EJTP 3, No. 12 (2006 37 45 Electronic Journal of Theoretical Physics Modified Moyal-Weyl Star product in a Curved Non Commutative space-time N.Mebarki,F.Khallili, M.Boussahel, and M.Haouchine Laboratoire de Physique Mathématique et Subatomique. Mentouri University,Constantine,ALGERIA Received 6 July 2006, Accepted 16 August 2006, Published 20 September 2006 Abstract: To generate gravitational terms in a curved noncommutative space-time, ne Moyal- Weyl star product as ell as Weyl ordering are defined. As an example, a complex scalar mass term action is considered. c Electronic Journal of Theoretical Physics. All rights reserved. Keyords: Quantum Groups, Non Commutative Geometry, Moyal-Weyl Star Product PACS (2006: 02.20.U, 11.10.Nx, 11.15.?q, 02.40.Gh 1. Introduction During the last to decades many efforts have been made to solve or at least to understand the reaming unsolved outstanding problems of theoretical physics by using ne ideas like quantum groups, deformation theory, noncommutative geometry etc.. 1] 9]. This may shed a light on the real microscopic geometry and structure of our universe. One approach, is to consider a noncommutative space-time here the dynamical variables become operators and therefore, the formalism of the quantum fields theories constructions must be modified. It turns out that in a flat space-time geometry, this amounts basically to replace the ordinary products by a Moyal-Weyl star products and taking into account the Weyl ordering 1] 10]. The goal of this paper is to consider a curved space-time (presence of a gravitational background and define the corresponding ne Moyal-Weyl star product and Weyl ordering. In section 2, e present our mathematical formalism and consider an example a mass term of a complex scalar field. Finally, in section 3, e dra our conclusions.
38 Electronic Journal of Theoretical Physics 3, No. 12 (2006 37 45 2. Formalism 2.1 T he ordinary Moyal W eyl product The Moyal-Weyl -product of any to smooth functions f and g such as 12] f (x = (2π 3 2 d 4 k e ikx f (k (1 g (x = (2π 3 2 d 4 k e ikx g (k (2 can be defined as follos: First e associate to f and g the Weyl operators W (f and W (g defined by W (f = (2π 3 2 W (g = (2π 3 2 d 4 k e ik x f (k (3 d 4 k e ik x g (k here x µ are non commuting operators satisfying x µ, x ν ] = iθ µν (4 Next e define the product W (f W (g as W (f W (g = (2π 3 2 (2π 3 2 d 4 k d 4 p e ik x e ip x f (k g (p (5 Using the C-B-H formula, the Weyl product W (f W (g reads : W (f W (g = (2π 3 2 (2π 3 2 d 4 k d 4 p e ik x+ip x i 2 k µp ν θ µν f (k g (p = W (f g (6 Where f g is a ne classical function defined by: (f g (x = e i 2 θµν x µ y ν f (x g (y (7 This is the ordinary Moyal -product. To the second ordre in θ reads: The Moyal -product (f g (x = f (x g (x + i 2 θµν x µ f (x x + i 2 θµν i 2 θαβ x µ g (x + (8 ν x f (x g (x +... α x ν xβ
Electronic Journal of Theoretical Physics 3, No. 12 (2006 37 45 39 Notice here, that the operators x µ are only defined modulo terms hich vanish at the classical limit, for example x µ and x µ + Σ µ αβ xα x β Σ µ αβ xβ x α are equal but the corresponding non commutating operators are not except if Σ µ αβ is symmetric x µ x µ + Σ µ αβ xα x β Σ µ αβ xβ x α = x µ + iθ αβ Σ µ αβ (9 2.2 T he deformed Moyal W eyl product Here, by deforming the ordinary Moyal -product, e propose a ne Moyal-Weyl -product hich take in consideration the missing terms cited