Group theoretical derivation of the minimal coupling principle

Area of Science:

• Theoretical Physics
• Quantum Mechanics
• Group Theory

Background:

• Group theoretical methods, established by Bargmann, Mackey, and Wigner, provide a deductive framework for the quantum theory of free particles.
• Galileian transformations form the symmetry group for free particles within this established framework.
• Extending these methods to interacting particles presents challenges due to the loss of symmetry.

Purpose of the Study:

• To extend established group theoretical methods to the quantum theory of interacting particles.
• To overcome the obstacles associated with symmetry loss in interacting systems.
• To explore the implications of these methods for the wave equations of interacting particles.

Main Methods:

• Extension of Bargmann, Mackey, and Wigner's group theoretical framework.
• Analysis of symmetry properties under Galileian transformations for interacting particles.
• Investigation of first-order invariance properties characterizing particle interactions.

Main Results:

• The developed approach successfully overcomes symmetry loss issues for interacting particles.
• Specific forms of the wave equation for interacting particles are derived.
• The minimal coupling principle's wave equation is shown to be one possible outcome.
• The framework accommodates the possibility of novel, yet undiscovered, wave equations.

Conclusions:

• Group theoretical methods offer a robust framework for quantum theory of interacting particles.
• Symmetry principles under Galileian transformations are key to characterizing interactions.
• This work opens avenues for discovering new forms of wave equations in quantum mechanics.