Semi-simple extension of the (super)Poincaré algebra

0.09 MB·English

by  Dmitrij V. Soroka

Semi-simple extension of the (super) Poincar´e
algebra
Dmitrij V. Soroka∗ and Vyacheslav A. Soroka†
Kharkov Institute of Physics and Technology,
1, Akademicheskaya St., 61108 Kharkov, Ukraine
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Abstract
n
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J A semi-simple tensor extension of the Poincar´e algebra is proposed for the ar-

5 bitrary dimensions D. A supersymmetric also semi-simple generalization of this

extension is constructed in the D = 4 dimensions. This paper is dedicated to the
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v memory of Anna Yakovlevna Gelyukh.
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5 PACS: 02.20.Sv; 11.30.Cp; 11.30.Pb
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5 Keywords: Poincar´e algebra, Tensor,Extension, Casimiroperators, Supersymmetry
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∗E-mail: [email protected]
†E-mail: [email protected]
1 Introduction
In the papers [1, 2, 3, 4, 5, 6] the Poincar´e algebra for the generators of the rotations M
ab
and translations P in D dimensions
a
[M ,M ] = (g M +g M )−(c ↔ d),
ab cd ad bc bc ad
[M ,P ] = g P −g P ,
ab c bc a ac b
[P ,P ] = 0 (1.1)
a b
has been extended by means of the second rank tensor generator Z in the following way:
ab
[M ,M ] = (g M +g M )−(c ↔ d),
ab cd ad bc bc ad
[M ,P ] = g P −g P ,
ab c bc a ac b
[P ,P ] = cZ ,
a b ab
[M ,Z ] = (g Z +g Z )−(c ↔ d),
ab cd ad bc bc ad
[P ,Z ] = 0,
a bc
[Z ,Z ] = 0, (1.2)
ab cd
where c is some constant1.
Such an extension makes common sense, since it is homomorphic to the usual Poincar´e
algebra (1.1). Moreover, in the limit c → 0 the algebra (1.2) goes to the semi-direct sum
of the commutative ideal Z and Poincar´e algebra (1.1).
ab
It is remarkable enough that the momentum square Casimir operator of the Poincar´e
algebra under this extension ceases to be the Casimir operator and it is generalized by
adding the term linearly dependent on the angular momentum
PaP +cZabM d=ef X hklX , (1.3)
a ba k l
where X = {P ,Z ,M }. Dueto this fact, anirreducible representation of the extended
k a ab ab
algebra (1.2) has to contain the fields of the different masses [4, 7]. This extension with
non-commuting momenta has also something in common with the ideas of the papers [8,
9, 10] and with the non-commutative geometry idea [11].
1
Notethat,toavoidthedoublecountundersummationoverthepairantisymmetricindices,weadopt
the rules which illustrated by the following example:
c 1
[Pa,Pb]=cZab = (δacδbd−δadδbc)Zcd = fabcdZcd = fabcdZcd,
2 2
c<d
X
where fabcd are structure constants, and so on.
1
It isinteresting tonotethat inspite ofthefactthat thealgebra(1.2)isnotsemi-simple
and therefore has a degenerate Cartan-Killing metric tensor nevertheless there exists
another non-degenerate invariant tensor h in adjoint representation which corresponds
kl
to the quadratic Casimir operator (1.3), where the matrix hkl is inverse to the matrix h :
kl
hklh = δk .
lm m
There are other quadratic Casimir operators
c2ZabZ , (1.4)
ab
c2ǫabcdZ Z . (1.5)
ab cd
Notethat theCasimir operator(1.5), dependent ontheLevi-Civita tensor ǫabcd, is suitable
only for the D = 4 dimensions.
