PoS(HRMS)024 http://pos.sissa.it/ † ∗ [email protected] Speaker. University of Maryland Preprint PP-11-001 I write about Héctor, his contributions tosion the of early quantum work statistics in the model, and a general discus- ∗ † Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c

Quarks, Strings and the Cosmos -August Héctor 09-11, Rubinstein 2010 Memorial Symposium AlbaNova )Stockholm) Sweden O.W. Greenberg Center for Fundamental , Department ofUniversity Physics of Maryland, College Park, MDand Helsinki 20742-4111, Institute USA for Physics P.O.Box 64 (Gustaf Hällströmin katu 2) FIN-00014 University of Helsinki, Finland E-mail: and quantum statistics PoS(HRMS)024 and 2 ρ m = 2 ω m O.W. Greenberg 3 to his aces. This ] and Murray Gell- / 1 . Experiment showed that ρπ → φ over ¯ K K → 2 φ was suppressed by over two orders of magnitude. This was an important motivation for ] in 1964. Zweig focussed on the constituent aspect of what he called “aces.” Gell-Mann 2 ρπ To make baryons out of 3 aces, Zweig had to assign baryon number 1 Zweig was struck by the dominance of Héctor and I first met at a summer workshop in Bebek, near Istanbul, Turkey. The workshop The tallest person above the left of the last row is Shelly Glashow, to his left is Nick Bur- Héctor and I stayed in touch after the Bebek workshop. We met often in Paris, where Héctor Héctor was brusque, outspoken, stimulating, energetic, fun to be with, and overall one-of-a- The first part of my talk will review the early days of the quark model, and Héctor’s early As we all know, the quark model was introduced by George Zweig [ → Mann [ φ Zweig. Zweig explained thisaces. by In making doing the this strangeerything Zweig and had that the to can disregard non-strange possibly Feynman’s mesonsGell-Mann’s dictum happen version from that of does, different this “in and rule the was with strongden the interactions “totalitarian the is principle” ev- compulsory.” maximum that “everything strength Zweig thatAllowed allowed drew is processes not quark by must forbid- have line unitarity.” some diagrams, quark lines “Zweig connecting diagrams,” the to initial and illustrate finalled his states. directly rules. to fractional electric charges for the aces. Zweig found mass relations, focussed on the algebraic properties of his “quarks.” photograph is at the endthere, of this but paper. he At is first turnedfull sight, around of Héctor talking energy is to or nowhere pose arguing to passively with be for somebody seen. a photo. behind Actually, him. he is Thatgoyne. is One Héctor, row too down, fromDell’Antonio, the Bruria left, Kaufman, are and Eduardo Arthur Caianiello, Jaffe,down among Joe others. is Dothan, On H.R. Sidney the Mani, Coleman, left and Gian-Fausto ofmentioned. on the David third the Fairlie row right is of near that theGiulio row, left Racah there end is is of near the Héctor, the fourth turned rightrow, Feza row. facing end Gürsey, Eugene of backwards the Wigner the as organizer is of next just near the row. the workshop, Louis left is Michel and at is the a end, bitliked and to to I go the am to right sitting Cafe of next Select the to Latin,in him. bottom which Moscow, was Helsinki, frequented Stockholm, by and people in fromvisit, Latin and Maryland, America. gave where the We for also denizans met some of years College Héctor Park the made news an from annual Israelkind, and honest Europe. to a fault, and full of vitality. We all miss him. 2. Outline of my talk contributions. Then Itwo will topics: shift to the a possiblethe general statistics precision discussion that experimental of tests quantum quantum for mechanicsremarks the statistics about and electron Héctor’s with quantum last and interests–cosmology emphasis field the and on theory photon. astroparticle physics. allow, I and will conclude with3. some brief Quarks and aces Quarks and quantum statistics 1. Introduction PoS(HRMS)024 , ) n 6 µ ( as a / from l p SU µ ] and in 6 and L O.W. Greenberg symmetry, with abstraction f lavor ) ] Many physicists were 3 , etc., in order to make 3 ( ) 12 SU ( ] I predicted the spectrum of ˜ , U s 8 , , ) d symmetry in which the quark is a 6 , , 6 u ( spin − U theory as a step towards a relativistic the- f lavor ] “It is fun to speculate about the way quarks ) 2 spin 6 ( − ] calculated the magnetic moment ratio, 5 3 SU f lavor ) 6 ( SU ], were the first to introduce spin in the quark model in an (his “”). He introduced “leptons” 4 ) 3 ] proved that the only way to incorporate an internal symmetry symmetry, to ( 7 2 for the quarks is the simplest way to account for baryons of spin SU spin / ) as a classification symmetry. 