Operator Algebras in Quantum Theory: A Practical Perspective
Abstract: Operators, the main mathematical objects in quantum theory, are rarely recognized as elements of an operator algebra. Operators that represent symmetries, on the other hand, are elements of a group that is one of the most consequential ideas in modern physics. In this talk I will introduce operator algebras as a practical alternative to symmetries and review their common irreducible representations (irrep) structure. Motivated by a variety of physical applications, I will show a convenient way of visualizing the irrep structure of an operator algebra and a practical, not necessarily numeric, algorithm that constructs it. I will end with a concrete example of a calculation that can be performed using operator algebra irrep decomposition.