Special Functions & Physics G. Dattoli ENEA FRASCATI A perennial marriage in spite of computers

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

Inclusion of negative arguments

Euler Beta Function Generalization of binomial

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

EULER 10 SWISS FRANCKS

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

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

Mesons and Barions

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

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

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

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

An example of the art of manipulating series

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

Integral representation for the Riemann Function

Planck law

Analytic continuation of the Riemann function Ac

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

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

…answer Consider the equation

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

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.

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

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

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

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

A further identity

Again dirty tricks Going back to Euler

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

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

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

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

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

Shift operators (Mac Laurin Series expansion)

Series Summation

We can do thinks more rigorously

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

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

Schwinger Was the first to realize a possible link between QFT diverging integrals and Ramanujan sums

Recursions

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

F.D. and renormalization

The Euler Dilatation operator

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

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

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

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

Zeta and prime numbers Euler!!!

A lot of rumours!!!

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

Evolution operator

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

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

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: 10-17 N Vuoto: 10-1-10-6 Pa

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Beta the way out …The Beta function once more More details upon request