Quantum science: new maths needed (Introduction)

by David Turell , Friday, March 31, 2017, 19:05 (2590 days ago) @ David Turell

An article on current quantum maths and the new maths needed:

https://www.quantamagazine.org/20170330-how-quantum-theory-is-inspiring-new-math/?utm_s...

"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”?

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'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. Its stimulating influence in mathematics will have a lasting and rewarding impact, whatever its final role in fundamental physics turns out to be. The number of disciplines that it touches is dizzying: analysis, geometry, algebra, topology, representation theory, combinatorics, probability — the list goes on and on. One starts to feel sorry for the poor students who have to learn all this!

"What could be the underlying reason for this unreasonable effectiveness of quantum theory? In my view, it is closely connected to the fact that in the quantum world everything that can happen does happen.

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"A striking example of the magic of quantum theory is mirror symmetry — a truly astonishing equivalence of spaces that has revolutionized geometry. The story starts in enumerative geometry, a well-established, but not very exciting branch of algebraic geometry that counts objects. For example, researchers might want to count the number of curves on Calabi-Yau spaces — six-dimensional solutions of Einstein’s equations of gravity that are of particular interest in string theory, where they are used to curl up extra space dimensions.

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"But a second ingredient was necessary to find the actual solution: an equivalent formulation of the physics using a so-called “mirror” Calabi–Yau space. The term “mirror” is deceptively simple. In contrast to the way an ordinary mirror reflects an image, here the original space and its mirror are of very different shapes; they do not even have the same topology. But in the realm of quantum theory, they share many properties. In particular, the string propagation in both spaces turns out to be identical. The difficult computation on the original manifold translates into a much simpler expression on the mirror manifold, where it can be computed by a single integral.

"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.

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"Take E = mc2, without a doubt the most famous equation in history. In all its understated elegance, it connects the physical concepts of mass and energy that were seen as totally distinct before the advent of relativity. Through Einstein’s equation we learn that mass can be transformed into energy, and vice versa. The equation of Einstein’s general theory of relativity, although less catchy and well-known, links the worlds of geometry and matter in an equally surprising and beautiful manner. A succinct way to summarize that theory is that mass tells space how to curve, and space tells mass how to move.

"Mirror symmetry is another perfect example of the power of the equal sign. It is capable of connecting two different mathematical worlds. One is the realm of symplectic geometry, the branch of mathematics that underlies much of mechanics. On the other side is the realm of algebraic geometry, the world of complex numbers. Quantum physics allows ideas to flow freely from one field to the other and provides an unexpected “grand unification” of these two mathematical disciplines.

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"Niels Bohr was very fond of the notion of complementarity. The concept emerged from the fact that, as Werner Heisenberg proved with his uncertainty principle, in quantum mechanics one can measure either the momentum p of a particle or its position q, but not both at the same time. Wolfgang Pauli wittily summarized this duality in a letter to Heisenberg dated October 19, 1926, just a few weeks after the discovery: “One can see the world with the p-eye, and one can see it with the q-eye, but if one opens both eyes, then one becomes crazy.'”

Comment: It seems there may be a maths role to fully understanding quantum mechanics. I think it is a view into the workings of God's consciousness.