Historical basis for: measurement <-> linear operator?

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SUMMARY

The discussion centers on the historical development of the relationship between measurement processes and linear operators, tracing its origins to Heisenberg's matrix mechanics in 1925. Key concepts include the linearity of expected values of random variables and the challenges Heisenberg faced regarding spectrum rays and observable transitions. The introduction of matrix multiplication allowed for the formulation of composition rules for transitions, while Hilbert spaces are linked to Dirac and Schrödinger's contributions. Additionally, the discussion highlights the relevance of C*-algebras and Positive Operator-Valued Measures (POVM) in the context of Heisenberg's work.

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What is the history of the concept that a measurement process is associated with a linear opeartor? Did it come from something in classical physics? Taking the expected value of a random variable is a linear operator - is that part of the story?
 
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in 1925 people were mainly interested in the way things add (like waves)
heisenberg faced another problem with spectrum rays. there was an addition rule for energy but
no composition rules for the rays. the only thing that seemed real for him were the transitions (they can be observed) and he had doubts about tbe observability of something else. a matrix with its non diagonal terms describes possible transitions. he invented matrix multiplication and found the composition rule for transitions.
hilbert spaces are in the tradition of dirac and schroedinger
c* algebras, povm etc are in the tradition of Heisenberg.
 

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