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Space, Mass, and Quarks P. Żenczykowski Institute of Nuclear Physics Polish Academy of Sciences Kraków, Poland May 11, 2006 Physics provides DESCRIPTION of phenomena In order to describe phenomena we use language Over centuries we have found that a very useful description is provided by the language of mathematics Consequently, some people think that mathematics may lead us in our attempts to describe Nature This is erroneous Mathematics does not lead anywhere It just provides a tool which facilitates and quantifies the description and the idea behind it In reality we are almost totally blind, we have no maps of new territory - it was, it is, it always will be like that Perfect figure Pythagoras - the most perfect figure is a sphere Vision, intuition, aesthetics r2 2 F m1 m 2 r 2 F m1 m 2 r Spherically symmetric r2 - SO(3) scalar Macroscopic classical space and elementary particles At least some of the attributes of observed particles are closely connected with the properties of macroscopic space Examples: Spin – rotation Parity, chirality – space reflection P Existence of particles and antiparticles – C – CPT – (Antiparticles as particles moving backward in time) Other quantum numbers – internal symmetries? Flavour, colour? There were attempts – no connection seen so far NO GO theorems However: such NO GO theorems are valid only EXACTLY in the form they were proven In fact: there exists an enormous space of untried concepts, ways, and possibilites HYPOTHESIS: All attributes of elementary particles are connected with the properties of macroscopic classical arena on which physical processes are deemed to occur Roger Penrose: („Structure of spacetime”, 1968) I do not believe that a real understanding of the nature of elementary particles can ever be achieved without a simultaneous deeper understanding of the nature of spacetime Other physicists of similar opinions: Wheeler: „How could one believe any account of the foundation for the central structure of physics, spacetime, which proceeded without reference to the quantum,..?” Finkelstein: „inherent spacetime illusion” Einstein: „Time and space are modes by which we think, …” Nonrelativistic or relativistic description ? Spin – rotation - nonrelativistic Parity – reflection - nonrelativistic For example one may linearize Schrödinger equation 2mE=p2 Existence of left and right does not have much to do with Lorentz transformations and different transformations of L and R Invariance under reflections requires that p2 p 1 = 0 3 p –p Somebody may say: but relativistic Dirac equation not only yields 1 1 – 1proper gyromagnetic ratio g=2 [e (L + 2 S)B/m] but also leads to ANTIPARTICLES Left and Right, Dirac matrices - - no relativity needed Gyromagnetic: Levy-Leblond 1971 This is true. But it does NOT follow from relativity! Particles to antiparticles? Nonrelativistic description is enough: (Horzela, Kapuścik; P.Ż) In words: no relativity is needed in the idea of antiparticles as particles moving „backwards in time” In formulas: in linearized Schrödinger equation we must obtain the term (p – e A) 1 (p – e A) As for Dirac case: complex conjugation yields But p* = – p, and C * C-1 = ( C = – i 2 2 ) „e” (p* – e A) * hence – (p + e A) change of relative sign at This suggests that NONRELATIVISTIC description should be sufficient (at least at the beginning) when trying to extend our understanding of quantum numbers of elementary particles in terms of space concepts Mass Higgs mechanism: (+) renormalizability with massive gauge bosons () THEORY: just shifting the problem constant term in energy density of vacuum 55 orders of magnitude too large EXPERIMENT: not observed (?) PHILOSOPHY: tendency to „explain” phenomena in terms of material objects rather than abstract concepts („heat fluid”, …) Particle and quark masses Free particles mp, mn, m F=evxB F = ma THEORETICAL CALCULATIONS LQCD(mq) mp= fp,theory(mq,…) e/m, m theory used to extract mass i.e. again: theory used to extract mass Conceptually identical method actually used PROBLEM method NOT used Constituent, current quark masses constituent – current with interaction A q1 5q2 2 