Special Functions & Physics G. Dattoli ENEA FRASCATI

1. Special Functions & PhysicsG. DattoliENEA FRASCATI A perennial marriage in spite of computers

2. Euler Gamma FunctionDefined to generalize the factorial operation to non integers

3. Inclusion of negative arguments

4. Euler Beta FunctionGeneralization of binomial

5. Further properties BETA: if x, y are both non positive integers the presence of a double pole is avoided

6. EULER10 SWISS FRANCKS

7. Strings: the old (beautiful) timesand Euler & Veneziano • Half a century ago the Regge trajectory • Angular momentum of barions and mesons vs. squared mass

8. Old beautiful times… • The surprise is that all those trajectories where lying on a stright line • Where s is the c. m. energy and the angular coefficient has an almost universal value

9. Mesons and Barions

10. Strings: Even though not immediately evident this phenomenological observation represented the germ of string theories.The Potential binding quarks in the resonances was indeed shown to increase linearly with the distance. Meson-Meson Scattering • m-m

11. Veneziano just asked what is the simplest form of the amplitude yielding the resonance where they appear on the C.F. Plot, and the “natural” answer was the Euler B-Function

12. From the Dark… • An obscure math. Formula, from an obscure mathematicians of XVIII century… (quoted from a review paper by a well known theorist who, among the other things, was also convinced that the Lie algebra had been invented by a contemporary Chinese physicist!!!) • From an obscure math. formula to strings • “A theory of XXI century fallen by chance in XX century” • D. Amati

13. Euler-Riemann function… It apparently diverges for negative x but Euler was convinced that one can assign a number to any series

14. An example of the art of manipulating series

15. Divergence has been invented by devil, no…no… It is a gift by God

16. Integral representation for the Riemann Function

17. Planck law

18. Analytic continuation of the Riemann function • Ac

19. Analytic continuation & some digression on series • From the formula connecting half planes of the Riemann function we get

20. ..digression and answer • “Euler” proved the following theorem, concerning the sum of the inverse of the roots of the algebraic equation

21. …answer • Consider the equation

22. Casimir Force • Casimir effect a force of quantum nature, induced by the vacuum fluctuations, between two parallel dielectric plates

23. Virtual particles pop out of the vacuum and wander around for an undefined time and then pop back – thus giving the vacuum an average zero point energy, but without disturbing the real world too much.

24. Sensitive sphere. This 200-µm-diameter sphere mounted on a cantilever was brought to within 100 nm of a flat surface to detect the elusive Casimir force. Casimir: The Force of empty space

25. Casimir Calculation a few math • Elementary Q. M. yields diverging sum

26. Regularization & Normalization • We can explicitly evaluate the integral • What is it and why does it provide a finite result?

27. Are we now able to compute the Casimir Force? • Remind that • And that • And that

28. A further identity

29. Again dirty tricks • Going back to Euler

30. What is the meaning of all this crazy stuff? • The sum o series according to Ramanujian

31. Renormalization: Quos perdere vult Deus dementat prius • A simple example, the divergence from elementary calculus

32. The way out: A dirty trick ormathemagics • We subtract to the constants of integration • A term (independent of x) but with the same behaviour (divergence) when n=-1. • That’s the essence of renormalization subtract infinity to infinity. • We set

33. Dirty...Renormalization • Our tools will be: subtraction and evaluation of a limit

34. Is everything clear? • If so • prove that find a finite value for • The diverging series “par excellence”

35. Shift operators(Mac Laurin Series expansion)

36. Series Summation

37. We can do thinks more rigorously

38. Jacob Bernoulli and E.R.F.Ars coniectandi 1713 (posthumous)

39. Diverging integrals in QED • In Perturbative QED the problem is that of giving a meaning to diverging integrals of the type

40. SchwingerWas the first to realize a possible link between QFT diverging integrals and Ramanujan sums

41. Recursions

42. Self Energy diagrams • Feynman loops (DIAGRAMMAR!!! ‘t-Hooft-Veltman, Feynman the modern Euler) • Loops diagram are divergent • Infrared or ultraviolet divergence

43. F.D. and renormalization • a

44. The Euler Dilatation operator

45. Can the Euler-Riemann function be defined in an operational way? • We introduce a naive generalization of the E--R function

46. Can the E-R Function…?YES • The exponential operator , is a dilatation operator

47. More deeply into the nature of dilatation operators • So far we have shown that we can generate the E-R function by the use of a fairly simple operational identity

48. Operators and integral transforms • Let us now define the operator (G. D. & M. Migliorati • And its associated transform, something in between Laplace and Mellin

49. Zeta and prime numbersEuler!!!

50. A lot of rumours!!!