lie superalgebras and physical models my. brahim sedra ibn tofail university faculty of sciences,...
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LIE SUPERALGEBRAS AND PHYSICAL MODELS
My. Brahim SEDRA
Ibn Tofail University Faculty of sciences, Physics Department, LHESIR,
Kenitra
1
WORKSHOP DE RABAT 6-8 JUIN 2013
Acknowledgements
2
For invitation to present a talk.
3
A brief comment about
Supersymmetry
is requested !
Before that: What is the contexte?
1. Opening
4
Structure de l’atome
Noyau
Electron
Interaction électromagnétique
10-10
m
MécaniqueQuantique
5
Strucure du noyau
10-14
m
Neutron
Proton
Interaction forte
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Structure des nucléons
10-15
m
Proton :2 quarks up1 quark down
Neutron :1 quark up2 quarks down
Interaction forte
Classical physics is no longer valuable Quantum physics:
7
What happens at these very small scales of the matter?
Major Properties:
- Spin- Incertainty (Heisenberg Principles)- Duality: particles/waves
Quantum field theory!
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Also quantum physics is not enough!!
Mixture of quantum physics with relativity !
… String theory,
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that's the contexte
…
In nature there are bosons and fermions Bosons: particles having integer value of the spin Fermions: particles having half integer value of the spin Susy: a mechanism that associates to each boson a
fermion.
Susy assumes that in nature (universe) the number of bosonic states should be the same as the number of fermionic states.
By virtue of susy, bosons and fermions should have the same MASSE.
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Susy: What is it?
Susy is broken at the present scale of the univers !
Susy theory assumes also that the super partner of the electron is a boson called the selectron: m (e)=m(se).
However, there is no experimental (or observational) indication about the existence of the selectron.
The difficulty to observe the selectron can have two causes:1. The selectron is very heavy !or1. There should exists an unknown mechanism that makes a screen on
it.
Thus, the observation of the selectron requests higher technology .
C/C: Since the masse of partners is not equilibrate, the susy is broken.
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Interpretation !
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Boson de Higgs :
C’est une particule soupçonnée être à l'origine de l’attribution des masses à toutes les particules de l'univers physique
Comment 1
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On 4 July 2012:CERN has announced in a conference that a new bosonic particle has been identified .
Probably it’s the Higgs!
The CERN is not yet completely assured about it!(Des études complémentaires seront nécessaires pour déterminer si cette particule possède l'ensemble des caractéristiques prévues pour le boson de Higgs).
Comment 2
The importance of LSA in physics deals, among other, with the connection with supersymmetry (briefly described before).
In constructing supersymmetric integrable models, the request of integrability implies several solutions for the Cartan matrix Kij.
In contrast to the standard (bosonic) LA, we don't have a unique Cartan matrix in the LSA.
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2. What’s about Lie superalgebras?
Definition and Properties
:a of degree theis a where
1,
bygiven }[,bracket (supr) a with C)or (R field over the
LLL
space vector a is L (LSA) rasuperalgeb LieA 2
baabba
gradedZ
ba
10
15
0
1
L
L
a even, is a if 0
a odd, is a if 1
deg aa
The superbracket is shown to satisfy:a) The supersymmetry:
a) The super Jacobi Identity
Remark:The restriction of L to the even part gives a standard Lie algebra with satisfying the antisymmetry and the Jacobi identity.
abba ba ,1, degdeg
0,,1,,1,,1 acbbaccba abbcca
0L
baba ,,
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Superbracket and Physics !Consider:
and let B and F be Fermionic and Bosonic operators respectively such that
with
abba ba ,1, degdeg
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BFF
FFB
BBB
,
,
,
0deg
1deg
BB
FF
.21
,b and b operators lowering and raising of
set theas definedLSA Heisenberg thebe Let
jj
,...,r,j
LExample
.0b ,b
,0b ,b
,1.b ,b
kj
kj
kj
jk
)21(dimension
ofLSA a defines
,,1
Then
r
bb jj
r1,...,j
These operators satisfy the following relations
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., , , b, , b
and
jj 0110101
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3. What is new in LSA?
