Fractal Calculus (H)and some Applications

Nugzar MakhaldianiJoint Institute for Nuclear Research

Dubna, Moscow Region, Russiae-mail address: mnv@jinr.ru

If you are only a poet,You are not even that.

(Piet Hein)

Если ты всего лишь поэт,Поэт ли ты, наверно нет :)(Пит Хейн - Граф О’Мар)

IntroductionIn the Universe, matter has manly two geometric structures, homogeneous and hierarchical one.The homogeneous structures are naturally described by real numbers with an infinite numberof digits in the fractional part and usual archimedean metrics. The hierarchical structures aredescribed with p-adic numbers with an infinite number of digits in the integer part and non-archimedean metrics.

A discrete, finite, regularized, version of the homogenous structures are homogeneous latticeswith constant steps, and distance rising as arithmetic progression. The discrete version of thehierarchical structures is hierarchical lattice-tree with scale rising in geometric progression.

There is an opinion that present day theoretical physics needs (almost) all mathematics, and theprogress of modern mathematics is stimulated by fundamental problems of theoretical physics.

I this talk,I would like to give a short history, background, and some applications of the Fractalcalculus (FC). Some speculations on the fine structure constants and the prime numbers arealso given.

Real, p - adic and q - uantum fractal calculus

Every (good) school boy/girl knows what is

dn

dxn= ∂n = (∂)n, (1)

but what is its following extension

dα

dxα= ∂α , α ∈ < ? (2)

Euler, ... Liouville, ... Holmgren, ...

Let us consider the integer derivatives of the monomials

dn

dxnxm = m(m− 1)...(m− (n− 1))xm−n, n ≤ m,

=Γ(m+ 1)

Γ(m+ 1− n)xm−n. (3)

L.Euler (1707 - 1783) invented the following definition of the fractal derivatives,

dα

dxαxβ =

Γ(β + 1)Γ(β + 1− α)

xβ−α. (4)

J.Liouville (1809-1882) takes exponents as a base functions,

dα

dxαeax = aαeax. (5)

J.H. Holmgren invented (in 1863) the following integral transformation,

D−αc,x f =1

Γ(α)

x∫c

|x− t|α−1f(t)dt. (6)

It is easy to show that

D−αc,x xm =

Γ(m+ 1)Γ(m+ 1 + α)

(xm+α − cm+α),

D−αc,x eax = a−α(eax − eac), (7)

so, c = 0, when m + α ≥ 0, in Holmgren’s definition of the fractal calculus, corresponds tothe Euler’s definition, and c = −∞, when a > 0, corresponds to the Liouville’s definition.Holmgren’s definition of the fractal calculus reduce to the Euler’s definition for finite c, and to

the Liouvill’s definition for c = ∞,

D−αc,x f = D−α0,xf −D

−α0,c f,

D−α∞,xf = D−α−∞,xf −D−α−∞,∞f. (8)

We considered the following modification of the c = 0 case [Makhaldiani,2003],

D−α0,xf =|x|α

Γ(α)

1∫0

|1− t|α−1f(xt)dt

=|x|α

Γ(α)B(α, ∂x)f(x) = |x|α Γ(∂x)

Γ(α+ ∂x)f(x),

f(xt) = txd

dx f(x). (9)

As an example, consider Euler B-function,

B(α, β) =∫ 1

0dx|1− x|α−1|x|β−1 = Γ(α)Γ(β)D−α01 D

−β0x 1 =

Γ(α)Γ(β)Γ(α+ β)

(10)

We can define also FC as

Dαf = (D−α)−1f =Γ(∂x+ α)

Γ(∂x)(|x|−αf),

∂x = δ + 1, δ = x∂ (11)

For the Liouville’s case,

Dα−∞,xf = (D−∞,x)αf = (∂x)αf, (12)

∂−αx f =1

Γ(α)

∫ ∞0

dttα−1e−t∂xf(x) =1

Γ(α)

∫ ∞0

dttα−1f(x− t)

=1

Γ(α)

∫ x−∞

dt(x− t)α−1f(t) = D−α−∞,xf. (13)

