Special Functions

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Special Functions PhysicsG. DattoliENEA
FRASCATI

• A perennial marriage in spite of computers

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Euler Gamma FunctionDefined to generalize the
factorial operation to non integers
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Inclusion of negative arguments
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Euler Beta FunctionGeneralization of binomial
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Further properties
BETA if x, y are both non positive integers the
presence of a double pole is avoided
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EULER10 SWISS FRANCKS
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Strings the old (beautiful) timesand Euler
Veneziano

• Half a century ago the Regge trajectory
• Angular momentum of barions and mesons vs.
  squared mass

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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

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Mesons and Barions
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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

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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
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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

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Euler-Riemann function

• It apparently diverges for negative x but
  Euler was convinced that one can assign a number
  to any series

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An example of the art of manipulating series
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Divergence has been invented by devil, nono It
is a gift by God

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Integral representation for the Riemann Function
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Planck law
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Analytic continuation of the Riemann function

• Ac

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Analytic continuation some digression on series

• From the formula connecting half planes of the
  Riemann function we get

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..digression and answer

• Euler proved the following theorem, concerning
  the sum of the inverse of the roots of the
  algebraic equation

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answer

• Consider the equation

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Casimir Force

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

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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.
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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.
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Casimir Calculation a few math

• Elementary Q. M. yields diverging sum

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Regularization Normalization

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

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Are we now able to compute the Casimir Force?

• Remind that
• And that
• And that

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A further identity
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Again dirty tricks

• Going back to Euler

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What is the meaning of all this crazy stuff?

• The sum o series according to Ramanujian

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Renormalization Quos perdere vult Deus dementat
prius

• A simple example, the divergence from elementary
  calculus

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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.
• Thats the essence of renormalization subtract
  infinity to infinity.
• We set

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Dirty...Renormalization

• Our tools will be subtraction and evaluation of
  a limit

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Is everything clear?

• If so
• prove that
• find a finite value for
• The diverging series par excellence

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Shift operators(Mac Laurin Series expansion)
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Series Summation
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We can do thinks more rigorously
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Jacob Bernoulli and E.R.F.Ars coniectandi 1713
(posthumous)
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Diverging integrals in QED

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

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SchwingerWas the first to realize a possible
link between QFT diverging integrals and
Ramanujan sums
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Recursions
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Self Energy diagrams

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

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F.D. and renormalization

• a

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The Euler Dilatation operator
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Can the Euler-Riemann function be defined in an
operational way?

• We introduce a naive generalization of the E--R
  function

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Can the E-R Function?YES

• The exponential operator , is a dilatation
  operator

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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

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Operators and integral transforms

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

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Zeta and prime numbersEuler!!!
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A lot of rumours!!!
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Hermitian and non Hermitian operators

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

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Evolution operator
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Riemann hypothesis

• RH The non trivial zeros are on the critical
  line

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The Riemann hypothesisThe 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

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Lavoro di Umar Mohideen e suoi collaboratori
alluniversità 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.
Lattrazione 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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EULER-BERNOULLI
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Beta the way out

• The Beta function once more
• More details upon request