2 ( ) 3 ( SU SU that agreed well with experiment. Now we understand that Zweig’s aces are the ) 2 ρ m + 2 φ m a fully relativistic theory. Some physicists hoped that a fully relativistic version of ) The response of the physics community to quarks or aces was mixed. Zweig took them as Here the 3 of parafermi statistics is the same 3 as the 3 of color. Parafermi statistics of order 3 What worked was to take quarks as real, but change their statistics. I realized that the Pauli Feza Gürsey and Luigi Radicati [ Gürsey and Radicati presented the Gell-Mann’s path to quarks went through models from which he abstracted symmetries. He = ( 6 ( ? 2. Gürsey and Radicati went much further. They unified the 2 K / deeply skeptical of the quark model, and even more of color. Oppenheimer’s response to me when 4. Mixed response of physics community real. Gell-Mann’s attitude was ambivalent.from Gell-Mann’s CERN response was “Oh, to the Zweig concrete when quark Zweig model. returned That’s for blockheads!” [ is equivalent to color spin up and spin down, 6. With this larger symmetryoctet they and recovered decuplet the and Gell-Mann–Okubo found massA. a relations Baqi new for Bég, relation the Benjamin between baryon Lee the and masses Abraham of Pais the [ octet and decuplet. Mirza SU excited states of baryons usingsymmetric parafermi state quarks in of the order visible 3needed and degrees for the the of Pauli fact freedom, principle that space, accounted 3 for spin quarks by and the can flavor, parafermi be with statistics, in the a antisymmetry exclusion principle is valid if quarks are parafermions of order 3. [ with Poincaré symmetry is supersymmetry. That was not to be the way. would behave if they were physical particlesas of they finite would mass be (instead in of thereassure purely limit mathematical us of entities of infinite the mass). non-existencea ... of Lagrangian real ” quarks.” field He theory. Gell-Mann continued with usedtheory. He “A the search used method In ... the of another would symmetries paper help ofin to he the French wrote, cuisine: Lagrangian “We to a compare infer piecediscarded.” this symmetries of process pheasant of to meat the a is cooked method between sometimes two employed slices of veal, which are then 1 using this symmetry, another success. Gürsey andsymmetric Radicati quark assigned the state. ground state This baryonsprinciple. assignment to the was paradoxical, because it violated the Pauli exclusion ory. Several authors explored higher symmetries, such as would wrap up . No-goHaag, theorems, Lopuszanski culminating and in Sohnius Coleman [ and Mandula [ intrinsic way. Choosing spin 1 went from global symmetry to Quarks and quantum statistics m constituent quarks of the quark model. tool to construct unitary spin. In hischarges. 1964 paper In he his proposed quark the model quark paper, model Gell-Mann with wrote fractional [ electric PoS(HRMS)024 is V H where , O.W. Greenberg V H + 0 H = H ] ] 12 13 ] ] 14 10 ] that states of several identical particles are 16 ] 4 11 ] 15 ], “At that time (1967) I didn’t have any faith in the existence of quarks.” 9 results; nonetheless he was outspoken and unambiguous that quarks are real. ) 6 ( SU Meson Decays in the Quark Model” [ + and ) Messiah’s “symmetrization postulate” (SP), [ The Hamiltonian (and all observables) must commute with permutations of the identical par- I will describe three issues related to quantum statistics: In contrast to Gell-Mann, Oppenheimer, Weinberg and many other physicists, Héctor whole- 3 ( small and does not commuteterm with that violates permutations, CP in or analogydistinguishability other with of symmetries. the the A way identical small particles. you statistics-violatinglike could term Also, red add would you electrons destroy a cannot or the small introduce blue in- Violating a electrons, statistics new by because degree a that of small would amount freedom, double requires something the subtle. pair5.2 production cross What section. allows either symmetric or antisymmetric underticles permutations, only is occur equivalent in to one-dimensionalmechanics stating representations that allows of identical states the par- not symmetric obeying group.is the an However, symmetrization quantum additional postulate; assumption theversion that of symmetrization is quantum postulate not mechanics without a the basic symmetrization postulate, principle the of concept quantum of “ray” mechanics. must be To formulate a “Spin 2 “Electromagnetic Properties of in the“Nucleon-Antinucleon Quark Annihilation Model” in [ the Quark Model”“A [ Dynamical Quark Model for Hadrons.” [ ticles; otherwise they would not beIt treated in does an not equivalent way. depend Thisstatistics is on a you locality, very cannot primitive or just condition. covariance, add a or small quantum term field that theory. violates statistics, In say particular, to violate 5.1 Difficulty to violate statistics by a small amount (1) Difficulties to make a model(2) with Theoretical small possibilities violations for of statistics statistics, in(3) three Types space of dimensions, experimental tests. Quarks and quantum statistics I showed him my paperSteven in Weinberg wrote 1964 [ was “Greenberg, it’s beautiful, but I don’theartedly believe embraced a the word idea of of it.” that quarks as required real quarks objects. to He be1966, immediately physical Héctor started wrote particles, developing or not models co-authored mere 6“Electromagnetic papers mathematical Mass on constructs. Differences the in quark the In model: Quark the Model” one [ year, “Dynamical Derivation of Baryon Masses in the Quark Model” [ This impressive output makes clear that Hectornot completely just invested in a the mechanism idea to thatSU get quarks are group real, theory results.This He reflected did both realize his physics that judgement quarks and could his be forthright used and enthusiastic to personality. get 5. Quantum statistics PoS(HRMS)024 (5.1) fermi → ] can con- ] (In fewer 19 22 . This system O.W. Greenberg e is odd, the dyon violates the kl π δ 2 / = bose (fermi) theories with anoma- † l eg p ] and infinite statistics. [ a k a 21 ,[ = p } + ] j † ] also emphasized: States in inequivalent rep- a , 18 k 5 a + [ − ] j † a , k and a charged particle with electric charge a [ g { ) ] can occur in 3 or more space dimensions: parabose statistics of 2 can be represented as sums of / 20 1 p ( bose transitions to the opposite type, but cannot produce bose → ] parafermi statistics of integer order 0, one can calculate all matrix elements of products of annihilation and creation 21 = i ,[ 0 p | fermi or bose k a 1 corresponds to the usual bose and fermi theories. These parastatistics theories have → = ] and that R. Amado and H. Primakoff [ p 17 This incoherence leads to a primitive superselection rule that Messiah and I first pointed In theories with noncommutative spacetime coordinates the Drinfel’d “twists” [ Three types of statistics [ Parastatistics theories can be formulated as local relativistic quantum field theories. Parabose One can find the commutation relations for infinite statistics by averaging the commutation vert fermi integer order This algebra is called the Cuntzcondition, algebra in the mathematical literature. With the Fock-like vacuum out [ Quarks and quantum statistics generalized to the concept of “generalized ray,” inthe which ambiguity the phase of ambiguity a of unitary a rayIndeed, representation is the of replaced usual an by phase irreducible ambiguity representation isresentation of a of the unitary the representation symmetric symmetric of group. a group. one-dimensionalcannot States irreducible be in rep- coherent, inequivalent i.e. representations cannot of interfere, the because symmetric no group observable connects such states. resentations of the symmetric groupare are that separated no by transitions this cancan superselection occur occur rule. between between bosonic Corollaries SP-obeying and of andfrom fermionic this SP-violating the states. meaning states, I of and emphasize identical no that particles,properties this transitions and of superselection does quantum rule not field follows require theory Lorentz covariance, locality, or other transitions or vice versa. 5.3 Theoretical possibilities for quantum statistics than 3 dimensions, fractional statisticsbose (anyons) statistics occur can for occur.) particles having Theodd-half integer integer rule statistics, spin also we and holds fermi all in statistics the know obviousis occur and way for an far love, particles parabose exception, and that having not parafermi statistics. yetmonopole, There discovered in with nature, magnetic the charge dyon. The dyon is a composite of a a magnetic (parafermi) theories of order relations for bose and fermi statistics, has angular momentum stored inspin-statistics its connection electromagnetic for field. the charged If particle in 3 dimensions. 