In order to fully understand the quantum world, we may have to develop a new realm of mathematics.
Mathematics might be more of an environmental science than we realize. Even though it is a search for eternal truths, many mathematical concepts trace their origins to everyday experience. Astrology and architecture inspired Egyptians and Babylonians to develop geometry. The study of mechanics during the scientific revolution of the 17th century brought us calculus.
.. The bizarre world of quantum theory — where things can seem to be in two places at the same time and are subject to the laws of probability — not only represents a more fundamental description of nature than what preceded it, it also provides a rich context for modern mathematics. Could the logical structure of quantum theory, once fully understood and absorbed, inspire a new realm of mathematics that might be called “quantum mathematics”?
.. Ideas that originate in particle physics have an uncanny tendency to appear in the most diverse mathematical fields. This is especially true for string theory.
.. in the quantum world everything that can happen does happen.
.. Mirror symmetry illustrates a powerful property of quantum theory called duality: Two classical models can become equivalent when considered as quantum systems, as if a magic wand is waved and all the differences suddenly disappear. Dualities point to deep but often mysterious symmetries of the underlying quantum theory. In general, they are poorly understood and an indication that our understanding of quantum theory is incomplete at best.
.. A succinct way to summarize that theory is that mass tells space how to curve, and space tells mass how to move.
.. It is comforting to see how mathematics has been able to absorb so much of the intuitive, often imprecise reasoning of quantum physics and string theory, and to transform many of these ideas into rigorous statements and proofs.