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(This message was added in version 6.7.0.) in /home4/scienrds/scienceandnerds/wp-includes/functions.php on line 6114Source:https:\/\/www.quantamagazine.org\/a-new-kind-of-symmetry-shakes-up-physics-20230418\/#comments<\/a><\/br> Seiberg and his colleagues imagined the one-dimensional string as being surrounded by a surface, a two-dimensional plane, so that it looked like a line drawn on a sheet of paper. Instead of measuring charge along the string, they described a method for measuring the total charge across the surface surrounding the string.<\/p>\n \u201cThe really new thing is you emphasize the charged object, and you think about [surfaces] that surround it,\u201d Schafer-Nameki said.<\/p>\n The four authors then considered what happens to the surrounding surface as the system evolves. Maybe it warps or twists or otherwise changes from the completely flat surface they measured originally. Then they demonstrated that even as the surface deforms, the total charge along it remains the same.<\/p>\n That is, if you measure charge at every point on a piece of paper, then distort the paper and measure again, you\u2019ll get the same number. You can say that charge is conserved along the surface, and since the surface is indexed to the string, you can say it\u2019s conserved along the string, too \u2014 regardless of what kind of string you started with.<\/p>\n \u201cThe mechanics of a superconducting string and a strong-force string are completely different, yet the mathematics of these strings and the conservation [laws] are exactly the same,\u201d Seiberg said. \u201cThat\u2019s the beauty of this whole idea.\u201d<\/p>\n The suggestion that a surface remains the same \u2014 has the same charge \u2014 even after it\u2019s deformed echoes concepts from the mathematical field of topology<\/a>. In topology, mathematicians classify surfaces according to whether one can be deformed into the other without any ripping. According to this viewpoint, a perfect sphere and a lopsided ball are equivalent, since you can inflate the ball to get the sphere. But a sphere and an inner tube are not, as you\u2019d have to gash the sphere to get the inner tube.<\/p>\n Similar thinking about equivalence applies to surfaces around strings \u2014 and by extension, the quantum field theories inside of which those surfaces are drawn, Seiberg and his co-authors wrote. They referred to their method of measuring charge on surfaces as a topological operator. The word \u201ctopological\u201d conveys that sense of overlooking insignificant variations between a flat surface and a warped one. If you measure the charge on each, and it comes out the same, you know that the two systems can be smoothly deformed into each other.<\/p>\n Topology allows mathematicians to look past minor variations to focus on fundamental ways in which different shapes are the same. Similarly, higher symmetries provide physicists with a new way of indexing quantum systems, the authors concluded. Those systems may look completely different from each other, but in a deep way they might really obey the same rules. Higher symmetries can detect that \u2014 and by detecting it, they allow physicists to take knowledge about better-understood quantum systems and apply it to others.<\/p>\n \u201cThe development of all these symmetries is like developing a series of ID numbers for a quantum system,\u201d said Shu-Heng Shao<\/a>, a theoretical physicist at Stony Brook University. \u201cSometimes two seemingly unrelated quantum systems turn out to have the same set of symmetries, which suggests they might be the same quantum system.\u201d<\/p>\n Despite these elegant insights about strings and symmetries in quantum field theories, the 2014 paper didn\u2019t spell out any dramatic ways of applying them. Equipped with new symmetries, physicists might hope to be able to answer new questions \u2014 but at the time, higher symmetries were only immediately useful for re-characterizing things physicists already knew. Seiberg recalls being disappointed that they couldn\u2019t do more than that.<\/p>\n \u201cI remember going around thinking, \u2018We need a killer app,\u2019\u201d he said.<\/p>\n To write a killer app, you need a good programming language. In physics, mathematics is that language, explaining in a formal, rigorous way how symmetries work together. Following the landmark paper, mathematicians and physicists started by investigating how higher symmetries could be expressed in terms of objects called groups, which are the main mathematical structure used to describe symmetries.<\/p>\n A group encodes all the ways the symmetries of a shape or a system can be combined. It establishes the rules for how the symmetries operate and tells you what positions the system can end up in following symmetry transformations (and which positions, or states, can never occur).<\/p>\n Group encoding work is expressed in the language of algebra. In the same way that order matters when you\u2019re solving an algebraic equation (dividing 4 by 2 is not the same as dividing 2 by 4), the algebraic structure of a group reveals how order matters when you\u2019re applying symmetry transformations, including rotations.<\/p>\n \u201cUnderstanding algebraic relationships between transformations is a precursor to any application,\u201d said Clay C\u00f3rdova<\/a> of the University of Chicago. \u201cYou can\u2019t understand how the world is constrained by rotations until you understand \u2018What are rotations?