Mathematics of Classical and Quantum Physics

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A student of physics once asked a famous expert how much mathematics a physicist needs to know. Until last year, algebraic geometry never rose to the top of the list. What changed? I realized that algebraic geometry is connected to the relation between classical and quantum physics. Classical physics is the physics of Newton, where we imagine that we can measure everything with complete precision, at least in principle.

And in classical mechanics, we assume we can measure this arrow precisely.

List of textbooks on classical mechanics and quantum mechanics

In quantum mechanics—a more accurate description of reality—this turns out not to be true. For example, if we know how far this arrow points in the x direction, we cannot know how far it points in the y direction. This uncertainty is too small to be noticeable for a spinning basketball, but for an electron it is important: Physicists had only a rough understanding of electrons until they took this into account. Immanuel Kant famously argued that lying is a parasite on truth-telling. But parasites must be self-limiting: Killing your host is an evolutionary dead-end, so a successful parasite must keep from pushing its growth to the utmost.

Lying, then, is often That is, they start with the classical description of some physical system, and they want to figure out the quantum description. There is no fully general and completely systematic procedure for doing this.

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This should not be surprising: The two world views are so different. The most systematic ones apply to a very limited selection of physics problems. For example, sometimes in classical physics we can describe a system by a point in a variety. This is not something one generally expects, but it happens in plenty of important cases. For example, consider a spinning object: If we fix how long its angular momentum arrow is, the arrow can still point in any direction, so its tip must lie on a sphere. Thus, we can describe a spinning object by a point on a sphere. When a classical physics problem is described by a variety, some magic happens.

The process of quantization becomes completely systematic—and surprisingly simple.

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The classical and quantum approaches to physics become tightly linked, and one can take ideas from either approach and see what they say about the other one. Even better, in this situation, many of the basic theorems about algebraic geometry can be seen as facts about quantization.

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Richard Feynman once said that, for him to make progress on a tough physics problem, he needed to have some sort of special angle on it:. I have to think I have a little better chance than the other guys, for some reason. Now I can read Hartshorne, translate some of the results into facts about physics, and feel I have a chance at understanding this stuff.

John Baez is a professor of mathematics at the University of California, Riverside and a visiting researcher at the Centre for Quantum Technologies in Singapore. He blogs about math, science, and environmental issues at Azimuth. Follow him on Twitter johncarlosbaez. Any additional software will be provided, or is generally freely available. However, some activities may have more specific requirements. For this reason, you will need to be able to install and run additional software on a device that meets the requirements below.

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How special relativity explains why moving fast propels you into the future. How an understanding of sound waves and electromagnetic waves explains why musical instruments emit a range of noises and why butterfly wings display a range of colours. How telescopes and microscopes work allowing us to see the very distant and the very small. How the properties of gases and solids are described, both from a macroscopic and a microscopic perspective, to explain, for example, hot air balloons and the structure of diamonds.

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