Two type of Simple Roots
DIFFERENT DYNKIN DIAGRAMM !!DIFFERENT DYNKIN DIAGRAMM !!
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LSA with odd simple roots play an important role in Susy Integrable models.
These integrable models are defined through a zero curvature condition
0, ADAD
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There are classes of LSA whose Cartan matrices lead to integrable models in such way that the simple roots is chosen to be purely fermionic (odd).
The constraint of integrability, leads to some explicit solutions of the Cartan matrix. As an example
The important result (Literature):
A(n|n-1)=sl(n+1|n),
B(n|n)=osp(2n+1|2n)
B(n-1|n)=osp(2n-1|2n),
D(n+1|n)=osp(2n+2|2n)
D(n|n) =osp(2n|2n),
D(2|1; )
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These are LSA SPIN OF CONSERVED CURRENTS
Sl(n+1/n)
osp(1/2) 3/2
osp(3/2) 3/2, 2
osp(2n-1/2n), n≥2
osp(2n+1/2n), n≥2
Osp(2/2)≈sl(2/1) 1, 3/2
Osp(2n/2n) , n≥2
12,...,3,22
1n
14,44,54,...,12,11,8,7,4,32
1 nnn
nn 4,14,...,12,11,8,7,4,32
1
nnnn ,14,44,54,...,12,11,8,7,4,32
1
Osp(2n+2/2n)
D(2/1,a) 3/2, 3/2,2
2
1,4,14,...,12,11,8,7,4,3
2
1
nnn
4. How things work in physics? Integrable models are systems of non linear
differential equations . Solving these equations is not an easy job. To avoid the non linearity, we use:
The famous: Lax technique The principal idea of the LT :
We start from a non linear diff. Equation with some fixed degrees of freedom.
We assume the existence of a Lax pair, defined in some Lie algebra structure.
If the Lax pair exists, the integrability is assured.
Operators belonging to some
Lie algebra structure
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Bosonic case – To illustrate the previous arguments, let’s consider the
following physical model: The 2d Conformal Liouville Field Theory
The equation of motion is
This is a n.l.d.eq. That can be solved by the following Lax pair:
with
0)2exp(2
field. bosonicscalar a is where
)2exp(2
zdS
ehzA
fzA
2exp
eeh
ffh
hfe
su2,
2,
,
)2(
We underline that the Lax pair satisfy the zero curvature condition
0, zzzzzz AAAAF
Fermionic case – The super(symmetric) case consists in considering
similar steps: The 2d super Liouville Field Theory
variablesGrassmann the -
sderivativesuper thespinors, are DD, -
,superfield is -
:where
)exp(22
DDzddS
As in the bosonic case, the Lax pair exists in this case in order to ensure the integrability of the model.
The Lie symmetry is given by the Superalgebras
Osp(1|2)
root) simple (one
:
5dim
1
,,,)21( 111022
DDiagramDynkin
Rank
fefehosp
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MBS (and collaborators):
http://inspirehep.net/search: M.B.Sedra.
Results
More on Lie superalgebras and Physical Models:MBS (Thèse de doctorat d’Etat 1995) Et references dedans
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References
1. M.Scheunert, The theory of Lie superalgebras, Lecture Notes in math (1979);
2. J.Wess and J. Bagger, supersymmetry and supergravity, princeton series in physics, 1983,
3. H. Nohara and all, Toda field theories, CFT (1990, 1991)4. M.B. Sedra,
• ADSTP (2011), with K. Bilal, A. Boukili, M. Nach
• CJP, (2009) with A. Boukili, A. Zemate.......
• Nucl.Phys. B513:709-722,1998• J.Math.Phys.37:3483-3490,1996. • Mod.Phys.Lett.A9:3163-3174,1994, • Mod.Phys.A9:1994. • Class.Quant.Grav.10:1937-1946, 1993. • J.Math.Phys.35, 3190,1993
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Thanks
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