Physics Letters B 707 (2012) 160–162

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Physics Letters B

www.elsevier.com/locate/physletb

Gauge semi-simple extension of the Poincaré group

Dmitrij V. Soroka, Vyacheslav A. Soroka ∗

Kharkov Institute of Physics and Technology, 1, Akademicheskaya St., 61108 Kharkov, Ukraine

a r t i c l e i n f o a b s t r a c t

Article history:Received 1 April 2011Received in revised form 30 June 2011Accepted 1 July 2011Available online 8 July 2011Editor: M. Cvetic

Keywords:Poincaré algebraTensorExtensionCasimir operatorsGauge group

An approach to the cosmological term problem is proposed, using the gauge semi-simple tensor extensionof the D-dimensional Poincaré group as a basis.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Recently, de Azcarraga, Kamimura and Lukierski [1] have pro-posed an approach to the cosmological term problem, based onthe tensor extension of the Poincaré algebra with the generators ofrotations Mab and translations Pa [2–20]

[Mab, Mcd] = (gad Mbc + gbc Mad) − (c ↔ d), (1.1)

[Mab, Pc] = gbc Pa − gac Pb, (1.2)

[Pa, Pb] = c Zab, (1.3)

[Mab, Zcd] = (gad Zbc + gbc Zad) − (c ↔ d), (1.4)

[Pa, Zbc] = 0, (1.5)

[Zab, Zcd] = 0. (1.6)

Here Zab is the tensor generator, gab is the constant Minkowskimetric and c is a certain constant.

In our Letter we present another approach to the problem,based on the gauge semi-simple tensor extension of the D-dimensional Poincaré group, whose Lie algebra has the followingform [14,17,21]:

[Zab, Pc] = Λ

3c(gbc Pa − gac Pb), (1.7)

* Corresponding author.E-mail addresses: dsoroka@kipt.kharkov.ua (D.V. Soroka),

vsoroka@kipt.kharkov.ua (V.A. Soroka).

0370-2693/$ – see front matter © 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2011.07.003

[Zab, Zcd] = Λ

3c

[(gad Zbc + gbc Zad) − (c ↔ d)

], (1.8)

whereas the form of the other permutation relations (1.1)–(1.4) re-mains unchanged. Λ is some constant.

The Lie algebra given by relations (1.1)–(1.4) and (1.7)–(1.8) hasthe following quadratic Casimir operator:

P a Pa + c Zab Mba + Λ

6Mab Mab

def= Xkhkl Xl, (1.9)

where Xk = {Pa, Mab, Zab} is a set of generators for the Lie algebraunder consideration (1.1)–(1.4) and (1.7)–(1.8) and the tensor hkl isinvariant with respect to the adjoint representation

hkl = UkmUl

nhmn. (1.10)

The inverse tensor hkl (hklhlm = δkm) is invariant with respect to

the co-adjoint representation

hkl = hmnUmkUn

l. (1.11)

2. Gauge-invariant Lagrangian

Let us consider the gauge group corresponding to the Lie alge-bra (1.1)–(1.4) and (1.7)–(1.8). To this end, we introduce the gauge1-form

A = Ak Xk = dxμ

(eμ

a Pa + 1

2ωμ

ab Mab + 1

2Bμ

ab Zab

)(2.1)

with the following gauge transformation:

D.V. Soroka, V.A. Soroka / Physics Letters B 707 (2012) 160–162 161

A′ = G−1 dG + G−1 AG, (2.2)

where G is the group element corresponding to the Lie algebra(1.1)–(1.4) and (1.7)–(1.8). Here xμ are the space–time coordinates,eμ

a is the vierbein, ωμab is the spin connection and Bμ

ab is thegauge field conforming to the tensor generator Zab .

The contravariant vector F k of the field strength 2-form

F = F k Xk = dA + A ∧ A = 1

2dxμ ∧ dxν Fμν (2.3)

is changed homogeneously under the gauge transformation

F ′k Xk = Ukl F l Xk = G−1 F k XkG. (2.4)

The field strength

Fμν = Fμνk Xk = ∂[μ Aν] + [Aμ, Aν ] (2.5)

has the following expansion:

Fμν = Fμνa Pa + 1

2Rμν

ab Mab + 1

2Fμν

ab Zab. (2.6)

Here we have

Fμνa = Tμν

a + Λ

3cB[μabeν]b, (2.7)

where

Tμνa = ∂[μeν]a + ω[μabeν]b (2.8)

is the torsion,

Rμνab = ∂[μων]ab + ω[μacων]cb (2.9)

is the curvature tensor and

Fμνab = ∂[μBν]ab + ω[μc[a Bν]b]

c

+ Λ

3cB[μca Bν]b

c + ce[μaeν]b (2.10)

is the component corresponding to the tensor generator Zab .The invariant Lagrangian is written as:

