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Exact, Broken, and Approximate Symmetries A.Zee University of California Santa Barbara, CA 6513 1

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Chadwick’s discovery of the small number ~ 0.00137 [ mass difference of neutron and proton /mass of neutron] led Heisenberg to open a window into internal space. Previously, symmetries of space and time 3

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At one time, people argued that there is something unseemly about exact global internal symmetries. If isospin were exact, to leave the action unchanged, we would have to rotate the proton and the neutron by the same amount throughout the universe. Now, within QCD, we know that isospin is not exact, and “merely” due to the fact that the up and down quark masses (whatever the mechanism responsible for them; their ratio is not particularly close to 1) happen to be much smaller than the QCD energy scale. 5

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Concepts in one area of theoretical physics migrating to another: a glorious history Now that the Higgs particle(s) have been discovered, Ginzburg-Landau (1950) Will we have the analog of BCS (1957)? The universal versus the specific in theoretical physics 7

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Search for the Universal Fermat: Least time principle Euler-Lagrange: Action principle What do we “gain”? Dirac-Feynman path integral formulation of quantum mechanics offers the most natural path to quantum field theory The action principle permeates modern theoretical physics Hilbert almost beat Einstein to the punch! 8

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The deep concept behind all this --- spontaneous symmetry breaking --- dating back to Heisenberg and ferromagnets, was first recognized as such in this context by Nambu. My former “boss” (twice over!) Bob Schrieffer (26 in 1957) told me that when he gave his talk at the University of Chicago, the old guys, in particular Wentzel (59 in 1957), gave him a hard time, but Nambu (36 in 1957) the youngster saw the profound implications. 10

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Two naturalness problems in physics: The cosmological constant problem & the Higgs mass We may have lost the right to talk about naturalness Another possible analogy for the cosmological constant paradox: A civilization on a very large planet with a thick cloud cover Physics could have developed to a high level. With new technology the rate of ships disappearing over the horizon, postulated by some eminent theorists to be mathematically zero, was finally measured to be tiny but not zero. But using the known physics, people are unable to calculate this rate. ……. 12

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What if ? Suppose superconductivity had not been discovered. (Perhaps refrigeration technology is particularly poor in this civilization.). What would have happened to electroweak unification? 13

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Some history Higgs’ first paper mentioned two confusing papers: one by Klein and Lee showing that Goldstone’s theorem could be avoided in a non-relativistic theory because an additional vector becomes available, and a subsequent paper by Gilbert arguing against Klein and Lee showing that such a vector is absent in relativistic theories. Higgs pointed out that in gauge theories, gauge fixing introduced this vector. 15

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Private Higgs Observe that the rich and the powerful do not share chauffeurs Strikes me as strange if the top quark and the electron have to share the same Higgs Various papers with Porto, Bentov, Yuan, and others; many implications for LHC phenomenology and a “natural” candidate for dark matter 17

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Stubble in Occam’s razor Ironically, the more unknown parameters a particle theory model has, the more wriggle room and the more difficult it is to eliminate it. Curse of the free parameters 18

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The Einstein-Hilbert action contains only one parameter G (in fact two, also Λ!) which was already known to Newton. A priori, no reason that it should be so. Suppose it had contained 17 previously unknown parameters instead. Then, in 1915, progress in physics would not have been nearly so dramatic. Experimentalists would have been kept busy for years, possibly decades, measuring these parameters. 19

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Not true that there were no numbers in particle physics to be explained, as some people said. The CKM and the neutrino mixing matrices are the two most expensive matrices since linear algebra was invented. It may well be that these numbers are different in the next village in the landscape, but it is not known where one can buy a bus ticket to the next village to verify this rumor. Hard to imagine how neutrino mixing angles could affect galaxy formation the way the cosmological constant can. 20

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Neutrino mixing Tetrahedral group T, double cover T’, etc etc etc Mixing presumably has to do with the deep mystery of why three families (Feynman: “If you want to be king…..”) vast literature, many many authors: E. Ma, ……, BenTov & Zee, Hartman & Zee (Frobenius groups), …… 21

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Dream of replicating Heisenberg’s and Gell-Mann’s successes have not come to pass. Ratio of masses and transition amplitudes not O(1). Family symmetry: continuous (e.g. SO(3), SU(3)), gauged, discrete Early literature: for example Wilczek and Zee (1977, 1978), many others…. 22

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Impose a symmetry on the action, and hope to “fix” the physics Einstein invented this way of doing theoretical physics 24

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25 From my book Fearful Symmetry (Magische Symmetrie in the German speaking world)

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26 Einstein’s schema (some would say possibly a bad influence on theoretical physics)

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Has the “traditional use” of symmetry in particle physics been exhausted? Again, the roadmap for possible new avenues may be provided by condensed matter physics. Quantum Hall fluid: a new kind of order topological rather than Ginzburg-Landau 27

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For example, the long distance physics of certain classes of quantum Hall fluids is described by Order is not characterized by a local field, but rather by fractional spin, statistics, and ground state degeneracy on genus g surfaces, all determined by the matrix K. Also, chiral spin liquid (Wen, Wilczek, & Zee, …) 28

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Thus far, perhaps disappointingly, this has not been turned into a major new paradigm for particle physics the way Ginzburg-Landau has been. The central role of symmetry in modern theoretical physics 32

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Symmetry as a reflection of the human mind? Physicists in love with their own Creation! what physicists are capable of understanding Jean-Léon Gérome 33

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Superconductivity at 100: some reflections A.Zee Institute for Theoretical Physics University of California Santa Barbara, CA Invited talk at the meeting of the American Physical Society, May 1, 2011 37

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Ginzburg-Landau (1950), Bardeen-Cooper-Schrieffer (1957) Energy of constant magnetic field in superconductor scales faster than volume since Meissner-Ochensfeld effect (1933) => “Higgs mechanism” (1963-64) 38

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Concepts in one area of theoretical physics migrating to another: a glorious history Some examples Diffraction: water wave => sound wave, electromagnetic wave, quantum wave Eigenfrequency: Vibrating string => quantization Spontaneous symmetry breaking: spin wave in ferromagnet => pion as Nambu-Goldstone boson 39

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I end with an extremely lame concluding remark 40

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What does the order parameter have to do with a field? Quantum fields as highly singular operators. “Don’t mess with fields!” Shackles of Feynman diagrams Students were taught quantum field theory as sums of perturbative diagrams (as late as early 1970s or perhaps even now in some places) 41

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Would non-relativistic conclusions hold in a relativistic context? In hindsight, it is clear that the addition of time does not change anything essential, but only in hindsight! Indeed, Higgs’ first paper was in response to some confusion on this issue. See later. 42

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What does free energy have to do with the action of a relativistic theory? Both functionals of fields (order parameters) But the way we learn about free energy, all tied up with mysterious concepts like temperature and entropy, would have prevented us from seeing the connection. Lesson is to look for similar objects from different areas of physics? 45

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What does the London penetration length have to do with the mass of a Yang-Mills gauge boson? Compton wavelength (1922) Conceivably, someone could have seen the connection, but it would have taken a stroke of genius. As far as I know, nobody suggested anything remotely like that. Even Landau, who straddled condensed matter and particle physics, did not see the connection. 46

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Qualitative understanding of mixing matrices *** Tled. Itor. 47

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