\u2019\u201d<\/p>\n By investigating those relationships, two separate teams \u2014 one involving C\u00f3rdova and Shao and one that includes researchers at Stony Brook and the University of Tokyo \u2014 discovered that even in realistic quantum systems, there are non-invertible\u00a0symmetries that fail to conform to the group structure, a feature that every other important type of symmetry in physics fits into. Instead, these symmetries are described by related objects called categories which have more relaxed rules for how symmetries can be combined.<\/p>\n For example, in a group, every symmetry is required to have an inverse symmetry \u2014 an operation that undoes it and sends the object it acts on back to where it started. But in separate<\/a> papers<\/a> published last year, the two groups showed that some higher symmetries are non-invertible, meaning once you apply them to a system, you can\u2019t get back to where you started.<\/p>\n This non-invertibility reflects the way that a higher symmetry can transform a quantum system into a superposition of states, in which it is probabilistically two things at once. From there, there\u2019s no road back to the original system. To capture this more complicated way higher symmetries and non-invertible symmetries interact, researchers including Johnson-Freyd have developed a new mathematical object called a higher fusion category.<\/p>\n \u201cIt\u2019s the mathematical edifice that describes the fusions and interactions of all these symmetries,\u201d C\u00f3rdova said. \u201cIt tells you all the algebraic possibilities for how they can interact.\u201d<\/p>\n Higher fusion categories help to define the non-invertible symmetries that are mathematically possible, but they don\u2019t tell you which symmetries are useful in specific physical situations. They establish the parameters of a hunt on which physicists then embark.<\/p>\n \u201cAs a physicist the exciting thing is the physics we get out of it. It shouldn\u2019t just be math for the sake of math,\u201d Schafer-Nameki said.<\/p>\n Equipped with higher symmetries, physicists are also reevaluating old cases in light of new evidence.<\/p>\n For example, in the 1960s physicists noticed a discrepancy in the decay rate of a particle called the pion. Theoretical calculations said it should be one thing, experimental observations said another. In 1969, two<\/a> papers<\/a> seemed to resolve the tension by showing that the quantum field theory which governs pion decay does not actually possess a symmetry that physicists thought it did. Without that symmetry, the discrepancy disappeared.<\/p>\n But last May, three physicists proved<\/a> that the 1969 verdict was only half the story. It wasn\u2019t just that the presupposed symmetry wasn\u2019t there \u2014 it was that higher symmetries were. And when those symmetries were incorporated into the theoretical picture, the predicted and observed decay rates matched exactly.<\/p>\n \u201cWe can reinterpret this mystery of the pion decay not in terms of the absence of symmetry but in terms of the presence of a new kind of symmetry,\u201d said Shao, a co-author of the paper.<\/p>\n Similar reexamination has taken place in condensed matter physics. Phase transitions occur when a physical system switches from one state of matter to another. At a formal level, physicists describe those changes in terms of symmetries being broken: Symmetries that pertained in one phase no longer apply in the next.<\/p>\n But not all phases have been neatly described by symmetry-breaking. One, called the fractional quantum Hall effect, involves the spontaneous reorganization of electrons, but without any apparent symmetry being broken. This made it an uncomfortable outlier within the theory of phase transitions. That is, until a paper in 2018<\/a> by Xiao-Gang Wen<\/a> of the Massachusetts Institute of Technology helped establish that the quantum Hall effect does in fact break a symmetry \u2014 just not a traditional one.<\/p>\n \u201cYou can think of [it] as symmetry-breaking if you generalize your notion of symmetry,\u201d said Ashvin Vishwinath<\/a> of Harvard University.<\/p>\n These early applications of higher and non-invertible symmetries \u2014 to the pion decay rate, and to the understanding of the fractional quantum Hall effect \u2014 are modest compared to what physicists anticipate.<\/p>\n In condensed matter physics, researchers hope that higher and non-invertible symmetries will help them with the fundamental task of identifying and classifying all possible phases of matter<\/a>. And in particle physics, researchers are looking to higher symmetries to assist with one of the biggest open questions of all: what principles organize physics beyond the Standard Model.<\/p>\n \u201cI want to get the Standard Model out of a consistent theory of quantum gravity, and these symmetries play a critical role,\u201d said Mirjam Cvetic<\/a> of the University of Pennsylvania.<\/p>\n It will take a while to fully reorient physics around an expanded understanding of symmetry and a broader notion of what makes systems the same. That so many physicists and mathematicians are joining in the effort suggests they think it will be worth it.<\/p>\n \u201cI have not yet seen shocking results that we didn\u2019t know before, but I have no doubt it\u2019s quite likely this will happen, because this is clearly a much better way of thinking about the problem,\u201d Seiberg said.<\/p>\n
\nA New Kind of Symmetry Shakes Up Physics<\/br>
\n2023-04-19 21:58:06<\/br><\/p>\nEquivalent Surfaces<\/strong><\/h2>\n
From New Symmetries to New Mathematics<\/strong><\/h2>\n
Early Applications<\/strong><\/h2>\n