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

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. { Q , Q  } = -2i (   )   P  [ M , Q  ] = -1/4 ([  ,  ])    Q , [ P , Q  ] = 0 What is supersymmetry supersymmetry = symmetry generated by a (Lie) superalgebra ? Poincaré supersymmetry! = HLS theorem – source of confusion? ruled out in LHC?

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

5. Simplicity as motivation? [ M , M  ] = i (  M  +   M  -   M  -   M  ), [ M , P ] = i (   P    P ), [ P , P ] = 0    1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1 Poincaré space-time: mass (momentum), spin Something else? 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,...)? Parabose algebra: + supersymmetry: [ M , Q  ] = -1/4 ([  ,  ])    Q , [ P , Q  ] = 0, { Q , Q  } = -2i (   )   P  + conformal symmetry: [ M , S  ] = -1/4 ([  ,  ])    S , { S , S  } = -2i (   )   K , [ K , S  ] = 0, + tens of additional relations

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

10. Generalized conformal superalgebra Choice of basis + bosonic part of algebra Connection with standard conformal algebra: Y 1 = Y 2 = N 11 = N 21 = P 11 = P 21 = K 11 = K 21 ≡ 0 {Q ,Q   }={Q ,Q   }={S ,S   }={S ,S   }= 0

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 s 1, s 2, s 3, 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 In general, d has continuous and discrete parts of spectrum: –continuous: d > d 1 ← LW Verma module is irreducible –discrete: d = d 1, d 2, d 3,… d k ← 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) expressions that must vanish and thus turn into equations of motion within a representation

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. s 1 =s 2 =s 3 =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 e.g. this one will turn into and massless Dirac equation! In free theory (at least) should be no motion equations put by hand

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

16. s 1 =0, s 2 >0, s 3 ≥0 (2 rows) d = 2 + s 2 /2 + s 3 /2, d = 5/2 + s 2 /2 + s 3 /2, d > 5/2 + s 2 /2 + s 3 /2  2 discrete families of 2-rows UIRs

17. s 1 >0, s 2 ≥ 0, s 3 ≥0 (3 rows) d = 3 + s 1 /2 + s 2 /2 + s 3 /2, d > 3 + s 1 /2 + s 2 /2 + s 3 /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 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 Now we have only ordinary bose operators and everything commutes!

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 + s 3 /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: s3s3

22. “Inner” SU(2) action Operators: generate an SU(2) group that commutes with action of the Poincare (and conformal) generators. Together with the Y 3 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. Other “families” are obtained by increasing p: –d = 3/2 + s 3 /2, p = 3, –d = 2 + s 3 /2, p = 4 Spaces of these UIRs are only subspaces of p = 3 and p = 4 Green’s ansatz spaces 1-row, other UIRs s3s3 s3s3

24. 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: 2-rows UIRs

25. 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: 3-rows UIRs

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 Q  and S  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, Y 3 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 Q  and S 

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

32. Massive states are labeled by Poincare numbers (mass, spin square, momentum, spin projection) but also Y 3 value, and q. numbers of SU(2) group generated by T 1, T 2 and T 3. 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 Isomorphic to sp(8) gen. rotations gen. Lorentz gen. Poincare gen. conf

37. Symmetry breaking N 11 N 12 N 13 N 21 N 22 N 23 N 31 N 32 N 33 J 1 J 2 J 3 P0P0 D K0K0 Y1Y2Y3Y1Y2Y3 P 11 P 12 P 13 P 21 P 22 P 23 P 31 P 32 P 33 K 11 K 12 K 13 K 21 K 22 K 23 K 31 K 32 K 33 {Q,S} operators {S,S} operators {Q,Q} operators

38. Symmetry breaking P0P0 K0K0 {Q,S} operators {S,S} operators {Q,Q} operators C(1,3) conformal algebra N 1 N 2 N 3 J 1 J 2 J 3 P 1 P 2 P 3 D K 1 K 2 K 3 Y3Y3 Potential ~(Y 3 ) 2 ?