special functions & physics g. dattoli enea frascati a perennial marriage in spite of computers
Post on 04-Jan-2016
213 Views
Preview:
TRANSCRIPT
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(
),(
top related