Download presentation

Presentation is loading. Please wait.

Published byGeorgia Rodgers Modified about 1 year ago

1
**Special Functions & Physics G. Dattoli ENEA FRASCATI**

A perennial marriage in spite of computers

2
**Euler Gamma Function Defined to generalize the factorial operation to non integers**

3
**Inclusion of negative arguments**

4
**Euler Beta Function Generalization of binomial**

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

6
EULER 10 SWISS FRANCKS

7
**Strings: the old (beautiful) times and 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
**Casimir: The Force of empty space**

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

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 or mathemagics**

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
Schwinger Was 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**

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 numbers Euler!!!**

50
A lot of rumours!!!

51
**Hermitian and non Hermitian operators**

The operator is not Hermitian The Hamiltonian Is Hermitian (at least for physicist)

52
Evolution operator

53
Riemann hypothesis RH: The non trivial zeros are on the critical line:

54
**The Riemann hypothesis:The Holy Graal of modern Math**

What is the point of view of physicists? The Berry-Keating conjecture: …zeros Coincide with the spectrum of the Operator: namely

55
**Lavoro di Umar Mohideen e suoi collaboratori all’università di California a Riverside**

Strumento utilizzato: microscopio a forza atomica Una sfera di polistirene 200 µm di diametro ricoperta di oro (85,6 nm) attaccata alla leva di un microscopio a forza atomica, ad una distanza di 0.1 µm da un disco piatto coperto con gli stessi materiali. L’attrazione tra sfera e disco ricavata dalla deviazione di un fascio laser. Differenza tra dato seprimentale e valore teorico entro 1%. Sensibilità: N Vuoto: Pa

56
EULER-BERNOULLI

57
**Beta the way out …The Beta function once more**

More details upon request

Similar presentations

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google