Posts Tagged ‘symmetry’
“Reality favors symmetry”*…
Emmy Noether showed that fundamental physical laws are themselves a consequence of simple symmetries. As Shalma Wegsman explains, a century later, her insights continue to shape physics…
In the fall of 1915, the foundations of physics began to crack. Einstein’s new theory of gravity seemed to imply that it should be possible to create and destroy energy, a result that threatened to upend two centuries of thinking in physics.
Einstein’s theory, called general relativity, radically transformed the meaning of space and time. Rather than being fixed backdrops to the events of the universe, space and time were now characters in their own right, able to curve, expand and contract in the presence of matter and energy.
One problem with this shifting space-time is that as it stretches and shrinks, the density of the energy inside it changes. As a consequence, the classical energy conservation law that previously described all of physics didn’t fit this framework. David Hilbert, one of the most prominent mathematicians at the time, quickly identified this issue and set out with his colleague Felix Klein to try to resolve this apparent failure of relativity. After they were stumped, Hilbert passed the problem on to his assistant, the 33-year-old Emmy Noether.
Noether was an assistant in name only. She was already a formidable mathematician when, in early 1915, Hilbert and Klein invited her to join them at the University of Göttingen. But other faculty members objected to hiring a woman, and Noether was blocked from joining the faculty. Regardless, she would spend the next three years prodding the fault line separating physics and mathematics, eventually setting off an earthquake that would shake the foundations of fundamental physics.
In 1918, Noether published the results of her investigations in two landmark theorems. One made sense of conservation laws in small regions of space, a mathematical feat that would later prove important for understanding the symmetries of quantum field theory. The other, now just known as Noether’s theorem, says that behind every conservation law lies a deeper symmetry.
In mathematical terms, a symmetry is something you can do to a system that leaves it unchanged. Consider the act of rotation. If you start with an equilateral triangle, you’ll find that you can rotate it by multiples of 120 degrees without changing how it looks. If you start with a circle, you can rotate it by any angle. These actions without consequences reveal the underlying symmetries of these shapes.
But symmetries go beyond shape. Imagine you do an experiment, then you move 10 meters to the left and do it again. The results of the experiment don’t change, because the laws of physics don’t change from place to place. This is called translation symmetry.
Now wait a few days and repeat your experiment again. The results don’t change, because the laws of physics don’t change as time passes. This is called time-translation symmetry.
Noether started with symmetries like these and explored their mathematical consequences. She worked with established physics using a common mathematical description of a physical system, called a Lagrangian.
This is where Noether’s insight went beyond the symbols on the page. On paper, symmetries seem to have no impact on the physics of the system, since symmetries don’t affect the Lagrangian. But Noether realized that symmetries must be mathematically important, since they constrain how a system can behave. She worked through what this constraint should be, and out of the mathematics of the Lagrangian popped a quantity that can’t change. That quantity corresponds to the physical property that’s conserved. The impact of symmetry had been hiding beneath the equations all along, just out of view.
In the case of translation symmetry, the system’s total momentum should never change. For time-translation symmetry, a system’s total energy is conserved. Noether discovered that conservation laws aren’t fundamental axioms of the universe. Instead, they emerge from deeper symmetries.
The conceptual consequences are hard to overstate. Physicists of the early 20th century were shocked to realize that a system that breaks time-translation symmetry can break energy conservation along with it. We now know that our own universe does this. The cosmos is expanding at an accelerating rate, stretching out the leftover light from the early universe. The process reduces the light’s energy as time passes…
… Noether’s theorem has shaped the quantum world too. In the 1970s, it played a big role in the construction of the Standard Model of particle physics. The symmetries of quantum fields dictate laws that restrict how fundamental particles behave. For instance, a symmetry in the electromagnetic field forces particles to conserve their charge.
The power of Noether’s theorem has inspired physicists to look toward symmetry to discover new physics. Over a century later, Noether’s insights continue to influence the way physicists think…
“How Noether’s Theorem Revolutionized Physics,” from @shalmawegs in @QuantaMagazine.
* Jorge Luis Borges
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As we contemplate cosmology, we might send insightful birthday greetings to the man who “wrote the book” on perspective, Leon Battista Alberti; he was born on this date in 1404. The archetypical Renaissance humanist polymath, Alberti was an author, artist, architect, poet, priest, linguist, philosopher, cartographer, and cryptographer. He collaborated with Toscanelli on the maps used by Columbus on his first voyage, and he published the the first book on cryptography that contained a frequency table.
But he is surely best remembered as the author of the first general treatise– Della Pictura (1434)– on the the laws of perspective, which built on and extended Brunelleschi’s work to describe the approach and technique that established the science of projective geometry… and fueled the progress of painting, sculpture, and architecture from the Greek- and Arabic-influenced formalism of the High Middle Ages to the more naturalistic (and Latinate) styles of Renaissance.