above and hich generate gravitational terms to the order θ 2. To any smooth function f e associate the Weyl operator W (f f (x = (2π 3 2 d 4 x e ikx f (k W (f = (2π 3 2 d 4 k e ik X f (k (10 here X µ are non commuting operators associated to the folloing classical variables X µ = x µ + Γ µ αβ xα x β 1 2 Γµ ρλ Γρ αβ xα x β x λ (11 here Γ µ αβ (x = Γµ βα (x is the symmetric affine connection. The non commuting operators X µ are defined by a symmetrization procedure: X µ = x µ + ( Γµ αβ xα x β 1 ( Γµ 2 ρλ Γ ρ αβ xα x β x λ Where the Weyl ordering is defined by: ith (12 αβ xα x β = ( Σ µ αβ xα x β 2 x α Σµ αβ xβ + x α x β Σµ αβ / g (13 and direct simplifications (see Appendix give: Where e have used the fact that Σ µ αβ g = det g µν (14 αβ xα x β = iθ βλ iθ ασ σ λ Σµ αβ / g (15 is symmetric, and: x µ, x ν ] = iθ µν and ] x µ, f (x = iθ µν ν f (x (16 Thus the second term in eq.( 15 reads:
40 Electronic Journal of Theoretical Physics 3, No. 12 (2006 37 45 ( Γµ αβ xα x β = θ βλ θ ασ σ λ Γµ αβ / g (17 It is orth to mention that e can get the same result if e define the Weyl ordering as: Where ( Γµ αβ xα x β = ( Γ µ αβ xα x β x α Γ µ αβ xβ + Γ µ αβ x α x β / g (18 x α x β x α, x β] (19 and ] x µ f (x x µ, f (x (20 No using these relations and the fact that Γ µ αβ eq.(18 can be reritten as x α x β = Γ µ αβ x α, x β] = iθ αβ Γµ αβ (21 ( Γµ αβ xα x β = θ βλ θ ασ σ λ Γµ αβ / g Which is the same result as above. Similarly, one can define the Weyl ordering αβλ xα x β x λ as: αβλ xα x β x λ = ( Σ µ αβλ xα β x λ + Σ µ αβλ xβ α x λ + Σ µ αβλ xλ α x β / g (22 and straightforard simplifications by using the fact that Σ µ αβλ is symmetric ith respect to αβ give: αβλ xα x β x λ = 2iθ βλ iθ ασ σ Σµ αβλ / g (23 and one can deduce that: ( Γµ ρλ Γ ρ αβ xα x β x λ = 2iθ βλ iθ ασ σ ( Γµ ρλ Γ ρ αβ / g (24 and the noncommuting variable X µ can be reritten as: X µ = x µ + ( Γµ αβ xα x β 1 ( Γµ 2 ρλ Γ ρ αβ xα x β x λ here after the corresponding expressions substitutions reads: (25 X µ = x µ + 1 2 iθβλ iθ ασ σ Rµαβλ (x / g (26 Here R µ αβλ stands for the Riemann curvature tensor defined as:
Electronic Journal of Theoretical Physics 3, No. 12 (2006 37 45 41 Using the C-B-H formula, one can rite: R µ αβλ (x = β Γ µ αλ λ Γ µ αβ + Γ µ ρβ Γ ρ αλ Γ µ ρλ Γ ρ αβ (27 e ik Xe ip X = e ik X+ip X+ 1 2ik X, ip X]+... = e ik x+ip x+ik µ x µ +ip ν x ν i 2 θµν k µ p ν... (28 ith Thus, The Moyal-Weyl -product reads: x µ = 1 2 iθβλ iθ ασ σ Rµαβλ / g (29 W (f W (g = (2π 3 2 (2π 3 2 here d 4 kd 4 pe ik x+ip x+ikµ xµ +ip ν x ν i 2 θµν k µp ν f (k g (p = W (f g (30 (f g (x = e xµ and to the second order in θ 2, one obtains: x µ + x ν y ν + i 2 θµν x µ y ν f (x g (y] x=y (31 (f g (x = f (x g (x + x µ µ (f (x g (x + i 2 θµν x µ f (x hich can be reritten as + i 2 θµν i 2 θαβ x µ x α f (x x ν g (x (32 xν g (x +... xβ here (f g (x = f (x g (x + i 2 θµν x µ f (x x i 2 θµν i 2 θαβ x µ g (x + (33 ν x f (x g (x +.. α x ν xβ x µ = x µ + x µ = x µ + 1 2 iθβλ iθ ασ σ R µ αβλ / g. (34 Notice that one can add to the expression of Xµ the term Γ µ αβ Γ ρ ρλ xα x β x λ. Hoever, if e require that the Weyl ordering of a product of non coupled terms like Σ µ αβλ...ϱ xα x β x λ... x ϱ and Λ ν πστ...κ x π x σ x τ... x κ factorizes i.e.: W αβλ...ϱ xα x β x λ... x ϱ Λν πστ...κ x π x σ x τ... x κ = W αβλ...ϱ xα x β x λ... x ϱ W ( Λν πστ...κ x π x σ x τ... x κ Then, this term does not contribute since (35
