Quantum Mechanical Approach to the Khintchine and Bochner Criteria for Characteristic Functions

Area of Science:

• Mathematical Physics
• Probability Theory
• Quantum Mechanics

Background:

• Standard probability theory is widely accepted as a probabilistic framework.
• Quantum mechanics, while a probability theory, employs distinct methodologies.
• Fundamental aspects of probability theory lack a quantum mechanical perspective.

Purpose of the Study:

• To apply quantum mechanics methods to understand standard probability theory.
• To demonstrate the emergence of wave functions and operators within probability theory.
• To generalize existing criteria for characteristic functions using quantum mechanical insights.

Main Methods:

• Generalizing the Khintchine and Bochner criteria for characteristic functions.
• Employing methods from quantum mechanics to analyze probability theory.
• Developing a framework that connects quantum formalism with probabilistic concepts.

Main Results:

• Wave functions and operators are shown to be relevant in standard probability theory.
• Generalized Khintchine and Bochner criteria are established.
• Quantum mechanics provides a clarifying lens for these probabilistic criteria.

Conclusions:

• Quantum mechanics can illuminate fundamental aspects of probability theory.
• The generalization of criteria offers new perspectives on characteristic functions.
• A deeper connection between quantum mechanics and probability theory is established.