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Chapter 12:
Advanced Operator Algebra
The operator algebra was introduced in chapter 4 as a shorthand notation to simplify the analysis of optical systems. The reason that this can be done so effectively is that these operators are rooted in a rigorous mathematical structure. This mathematical structure can be derived from an analogy with quantum theoretical operators in combination with the ray matrix representation of geometrical optics. This chapter outlines the mathematical basis for the operator algebra and leads to a more extended framework which further simplifies the use of operators for complex systems and provides additional insight and applications. Further extensions can be found in Refs. [21–27] while a comprehensive rigorous analysis is given in Ref. [28]. In the next two sections we review the matrix theory of geometrical optics and the quantum mechanical operators relevant to our subject. After that, we present the group background of the operator algebra and provide some interesting applications of the enhanced theory.
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