As an example, let us consider integer derivatives, α = −n,

Dn0xf =1xn

Γ(∂x)Γ(−n+ ∂x)

f

= x−n(−n+ ∂x)(−n+ 1 + ∂x)...(−1 + ∂x)f = ...= x−n(−n+ 1 + x∂)xn−1∂n−1f = ∂nf = f (n). (14)

The integrals can be calculated as

D−nf = (D−1)nf, (15)

where

D−1f = xΓ(∂x)

Γ(1 + ∂x)f = x

1∂xf = x(∂x)−1f

= (∂)−1f =∫ x

0dtf(t). (16)

As another example, let us consider Weierstrass C.T.W. (1815 - 1897) fractal function

f(t) =∑n≥0

anei(bnt+ϕn), a < 1, ab > 1. (17)

For fractals we have no integer derivatives,

f (1)(t) = i∑

(ab)nei(bnt+ϕn) = ∞, (18)

but the fractal derivative,

f (α)(t) =∑

(abα)nei(bnt+πα/2+ϕn), (19)

when abα = a′ < 1, is another fractal (17).

p - adic fractal calculus

p-adic analog of the fractal calculus (6) ,

D−αx f =1

Γp(α)

∫Qp

|x− t|α−1p f(t)dt, (20)

where f(x) is a complex function of the p-adic variable x, with p-adic Γ–function

Γp(α) =∫Qp

dt|t|α−1p χ(t) =1− pα−1

1− p−α, (21)

was considered by V.S. Vladimirov [Vladimirov,1988].

The following modification of p-adic FC is given in [Makhaldiani,2003]

D−αx f =|x|αp

Γp(α)

∫Qp

|1− t|α−1p f(xt)dt

= |x|αpΓp(∂|x|)

Γp(α+ ∂|x|)f(x). (22)

Last expression is applicable for functions of type f(x) = f(|x|).

Fractal qalculus

Another important mathematical structure is q-calculus (qalculus), [Gasper,Rahman,1990]. Thebasic object of this calculus is q-derivative

Dqf(x) =f(x)− f(qx)

(1− q)x=

1− qx∂

(1− q)xf(x), (23)

where either 0 < q < 1 or 1 < q

numbers q = p = 2, 3, 5, ..., 29, ..., 137, ... This is an algebra-analytic quantization of the q-calculus and corresponding physical models. Note also, that p-adic calculus is the natural toolfor the physical models defined on the fractal( space)s like Bete lattice ( or Brua-Tits trees, inmathematical literature) [Makhaldiani,1990].

Fractal finite - difference calculus

Usual finite difference calculus is based on the following (left) derivative operator

D−f(x) =f(x)− f(x− h)

h= (

1− e−h∂

h)f(x). (26)

We define corresponding fractal calculus as

Dα−f(x) = (D−)αf(x). (27)

In the case of α = −1, we have usual finite difference sum as regularization of the Riemannintegral

D−1− f(x) = h(f(x) + f(x− h) + f(x− 2h) + ...). (28)

(I believe that) the fractal calculus (and geometry) are the proper language for the quantume(field) theories, and discrete versions of the fractal calculus are proper regularizations of the

fractal calculus and field theories, [Makhaldiani].

Hypergeometric functions

One of the interesting applications of the new calculus is the following representation of thehypergeometric functions

F (α, β; γ;x) =Γ(γ)

Γ(β)Γ(γ − β)

·∫ 1

0dttβ−1(1− t)γ−β−1(1− tx)−α

=Γ(γ)Γ(β)

Dβ−γ0,1 (tβ−1(1− tx)−α)

=Γ(γ)

Γ(β)Γ(γ − β)

∫ 10dttβ+x∂x−1(1− t)γ−β−1(1− x)−α

=Γ(γ)Γ(β)

Γ(β + x∂x)Γ(γ + x∂x)

(1− x)−α

=∑n≥0

(α)n(β)n(γ)nn!

xn

=Γ(γ)Γ(β)

Γ(β + x∂x)Γ(γ + x∂x)

