* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download slides
Dirac equation wikipedia , lookup
Higgs mechanism wikipedia , lookup
Quantum chromodynamics wikipedia , lookup
Second quantization wikipedia , lookup
Matter wave wikipedia , lookup
Hilbert space wikipedia , lookup
Bell's theorem wikipedia , lookup
Quantum state wikipedia , lookup
Spin (physics) wikipedia , lookup
Lie algebra extension wikipedia , lookup
Wave function wikipedia , lookup
Self-adjoint operator wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Atomic theory wikipedia , lookup
Elementary particle wikipedia , lookup
Vertex operator algebra wikipedia , lookup
Identical particles wikipedia , lookup
Scalar field theory wikipedia , lookup
Bra–ket notation wikipedia , lookup
Two-dimensional conformal field theory wikipedia , lookup
Quantum group wikipedia , lookup
Compact operator on Hilbert space wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
(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 • 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 What is supersymmetry {Qa, Qb} = -2i (gm)ab Pm, generated b [Msymmetry , Q ] = -1/4 ([g mn a m, gn])a Qb, (Lie) superalgebra [by Pm, Qa a] = 0 supersymmetry = ? ruled out in LHC? = Poincaré supersymmetry! HLS theorem – source of confusion? But what else? {Qa,Qb }=0 {Qa,Qb }= 0 • in 4 spacetime dimensions: Tensorial central charges • in 11 spacetime dimensions: this is known as M-theory algebra • can be extended to super conformal case + supersymmetry: [Mmn, Qa] = -1/4 ([gm, gn])ab Qb, [Pm, Qa] = 0, {Qa, Qb} = -2i (gm)ab Pm Simplicity as motivation? Poincaré space-time: Something Parabose algebra: else? + conformal symmetry: [Mmn, Mlr] = i (hnl Mmr + hmr[Mmn, Sa] = -1/4 ([gm, gn]) Mnl - hml Mnr - hnr Mml), {Sa, Sb} = -2i (gm)ab Km, [Mmn, Pl] = i (- hnl Pm + hml Pn), [Km, Sa] = 0, [Pm, Pn] = 0 hmn = 1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1 bS , a b + tens of additional relations spin • mass (momentum), • usual massless particles • “charged” particles carrying SU(2) x U(1) numbers • “elementary” composite particles from up to 3 charged subparticles mass (momentum), spin • a sort of parity asymmetry • ….(flavors, ...)? 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: Close relation to orthosymplectic superalgebra • Operators form osp(1|2n) superalgebra. • osp generalization of supersymmetry first analyzed by C. Fronsdal back inFrom 1986 now on n = 4 • 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. Change of basis - step 1 of 2 • Switch to hermitian combinations consequently satisfying “para-Heisenberg” algebra: 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: , Generalized conformal superalgebra Connection with standard conformal algebra: Y1 = Y2 = N11 = N21 = P11 = P21 = K11 = K21Choice ≡ 0 of basis {Qa,Qb }={Qa,Qb }={Sa,Sb }={Sa,Sb }= 0 + bosonic part of algebra 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! Allowed d values expressions • In general, d has continuous and discrete parts ofthat must vanish and thus turn spectrum: into equations of motion within representation – continuous: d > d1 ← LW Verma module is airreducible – 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) 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 s1=s2=s3=0 (zero rows) e.g. this one will turn into and massless Dirac equation! • d = 0, trivial UIR • d = 1/2, these vectors are of zero (Shapovalov) norm, and thus must be factored out, i.e. set to zero to get UIR • d = 1, • d = 3/2, • d > 3/2 3 discrete “fundamentally scalar” UIRs In free theory (at least) should be no motion equations put by hand 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). 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 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) 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 Green’s ansatz representations • Now we have only ordinary Green’s ansatz of order p (combined with Klain’s transformation): bose operators and everything commutes! • 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 “Fundamentally scalar UIRs” • d = 1/2 p = 1 – this parabose UIR is representation of ordinary bose operators – singular vector identically vanishes • d=1p=2 – vacuum state is multiple of ordinary bose vacuums in factor spaces: • d = 3/2 p = 3 – vacuum: 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”: s3 • All operators will annihilate this state: “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. 1-row, other UIRs • Other “families” are obtained by increasing p: – d = 3/2 + s3/2, p = 3, s3 – d = 2 + s3/2, p = 4 s3 • Spaces of these UIRs are only subspaces of p = 3 and p = 4 Green’s ansatz spaces 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: 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: 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) Simplest nontrivial UIR - p=1• Parabose operators act as bose operators and supersymmetry generators Qa and 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. 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: Simplest nontrivial UIR - p=1• introduce “field states” as vector coherent states: source of equations of motion can be traced back to the corresponding • derive familiar results: singular vector 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 Qa and Sa. Space p=2 • Fiertz identities: • where: • only the third term vanishes, leaving two mass terms! Dirac equation is affected. 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. 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! Thank you for your attention! A simple relation in a complicated basis Algebra of anticommutators gen. rotations gen. Lorentz gen. Poincare gen. conf Isomorphic to sp(8) Symmetry breaking J1 D J3 {Q,S} Y1 N11 N12 N13 Y2 N21 N22 N23 Y3 N31 N32 N33 P0 {Q,Q} J2 operators K0 P11 P12 P13 K11 K12 K13 P21 P22 P23 K21 K22 K23 P31 P32 P33 K31 K32 K33 operators {S,S} operators Symmetry breaking J1 D J2 J3 {Q,S} operators ? Potential ~(Y3)2 P0 {Q,Q} operators P1 Y3 N1 N2 C(1,3) conformal algebra N3 K0 P2 P3 {S,S} K1 operators K2 K3