

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Area of Science:

  • Quantum Mechanics
  • Mathematical Physics
  • Probability Theory

Background:

  • The ontological interpretation of quantum mechanics hinges on understanding probability.
  • Trajectory interpretations like Bohmian mechanics and stochastic mechanics have explored quantum probability.
  • Extending probability to complex spaces presents challenges in defining complex trajectories and probabilities.

Purpose of the Study:

  • To develop a framework for complex probability in quantum mechanics.
  • To investigate the generation of complex trajectories and define complex probability.
  • To establish the relationship between complex probability and standard quantum probability.

Main Methods:

  • Applied the optimal quantum guidance law to derive a stochastic differential equation for complex plane motion.
  • Solved the complex stochastic differential equation to generate an ensemble of complex quantum random trajectories.
  • Utilized the complex Fokker-Planck equation to verify the derived complex probability distribution.

Main Results:

  • Derived a stochastic differential equation governing particle motion in the complex plane.
  • Formed a complex probability distribution ρc(t,x,y) from an ensemble of complex quantum random trajectories.
  • Verified the complex probability distribution using the complex Fokker-Planck equation.

Conclusions:

  • The proposed complex probability framework integrates both quantum probability (Ψ²) and classical probability.
  • Both quantum and classical probabilities can be derived from the complex probability distribution ρc(t,x,y) through different statistical methods.
  • This unified approach offers a novel perspective on the interpretation of quantum mechanics.