On Characterization of Non-commutative Minkowski Space Time

Several field theory models on κspace time have been constructed , using different techniques such as by F. Mejer and J Lukierski specially in, scalar field theory on κ-Minkowski space. It has been shown by S. Tanimutra that in the flat space time, Feynman approach and minimal coupling method are equivalent which is usefull to derive the general equation of motion for a charged particle. We discuss here the applicability of Feynman approach for the case of general relativity, resulting in the derivation of the geodesic equation. We generalize the procedure for κ-Minkowski space. We obtain here corrections to the geodesic equation due to the κ-deformation of space time , up to the first order in the deformation parameter. WE know that a relativistic particle of mass m and electric charge e is described by in 4D-Minkowski space, where in 4DMinkowski space, where τ is a parameter. Let us write the following relations Material Science Research India Vol. 9(1), 123-127 (2012)


INTRODUCTION
Several field theory models on κspace time have been constructed , using different techniques such as by F. Mejer and J Lukierski specially in, scalar field theory on κ-Minkowski space.It has been shown by S. Tanimutra that in the flat space time, Feynman approach and minimal coupling method are equivalent which is usefull to derive the general equation of motion for a charged particle.We discuss here the applicability of Feynman approach for the case of general relativity, resulting in the derivation of the geodesic equation.We generalize the procedure for κ-Minkowski space.We obtain here corrections to the geodesic equation due to the κ-deformation of space time , up to the first order in the deformation parameter.WE know that a relativistic particle of mass m and electric charge e is described by in 4D-Minkowski space, where in 4D-Minkowski space, where τ is a parameter.Let us write the following relations ... (1) Where is the canonical momentum operator, the force, the electromagnetic strength tensor, is a gauge field and is an arbitrary function odf x.We deal with gravity only, which do not require the gauge field.We onlyevaluate particles with no electric charge, which for a neutral particle we obtain the following equations.
.. (2) , to find the correct geodesic equation.M. Montesinos obtained that the generalization from flat to curved space by taking Eq.( 1) as valid in a local Lorenz frame of reference and effect of gravity is brought in by replacing the Minkowskian metric with an arbitrary metric .He has shown that this assumption leads to geodesic equation.We choose Let us assume that the 'metric' is a function of operator X which is a symmetric tensor.

Theorem (Generalization of commutative space time )
Let be a new position operator and is the corresponding conjugate momenta, , then solution of equation may be obtain in terms of the operators given in (2).Proof We construct operators X and P expressed as follows.Where .Combining equations ( 5) and ( 2) Where .Also we get Combining equations ( 3), ( 5), ( 6), we obtain the equation ... (7) By choosing the operator X such that is ,LHS of ( 7 Where all Jacobi identities are satisfied.Eq. ( 6) is similar to the Christoffel symbol in general relativity, and Eq. ( 8) is similar to the geodesic equation.Let us assume that 'metric' is invertible and define an inverse of the symmetric tensor g μα by the following relation Combining the equations ( 4), ( 7) and ( 9 By an appropriate application of (14), we find that this construction satisfioes all Jacobi identities and Eq. ( 15) is also satisfied to all orders in a.
. Hence , get its modified form as follows.
An alternative construction of the operator Pμ up to the first order in the deformation parameter may be obtained by differentiating Eq. ( 17) with respect to τ.
It satisfies antisymmetric part of .
We write it as follows Where and .
Combining Eq. ( 16) and (17), We get …(18) By taking the limit a →0, we obtain Hence , the case of up to the first order in the deformation parameter a i.e.
Here and we get the following relations

…( 4 )
Where and satisfy relation.By taking the derivative with respect to of Eqs.(4), we obtain the following relation....(5) ) reduces to the equation We now integrate Eq. (6) over which gives the relation By taking we get ...(8) only as a symmetric tensor with an inverse defined in Eq. (10).Theorem (Derivation of geodric equation in k-Minkowski space time )We derive the geodesic equation for a particle moving in the non-commutative curved space time and analyze the κ-Minkowski deformations of gravity.The methods taken here use κ-Minkowskideformations on the flat space time.It is further generalized to κ-deformed space time with arbitrary metric.ProofLet us define κ-Minkowski space by the relation Operators care expressed in terms of operators x and using the derivations due to J. Lukerki and H. Ruegg.We get Where satisfy the identity …(12) Let us solve Eq. (11) up to the first order in deformation parameter a, which gives Where parameters of the realization α, β, and γ satisfy the a constraint R β ∈ Let us define an operator which commutes with , i.e.only first order in the modified form of give and f( ) up to the first order in The canonical momentum operator ( in e = 0 case ) as derived by the relation ….(14) Hence , we conclude that construction due to E. HariKumar via Feynman approach satisfy all Jacobi identities.The condition that comes by taking the derivative of Eq. (12) with respect to τ is Hence the geodesic equation in the κ-Minkowski space time is established.Theorem ( κ-dependent corrections to the geodesic equation ) Proof Let us choose the conjugate pairs (x , p ) and for flat non-commutative space time, ( , ) in commutative and non-commutative space respectively.It is shown that all the operators in the flat non-commutative space time are expressed in terms of x , p and deformation parameter a.For the non-commutative space time with curvature.Let us construct it as functions of x , p and deformation parameter .In case of neutral particles , conjugate momenta is given by .We derive the corrections to the geodesic equation due to the κdeformation of space time.Let us consider the relation.Where And satisfies Eq. (12).satisfy all the Jacobi Indentifies and the relation But in the limit a → 0, it implies that By taking limit we get the following relations Combining these equation, and comparing it with (13) & (14) substitute with a function that commutes with .That is with .Thus , we obtain the required form …(16)