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(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=1p=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