5.3.1 Parastatistics lous relative commutation relations via thecan Green also ansatz. be Messiah treated and without I using showed therepresentation that Green of these ansatz, the and theories Green that trilinear the Green commutationcase, ansatz rules can that be has derived a for any Fock-likegross vacuum violations state. of In the each usual statistics and do not provide5.3.2 a model Infinite for statistics small violations of statistics. PoS(HRMS)024 (5.2) (5.3) . For q ] repeated 25 This improve- 1 that occur in O.W. Greenberg 29 ± , occur with equal − is needed and none n S 10 † a × 1 the more symmetric † . a 6 kl δ → or ≤ q = F v aa k a rather than the † l q qa − † l S a ρ k F . This bound was improved by the VIP a occur, with weights dependent on v 26 n = + − S A + ] 10 ρ † l ) ] in 1947. They pointed out that if the electrons a × 6 F , v k 7 24 . a [ 1 − q 1 ≤ 2 − F = ( 1 v ρ + 0 all representations occur with equal weight. Quons give a − 1. An analog of Wick’s theorem holds. In contrast to the bose ] ] in (2010), which gave the bound † l = ± a 26 , q k = ] a [ q q 23 2 + quons, all representation of 1 n 1 the more antisymmetric representations dominate, and for There are four main types of tests of quantum statistics: The pioneering experiment to search for transitions between anomalous states was carried out For a state with To generalize infinite statistics as described by the Cuntz algebra, take a convex sum of the → − This is called thecondition quon suffice to algebra. calculate all matrix Again, elements, the and no quon relation commutation on relations and the Fock vacuum this type of experiment running athe 30 A slightly current displaced in x-rays. a thinwith They copper high strip had precision. for Avogadro’s With one number the month on two-body and density their looking matrix for side in and the found form a null result ment came from three factors, (i) using CCD detectors that give greater sensitivity, (ii) running (i) Transitions between anomalous states, (ii) Accumulations in anomalous states, (iii) Deviations from the usual statistical(iv) properties Stability of of identical matter. particles, by M. Goldhaber and G. Scharff-Goldhaber [ small violation of statistics bycomes producing from a the smallness mixed of density the matrix. mixture of The “abnormal” smallness states. of5.4 the violation Experimental tests of quantum statistics in beta decay were notelectrons quantum-mechanically could identical fall to into the the electronslet K in the shell atoms, beta of then decay an electrons beta from atom decay alooked in naturally for violation occurring the radioactive of source x-rays the fall from Pauli onqualitative transitions exclusion a bound block into principle. on of the They lead violations K and of shell. the exclusion They principle. did E. not Ramberg find and such G. x-rays Snow and [ found a can be imposed, except for and fermi cases, the termsthe have bose coefficients and that fermi cases. are [ powers of q representations dominate. At bose and fermi commutation relations, Quarks and quantum statistics operators. One does not needno a relation such with relation only can creation oroperators be only acting imposed annihilation on consistently. operators. the Further, vacuum The“quantum is Boltzmann” norm 1. statistics. of The All every Gibbs representations monomialweights. correction of in factor the is symmetric the group, absent annihilation here. This statistics is Ramberg and Snow foundexperiment the of bound Bartalucci, et al [ PoS(HRMS)024 . 11 − 10 × 0 . O.W. Greenberg 4 ≤ ] looked for bose-einstein allowed ν 27 ] studied the possibility of violations forbidden/ 28 ν and Electromagnetic Interactions. Phys. Rev. ) 6 ( 7 SU bose transitions between two-electron states that obey the exclusion → Lett. 13: 514, erratum 650. TheElectromagnetic magnetic Properties moment of ratio Baryons was in found theRev. Lett. independently Supermultiplet 13: by Scheme Sakita, of 643. B. Elem,entary (1964) Particles. Phys. 8419 TH 412. Rev. Lett. 13: 173. Here is the photo of the Bebek summer institute, as well as a photo of Héctor taken in 2008. We conclude that high-precision tests have ruled out violations of statistics with techniques Héctor had the wide ranging interests, and the perspicacity,Héctor to had, turn as did to many astrophysics of and us here cos- today, the good luck to live through the wonderful years in I thank Dr. Dan-Olof Riska for his hospitality at the Helsinki Institute for Physics where I In two papers in 2010, A.P. Balachandran, et al, [ High-precision tests of bose-einstein statistics for photons are more difficult because