It has also been shown that for the dimensions D = 2,3,4 the extended Poincar´e
algebra (1.2) allows the following supersymmetric generalization:
{Q ,Q } = −d(σabC) Z ,
κ λ κλ ab
[M ,Q ] = −(σ Q) ,
ab κ ab κ
[P ,Q ] = 0,
a κ
[Z ,Q ] = 0 (1.6)
ab κ
with the help of the super-translation generators Q . In (1.6) C is a charge conjugation
κ
matrix, d is some constant and σ = 1[γ ,γ ], where γ is the Dirac matrix. Under this
ab 4 a b a
supersymmetric generalization the quadratic Casimir operator (1.3) is modified into the
following form:
c
PaP +cZabM − Q (C−1)κλQ , (1.7)
a ba κ λ
2d
while the form of the rest quadratic Casimir operators (1.4), (1.5) remains unchanged.
In the present paper we propose another possible semi-simple tensor extension of
the Poincar´e algebra (1.1) and for the case D = 4 dimensions we give a supersymmetric
generalizationofthisextension. Inthelimitthissupersymmetrically generalizedextension
go to the Lie superalgebra (1.2), (1.6).
2 Semi-simple tensor extension
Let us extend the Poincar´e algebra (1.1) in the D dimensions by means of the tensor
generator Z in the following way:
ab
[M ,M ] = (g M +g M )−(c ↔ d),
ab cd ad bc bc ad
[M ,P ] = g P −g P ,
ab c bc a ac b
2
[P ,P ] = cZ ,
a b ab
[M ,Z ] = (g Z +g Z )−(c ↔ d),
ab cd ad bc bc ad
4a2
[Z ,P ] = (g P −g P ),
ab c bc a ac b
c
4a2
[Z ,Z ] = [(g Z +g Z )−(c ↔ d)], (2.1)
ab cd ad bc bc ad
c
where a and c are some constants. This Lie algebra, when the quantities P and Z are
a ab
taken as the generators of a homomorphism kernel, is homomorphic to the usual Lorentz
algebra. It is remarkable that the Lie algebra (2.1) is semi-simple in contrast to the
Poincar´e algebra (1.1) and extended Poincar´e algebra (1.2).
The extended Lie algebra (2.1) has the following quadratic Casimir operators:
C = PaP +cZabM +2a2MabM d=ef X HklX , (2.2)
1 a ba ab k 1 l
C = c2ZabZ +8a2(cZabM +2a2MabM ) d=ef X HklX , (2.3)
2 ab ba ab k 2 l
C = ǫabcd[c2Z Z +8a2(cZ M +2a2M M )]. (2.4)
3 ab cd ba cd ab cd
Note that in the limit a → 0 the algebra (2.1) tend to the algebra (1.2) and the quadratic
Casimir operators (2.2), (2.3) and (2.4) are turned into (1.3), (1.4) and (1.5), respectively.
The symmetric tensor
Hkl = sHkl +tHkl = Hlk (2.5)
1 2
with arbitrary constants s and t is invariant with respect to the adjoint representation
Hkl = HmnU kU l.
m n
Conversely, if we demand the invariance with respect to the adjoint representation of the
second rank contravariant symmetric tensor, then we come to the structure (2.5) (see also
the relation (32) in [6]).
The semi-simple algebra (2.1)
[X ,X ] = f mX
k l kl m
has the non-degenerate Cartan-Killing metric tensor
g = f nf m,
kl km ln
which is invariant with respect to the co-adjoint representation
g = U mU ng .
kl k l mn
3
With the help of the inverse metric tensor gkl: gklg = δk we can construct the quadratic
lm m
Casimir operator which, as it turned out, has the following expression in terms of the
quadratic Casimir operators (2.2) and (2.3):
1 3−2D
X gklX = C + C , (2.6)
k l 8a2(D −1) " 1 8a2(D −2) 2#
that corresponds to the particular choice of the constants s and t in (2.5).