Γ(α+ x∂)Γ(α)

ex, (29)

(α)δex =Γ(α+ x∂)

Γ(α)ex =

1Γ(α)

∫ ∞0

dttα−1+x∂xe−tex

=1

Γ(α)

∫ ∞0

dttα−1e−tetx =∑n≥0

(α)nn!

xn

=1

Γ(α)

∫ ∞0

dttα−1e−(1−x)t = (1− x)−α. (30)

So

F (α, β; γ;x) =Γ(α+ x∂)

Γ(α)Γ(β + x∂)

Γ(β)Γ(γ)

Γ(γ + x∂)ex

=(α)δ(β)δ

(γ)δex (31)

with obvious any parameter generalization

F (α1, ...αp;β1, ...βq;x) =Γ(α1 + x∂) · · · Γ(αp + x∂)

Γ(α1) · · · Γ(αp)Γ(β1) · · · Γ(βq)

Γ(β1 + x∂) · · · Γ(βq + x∂)ex

=∑n≥0

(α1)n · · · (αp)n(β1)n · · · (βq)nn!

xn

=(α1)δ · · · (αp)δ(β1)δ · · · (βq)δ

ex (32)

n-dimensional hypergeometric functions

Let us consider the following generalization of the previous integral representation of the hy-pergeometric function (29)

F (α, β1, β2; γ;x1, x2) =Γ(γ)

Γ(β1)Γ(β2)Γ(γ − β1 − β2)·∫

d2ttβ1−11 tβ2−12 (1− t1 − t2)

γ−β1−β2−1(1− t1x1 − t2x2)−α

=Γ(γ)

Γ(β1)Γ(β2)Γ(β1 + δ1)Γ(β2 + δ2)

Γ(γ + δ1 + δ2)(1− t1 − t2)−α;

(1− t1 − t2)−α =1

Γ(α)

∫ ∞0

duuα−1e−u(1−t1−t2)

=∑

n1,n2≥0(α)n1+n2t

n11 t

n22

=(α)δ1+δ2

(α)δ1(α)δ2(1− t1)−α(1− t2)−α

= (α)δ1+δ2et1+t2 (33)

So

F (α, β1, β2; γ;x1, x2) =(α)δ1+δ2

(α)δ1(α)δ2

(γ)δ1(γ)δ2(γ)δ1+δ2

F (α, β1; γ, x1)F (α, β2; γ;x2)

=∑n1,n2

(α)n1+n2(β1)n1(β2)n2(γ)n1+n2n1!n2!

xn11 xn22 (34)

where

1 ≥ t1 + t2 ≥ 0, t1, t2 ≥ 0, δ ≡ x∂ (35)

It is obvious, n-dimensional generalization of these formulas.

Lauricella Hypergeometric functions (LFs)

For LFs (see, e.g. [Miller,1977]), we find the following formulas

FA(a; b1, ..., bn; c1, ..., cn; z1, ..., zn)

=(a)δ1+...+δn(b1)δ1 ...(bn)δn

(c1)δ1 ...(cn)δnez1+...+zn

=(a)δ1+...+δn

(a1)δ1 ...(an)δnF (a1, b1; c1; z1)...F (an, bn; cn; zn)

= T−1(a)Fn

= Σm≥0(a)m1+...+mn(b1)m1 ...(bn)mn

(c1)m1 ...(cn)mn

zm11m1!

...zmnnmn!

,

|z1|+ ...+ |zn| < 1;FB(a1, ..., an; b1, ..., bn; c; z1, ..., zn)

=(a1)δ1 ...(an)δn(b1)δ1 ...(bn)δn

(c)δ1+...+δnez1+...+zn

=(c1)δ1 ...(cn)δn(c)δ1+...+δn

F (a1, b1; c1; z1)...F (an, bn; cn; zn)

= T (c)Fn

= Σm≥0(a1)m1 ...(an)mn(b1)m1 ...(bn)mn

(c)m1+...+mn

zm11m1!

...zmnnmn!

