New Challenges for Classical and Quantum Probability.

Area of Science:

• Quantum mechanics
• Classical probability and statistics

Background:

• Classical random variables with all moments can generate quantum theories.
• This quantum-type formalism extends to most classical probability and statistics applications.
• Existing quantum mechanics, based on Gaussian or Poisson variables, has clear interpretations for operators like momentum.

Purpose of the Study:

• To explore the implications of classical random variables generating quantum theories.
• To address the challenge of interpreting quantum notions in classical contexts.
• To investigate the interpretation of conjugate momentum operators for non-Gaussian/non-Poisson classical random variables.

Main Methods:

• The study is primarily theoretical and expository.
• It involves analyzing the mathematical framework connecting classical random variables to quantum formalisms.
• Historical perspective is used to contextualize recent developments.

Main Results:

• Any classical random variable with all moments yields a quantum theory.
• This theory aligns with standard quantum mechanics for Gaussian and Poisson cases.
• A key challenge is the classical interpretation of quantum concepts like entanglement and conjugate momentum for broader classes of random variables.

Conclusions:

• The discovery necessitates a quantum-type formalism in broad areas of classical probability and statistics.
• New interpretations are required for quantum notions when applied to diverse classical contexts.
• Understanding the conjugate momentum for non-Gaussian/non-Poisson variables is a critical open question.