m A q1 p 1 p 2 5q2 m1 m2q1 5q2 external quarks current: 2 2 m , mK mu md , md ms Nothing in common with mp = fp,theory(mq,…) Constituent – e.g. magnetic moments Free Dirac quarks again Free Dirac equation used Quarks on mass shell Plane, infinite waves No confinement CONCEPTUAL NONSENSE Quark mass – internal quarks Trees Loops WRHD s u u charmed quark mass s d d s Gaillard, Lee (1974) 1.5-2.0 GeV No quark model pole Quark mass through propagator No: Zweig Yes: Gell-Mann SU(3)L SU(3)R meson formfactors (Schwinger – magnetic moments) DEEPLY 1) Conceptual problems & internal inconsistencies DISSATISFYING: 2) Point 1 either unnoticed or „swept under the carpet” Mass problem Quark mass - no standard propagators (trees) - some meaning in loops? - extension of concept ? detach concept of mass from concept of standard propagator broader structure needed One basic constant needed: e.g. m0 – other masses from (unknown) theory through dimensional mass ratios If (c), ħ added – the basic constant may be of dimension GeV/cm ( Fundamental length, momentum) M. Born -1949 – „Reciprocity theory of elementary particles” „I think that the assumption of the observability of the 4-dimensional distance of two events inside atomic dimensions (no clocks or measuring rods) is an extrapolation… …I am inclined to interpret the difficulties which QM encounters in describing elementary particles and their interactions as indicating the failure of that assumption There is of course a quantity analogous to R=t2-x2, namely P=E2-p2 , not continuous (square of rest mass). A determination of P is not a real measurement, but a choice between a number of values corresponding to the particles… It looks, therefore, as if the distance P in momentum space is capable of an infinite number of discrete values which can be roughly determined, while the distance R in coordinate space is not an observable quantity at all This LACK OF SYMMETRY seems to me very strange and rather improbable.” Born’s principle of reciprocity („odwzajemnianie, wzajemność”) Laws of nature are invariant under reciprocity transformation: xk pk, pk -xk H x k pk , k p H xk xk , pl i kl Lkl xk pl xl pk Arena of Physics Macroscopic space on which (classical) processes are described Usually adopted description: arena = spacetime Hamiltonian formalism: independent position and momentum coordinates CONJECTURE: use NONRELATIVISTIC PHASE SPACE as the arena on which processes are to be described introduce constant [GeV/cm] to permit SYMMETRIC treatment of momentum and position Further arguments (nonrelativistic) In classical Hamiltonian formalism – time occupies a distinguished position – it is a parameter upon which p and x depend In quantum mechanics p and x are operators, while time is still a c-number parameter Relativistic field theory unites relativity and quantum physics, but does this in a formal way, argued by many to be unsatisfactory (Wigner, Dirac, Chew, Finkelstein, Penrose) Instantaneous reduction of a nonlocal state does not seem to be in accord with the spirit of relativity (instantaneous in which Lorentz frame?) This highlights essential difference between time and space, In spite ot these notions being united in the concept of Minkowskian spacetime (description!) Local Lorentz-equivalent frames are not fully equivalent physically (background radiation defines a preferred frame) Further physical arguments (GeV/cm) 1) Description in terms of spacetime followed from the observation that Maxwell equations are form-invariant under Lorentz transformations, which transform time space, with [c]=cm/s being the dimensional constant permitting these transformations to be effected 2) String-like properties of hadrons (with energy proportional to the length of the string) suggests introduction of []=GeV/cm; dual string model of 60’s-70’s: relevant constant is [’] = GeV - 2 (slope of Regge trajectories) (add [ħc] = GeV cm); nowadays: string-like properties of hadrons - confined flux tubes, here: more fundamental in origin whether is related to 1/’ 1 GeV2 or m2Planck 10 38 GeV2 is irrelevant at the moment Constants (phase space, GeV/cm) and ħ (quantum, GeVcm) suffice to set mass scale. Thus, the