,

|z1| < 1, ..., |zn| < 1;FC(a; b; c1, ..., cn; z1, ..., zn)

=(a)δ1+...+δn(b)δ1+...+δn

(c1)δ1 ...(cn)δnez1+...+zn

=(a)δ1+...+δn(b)δ1+...+δn

(a1)δ1 ...(an)δn(b1)δ1 ...(bn)δnF (a1, b1; c1; z1)

...F (an, bn; cn; zn)

= T−1(a)T−1(b)Fn = T−1(b)FA

= Σm≥0(a)m1+...+mn(b)m1+...+mn

(c1)m1 ...(cn)mn

zm11m1!

...zmnnmn!

,

|z1|1/2 + ...+ |zn|1/2 < 1;FD(a; b1, ..., bn; c; z1, ..., zn)

=(a)δ1+...+δn(b1)δ1 ...(bn)δn

(c)δ1+...δnez1+...+zn

=(a)δ1+...+δn(c1)δ1 ...(cn)δn(a1)δ1 ...(an)δn(c)δ1+...δn

F (a1, b1; c1; z1)

...F (an, bn; cn; zn)= T−1(a)T (c)Fn = T (c)FA = T−1(a)FB

= Σm≥0(a)m1+...+mn(b1)m1 ...(bn)mn

(c1)m1 ...(cn)mn

zm11m1!

...zmnnmn!

,

|z1| < 1, ..., |zn| < 1. (36)

It is interesting problem to find integral representations of these hypergeometric functions. Asa first step in this direction, note that previous example of two dimensional hypergeometricfunction was LFs - FD for n = 2, so for general n, we have

FD(a, b1, ...bn; c;x1, ..., xn) =

Γ(c)Γ(b1)...Γ(bn)Γ(c− b1 − ...− bn)

·∫dnttb1−11 ...t

bn−1n (1− t1 − ...− tn)c−b1−...−bn−1·

(1− t1x1 − ...− tnxn)−a (37)

Next step is to find operators that transform one LF to another and express them in terms ofintegral transformation of the FC. We have

FN = TNDFD, N = A,B,C,

TAD = T−1(c) =(c)δ1+...+δn

(c1)δ1 ...(cn)δn

TBD = T (a) =(a1)δ1 ...(an)δn

(a)δ1+...δnTCD = T−1(c)T−1(b) (38)

These operators reduce to the following one

(a)δf(x) =Γ(a+ δ)

Γ(a)f(x)

=1

Γ(a)

∫ ∞0

dtta−1e−tf(tx);

(a)−1δ =Γ(a)

Γ(δ + 1)x1−aD1−a;

(a)−1δ1+...+δn = (a+ δ1 + ...+ δn−1)−1δn...(a)−1δ1 (39)

Monomials

Ψ(x1, ..., xn) = xm11 ...xmnn , (40)

are eigenfunctions of the operator T (a)

T (a)Ψ(x1, ..., xn) =Γ(a)

Γ(a1)...Γ(an)Γ(a1 +m1)...Γ(an +mn)

Γ(a+m1 + ...+mn)Ψ (41)

For generalized Euler B-function we have

Bn(a1, ..., an) =Γ(a1)...Γ(an)

Γ(a1 + ...+ an)

=∫ 1

0dt1...dtnt

a1−11 ...t

an−1n δ(t1 + ...+ tn − 1), (42)

so, for analytic functions f(x1, ..., xn), when a = a1 + ...+ an,

T (a)f =1

Bn(a1, ..., an)

∫ 10dtnta1−11 ...t

an−1n

·δ(t1 + ...+ tn − 1)f(t1x1, ..., tnxn). (43)

Using this Bn− bien transformation, we obtain an integral formula for FB

FB(a1, ..., an; b1, ..., bn; c; z1, ..., zn) =Γ(a1 + a2 + ...+ an)Γ(a1)Γ(a2)...Γ(an)

·∫ 1

0dntta1−11 ...t

an−1n δ(t1 + t2 + ...+ tn − 1)

·FD(a1 + ...+ an; b1, ..., bn; c; t1x1, ..., tnxn)

=Γ(a1 + ...+ an)Γ(c)