“In the beginning the Universe was created. This has made a lot of people very angry and been widely regarded as a bad move.”*…
In the face of cosmologists who are trying to combine string theory with the theory of cosmic inflation to understand the universe-as-we-find-it, Neil Turok suggests a much simpler explanation. Frank Landymore reports…
Our understanding of the universe, as advanced as it is, remains riddled with paradoxes and huge question marks. Physicists have come up with some pretty heady ideas to explain them — we’ll get to those later — but there just might be a far “simpler” solution to all those holes in cosmology.
As Higgs Chair of Theoretical Physics at the University of Edinburgh Neil Turok explains in an essay for The Conversation, there could be a “mirror” universe that existed before the Big Bang and is a reflection of our own, moving backward in time.
It’s a trippy concept to wrap your head around, but simpler on the physics side of things. It would neatly balance out some of the asymmetries we observe in the universe, provide an answer to dark matter, and supplant some of what Turok would characterize as clumsier leading theories in cosmology, like cosmic inflation and string theory.
“Picturing the big bang as a mirror neatly explains many features of the universe which might otherwise appear to conflict with the most basic laws of physics,” wrote Turok, who published his team’s findings in the journal Annals of Physics. “The progress we have already made convinces me that, in all likelihood, there are alternatives to the standard orthodoxy — which has become a straitjacket we need to break out of.”
The physical laws of the universe should exhibit charge, parity, and time reversal — collectively known as CPT — symmetry, which essentially means every physical interaction can be mirrored. So to break down its implications: every particle should have an anti-particle of the opposite charge, every space has its inversion, and time can be reversed.
Except that’s not what we actually observe. Time only goes forward, and there are more particles than anti-matter particles. As far as we can tell, our universe is not symmetrical.
But: “Our mirror hypothesis restores the symmetry of the universe,” Turok argued. He compared it to looking at your reflection: “The combination of you and your mirror image are more symmetrical than you are alone.”
Extrapolating our universe backward in time through the Big Bang, “we found its mirror image, a pre-bang universe in which (relative to us) time runs backward and antiparticles outnumber particles,” Turok wrote.
This could also solve the mystery of dark matter, an invisible substance thought to make up 85 percent of all matter in the universe. Under the mirror hypothesis, weak, subatomic particles called neutrinos would be the ideal candidate to explain it.
Since we’ve only observed left-handed neutrinos, perhaps yet-unseen right-handed could even exist in the mirror universe.
What’s more, this could also tidily explain why the universe appears to be so uniform and flat. The prevailing theory is that a period of accelerated, faster-than-light expansion called cosmic inflation was responsible for shaping how the universe is today — but we’re yet to observe the large gravitational waves this would have produced.
With a handy mirror universe, however, “statistical arguments explain why the universe is flat and smooth and has a small positive accelerated expansion, with no need for cosmic inflation,” Turok wrote.
Of course, there’s a lot more needed to bear out this intriguing hypothesis. But Turok argues that, even if disproven, it demonstrates that there could be more straightforward explanations than what the Standard Model offers…
A mirror of our own, going backwards in time: “Physicist Says There’s Another Universe Hiding Behind the Big Bang,” from @futurism.
Turok’s full essay– longer and more detailed, but very accessible– is here.
* Douglas Adams, The Restaurant at the End of the Universe
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As we size up symmetry, we might send lofty birthday greetings to another ponderer of the cosmos, Carl Edward Sagan; he was born on this date in 1934. An astronomer, cosmologist, astrophysicist, astrobiologist (his contributions were central to the discovery of the high surface temperatures of Venus), he is best remembered as a popularizer of science– via books like The Dragons of Eden, Broca’s Brain and Pale Blue Dot, and the award-winning 1980 television series Cosmos: A Personal Voyage (which he narrated and co-wrote), the most widely-watched series in the history of American public television (seen by at least 500 million people across 60 different countries).
He is also remembered for his contributions to the scientific research of extraterrestrial life, including experimental demonstration of the production of amino acids from basic chemicals by radiation.
(Readers can enjoy a loving riff on Cosmos here.)
“Two obsessions are the hallmarks of Nature’s artistic style: Symmetry- a love of harmony, balance, and proportion [and] Economy- satisfaction in producing an abundance of effects from very limited means”*…
Life is built of symmetrical structures. But why? Sachin Rawat explores…
Life comes in a variety of shapes and sizes, but all organisms generally have at least one feature in common: symmetry.
Notice how your left half mirrors the right or the radial arrangement of the petals of a flower or a starfish’s arms. Such symmetry persists even at the microscopic level, too, in the near-spherical shape of many microbes or in the identical sub-units of different proteins.
