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New Dirac Delta function based methods - Kempf, Jackson, Morales Marco Knipfer Goethe Universitaet Frankfurt knipfer@fias.uni-frankfurt.de July 30, 2014 Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 1 / 23 The paper I am talking about

  1. New Dirac Delta function based methods - Kempf, Jackson, Morales Marco Knipfer Goethe Universitaet Frankfurt knipfer@fias.uni-frankfurt.de July 30, 2014 Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 1 / 23

  2. The paper I am talking about Achim Kempf, David M. Jackson, Alejandro H. Morales New Dirac Delta function based methods with applications to perturbative expansions in quantum field theory and beyond http://arxiv.org/abs/1404.0747 Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 2 / 23

  3. Overview Introduction 1 New way of writing the Dirac Delta New way of Fourier transforming New integration method Application 2 Quantum Field Theory Application to our work? New delta, blurring, deblurring and QFT 3 Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 3 / 23

  4. Fourier Transform Convention � 1 g ( x ) e ixy dx � g ( y ) := √ (1) 2 π Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 4 / 23

  5. Fourier Transform Convention � 1 g ( x ) e ixy dx � g ( y ) := √ (1) 2 π � 1 g ( y ) e − ixy dx √ g ( x ) := � (2) 2 π Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 4 / 23

  6. New Dirac Delta � δ ( x ) = 1 e ixy dy (3) 2 π � = 1 1 g ( y ) g ( y ) e ixy dy (4) 2 π 1 1 1 g ( y ) e ixy dy = √ √ (5) g ( − i ∂ x ) 2 π 2 π � �� � � g ( x ) Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 5 / 23

  7. New Dirac Delta � δ ( x ) = 1 e ixy dy (3) 2 π � = 1 1 g ( y ) g ( y ) e ixy dy (4) 2 π 1 1 1 g ( y ) e ixy dy = √ √ (5) g ( − i ∂ x ) 2 π 2 π � �� � � g ( x ) 1 1 √ δ ( x ) = g ( − i ∂ x ) � g ( x ) , (6) 2 π Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 5 / 23

  8. New Dirac Delta � δ ( x ) = 1 e ixy dy (3) 2 π � = 1 1 g ( y ) g ( y ) e ixy dy (4) 2 π 1 1 1 g ( y ) e ixy dy = √ √ (5) g ( − i ∂ x ) 2 π 2 π � �� � � g ( x ) 1 1 √ δ ( x ) = g ( − i ∂ x ) � g ( x ) , (6) 2 π where g must be a “sufficiently well-behaved” function, e.g. a Gaussian, then Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 5 / 23

  9. New Dirac Delta Example: g is a Gaussian, then � ( − σ 2 ∂ 2 x ) n ∞ 1 e − x 2 / 2 σ , δ ( x ) = √ (7) n ! 2 πσ n =0 Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 6 / 23

  10. New Dirac Delta Example: g is a Gaussian, then � ( − σ 2 ∂ 2 x ) n ∞ 1 e − x 2 / 2 σ , δ ( x ) = √ (7) n ! 2 πσ n =0 can be truncated → approximation of δ ( x ). Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 6 / 23

  11. New way of Fourier transforming 1 1 √ δ ( x ) = g ( − i ∂ x ) � g ( x ) , (8) 2 π Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 7 / 23

  12. New way of Fourier transforming 1 1 √ δ ( x ) = g ( − i ∂ x ) � g ( x ) , (8) 2 π can be inverted √ g ( x ) = 2 π g ( − i ∂ x ) δ ( x ) . (9) � Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 7 / 23

  13. New way of Fourier transforming 1 1 √ δ ( x ) = g ( − i ∂ x ) � g ( x ) , (8) 2 π can be inverted √ g ( x ) = 2 π g ( − i ∂ x ) δ ( x ) . (9) � Benefits? Maybe easier to compute Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 7 / 23

  14. New way of Fourier transforming 1 1 √ δ ( x ) = g ( − i ∂ x ) � g ( x ) , (8) 2 π can be inverted √ g ( x ) = 2 π g ( − i ∂ x ) δ ( x ) . (9) � Benefits? Maybe easier to compute Put in approximation of δ ( x ) → approximative FT Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 7 / 23

  15. Second version of the FT First one: √ � g ( x ) = 2 π g ( − i ∂ x ) δ ( x ) (10) Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 8 / 23

  16. Second version of the FT First one: √ � g ( x ) = 2 π g ( − i ∂ x ) δ ( x ) (10) Second one (proof easy, see paper): √ 2 π e ixy δ ( i ∂ x − y ) g ( x ) g ( y ) = � (11) Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 8 / 23

  17. Second version of the FT First one: √ � g ( x ) = 2 π g ( − i ∂ x ) δ ( x ) (10) Second one (proof easy, see paper): √ 2 π e ixy δ ( i ∂ x − y ) g ( x ) g ( y ) = � (11) How can this be used to integrate? Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 8 / 23

