The physical meaning of the integration of Dirac's symbols

The introduction of the quantum was led by Planck's reluctant move in 1900 to "guesstimate" the theory to match the experimental curve of blackbody radiation (curve fitting). This move then quickly became a flag summoning a storm. Many streams gathered into a river, riding on the wind and moving forward with great force, eventually resulting in the prevailing popularity of quantum mechanics today. It was the collective wisdom of a few revolutionary people who complemented each other, or it was the times that create heroes， or it was both. Planck's discrete energy view of the world was gradually formed when he derived the theory to fit experimental results. Dirac pointed out that the great physicists, such as Newton and Einstein, deduced their laws of nature from a "top- down" approach based on their fundamental beliefs. Dirac's own belief was that in some cases the beauty of equations is more important than experimental results, because experiments may involve measurement errors.
The fashion of all things quantum has naturally attracted different opinions, but only those who have actually "planted trees" in the quantum orchard may have a true understanding. After more than 50 years of theoretical exploration, the author's contributions to the development of quantum mechanics, in particular his theory of the integration of Dirac's symbols within an ordered product of operators, has not only developed the mathematics of quantum mechanics, providing Einstein with a representation of the entangled state, but has also incorporated the probability hypothesis of quantum mechanics into the ordered operators' normal distribution, a form of statistical distribution. The present paper should be helpful for quantum experimentalists and theorists to enrich their conceptions, and should also be suitable for all those interested in quantum optics and quantum statistics.