Gravitation and Gauge Theory in the Non Commutative Geometry Formalism

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Non-commutative gravity and applications

The standard concept of space-time as a geometric manifold is based on the notion of a manifold whose points are locally labeled by a finite number of real coordinates. However, it is generally believed that this picture of space-time as a manifold should break down at very short distances of the order of the Planck length. This implies that the mathematical concepts of High energy physics has to be changed or more precisely our classical geometric concepts may not be well suited for the description of physical phenomenon at short distances. Noncommutativity is a mathematical concept expressing uncertainty in quantum mechanics, where it applies to any pair of conjugate variables such as position and momentum.

Thought out this work, we have first constructed generalized infinitesimal local coordinates and Lorentz transformations which preserve the non-commutative coordinates canonical commutation relations. It turns out that the form of the latter, looks like classical inhomogeneous Lorentz transformations. This suggests that non-commutativity induces gravity [31]. Second, and contrary to ref. [21], we do not have unimodular theory of gravity. Third, using Seiberg-Witten fields and Moyal product, we have generalized the equations of motion and Noether theorem. Fifth, we have constructed an invariant action of a pure non-commutative gravity with a symmetric non-commutative metric and a Seiberg-Witten spinor matter field in interaction with a Seiberg-Witten electromagnetic field potential in a curved non-commutative space-time with respect to modified infinitesimal local Poincar´e and U(1) transformations. As an application to the generalized field equations, we have derived the modified Dirac equation and studied the particle creation process in a non-commutative space-time anisotropic Bianchi I universe. After straightforward calculations, using the Bogouliubov transformations and the quasi-classical limit for identifying the positive and negative frequency modes, we have deduced the corresponding number density as a function of θ. We have studied both the weak and strong field limits and obtained under certain circumstances a Fermi-Dirac like distribution. In the weak electric field approximation certain thermodynamic quantities like the total number of the created particles, internal and grand potential per a unit volume were calculated. Moreover, and as in the ordinary case [30], we have found also that in this limit the gravitational density decreases faster than the total number Nˆ. Consequently, the particle creation mechanism effectively isotropizes in the non commutative space-time of 43 an anisotropic Bianchi I universe in the presence of an electric field. It is worth to mention that in the limit θ → 0 and high energies our results coincide with those or ref. [30]. As a conclusion, the non commutativity plays the same role as the electric field and gravity and contribute to the pair creation process.

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1 Introduction 
1.1 Why Gauge theories ?
1.2 Why Non-commutative Geometry?
1.3 Plan of thesis
2 Non-commutative Gauge ¯eld theory and Siberg-Witten maps 
2.1 Non-commutative Gauge ¯eld theory
2.1.1 Generalized in¯nitesimal general coordinate transformations
2.1.2 Generalized in¯nitesimal local Lorentz transformations
2.1.3 Non-commutative gauge transformations
2.2 Seiberg-Witten maps
3 Non-commutative gravity and applications 
3.1 Introduction
3.2 QED action in a curved non commutative space-time
3.2.1 Generalized ¯eld equations and Noether theorem
3.2.2 Generalized Dirac equation and particle creation process
3.2.3 The weak ¯eld approximation
3.2.4 The strong ¯eld approximation:
3.3 Conclusion
4 Gauge Gravity in Non-commutative De Sitter Space-time and Pair creation
4.1 Introduction
4.2 Generalized Dirac equation and particle creation process
4.3 Conclusion
5 Non-commutative Minimal Super-symmetric Standard Model 
5.1 Introduction
5.2 Minimal Super-symmetric Standard Model
5.3 Conclusion

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