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Particle content of models with parabose spacetime symmetry

(Also called: generalized conformal supersymmetry with tensorial central charges; conformal M-algebra; osp spacetime supersymmetry). Particle content of models with parabose spacetime symmetry. Igor Salom Institute of physics, University of Belgrade. Talk outline.

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Particle content of models with parabose spacetime symmetry

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  1. (Also called: generalized conformal supersymmetry with tensorial central charges; conformal M-algebra; osp spacetime supersymmetry) Particle content of models with parabosespacetime symmetry Igor Salom Institute of physics, University of Belgrade

  2. Talk outline • What is this supersymmetry? • Connection with Poincaré (and super-conformal) algebras and required symmetry breaking • Unitary irreducible representations • What are the labels and their values? • How can we construct them and “work” with them? • Simplest particle states: • massless particles without “charge” • simplest “charged” particles

  3. What is supersymmetry {Qa,Qb} = -2i (gm)abPm, [Mmn,Qa] = -1/4 ([gm, gn])abQb, [Pm,Qa] = 0 ? supersymmetry = symmetry generated by a (Lie) superalgebra ruled out in LHC? Poincaré supersymmetry! = HLS theorem – source of confusion?

  4. But what else? in 4 spacetime dimensions: in 11 spacetime dimensions: this is known as M-theory algebra can be extended to super conformal case {Qa,Qb}=0 {Qa,Qb}= 0 Tensorial central charges

  5. + supersymmetry: [Mmn,Qa] = -1/4 ([gm, gn])abQb, [Pm,Qa] = 0, {Qa,Qb} = -2i (gm)abPm + conformal symmetry: [Mmn,Sa] = -1/4 ([gm, gn])abSb, {Sa,Sb} = -2i (gm)abKm, [Km,Sa] = 0, + tens of additional relations Simplicity as motivation? Poincaré space-time: Something else? Parabose algebra: [Mmn,Mlr] = i (hnl Mmr + hmr Mnl - hml Mnr - hnr Mml), [Mmn,Pl] = i (-hnl Pm +hml Pn), [Pm,Pn] = 0 hmn = • mass (momentum), spin • usual massless particles • “charged” particles carrying SU(2) x U(1) numbers • “elementary” composite particles from up to 3 charged subparticles • a sort of parity asymmetry • ….(flavors, ...)? 1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1 mass (momentum), spin

  6. Parabose algebra • Algebra of n pairs of mutually adjoint operators satisfying: and relations following from these. • Generally, but not here, it is related to parastatistics. • It is generalization of bose algebra:  ,

  7. Close relation to orthosymplectic superalgebra • Operators form osp(1|2n) superalgebra. • osp generalization of supersymmetry first analyzed by C. Fronsdal back in 1986 • Since then appeared in different context: higher spin models, bps particles, branes, M-theory algebra • mostly n=16, 32 (mostly in 10 or 11 space-time dimensions) • we are interested in n = 4 case that corresponds to d=4. From now on n = 4

  8. Change of basis- step 1 of 2 - • Switch to hermitian combinations consequently satisfying “para-Heisenberg” algebra:

  9. Change of basis - step 2 of 2 - • define new basis for expressing parabose anticommutators: • we used the following basis of 4x4 real matrices: • 6 antisymmetric: • 10 symmetric matrices: , , ,

  10. Generalized conformal superalgebra Connection with standard conformal algebra: Y1 = Y2 = N11 = N21 = P11 = P21 = K11 = K21 ≡ 0 {Qa,Qb}={Qa,Qb}={Sa,Sb}={Sa,Sb}= 0 Choice of basis + bosonic part of algebra

  11. Unitary irreducible representations • only “positive energy” UIRs of osp appear in parabose case, spectrum of operator is bounded from below. Yet, they were not completely known. • states of the lowest E value (span “vacuum” subspace) are annihilated by all , and carry a representation of SU(n) group generated by (traceless) operators . • thus, each parabose UIR is labeled by an unitary irreducible representation of SU(n), labels s1, s2, s3, and value of a (continuous) parameter – more often it is so called “conformal weight” d than E. • allowed values of parameter d depend upon SU(n) labels, and were not completely known – we had to find them!

  12. Allowed d values expressions that must vanish and thus turn into equations of motion within a representation • In general, d has continuous and discrete parts of spectrum: • continuous: d > d1← LW Verma module is irreducible • discrete: d = d1, d2, d3,… dk← submodules must be factored out • points in discrete spectrum may arrise due to: • singular vectors ← quite understood, at known values of d • subsingular vectors ←exotic, did require computer analysis! • Discrete part is specially interesting for (additional) equations of motion, continuous part might be nonphysical (as in Poincare case)

  13. Verma module structure • superalgebra structure: osp(1|2n) root system, positive roots , defined PBW ordering • – lowest weight vector, annihilated by all negative roots • Verma module: • some of vectors – singular and subsingular – again “behave” like LWV and generate submodules • upon removing these, module is irreducible

  14. e.g. this one will turn into and massless Dirac equation! s1=s2=s3=0(zero rows) • d = 0, trivial UIR • d = 1/2, • d = 1, • d = 3/2, • d > 3/2  3 discrete “fundamentally scalar” UIRs these vectors are of zero (Shapovalov) norm, and thus must be factored out, i.e. set to zero to get UIR In free theory (at least) should be no motion equations put by hand

  15. s1=s2=0, s3>0(1 row) this UIR class will turn out to have additional SU(2)xU(1) quantum numbers, the rest are still to be investigated • d = 1 + s3/2, • d = 3/2 + s3/2, • d = 2 + s3/2, • d > 2 + s3/2  3 discrete families of 1-row UIRs, in particular 3 discrete “fundamental spinors” (first, i.e. s3=1 particles).

