Generating functionals for harmonic expectation values of paths with fixed end points: Feynman diagrams for...

Area of Science:

• Quantum mechanics
• Theoretical physics
• Statistical mechanics

Background:

• Calculating quantum-mechanical expectation values for complex systems can be challenging.
• Fluctuating paths and their properties are central to understanding quantum phenomena.
• Existing methods may not efficiently handle nonpolynomial interactions.

Purpose of the Study:

• Introduce a general class of generating functionals for calculating quantum-mechanical expectation values.
• Apply these functionals to systems with time-dependent parameters, such as the harmonic oscillator.
• Develop a smearing formula to account for quantum fluctuations in correlation functions.

Main Methods:

• Developed a general class of generating functionals for fluctuating paths with fixed endpoints.
• Explicitly calculated these generating functionals for a time-dependent harmonic oscillator.
• Derived a smearing formula for correlation functions of polynomial and nonpolynomial functions of time-dependent positions and momenta.

Main Results:

• The generating functionals provide a unified framework for calculating expectation values.
• The derived smearing formula effectively summarizes the impact of quantum fluctuations.
• The approach facilitates the derivation of generalized Wick rules and Feynman diagrams for nonpolynomial interactions.

Conclusions:

• The introduced generating functionals offer a powerful tool for quantum mechanical calculations.
• The smearing formula simplifies the analysis of correlation functions in complex systems.
• This work provides a foundation for studying nonpolynomial interactions within quantum field theory.