The abundance of symmetry in biological forms begs the question of whether symmetric designs provide an advantage. Any engineer would tell you that they do. Symmetry is crucial to designing modular, robust parts that can be combined together to create more complex structures. Think of Lego blocks and how they can be assembled easily to create just about anything.
However, unlike an engineer, evolution doesn’t have the gift of foresight. Some biologists suggest that symmetry must provide an immediate selective advantage. But any adaptive advantage that symmetry may provide isn’t by itself sufficient to explain its pervasiveness in biology across scales both great and small.
Now, based on insights from algorithmic information theory, a study published in Proceedings of the Natural Academy of Sciences suggests that there could be a non-adaptive explanation…
Symmetrical objects are less complex than non-symmetrical ones. Perhaps evolution acts as an algorithm with a bias toward simplicity: “Simple is beautiful: Why evolution repeatedly selects symmetrical structures,” from @sachinxr in @bigthink.
* Frank Wilczek (@FrankWilczek)
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As we celebrate symmetry, we might recall (speaking of symmetry) that it was on this date in 1963 that the Equal Pay Act of 1963 was signed into law by president John F. Kennedy. Aimed at abolishing wage disparity based on sex, it provided that “[n]o employer having employees subject to any provisions of this section [section 206 of title 29 of the United States Code] shall discriminate, within any establishment in which such employees are employed, between employees on the basis of sex by paying wages to employees in such establishment at a rate less than the rate at which he pays wages to employees of the opposite sex in such establishment for equal work on jobs[,] the performance of which requires equal skill, effort, and responsibility, and which are performed under similar working conditions, except where such payment is made pursuant to (i) a seniority system; (ii) a merit system; (iii) a system which measures earnings by quantity or quality of production; or (iv) a differential based on any other factor other than sex […].
Those exceptions (and lax enforcement) have meant that, 60 years later, women in the U.S. are still paid less than men in comparable positions in nearly all occupations, earning on average 83 cents for every dollar earned by a man in a similar role.
“For the moment we might very well can them DUNNOS (for Dark Unknown Nonreflective Nondetectable Objects Somewhere)”*…
When does one give up on a hypothesis?…
In 1969, the American astronomer Vera Rubin puzzled over her observations of the sprawling Andromeda Galaxy, the Milky Way’s biggest neighbour. As she mapped out the rotating spiral arms of stars through spectra carefully measured at the Kitt Peak National Observatory and the Lowell Observatory, both in Arizona, she noticed something strange: the stars in the galaxy’s outskirts seemed to be orbiting far too fast. So fast that she’d expect them to escape Andromeda and fling out into the heavens beyond. Yet the whirling stars stayed in place.
Rubin’s research, which she expanded to dozens of other spiral galaxies, led to a dramatic dilemma: either there was much more matter out there, dark and hidden from sight but holding the galaxies together with its gravitational pull, or gravity somehow works very differently on the vast scale of a galaxy than scientists previously thought.
Her influential discovery never earned Rubin a Nobel Prize, but scientists began looking for signs of dark matter everywhere, around stars and gas clouds and among the largest structures in the galaxies in the Universe…
But… over the past half century, no one has ever directly detected a single particle of dark matter. Over and over again, dark matter has resisted being pinned down, like a fleeting shadow in the woods. Every time physicists have searched for dark matter particles with powerful and sensitive experiments in abandoned mines and in Antarctica, and whenever they’ve tried to produce them in particle accelerators, they’ve come back empty-handed. For a while, physicists hoped to find a theoretical type of matter called weakly interacting massive particles (WIMPs), but searches for them have repeatedly turned up nothing.
With the WIMP candidacy all but dead, dark matter is apparently the most ubiquitous thing physicists have never found. And as long as it’s not found, it’s still possible that there is no dark matter at all. An alternative remains: instead of huge amounts of hidden matter, some mysterious aspect of gravity could be warping the cosmos instead…
Dark matter is the most ubiquitous thing physicists have never found; Ramin Skibba (@raminskibba) wonders if it isn’t time to consider alternative explanations: “Does dark matter exist?” in @aeonmag.
* Bill Bryson on dark matter, in A Short History of Nearly Everything (2003)
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As we interrogate the invisible, we might send observant birthday greetings to Val Logsdon Fitch; he was born on this date in 1923. A particle physicist, he shared the 1964 Nobel Prize in Physics with his collaborator James Cronin for their experiments proving that some subatomic reactions do not adhere to fundamental symmetry principles (and are therefore indifferent to the direction of time).
By examining the decay of K-mesons, they proved that a reaction run in reverse does not retrace the path of the original reaction, which showed that the reactions of subatomic particles are not indifferent to time. Thus the phenomenon of CP violation was discovered… and thus was demolished the faith that physicists had previously had that natural laws were universally governed by symmetry.
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