1. 1 Electromagnetic Casimir Effect in Krein Space M. Naseri Islamic Azad University, Kermanshah Branch, Kermanshah, IRAN E-mail:sepehr1976@yahoo.com

2. 2 Minimally Coupled Scalar Field in de Sitter Space There is a great deal of similarity between Infrared divergence appears in the linear quantum gravity and the minimally coupled scalar field in de Sitter space. M.V. Takook, Thèse de l’université Paris VI, 1997 Théorie quantique des champs pour des systèmes élémentaires “massifs” et de “masse nulle” sur Antoniadis I., Iliopoulos J., Tomaras T. N., Phys. Rev. Letters 56 (1986) 1319 Quantum Instability of de Sitter Space

3. 3 Minimally Coupled Scalar Field in de Sitter Space A covariant quantization of minimally coupled scalar field cannot be constructed by positive norm states alone. Allen B., Phys. Rev. D, 32 (1985) 3136 Vacuum states in de Sitter space

4. 4 Covariant Quantization and Negative Norm States To construct a covariant quantization of minimally coupled scalar field in ds space, a new method of field quantization (Krein Space Quantization) has been presented J.P. Gazeau, J. Renaud, M.V. Takook, Class. Quantum Grav. 17 (2000) 1415, gr-qc/9904023. S. De Bièvre, J. Renaud, Phys. Rev. D 57 (1998) 6230.

5. 5 Krein Space Quantization The field equation The two sets of solutions Where:

6. 6 Krein Space Quantization Where;

7. 7

8. 8 A direct consequence of these formulas is the positivity of the energy i.e. The Advantages of Krein Space Quantization And for the vacuum, this quantity vanishes.

9. 9 The Advantages of Krein Space Quantization Another consequence of this formula is a covariant two-point function, which is free of any infrared divergence. Gazeau J.P., Renaud J., Takook M.V., Class. Quant. Grav., 17(2000)1415, gr-qc/9904023. Takook M.V., Mod. Phys. Lett. A, 16(2001)1691, gr- qc/0005020.

10. 10 Advantages ofd this method of quantization can be seen in The other Advantages ofd this method of quantization can be seen in the following reports; the following reports; Gazeau J.P., Rouhani S. and Takook M.V., Linear covariant quantum gravity in de Sitter space, in preparation; Behroozi S. et al., Conformal invariance wave equations in de Sitter space, gr-qc/0512105. The Advantages of Krein Space Quantization

11. 11 Is it possible to generalize the free QFT in dS method to the interaction field?

12. 12 Generalization of The Krein QFT λ Φ 4 theory in Minkowski space- time in one-loop approximation is automatically renormalized Mohammad Vahid Takook,Int.J.Mod.Phys. E14 (2005) 219-224

13. 13 Casimir Effect In its simplest case the Casimir effect is the reaction of the vacuum of the quantized electromagnetic field to changes in external conditions like conducting surfaces.

14. 14 Casimir Effect

15. 15 QFT Casimir Effect Calculation

16. 16 Regularization & Renormalization

17. 17 Casimir Effect Calculation in Krein QFT

18. 18 Commutation Rules Creation and annihilation operators are constrained to obey the following commutation rules

19. 19 Commutation Rules & The Vacuum The vacuum is defined by:

20. 20 Casimir Energy & Casimir Force

21. 21 Conclusions The problem of singularity in QFT (the regularization and renormalization procedure) is an anomaly. This problem may be resolved in: Quantum Field Theory with the N.N.S.