  16. s1=0, s2>0, s3 ≥0(2 rows) • d = 2 + s2/2 + s3/2, • d = 5/2 + s2/2 + s3/2, • d > 5/2 + s2/2 + s3/2  2 discrete families of 2-rows UIRs

  17. s1>0, s2 ≥ 0, s3 ≥0(3 rows) • d = 3 + s1/2 + s2/2 + s3/2, • d > 3 + s1/2 + s2/2 + s3/2  single discrete familiy of 3-rows UIRs (i.e. discrete UIR is determined by Young diagram alone)

  18. How to do “work” with these representations? • solution: realize UIRs in Green’s ansatz! • automatically: (sub)singular vectors vanish, unitarity guaranteed • for “fundamentally scalar” (unique vacuum) UIRs Greens ansatz was known • we generalized construction for SU nontrivial UIRs

  19. Green’s ansatz representations Now we have only ordinary bose operators and everything commutes! • Green’s ansatz of order p (combined with Klain’s transformation): • we introduced 4p pairs of ordinary bose operators: • and “spinor inversion” operators that can be constructed as 2 pi rotations in the factor space: • all live in product of p ordinary 4-dim LHO Hilbert spaces: • p=1 is representation of bose operators

  20. “Fundamentally scalar UIRs” • d = 1/2  p = 1 • this parabose UIR is representation of ordinary bose operators • singular vector identically vanishes • d = 1  p = 2 • vacuum state is multiple of ordinary bose vacuums in factor spaces: • d = 3/2  p = 3 • vacuum:

  21. 1-row, d = 1 + s3/2 UIR This class of UIRs exactly constitutes p=2 Green’s ansatz: • Define: • – two independent pairs of bose operators • are “vacuum generators”: • All operators will annihilate this state: s3

  22. “Inner” SU(2) action • Operators: generate an SU(2) group that commutes with action of the Poincare (and conformal) generators. • Together with the Y3 generated U(1) group, we have SU(2) x U(1) group that commutes with observable spacetime symmetry and additionally label the particle states.

  23. 1-row, other UIRs • Other “families” are obtained by increasing p: • d = 3/2 + s3/2, p = 3, • d = 2 + s3/2, p = 4 • Spaces of these UIRs are only subspaces of p = 3 and p = 4 Green’s ansatz spaces s3 s3

  24. 2-rows UIRs • Two “vacuum generating” operators must be antisymmetrized  we need product of two p=2 spaces. • To produce two families of 2-rows UIRs act on a natural vacuum in p=4 and p=5 by:

  25. 3-rows UIRs • Three “vacuum generating” operators must be antisymmetrized  we need product of three p=2 spaces. • Single family of 3-rows UIRs is obtained by acting on a natural vacuum in p=6 by:

  26. Conclusion so far • All discrete UIRs can be reproduced by combining up to 3 “double” 1-row spaces (those that correspond to SU(2)xU(1) labeled particles)

  27. Simplest nontrivial UIR- p=1- • Parabose operators act as bose operators and supersymmetry generators Qaand Sb satisfy 4-dim Heisenberg algebra. • Hilbert space is that of 4-dim nonrelativistic quantum mechanics. We may introduce equivalent of coordinate or momentum basis: • Yet, these coordinates transform as spinors and, when symmetry breaking is assumed, three spatial coordinates remain.

  28. Simplest nontrivial UIR- p=1- • Fiertz identities, in general give: • where: • since generators Q mutually comute in p=1, all states are massless: • in p=1, Y3 becomes helicity: • states are labeled by 3-momentum and helicity:

  29. Simplest nontrivial UIR- p=1- • introduce “field states” as vector coherent states: • derive familiar results: source of equations of motion can be traced back to the corresponding singular vector

  30. Next more complex class of UIR: p=2 • Hilbert space is mathematically similar to that of two (nonidentical) particles in 4-dim Euclidean space • However, presence of inversion operators in complicates eigenstates. • In turn, mathematically most natural solution becomes to take complex values for Qaand Sa.

  31. Space p=2 • Fiertz identities: • where: • only the third term vanishes, leaving two mass terms! Dirac equation is affected.

  32. Space p=2 • Massive states are labeled by Poincare numbers (mass, spin square, momentum, spin projection) but also Y3 value, and q. numbers of SU(2) group generated by T1 , T2 and T3. • square of this “isospin” coincides with square of spin. • Similarly, massless states also have additional U(1)xSU(2) quantum numbers.

  33. Conclusion Simple in statement but rich in properties Symmetry breaking of a nice type Promising particle structure Many predictions but yet to be calculated A promising type of supersymmetry!

  34. Thank you for your attention!

  35. A simple relation in a complicated basis

  36. Algebra of anticommutators gen. rotations Isomorphic to sp(8) gen. Lorentz gen. Poincare gen. conf

  37. Symmetry breaking J1 J2J3 {Q,S}operators D Y1 Y2 Y3 N11 N12N13 N21 N22N23 N31 N32N33 P0 K0 K11 K12K13 K21 K22K23 K31 K32K33 P11 P12P13 P21 P22P23 P31 P32P33 {S,S}operators {Q,Q}operators

  38. Symmetry breaking J1 J2J3 D {Q,S}operators ? C(1,3) conformal algebra Potential ~(Y3)2 Y3 N1 N2N3 P0 K0 K1 K2K3 P1 P2P3 {S,S}operators {Q,Q}operators

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