problem of mass may be addresssed. „Perfect figure” in phase space Generalization: 2 p x 2 beyond Born reciprocity beyond sphere in 3 dimensions O(6) Require Poisson brackets (commutators) to be form-invariant U(1) SU(3) local? „global” Perhaps, but: further conceptual shifts needed Generators SU(3) F1 p1 p2 x1x2 F2 x1 p2 x2 p1 ... F5 x3 p1 x1 p3 ... F8 ... U(1) R=… Reciprocity transformations, Standard reflections p -x - p xp-x Standard rotations + … So far p and x treated completely symmetrically: One cannot really say what is momentum and what is position Standard mass: momentum-position distinction For individual objects separated by large distances: energy of observed FREE objects (elem. particles) is defined by their mass and momenta (either via relativistic or nonrelativistic formula) THE STANDARD CONCEPT OF MASS may be said to be directly ASSOCIATED with the concept of momentum (p), NOT position (x) Recall: Born The six-dimensional vector (p1,p2,p3,x1,x2,x3) is divided into two triplets of canonically conjugated variables in such a way that ONE of the triplets is ASSOCIATED with the concept of „mass” Generalization of the concept of mass Is such a division into (p1,p2,p3) and (x1,x2,x3) unique? Gen.momentum SU(3) gen.position (p1,p2,p3) (x1,x2,x3) (p1,x2,-x3) (-x1,p2,x3) (x1,-x2,p3) (x1,-p2,p3) (p1,x2,-p3) (-p1,p2,x3) Odd # of p’s SO(3) SO(3) Even # of p’s Generalizarion through SU(3), not U(1) U(1) Concept of mass may be ASSOCIATED with these 1 + 3 possibilities Rotational (translational) invariance ? Since each of three new choices violates rotational invariance – - an object with mass associated with any one of these choices CANNOT belong as individual object to our rotationally invariant macroworld However, these objects could belong to macroworld as unseparable components of composite objects, provided the latter are constructed in such a way, that all appropriate invariance conditions are satisfied CONJECTURE: QUARKS Argument that strong interactions are rotationally etc. invariant is NOT a valid one: What we ALWAYS see is the interaction of external probes (photons, W,Z bosons) with colour-singlet currents. Colour-singlet current has quantum numbers of a colourless meson, and from the point of photon, etc. behaves like a hadron – not an individual quark (e.g. VMD) In other words quarks have to conspire in such a way that the resulting composite object – hadron - behaves in a „normal” way Toy model Apply SU(3) to Dirac equation (simpler than non-rel. Schródinger) (linearize p2+x2+m2) H D Ak pk Bm Ak k 0 k 1 0 B 0 0 3 0 Bk .......... 0 2 k Ak , Al Bk , Bl 2 kl Ak , Bl Ak , B Bk , B 0 BB 1 H R A1 p1 B2 x2 B3 x3 Bm H G B1x1 A2 p2 B3 x3 Bm H B A1 p1 B2 x2 B3 x3 Bm H = HR+HG+HB Rotational invariance restored Charge conjugation HD=Ak(pk- e A k)+Bm+e A 0 Explicit particles Antiparticles through reinterpretation c.c.: i - i, x x, t t, H - H, p - p, Ak A*k, B B*, A A, m m, e e, -H’D= A*k(-pk- eA k)+B*m+e A 0 HD=Ak(pk+ eA k)+Bm-e A 0 HR=A1(p1- e A 1)+B2x2+B3x3+Bm quark C=-i 222 CB*C-1=-B CA*kC-1=A k CB*kC-1=B k HR=A1(p1+e A 1)-B2x2-B3x3+Bm antiquark When particle goes into antiparticle, physical quantities change as follows: E E, ,p p, x - x, e - e, m m, (and i - i) Simple quark-antiquark system H R H R A1 ( p1 p1 ) B2 ( x2 x2 ) B3 ( x3 x3 ) B 2m Translation and rotation invariant after summation over quarks, antiquarks of all three types Spin 0 and 1 (tensor product kl0) Comments: * additivity of quark charges * additivity of quark masses * appearance of a „string” * intrinsic angular momentum of a quark: p1 x2 etc. * objects exhibiting well-defined properties of one kind do not have well-defined properties of another kind * vague similarity to RISHON scheme of Harari (1979) Two rishons (charge): T(+1/3), V(0) e+ T T T VVV u TTV TVT VTT d VVT VTV TVV Last transparency Mirage only? I believe not… CONCEPTS OF PHYSICS: „Physics is based on well-founded concepts, and its progress relies heavily on the creation of new concepts. Our understanding of Physics deepens as we gain new perspectives on already existing notions. Therefore, Physics continuously requires both new concepts and new perceptions of the old ones.”