Emmy Noether and her wonderful theorem

[ This blog is dedicated to tracking my most recent publications. Subscribe to the feed to keep up with all the science stories I write! ]

Mathematician to know: Emmy Noether

Noether’s theorem is a thread woven into the fabric of the science

For Symmetry Magazine:

We are able to understand the world because it is predictable. If we drop a rubber ball, it falls down rather than flying up. But more specifically: if we drop the same ball from the same height over and over again, we know it will hit the ground with the same speed every time (within vagaries of air currents). That repeatability is a huge part of what makes physics effective.

The repeatability of the ball experiment is an example of what physicists call “the law of conservation of energy.” An equivalent way to put it is to say the force of gravity doesn’t change in strength from moment to moment.

The connection between those ways of thinking is a simple example of a deep principle called Noether’s theorem: Wherever a symmetry of nature exists, there is a conservation law attached to it, and vice versa. The theorem is named for arguably the greatest 20th century mathematician: Emmy Noether.

So who was the mathematician behind Noether’s theorem? [Read the rest at Symmetry…]

I'm in a magazine!

I’m in a magazine!

Physics is largely a matter of finding patterns in natural processes and translating that to mathematical expression. That’s a horribly oversimplified view, of course, but there’s no question that physics (and other branches of science) seeks to find symmetries. The huge successes of modern particle physics have largely arisen from identifying symmetries — and when those symmetries break down. To cite just one: physicists understand the weak force, which governs neutrinos and processes like nuclear beta decay, using a mathematical symmetry. That symmetry isn’t perfect, however, and one outward manifestation of that imperfection is the Higgs boson.

This pattern-seeking behavior among physicists is the theme of Dave Goldberg’s book The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality. I reviewed the book for Physics World, which marks my first publication in a print magazine. (It also may be the first time The Decemberists were quoted in Physics World.) You can read my review online, though the site requires a free registration to do so. In brief, I enjoyed the book, but found a few problems with it as well.

Inevitably, Goldberg’s explanations vary in quality. I found his discussion of the Casimir and Unruh effects (weird quantum phenomena in the vacuum) to be very good introductions for non-specialists. He also provides an excellent summary of the problems facing attempts to unify the different forces of nature, and specifically the question of pro- ton decay. On the other hand, his explanation of Lagrangians and the principle of least action (both essential topics in a mathematical sense) falls short, since it requires him to define a lot of new terminology in just a few pages, most of it barely mentioned again. The book also misses an opportunity to explain how specific symmetries shaped the development of the Standard Model; while it outlines a few of the important symmetries (including parity or reflection symmetry, time-reversal, time-translation, and exchange of matter and antimatter) early on, it fails to bring them back into the picture when the Standard Model is discussed. [Read more…]

We are bound by symmetry

Typically, reversing the direction of time twice is the same as never reversing it at all. Think of running an old-fashioned filmstrip backward, then forward (not an unusual experience for those of us um…of a certain generation): the film will look the same as though you never ran it backward. However, a particular uranium compound, URu2Si2, may break that rule. In that sense, it behaves akin to a spinor, the mathematical description of particles like electrons, protons, and so forth. (For more on spinors,  read my earlier post at Galileo’s Pendulum.) This model could explain all the weird properties of the uranium compound, including its strange magnetic behaviors.

A new model may help resolve the confusion by proposing a different form of symmetry breaking. Ordinarily, if you reverse the direction of time (akin to running a movie backward), then reverse it again, everything comes back to normal. For the particular uranium-rubidium-silicon compound at issue, Premala Chandra, Piers Coleman, and Rebecca Flint argued that symmetry is broken: it will not behave normally even under double time reversal. While a literal double reversing of time isn’t possible in the lab, the broken symmetry has a measurable consequence in the distortion of electron orbits in the uranium. If confirmed, this hypothesis could resolve a thirty-year-old mystery. [Read more…]

When the reverse of reverse isn’t forward: weird symmetry in uranium compound

Macroscopic processes are usually not completely reversible: you can’t unmix or unbake cake, and perfume released from a bottle won’t spontaneously recollect. However, these phenomena involve huge numbers of particles. On the level of individual particles in elementary physics, the direction of time doesn’t matter to the forces involved. The exception to this is the weak force, one of the four fundamental forces of nature. However, no experiment thus far had been able to demonstrate time asymmetry unambiguously until now.

New results from the BaBar detector at the Stanford Linear Accelerator Center (SLAC) have uncovered this asymmetry in time. Researchers measured transformations of entangled pairs of particles, including the rates at which these transformations occurred. Through analyzing over 10 years of data, they found clear time-reversal asymmetry with an error of only one part in 1043, a clear discovery by any standard. These results are a strong confirmation of predictions of the Standard Model, filling in one of the final missing details of that theory. [Read more…]

Processes in particle physics demonstrate arrow of time