Special Functions & Physics

G. DattoliENEA FRASCATI

A perennial marriage in spite of computers

Euler Gamma FunctionDefined to generalize the factorial operation

to non integers

!)1()()1(

0)Re(,)(!0

1

0

nnxxx

xdtetxdtetn txtn

Inclusion of negative arguments

)()(

!lim)(

0

xmxxj

nnx n

j

x

n

)(

1

)sin()1(

xxx

Euler Beta FunctionGeneralization of binomial

),()1(

,)1(),(

11

0

1

1

0

)1(

yxBdtttte

deeyxI

yx

yx

1

1

11

)(

)()(),(

y

yx

xyx

yxyxB

Further properties

)(

)(

)(

1

!

)1(

)(

)(

)(

1

!

)1(),(

0

0

mx

x

mym

ny

y

nxnyxB

m

m

n

n

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

EULER10 SWISS FRANCKS

Strings: the old (beautiful) timesand 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

smsJ

ssJ

m

0

0

)(

,)(

21 GeV

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

4

1

22)3()1(

2)4()1(

2)2()1(

)(

,)(

,)(

iimutsppu

ppt

pps

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

))(),(())()((

))(())((),( tsB

ts

tstsA

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

1

1)(

nxn

x

An example of the art of manipulating series

4

1...4321)1(

)1(

1

)1()(1

1)1()(

2

1

10

STt

tntTt

ttT n

n

nt

n

s

n

EOS ...654321

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

4

1...4321 S

12

1)1(

12

11...321

32

2

)1(1

...642

...531

11

11

1

n

nn

n

SE

SE

EOS

nnEO

D

O

Integral representation for the Riemann Function

dex

A xAx 1

0)(

1

aa

n

n

1

1

0

dex

dexn

x

x

xn

nnx

0

1

1

011

1

1

)(

1

)(

11)(

Planck law

)4()4(1

),(

,

1

18),(

0

3

03

3

3

x

TK

h

e

x

L

UdTu

ec

hTu

Analytic continuation of the Riemann function

• Ac

22

)1(

)()()2

cos(2)1(

2

1

ssss ss

Analytic continuation & some digression on series

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

•

1222

1

1

2

1)2()2(

2

1)1(

2),(2

cos)(2)1(

n

ss

n

sss

ss

?11

122

n n

..digression and answer

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

0... 011

1 bxbxbxb nn

nn

0

1

1

1

b

b

x

n

s s

…answer

• Consider the equation

22

42753

0...!5!3

10...!7!5!3

0)sin(

sy

yyxxxxx

s

12

1)1(

6)2(

6

11)(

2

122

22

ss s

sy

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 Calculation a few math

• Elementary Q. M. yields diverging sum

1

2

222

22

1

)(

,2

1

nnyx

yxn

nn

AdkdkE

a

nkkc

EE

Regularization & Normalization

• We can explicitly evaluate the integral

• What is it and why does it provide a finite result?

1

3)3(n

n

n

ss

ss

s

nn

nyx

nsa

c

A

sE

AdkdkE

33

21

1

3

1

2

)(

22

1

A

EF

a

c

A

Eac )3(

6 3

2

Are we now able to compute the Casimir Force?

• Remind that

• And that

• And that

1

3)3(n

n

1444

1

1

8

!3)4()4(

8

1)3(

2),(2

cos)(2)1(

n

ss

n

sss

ss

A

EF

a

c

A

Eac )3(

6 3

2

A further identity

3333

1

3

0

33

)1(

1

1)1()(

)()1()(

EOST

tttT

tTttntT

n

n

n

tn

s

n

Again dirty tricks

• Going back to Euler

c

eaS

a

e

S

a

c

A

ES

EOS

EO

EOE

O

cc

c

22

2

2

3

2

43

33333333

3

333

333333

3

3333

,760

,

760120

1

21)3(

...654321

)3(

)(2...642

...531

What is the meaning of all this crazy stuff?

• The sum o series according to Ramanujian

,12

1)1(

1

n

n

Renormalization: Quos perdere vult Deus dementat prius

• A simple example, the divergence from elementary calculus

cxn

dxxnxI nn

1

1

1),(

cxxI )ln()1,(

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

1

1

n

cc

Dirty...Renormalization

• Our tools will be: subtraction and evaluation of a limit

cn

xxI

n

n

)1

1(lim)1,(

1

1

cxxIax

a x

x

)ln()1,()ln(1

lim 0

termfinite

Is everything clear?

