Download - Special Functions & Physics G. Dattoli ENEA FRASCATI A perennial marriage in spite of computers
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Special Functions & Physics
G. DattoliENEA FRASCATI
A perennial marriage in spite of computers
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Euler Gamma FunctionDefined to generalize the factorial operation
to non integers
!)1()()1(
0)Re(,)(!0
1
0
nnxxx
xdtetxdtetn txtn
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Inclusion of negative arguments
)()(
!lim)(
0
xmxxj
nnx n
j
x
n
)(
1
)sin()1(
xxx
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Euler Beta FunctionGeneralization of binomial
),()1(
,)1(),(
11
0
1
1
0
)1(
yxBdtttte
deeyxI
yx
yx
1
1
11
)(
)()(),(
y
yx
xyx
yxyxB
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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
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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
smsJ
ssJ
m
0
0
)(
,)(
21 GeV
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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
4
1
22)3()1(
2)4()1(
2)2()1(
)(
,)(
,)(
iimutsppu
ppt
pps
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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
))(),(())()((
))(())((),( tsB
ts
tstsA
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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
1
1)(
nxn
x
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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
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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
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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)(
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Planck law
)4()4(1
),(
,
1
18),(
0
3
03
3
3
x
TK
h
e
x
L
UdTu
ec
hTu
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Analytic continuation of the Riemann function
• Ac
22
)1(
)()()2
cos(2)1(
2
1
ssss ss
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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
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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
0... 011
1 bxbxbxb nn
nn
0
1
1
1
b
b
x
n
s s
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…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
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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
1
2
222
22
1
)(
,2
1
nnyx
yxn
nn
AdkdkE
a
nkkc
EE
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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
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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
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A further identity
3333
1
3
0
33
)1(
1
1)1()(
)()1()(
EOST
tttT
tTttntT
n
n
n
tn
s
n
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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
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What is the meaning of all this crazy stuff?
• The sum o series according to Ramanujian
,12
1)1(
1
n
n
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Renormalization: Quos perdere vult Deus dementat prius
• A simple example, the divergence from elementary calculus
cxn
dxxnxI nn
1
1
1),(
cxxI )ln()1,(
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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.• That’s the essence of renormalization subtract
infinity to infinity.• We set
1
1
n
cc
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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
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Is everything clear?
• If so• prove that
find a finite value for
• The diverging series “par excellence”
2!
n
n
n xn!)1(0
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Shift operators(Mac Laurin Series expansion)
)(!
)(!
)()(
)(
0
0
xfs
b
xfs
bxfebxf
s
s
s
sx
s
sb x
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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
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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
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Jacob Bernoulli and E.R.F.Ars coniectandi 1713 (posthumous)
)!1(
!
!
1)()(
1
0 nm
m
nBmxxf
m
nn
m
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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
sdivergenceIRelnegativem
sdivergenceUVsmallpositivem
kdkkI mm
arg,
,,
,2
,0
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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
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Recursions
...
),1()),0((2
1),3(
),1(),0(2
1),1(
,2
1)0(),0(
21,2
BaII
II
I
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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
k
k 0
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F.D. and renormalization
• a
),2()12()!2(
)(),1(2
,
,1
2
rmIrmar
B
mmIm
mI
rmr
r
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The Euler Dilatation operator
xefxfe
xexexs
nxe
xnxx
xnxx
x
x
x
nnn
s
ns
nx
npnpx
nnx
)(
)(!
,)(
0
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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
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
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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
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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
dxep
x
dx
xe
ppx
p
pD
0
1
0
1ˆ
)(
1)(
1),(
)(),1( pp
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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
dep
Z pDp
0
1ˆ
)(
1ˆ
),(),(ˆ
)()(
1)(ˆ 1
0
pmxmxZ
dxefp
xfZ
p
pp
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Zeta and prime numbersEuler!!!
p
sp
s1
1
1)(
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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)
xx
)2
1()(
2
1ˆ xxixppxH
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Evolution operator
xx D
x
eeeUˆ2
12
1
)(ˆ
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Riemann hypothesis
• RH: The non trivial zeros are on the critical line:
2
1ti
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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
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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
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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
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Beta the way out
• …The Beta function once more
• More details upon request
aD
nL
DyxB
2)()
2(
),(