Gauge semi-simple extension of the Poincaré group
aa r X i v : . [ h e p - t h ] A p r Gauge semi-simple extension of the Poincar´e group
Dmitrij V. Soroka ∗ and Vyacheslav A. Soroka † Kharkov Institute of Physics and Technology,1, Akademicheskaya St., 61108 Kharkov, Ukraine
Abstract
Based on the gauge semi-simple tensor extension of the D -dimensional Poincar´egroup another alternative approach to the cosmological term problem is proposed. PACS:
Keywords:
Poincar´e algebra, Tensor, Extension, Casimir operators, Gauge group ∗ E-mail: [email protected] † E-mail: [email protected] . Recently the approach to the cosmological constant problem based on the tensorextension of the Poincar´e algebra with the generators of the rotations M ab and translations P a [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19][ M ab , M cd ] = ( g ad M bc + g bc M ad ) − ( c ↔ d ) , (1)[ M ab , P c ] = g bc P a − g ac P b , (2)[ P a , P b ] = cZ ab , (3)[ M ab , Z cd ] = ( g ad Z bc + g bc Z ad ) − ( c ↔ d ) , (4)[ P a , Z bc ] = 0 , [ Z ab , Z cd ] = 0was given by de Azcarraga, Kamimura and Lukierski [20]. Here Z ab is a tensor generator, g ab is a constant Minkovski metric and c is some constant.In this paper we present another approach to the problem based on the gauge semi-simple tensor extension of the D -dimensional Poincar´e group which Lie algebra has thefollowing form [13, 21]: [ Z ab , P c ] = Λ3 c ( g bc P a − g ac P b ) , (5)[ Z ab , Z cd ] = Λ3 c [( g ad Z bc + g bc Z ad ) − ( c ↔ d )] , (6)whereas the form of the rest permutation relation (1)-(4) is not changed. Λ is someconstant.The Lie algebra (1)-(6) has the following quadratic Casimir operator: P a P a + cZ ab M ba + Λ6 M ab M ab def = X k h kl X l , where X k = { P a , M ab , Z ab } is a set of the generators for the Lie algebra under consideration(1)-(6) and the tensor h kl is invariant with respect to the adjoint representation h kl = U km U ln h mn . The inverse tensor h kl ( h kl h lm = δ km ) is invariant with respect to the co-adjoint represen-tation h kl = h mn U mk U nl . Let us consider a gauge group corresponding to the Lie algebra (1)-(6). To thisend we introduce a gauge 1-form A = A k X k = dx µ ( e µa P a + 12 ω µab M ab + 12 B µab Z ab )1ith the following gauge transformation: A ′ = G − dG + G − AG, where G is a group element corresponding to the Lie algebra (1)-(6). Here x µ are space-time coordinates, e µa is a vierbein, ω µab is a spin connection and B µab is a gauge fieldconforming to the tensor generator Z ab .A contravariant vector F k of the field strength 2-form F = F k X k = dA + A ∧ A = 12 dx µ ∧ dx ν F µν is transformed homogeneously under the gauge transformation F ′ k X k = U kl F l X k = G − F k X k G. The field strength F µν = F µν k X k = ∂ [ µ A ν ] + [ A µ , A ν ]has the following decomposition: F µν = F µν a P a + 12 R µν ab M ab + 12 F µν ab Z ab . Here F µνa = T µν a + Λ3 c B [ µab e ν ] b , where T µν a = ∂ [ µ e ν ] a + ω [ µab e ν ] b is a torsion, R µν ab = ∂ [ µ ω ν ] ab + ω [ µac ω ν ] cb is a curvature tensor and F µν ab = ∂ [ µ B ν ] ab + ω [ µc [ a B ν ] b ] c + Λ3 c B [ µca B ν ] bc + ce [ µa e ν ] b is a component corresponding to the tensor generator Z ab .An invariant Lagrangian has the following form: L = − e h kl F µνl F ρλk g µρ g νλ = e (cid:18) c R µν ab F ρλ ; ab + Λ6 c F µνab F ρλ ; ab − F µν a F ρλ ; a (cid:19) g µρ g νλ . Note that there exists a curious limit c → ∞ which results in L → L = (cid:18) R + Λ − T µν a T µν a (cid:19) e, where R = R µνab e aµ e bν is a scalar curvature, g µν = g ab e aµ e bν is a metric tensor, e = det e µa is a determinant of the vierbein and Λ is a cosmological constant. Thus, we have presented another alternative approach to the cosmological termproblem within the gauge semi-simple tensor extension of the Poincar´e group.2 eferenceseferences