42 Electronic Journal of Theoretical Physics 3, No. 12 (2006 37 45 ( Γρ = Γ ρ ρλ xλ / g = iθ λσ σ Γρ ρλ / g (36 No, using the fact that one deduce that: Γ ρ ρλ = λ log (g (37 ( Γρ ρλ xλ = iθ λσ σ Γρ ρλ / g = iθ λσ σ λ log (g / g = 0 (38 2.3 T he N oncommutative Action Let us calculate the mass term Φ + Φ here Φ is a complex scalar field in this ne noncommuting space-time: Eq.(39 can be reritten as: ( Φ + Φ (x = Φ + (x Φ (x + total derivative (39 ( Φ + Φ (x = Φ + (x + x Φ (x + x = Φ + (x Φ (x+ x µ µ Φ + (x Φ (x ] +total derivative (40 hich after direct simplifications, the action reads: I = d 4 x ( Φ + Φ (x = d 4 xφ + (x Φ (x+ 1 2 d 4 xθ βλ θ ασ R µ αβλ σ µ Φ + (x Φ (x ] g (41 2.4 2 d Gravity coupled to a scalar field Let us calculate this action in to dimensions, choosing θ 01 = +η, ith η 1, The gravitational term reads θ βλ θ ασ R µ αβλ σ µ Φ + (x Φ (x ] / g = θ βλ θ ασ R µαβλ µ σ Φ + (x Φ (x ] / g (42 hich can be simplified as: θ βλ θ ασ R µ αβλ σ µ Φ + (x Φ (x ] / g = 2η 2 R 0101 g µν µ ν Φ + (x Φ (x ] / g (43 In to dimensions, the scalar curvature:
Electronic Journal of Theoretical Physics 3, No. 12 (2006 37 45 43 is related to the component R 0101 by the relation : R = g µν R µν (44 Straightforard simplifications lead to: R 0101 = 1 gr (45 2 R 00 = g 11 R 0101 (46 and R 11 = g 00 R 0101 (47 R 01 = R 10 = g 01 R 0101 and consequently, the scalar curvature R reads R = 2g 1 R 0101 (48 Finally the action gets the form: I = d 2 xφ + (x Φ (x 1 2 η2 d 2 x grg µν µ ν Φ + (x Φ (x ] (49 3. Conclusion Thought this ork e have considered a noncommutative curved space-time in a gravitational background and define ne Moyal-Weyl star product and Weyl ordering at the order of θ 2 (θ is the order parameter of the noncommutative of the space timehere the geometric structure is included. As an example, e have considered the mass term of a complex field and sho explicitly the gravitation effect on the noncommutative-space time. (More studies are under investigations. Appendix e have αβ xα x β = ( αβ xα x α Σµ αβ x β x α αβ xβ x β Σµ αβ / g (A-1 hich can be reritten as: αβ xα x β == ( x x α β, Σ ] µ αβ x α, Σ ] µ αβ x β / g (A-2
44 Electronic Journal of Theoretical Physics 3, No. 12 (2006 37 45 using the relations eq.(16 e obtain: ( αβ xα x β = iθ βλ x α λ Σµ αβ λ Σ µ βα xα / ( g = iθ βλ x α λ Σµ αβ λ Σ µ αβ xα / g (A-3 thus, ] αβ xα x β = iθ βλ x α, λ Σµ αβ / g = iθ βλ iθ ασ σ λ Σµ αβ / g Acknoledgement (A-4 One of us (N. Mebarki is very grateful to Profs. P. Aurenche and F. Boudjemaa for the hospitality during the visit to LAPTH d Annecy.This ork is supported by the research grant N D2005/13/04
Electronic Journal of Theoretical Physics 3, No. 12 (2006 37 45 45 References 1] J.Madore, S.Schraml,P.Schupp and J.Wess, Eur.Phys.Jour.C16(2000161 2] M.R.Douglas and N.A.Nekrasov,Rev.Mod.Phys.73,(2001977. 3] J.K.Glikman and S.Noak, Int.Journ.Mod.Phys.A12,(2003299 4] X.Calmet,B.Jurco,J.Wess and M.Wohlgenannt, Eur.Phys.Jour.C23,(2002363. 5] X.Calmet,and M.Wohlgenannt,Phys.Rev.D68(2003025016. 6] J.Madore, An introduction to Noncommutetive geometry and its physical applications, Cambridge University Press1999. 7] F.Ardalan, H.Arafi and M.M.Sheikh-Jabbari JHEP(19999902. 8] N.Seiberg and E.Witten, JHEP(1999032. 9] M.Chaichan, A.Demitchev, P.Presnajder, M.M.Sheick-jabbari and A.Tureanu, Nucl.Phys.B611(2001383 10] M.M.Sheick-Jabbari,JHEP(1999015. 11] J.E.Moyal, Proc.Cambridge Phil. Soc.45,(194999.