The extended Poincar´e algebra (2.1) can be rewritten in the form:
[N ,N ] = (g N +g N )−(c ↔ d), (2.7)
ab cd ad bc bc ad
[L ,L ] = (g L +g L )−(C ↔ D), (2.8)
AB CD AD BC BC AD
[N ,L ] = 0, (2.9)
ab CD
where the metric tensor g has the following nonzero components:
AB
g = {g ,g = −1}. (2.10)
AB ab D+1D+1
The generators
c
N = M − Z (2.11)
ab ab ab
4a2
form the Lorentz algebra so(D −1,1) and the generators
c 1
L = {L = Z ,L = −L = P ,L = 0} (2.12)
AB ab 4a2 ab aD+1 D+1a 2a a D+1D+1
form the algebra so(D − 1,2)2. The algebra (2.7)-(2.9) is a direct sum so(D − 1,1) ⊕
so(D −1,2).
The quadratic Casimir operators N Nab, L LAB and ǫabcdN N of the algebra
ab AB ab cd
(2.7)-(2.9) are expressed in terms of the operators C (2.2), C (2.3) and C (2.4) in the
1 2 3
following way:
1
N Nab −L LAB = C , (2.13)
ab AB 2a2 1
1
N Nab = C , (2.14)
ab 16a4 2
1
ǫabcdN N = C . (2.15)
ab cd 16a4 3
2Note that in the case D =4 we obtain the anti-de Sitter algebra so(3,2).
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3 Supersymmetric generalization
In the case D = 4 dimensions the extended Poincar´e algebra (2.1) admits the following
supersymmetric generalization:
2a
{Q ,Q } = −d (γaC) P +(σabC) Z ,
κ λ κλ a κλ ab
c
(cid:20) (cid:21)
[M ,Q ] = −(σ Q) ,
ab κ ab κ
[P ,Q ] = a(γ Q) ,
a κ a κ
4a2
[Z ,Q ] = − (σ Q) , (3.1)
ab κ ab κ
c
where Q are the super-translation generators.
κ
Under such a generalization the Casimir operator (2.2) is modified by adding a term
quadratic in the super-translation generators
c
C˜ = PaP +cZabM +2a2MabM − Q (C−1)κλQ
1 a ba ab 2d κ λ
d=ef X HKLX , (3.2)
K 1 L
whereas theformoftherest quadraticCasimir operators(2.3)and(2.4)isnotchanged. In
(3.2) X = {P ,Z ,M ,Q } is a set of the generators for the also semi-simple extended
K a ab ab κ
superalgebra (2.1), (3.1).
The tensor
HKL = vHKL+wHKL = (−1)pKpL+pK+pLHLK (3.3)
1 2
is invariant with respect to the adjoint representation
HKL = (−1)(pK+pM)(pL+1)HMNU LU K,
N M
where p = p(K) is a Grassmann parity of the quantity K. In (3.3) v and w are arbitrary
K
constants and nonzero elements of the matrix HKL equal to the elements of the matrix
2
Hkl followed from (2.3). Again, by demanding the invariance with respect to the adjoint
2
representation of the second rank contravariant tensor HKL = (−1)pKpL+pK+pLHLK, we
come to the structure (3.3) (see also the relation (32) in [6]).
The semi-simple Lie superalgebra (2.1) (3.1) has the non-degenerate Cartan-Killing
metric tensor G (see the relation (A.5) in the Appendix A) which is invariant with
KL
respect to the co-adjoint representation
G = (−1)pK(pL+pN)U NU MG .
KL L K MN
With the use of the inverse metric tensor GKL
GKLG = δK
LM M
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we can construct the quadratic Casimir operator (see the relation (A.8) in the Appendix
A) which takes the following expression in terms of the Casimir operators (2.3) and (3.2):
1 9
X GKLX = C˜ − C , (3.4)
K L 20a2 1 32a2 2
(cid:18) (cid:19)
that meets the particular choice of the constants v and w in (3.3).