we don’t [6] Coleman, S. and Mandula, J. (1967). All Possible Symmetries Of The S Matrix. Phys.Rev. 159: 1251. [1] Zweig, G. (1964). An SU(3) Model for Strong Interaction Symmetry and Its[2] Breaking. CERN Gell-Mann, Report M. (1964). A Schematic Model[3] of Baryons Zweig, and G. Mesons. (2010). Phys. Memories Lett. of 8: Murray 214. [4] and the Gürsey, Quark F. Model. and Int.J.Mod.Phys.A Radicati, 25: L. 3863. (1964). Spin and Unitary Spin Independence of[5] Strong Interactions. Bég, Phys. M.A.B., Lee, B.W. and Pais, A. (1964). References now available. No violations of Lorentzfound. covariance have Extra been found. dimensions Superpartners haveLHC. have not not been been found. We have high hopesmology for where such the discoveries great discoveries, at dark the energy and dark matter, have beenwhich made the in standard model recent of years. elementary particlesto was developed. this He development. made significant We contributions are alland sorry with his that insight he in is his no new longer interests, with astrophysics us and inspire cosmology. us withAcknowledgments his contributions prepared this talk. principle. The mass scalecannot for be these observed transitions at is present. well above the Planck scale, so such transitions forbidden two-photon transitions in atomic barium.Yang bose-einstein These selection transtions rule. are They forbidden found by the the bound Landau- of statistics in theories“twists” with can noncommutative induce space bose coordinates. They found that the Drinfel’d have bound states with many photons. In 2010, D. English, et al, [ Quarks and quantum statistics the experiment underground with less background,statistics. and (iii) taking a longer run which give greater PoS(HRMS)024 O.W. Greenberg Meson Decays in the Quark + 8 the s Matrix. (1975). Nucl.Phys.B88: 257. Mesons. Phys. Rev. Lett. 13: 598. Phys.Rev.Lett. 17: 41. Model. Phys. Rev. Lett. 17: 420. Quark Model. Phys. Lett. 22: 208. 1608. Lett. 21: 447. Foundation. Phys. Rev. 136: B 248. 1338. earlier papers cited there. Commun.Math.Phys. 35: 49. small violations of fermi or bose statistics. Phys. Rev. D 43: 4111. parafields. J. Math. Phys. 51: 023530. Phys. Rev. 73: 1472. Principle. Phys.Lett.B 238: 438. Electrons. Found.Phys. 40: 765. 104: 253604. 105:051601; Non-Pauli Effects from Noncommutative Spacetimes. JHEP 1012:001. [7] Haag, R., Lopuszanski, J.T. and Sohnius, M. (1975). All Possible Generators[8] of Supersymmetries of Greenberg, O.W. (1964). Spin and Unitary-Spin Independence in a Paraquark Model of[9] Baryons and Weinberg, S. (2004). The Making of the Standard Model. arXiv:hep-ph/0401010v1. Quarks and quantum statistics [10] Rubinstein, H.R. and Scheck, F. (1966). Electromagnetic Mass Differences in the Quark[11] Model. Elitzur, M., Rubinstein, H.R., Stern, H. and Lipkin, H.J. (1966). Spin-2 [12] Federman, P., Rubinstein, H.R. and Talmi, I. (1966). Dynamical Derivation Of Baryon[13] Masses In Rubinstein, The H.R. (1967). Electromagnetic Properties of Hadrons in the Quark Model.[14] Phys.Rev. 154: Rubinstein, H.R. and Stern, H. (1966). Nucleon-antinucleon annihilation in the quark[15] model. Phys Rubinstein, H.R. (1966). A dynamical quark[16] model for Messiah, hadrons. A. Phys. (1962). Lett. Quantum 22: Mechanics, 210. Vol.[17] II. (North-Holland, Amsterdam), Messiah, p A.M.L. 585. and Greenberg, O.W. (1964). Symmetrization Postulate and Its Experimental [18] Amado, R.D. and Primakoff, H. (1980). Comments on testing the Pauli[19] principle. Phys. Drinfeld, Rev. C V.G. (1990). 22: Quasi-Hopf algebras, Lengingrad[20] Math. J. 1: Doplicher, S., 1419. Haag, R. and Roberts, J.E. (1974). Local observables and[21] particle statistics. 2 Green, and H.S. (1953). A generalized method[22] of field Greenberg, quantization. O.W. (1990). Phys. Example Rev. 90: of 270. infinite statistics. Phys. Rev. Lett. 64: 705;[23] (1991). Particles Greenberg, with O.W. and Mishra, A.K. (2010). Study of the vacuum matrix element[24] of products Goldhaber, of M. and Scharff-Goldhaber, G. (1948). Identification of beta-rays with atomic electrons. [25] Ramberg, E. and Snow, G.A. (1990). A New Experimental Limit On Small[26] Violation Of Bartalucci, The S. Pauli et al. (2010). The VIP Experimental Limit on the[27] Pauli Exclusion Principle English, Violation D., by et al. (2010). Spectroscopic Test of Bose-Einstein Statistics for[28] Photons. Phys. Rev. Balachandran, Lett. A.P. (2010). Non-Pauli Transitions From Spacetime Noncommutativity. Phys.Rev.Lett. PoS(HRMS)024 O.W. Greenberg 9 Quarks and quantum statistics