Particle content of models with Particle content of models with parabose spacetime symmetryparabose 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)

Talk outlineTalk 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

{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?

But what else?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

Simplicity as motivation?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

Parabose algebraParabose 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 Close relation to orthosymplectic superalgebraorthosymplectic 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

Change of basisChange of basis- step 1 of 2 -- step 1 of 2 -

• Switch to hermitian combinations

consequently satisfying “para-Heisenberg” algebra:

• define new basis for expressing parabose anticommutators:

• we used the following basis of 4x4 real matrices:

– 6 antisymmetric:

– 10 symmetric matrices:

Change of basis Change of basis - step 2 of 2 -- step 2 of 2 -

, ,

,

Generalized conformal Generalized conformal superalgebrasuperalgebra

Choice of basis

+ bosonic part of algebra

Connection with standard conformal algebra:

Y1 = Y2 = N11 = N21 = P11 = P21 = K11 = K21 ≡ 0

{Q,Q}={Q,Q}={S,S}={S,S}= 0

Unitary irreducible Unitary irreducible representationsrepresentations

• 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 Allowed dd values values

• 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)

expressions that must vanish and thus turn

into equations of motion within a representation

Verma module structureVerma 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

ss11=s=s22=s=s33=0=0

(zero rows)(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 handIn free theory (at least) should be no motion equations put by hand

ss11=s=s22=0, s=0, s33>0>0

(1 row)(1 row)

• 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).

this UIR class will turn out to have

additional SU(2)xU(1)

quantum numbers, the rest are still to be investigated

ss11=0, s=0, s22>0, s>0, s33 ≥0 ≥0

(2 rows)(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

ss11>0, s>0, s22 ≥ 0, s ≥ 0, s33 ≥0 ≥0

(3 rows)(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 How to do “work” with these representations? 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 representationsGreen’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!

““Fundamentally scalar Fundamentally scalar UIRs” 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:

1-row, 1-row, dd = 1 + s = 1 + s33/2 UIR/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

““Inner” SU(2) actionInner” 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.

• 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

1-row, other UIRs1-row, other UIRs

s3

s3

• 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 UIRs2-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:

3-rows UIRs3-rows UIRs

Conclusion so farConclusion 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 UIRSimplest nontrivial UIR- p=1-- 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.

Simplest nontrivial UIRSimplest nontrivial UIR- p=1-- 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 UIRSimplest nontrivial UIR- p=1-- 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

Next more complex Next more complex class of UIR: p=2class 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

• Fiertz identities:

• where:

• only the third term vanishes, leaving two mass terms! Dirac equation is affected.

Space p=2Space p=2

Space p=2Space 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.

ConclusionConclusion

• 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 A simple relation in a complicated basiscomplicated basis

Algebra of anticommutatorsAlgebra of anticommutators

Isomorphic to sp(8)

gen. rotations

gen. Lorentz

gen. Poincare

gen. conf

Symmetry breakingSymmetry breaking

N11 N12 N13

N21 N22 N23

N31 N32 N33

J1 J2 J3

P0

D

K0

Y1

Y2

Y3

P11 P12 P13

P21 P22 P23

P31 P32 P33

K11 K12 K13

K21 K22 K23

K31 K32 K33

{Q,S} operators

{S,S} operators

{Q,Q} operators

Symmetry breakingSymmetry breaking

P0 K0

{Q,S} operators

{S,S} operators

{Q,Q} operators

C(1,3) conformal algebra

N1 N2 N3

J1 J2 J3

P1 P2 P3

D

K1 K2 K3

Y3Potential

~(Y3)2

?