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XVII European Workshop on String Theory 2011 de Italy, Padua By:Farrin Payandeh University of Payame Noor, Tabriz,Iran September 2011 Dirac’s equation and the sea of negative energy. Standard quantum field theory leading to the infinites. Renormalization : Sweeping the infinites under the rug. EPR Paradox and Antiparticles. Quantum Mechanics in complex spacetime. The Miracle of creation. Conservation of Angular momentum(a.k.a. “The photon”). 2 Neglecting negative Energies: “At what cost?” 3 4 Case Study P. A. M. Dirac: Bakerian Lecture , Proceedings of the Royal Society A (1941) 4 Quantization of the minimally coupled scalar field and calculating the gravitaton two point function in de Sitter spacetime: Breaking of the covariance. (a) Appearance of the infrared divergence. (b) Covariant Quantization adding Negative Frequency States and Free of divergence graviton field operator in krein Space Field operator in Krein space: ( x ) 1 (p ( x ) n ( x )) 2 Krein QFT calculations in Minkowski spacetime Two point function in Krein space iGT ( x, x) 0 | T ( x) ( x) | 0 J1 ( 2m 2 0 ) 1 m2 GT ( x, x) Re G ( x, x) ( 0 ) ( 0 ) 8 8 2m 2 0 P F 2 0 ( x x )( x x ) 0 Feynman propagator G ( x, x) P F J ( 2m 2 ) iN ( 2m 2 ) im 2 K1 ( 2m 2 ( 0 ) ) d 4 k e ik.( x x) 1 m2 1 0 1 0 2 ( 0 ) ( 0 ) ( 0 ) (2 ) 4 k 2 m 2 i 8 8 4 2m 2 0 2m 2 ( 0 ) The divergence appears in the imaginary part of this eq. and the real part is convergent : m2 1 P lim Re GF ( x, x ) ( 0 ) 0 0 16 8 lim z 0 , J1 ( z ) 1 N ( z) 2 1 , lim 1 z 0 z 2 z z2 lim Re GFP ( x, x) 0 0 , lim z 0 K1 ( z ) 1 2 Z z z (1) s z J1 ( z ) 2 s 0 s!( s 1)! 2 z 2 N1 ( z ) 2 J1 ( z ) log 2 z K1 ( z ) 2 2s J1 (iz ) iN1 (iz ) 10 Divergent Integrals of QED in Krein Space General form of divergent integrals of QED is: and using integral representation of the causal propagator in momentum space: i is ( k 2 m2 i ) ds e k 2 m 2 i 0 we can see that the real part is convergent: (1) Convergent expressions of vacuum polarization and vertex graphs in krein space will be: † ( p, q, p) p ( p, q, p) 0 p ( p, q, p) 0 13 14 15 16 17 18 Negative-energy modes as a candidate solution to some problems in quantum physics? Quantum Field Theory with the N.F.S. , an answer to Feynman reply , “ A Nobel prize for sweeping the rushes (infinties) under the rug” ? 20 21 22 23 GHOSTBUSTING: Case Study Reviving quantum P. A. M. Dirac: Bakerian Lecture , Proceedings of the Royal Society A (1941) theories that were thought to be dead Thank you for your kind attention 24 24