• If so• prove that

find a finite value for

• The diverging series “par excellence”

2!

n

n

n xn!)1(0

Shift operators(Mac Laurin Series expansion)

)(!

)(!

)()(

)(

0

0

xfs

b

xfs

bxfebxf

s

s

s

sx

s

sb x

Series Summation

0

0

)(

0

1

1

,ˆ)0(!

1)ˆ(

,1

1

n

n

n

n

n

n

n

x

xe

e

Ofn

Of

xx

We can do thinks more rigorously

)(!

)(1

)(1

1)()(

0

11

0 0

xfn

Bxfe

xfe

xfenxf

nx

nnx

xx

n n

n

x

x

x

numbersBernoulliBn

kk

k

tB

k

tt

e

t

2 !2

11

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

)!1(

!

!

1)()(

1

0 nm

m

nBmxxf

m

nn

m

Diverging integrals in QED

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

sdivergenceIRelnegativem

sdivergenceUVsmallpositivem

kdkkI mm

arg,

,,

,2

,0

SchwingerWas the first to realize a possible link between QFT diverging integrals and

Ramanujan sums

),2()12()!2(

)(),1(2

,

,)22(

)1(

,)!2(2

,

,1

2

,

12,

1

2

10

rmIrmar

B

mmIm

mI

rm

ma

ar

BndppmI

rmr

r

rm

rmrm

r

r

n

mm

m

Recursions

...

),1()),0((2

1),3(

),1(),0(2

1),1(

,2

1)0(),0(

21,2

BaII

II

I

Self Energy diagrams

• Feynman loops (DIAGRAMMAR!!! ‘t-Hooft-Veltman, Feynman the modern Euler)

• Loops diagram are divergent

• Infrared or ultraviolet divergence

k

k 0

F.D. and renormalization

• a

),2()12()!2(

)(),1(2

,

,1

2

rmIrmar

B

mmIm

mI

rmr

r

The Euler Dilatation operator

xefxfe

xexexs

nxe

xnxx

xnxx

x

x

x

nnn

s

ns

nx

npnpx

nnx

)(

)(!

,)(

0

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

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

1

),(n

p

n

p

nnp

x

n

xpx

n

xxx

11

1)()(),(

1 xxxxnx p

xn

npx

x

xxnx p

x

1)(),(

dea a 1

0)(

1

xx

Dp

xD

dep

Z

ˆ

,)(

1ˆ0

1ˆ

Can the E-R Function…?YES

• The exponential operator , is a dilatation operator

)()(ˆ

xefxfe xD

dxep

x

dex

e

p

x

dx

xe

px

xZpx

p

p

pDp

0

1

1

0

0

1ˆ

)(

1)(

1)(

1

1ˆ),(

xDeˆ

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

dxep

x

dx

xe

ppx

p

pD

0

1

0

1ˆ

)(

1)(

1),(

)(),1( pp

Operators and integral transforms

• Let us now define the operator (G. D. & M. Migliorati

• And its associated transform, something in between Laplace and Mellin

dep

Z pDp

0

1ˆ

)(

1ˆ

),(),(ˆ

)()(

1)(ˆ 1

0

pmxmxZ

dxefp

xfZ

p

pp

Zeta and prime numbersEuler!!!

p

sp

s1

1

1)(

A lot of rumours!!!

Hermitian and non Hermitian operators

• The operator is not Hermitian

• The Hamiltonian

• Is Hermitian (at least for physicist)

xx

)2

1()(

2

1ˆ xxixppxH

Evolution operator

xx D

x

eeeUˆ2

12

1

)(ˆ

Riemann hypothesis

• RH: The non trivial zeros are on the critical line:

2

1ti

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:

namely0)

2

1( nEi

)2

1(ˆ xxiH

L’attrazione tra sfera e disco ricavata dalla deviazione di un fascio laser.

Differenza tra dato seprimentale e valore teorico entro 1%.

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

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.

Sensibilità: 10-17 NVuoto: 10-1-10-6 Pa

Strumento

utilizzato:

microscopio a forza

atomica

EULER-BERNOULLI

12

1

22

12

12

122

2

212

)!2(

2)1(1)(

,22)(

21)(

,)!2(

2)2(

k

k

kkk

kk

k

n

k

kk

zk

B

zzctg

kz

nz

z

zzctg

Bk

k

Beta the way out

• …The Beta function once more

• More details upon request

aD

nL

DyxB

2)()

2(

),(