L = − e

4hkl Fμν

l Fρλk gμρ gνλ

= e

4

(1

cRμν

ab Fρλ;ab + Λ

6c2Fμν

ab Fρλ;ab

− Fμνa Fρλ;a

)gμρ gνλ. (2.11)

With the use of the relations (2.7)–(2.10) the Lagrangian L (2.11)can be expressed in a more detailed form, which includes the con-tribution of three terms(

1

2R + Λ − 1

4Tμν

a T μνa

)e (2.12)

plus a few additional cumbersome terms, whose coefficients aredependent on the negative degrees of the constant c. In expres-sions (2.11), (2.12) R = Rμν

abeaμeb

ν is a scalar curvature, gμν =gabea

μebν is a metric tensor, e = det eμ

a is the determinant of thevierbein and Λ is a cosmological constant.

Note that in the limit c → ∞, the algebra (1.1)–(1.4) and (1.7)–(1.8) is reduced by rescaling

Pa → √c Pa (2.13)

to the algebra (1.1)–(1.6) with c = 1. In this case, the invariant La-grangian is

L = e

4

(2R + Rμν

ab F μνab − Tμν

a T μνa), (2.14)

where

Fμνab = ∂[μBν]ab + ω[μc[a Bν]b]

c. (2.15)

3. Another basis

The extended Poincaré algebra (1.1)–(1.4) and (1.7)–(1.8) can berewritten in the different basis [14,17,21]:

[Nab, Ncd] = (gad Nbc + gbc Nad) − (c ↔ d), (3.1)

[L AB , LC D ] = (g AD LBC + gBC L AD) − (C ↔ D), (3.2)

[Nab, LC D ] = 0, (3.3)

where the metric tensor g AB has the following nonzero compo-nents:

g AB = {gab, gD+1D+1 = −1}. (3.4)

The generators

Nab = Mab − 3c

ΛZab (3.5)

form the Lorentz algebra so(D − 1,1), and the generators

L AB = −LB A ={

Lab = 3c

ΛZab, LaD+1 =

√3

ΛPa

}(3.6)

form the algebra so(D − 1,2). The algebra (3.1)–(3.3) is a directsum so(D − 1,1)⊕ so(D − 1,2) of the D-dimensional Lorentz alge-bra and D-dimensional anti-de Sitter algebra, correspondingly.

The gauge 1-form (2.1) in this basis takes the form

A = 1

2dxμ

(ωμ

ab Nab + ΩμAB L AB

), (3.7)

where

Ωμab = ωμ

ab + Λ

3cBμ

ab (3.8)

and

ΩμaD+1 =

√Λ

3eμ

a. (3.9)

The field strength 2-form (2.5) assumes the following form:

Fμν = 1

2

(Rμν

ab Nab + FμνAB L AB

), (3.10)

where the quantity

FμνAB = ∂[μΩν] AB + Ω[μ ACΩν]C B (3.11)

comprises the components

Fμνab = Rμν

ab + Λ

3cFμν

ab (3.12)

and

FμνaD+1 =

√Λ

3Fμν

a. (3.13)

The quadratic Casimir operator (1.9) is expressed in the follow-ing way:

Λ

6

(Nab Nab − L AB L AB)

= 1 (Nabhab;cd Ncd + L ABhAB;C D LC D

), (3.14)

4

162 D.V. Soroka, V.A. Soroka / Physics Letters B 707 (2012) 160–162

where

hab;cd = Λ

3

(gac gbd − gad gbc) (3.15)

and

hAB;C D = Λ

3

(g AD g BC − g AC g B D)

. (3.16)

With the use of the inverse tensor components

hab;cd = 3

Λ(gac gbd − gad gbc) (3.17)

and

hAB;C D = 3

Λ(g AD gBC − g AC gB D) (3.18)

the invariant Lagrangian L (2.11) takes the following form:

L = − e

16

(hab;cd Rμν

ab Rμν;cd + hAB;C D FμνAB F μν;C D)

= 3e

8Λ

(Fμν

AB F μνAB − Rμν

ab Rμνab

). (3.19)

Finally, with the help of relations (2.7), (2.10), (3.12) and (3.13) wecome again to expression (2.11) for the Lagrangian L.

4. Resume

Thus, we have presented another approach to the cosmologicalterm problem in comparison with the possibility described in pa-per [1]. Our approach is based on the gauge semi-simple tensorextension of the Poincaré group.

Acknowledgements

We are greatly indebted to the referee for the constructive ad-vices. We also thank A.A. Zheltukhin for the useful discussion.

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