In the D = 4 case the extended superalgebra (2.1), (3.1) can be rewritten in the form
of the relations (2.7)-(2.9) and the following ones:
4a2d
{Q ,Q } = − (ΣABC) L , (3.5)
κ λ κλ AB
c
[L ,Q ] = −(Σ Q) , (3.6)
AB κ AB κ
[N ,Q ] = 0, (3.7)
ab κ
where
1
Σ = [Γ ,Γ ], Γ = {γ γ ,γ },
AB A B A a 5 5
4
{γ ,γ } = 2g , g = diag(−1,1,1,1),
a b ab ab
γ = γ γ γ γ .
5 0 1 2 3
ThegeneratorsN (2.11)formtheLorentzalgebraso(3,1)andthegeneratorsL (2.12),
ab AB
Q formtheorthosymplectic algebraosp(1,4). Weseethatsuperalgebra(2.7)-(2.9),(3.5)-
κ
(3.7) is a direct sum so(3,1)⊕osp(1,4).
In this case the Casimir operator (2.13) is modified by adding a term quadratic in the
super-translation generators
c 1
N Nab −L LAB − Q (C−1)κλQ = C˜ ,
ab AB 4a2d κ λ 2a2 1
while the form of the quadratic Casimir operators (2.14) and (2.15) is not changed.
4 Conclusion
Thus, we proposed the semi-simple second rank tensor extension of the Poincar´e algebra
in the arbitrary dimensions D and super Poincar´e algebra in the D = 4 dimensions. It is
very important, since under construction of the models it is more convenient to deal with
the non-degenerate space-time symmetry. We also constructed the quadratic Casimir
operators for the semi-simple extended Poincar´e and super Poincar´e algebras.
It is interesting to develop the models based on these extended algebras. The work in
this direction is in progress.
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Acknowledgments
We are gratefulto J.A. de Azcarraga for the valuable remark. One of the authors (V.A.S.)
thanks the administration of the Office of Associate and Federation Schemes of the Abdus
Salam ICTP for the kind hospitality at Trieste where this work has been completed. The
research of V.A.S. was partially supported by the Ukrainian National Academy of Science
and Russian Fund of Fundamental Research, Grant No 38/50-2008.
A Appendix: Properties of Lie superalgebra
Permutation relations for the generators X of Lie superalgebra are
K
[X ,X } d=ef X X −(−1)pKpLX X = f MX . (A.1)
K L K L L K KL M
Structure constants f M have the Grassmann parity
KL
p f M = p +p +p = 0 (mod 2), (A.2)
KL K L M
(cid:16) (cid:17)
following symmetry property:
f M = −(−1)pKpLf M (A.3)
KL LK
and obey the Jacobi identities
(−1)pKpMf Pf N = 0, (A.4)
KN LM
(KXLM)
where the symbol (KLM) means a cyclic permutation of the quantities K, L and M. In
therelations(A.1)-(A.4)anevery indexK takes either aGrassmann-even valuek (p = 0)
k
or a Grassmann-odd one κ (p = 1). The relations (A.1) have the following components:
κ
[X ,X ] = f mX ,
k l kl m
{X ,X } = f mX ,
κ λ κλ m
[X ,X ] = f µX .
k λ kλ µ
The Lie superalgebra possesses the Cartan-Killing metric tensor
G = (−1)pNf Nf M = (−1)pKpLG = (−1)pKG = (−1)pLG , (A.5)
KL KM LN LK LK LK
which components are
G = f nf m −f νf µ,
kl km ln kµ lν
G = f mf µ −f µf m,
κλ κµ λm κm λµ
G = 0.
kλ
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As a consequence of the relations (A.3) and (A.4) the tensor with low indices
f = f NG (A.6)
KLM KL NM
has the following symmetry properties:
f = −(−1)pKpLf = −(−1)pLpMf . (A.7)
KLM LKM KML
For a semi-simple Lie superalgebra the Cartan-Killing metric tensor is non-degenerate
and therefore there exists an inverse tensor GKL
G GLM = δM.
KL K
In this case, as a result of the symmetry properties (A.7), the quantity
X GKLX (A.8)
K L
is a Casimir operator
[X GKLX ,X ] = 0.
K L M
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