  18. Second version of the FT First one: √ � g ( x ) = 2 π g ( − i ∂ x ) δ ( x ) (10) Second one (proof easy, see paper): √ 2 π e ixy δ ( i ∂ x − y ) g ( x ) g ( y ) = � (11) How can this be used to integrate? � ∞ g ( x ) dx = 2 π � g (0) (12) −∞ Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 8 / 23

  19. New integration method using the new Dirac Delta � ∞ g ( x ) dx = 2 π � g (0) , (13) −∞ Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 9 / 23

  20. New integration method using the new Dirac Delta � ∞ g ( x ) dx = 2 π � g (0) , (13) −∞ so by using the above representations of � g � g ( x ) dx = 2 π g ( − i ∂ x ) δ ( x ) | x =0 (14) Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 9 / 23

  21. New integration method using the new Dirac Delta � ∞ g ( x ) dx = 2 π � g (0) , (13) −∞ so by using the above representations of � g � g ( x ) dx = 2 π g ( − i ∂ x ) δ ( x ) | x =0 (14) � g ( x ) dx = 2 πδ ( i ∂ x ) g ( x ) (15) Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 9 / 23

  22. What do we got so far? Two representations of FT √ � g ( x ) = 2 π g ( − i ∂ x ) δ ( x ) (16) √ 2 π e ixy δ ( i ∂ x − y ) g ( x ) g ( y ) = (17) � Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 10 / 23

  23. What do we got so far? Two representations of FT √ � g ( x ) = 2 π g ( − i ∂ x ) δ ( x ) (16) √ 2 π e ixy δ ( i ∂ x − y ) g ( x ) g ( y ) = (17) � Two representations of integrals from 0 to ∞ � g ( x ) dx = 2 π g ( − i ∂ x ) δ ( x ) | x =0 (18) � g ( x ) dx = 2 πδ ( i ∂ x ) g ( x ) (19) Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 10 / 23

  24. Overview Introduction 1 New way of writing the Dirac Delta New way of Fourier transforming New integration method Application 2 Quantum Field Theory Application to our work? New delta, blurring, deblurring and QFT 3 Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 11 / 23

  25. Quantum Field Theory I Generating functional (applying eqs. for FT) � � � � J φ d n x Z [ J ] = exp iS [ φ ] + i D φ , (20) Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 12 / 23

  26. Quantum Field Theory I Generating functional (applying eqs. for FT) � � � � J φ d n x Z [ J ] = exp iS [ φ ] + i D φ , (20) translates to Z [ J ] = Ne iS [ − i δ/δ J ] δ [ J ] (21) � φ J d n x δ ( i δ/δφ − J ) e iS [ φ ] = Ne i (22) Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 12 / 23

  27. Quantum Field Theory I Generating functional (applying eqs. for FT) � � � � J φ d n x Z [ J ] = exp iS [ φ ] + i D φ , (20) translates to Z [ J ] = Ne iS [ − i δ/δ J ] δ [ J ] (21) � φ J d n x δ ( i δ/δφ − J ) e iS [ φ ] = Ne i (22) Approximation by using approximative δ [ J ] Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 12 / 23

  28. Quantum Field Theory I Generating functional (applying eqs. for FT) � � � � J φ d n x Z [ J ] = exp iS [ φ ] + i D φ , (20) translates to Z [ J ] = Ne iS [ − i δ/δ J ] δ [ J ] (21) � φ J d n x δ ( i δ/δφ − J ) e iS [ φ ] = Ne i (22) Approximation by using approximative δ [ J ] “Usual pertubative expansion of Z [ J ] is not convergent and is at best asymptotic” Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 12 / 23

  29. Quantum Field Theory I Generating functional (applying eqs. for FT) � � � � J φ d n x Z [ J ] = exp iS [ φ ] + i D φ , (20) translates to Z [ J ] = Ne iS [ − i δ/δ J ] δ [ J ] (21) � φ J d n x δ ( i δ/δφ − J ) e iS [ φ ] = Ne i (22) Approximation by using approximative δ [ J ] “Usual pertubative expansion of Z [ J ] is not convergent and is at best asymptotic” This could lead to better convergence Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 12 / 23

  30. Quantum Field Theory II Generating functional (applying eqs. for integration) � J δ/δφ d n x δ [ φ ] | φ =0 Z [ J ] = Ne iS [ − i δ/δφ ]+ (23) � J φ d n x = N δ [ i δ/δφ ] e iS [ φ ]+ i (24) Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 13 / 23

  31. Application to our work To solve � � d n p 1 + β p n e ixp = f ( p ) e ixp d n p (25) Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 14 / 23

  32. Application to our work To solve � � d n p 1 + β p n e ixp = f ( p ) e ixp d n p (25) The possibilities √ g ( x ) = � 2 π g ( − i ∂ x ) δ ( x ) 1 Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 14 / 23

  33. Application to our work To solve � � d n p 1 + β p n e ixp = f ( p ) e ixp d n p (25) The possibilities √ g ( x ) = � 2 π g ( − i ∂ x ) δ ( x ) 1 √ 2 π e ixp δ ( i ∂ x − y ) g ( x ) g ( y ) = � 2 Marco Knipfer (FIAS) New Dirac Delta July 30, 2014 14 / 23

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