Γ(a1)...Γ(an)Γ(b1)...Γ(bn)Γ(c− b1 − ...− bn)

·∫ 1

0dtnta1−11 ...t

an−1n δ(t1 + t2 + ...+ tn − 1)

·∫ 1

0dnyyb1−11 ...y

bn−1n (1− y1 − ...− yn)c−b1−...−bn

·(1− t1x1y1 − ...− tnxnyn)−(a1+...+an) (44)

Riemann ζ - function

Let us consider Riemann ζ - function,

ζ =∑n≥1

n−s (45)

Note, that

x∂xn = nxn, (x∂)−sxn = n−sxn, n ≥ 1, (46)

so

(x∂)−s∑n≥1

xn =∑n≥1

xn

ns≡ ζ(s, x) = (H)−s x

1− x,

H = x∂ (47)

than

H−s =1

Γ(s)

∫ ∞0

dtts−1e−tH , qHf(x) = f(qx), (48)

so

ζ(s) = ζ(s, 1) =1

Γ(s)

∫ ∞0

dtts−1e−tx

1− e−tx|x=1

=1

Γ(s)

∫ ∞0

dtts−11

et − 1. (49)

Now we consider following finite sum

ZN (α) ≡N∑n=1

eαn =N∑n=1

qn =q(1− qN )

1− q= q[N ]q, q = eα. (50)

Using the Liouville’s case of the fractal calculus, we obtain a compact representation of thefollowing Riemann ζ - function’s motivated finite sums

ζN (s) =N∑n=1

n−s =dβ

dαβZN (α)|α→0 = Z(s)N (0), β = −s (51)

Field theory applications of FC

Let us consider the following action

S =12

∫Qv

dxΦ(x)DαxΦ, v = 1, 2, 3, 5, ..., 29, ..., 137, ... (52)

Q1 is real number field,Qp, p - prime, are p-adic number fields. In the momentum representation

S =12

∫Qv

duΦ̃(−u)|u|αv Φ̃(u), (53)

Φ(x) =∫Qv

duχv(ux)Φ̃(u),

D−αχv(ux) = |u|−αv χv(ux). (54)

The statistical sum of the corresponding quantum theory is

Zv =∫dΦe

− 12

∫ΦDαΦ

= det−1/2Dα

= (∏u

|u|v)−α/2. (55)

String theory applications

For (symmetrized, 4-tachyon) Veneziano amplitude we have (see, e.g. [Kaku,2000])

Bs(α, β) = B(α, β) +B(β, γ) +B(γ, α)

=∫ ∞−∞

dx|1− x|α−1|x|β−1,

α+ β + γ = 1 (56)

For the p-adic Veneziano amplitude we take

Bp(α, β) =∫Qp

dx|1− x|α−1p |x|β−1p =Γp(α)Γp(β)Γp(α+ β)

(57)

Now we obtain the N-tachyon amplitude using fractal calculus. We consider the dynamics ofparticle given by multicomponent generalization of the action (52), Φ → xµ. For the closed

trajectory of the particle passing through N points, we have

A(x1, x2, ..., xN ) =∫dt

∫dt1...

∫dtNδ(t− Σtn)

v(x1, t1;x2, t2)v(x2, t2;x3, t3)...v(xN , tN ;x1, t1)

=∫dx(t)Π(

∫dtnδ(xµ(tn)− xµn))exp(−S[x(t)])

=∫

Π(dkµnχ(knxn))Ã(k)exp(−S), (58)

where

Ã(k) =∫dxV (k1)V (k2)...V (kN )exp(−S),

V (kn) =∫dtχ(−knx(t)) (59)

-vertex function.

Motion equation

Dαxµ − iΣkµnδ(t− tn) = 0, (60)

in the momentum representation

|u|αx̃µ(u)− iΣnkµnχ(−utn) = 0 (61)

have the solution

x̃µ(u) = iΣkµnχ(−utn)|u|α

, u 6= 0, (62)

the constraint

Σnkn = 0, (63)

and the zero mod x̃µn(0), which is arbitrary. Integration in (58) with respect to this zero modgives the constraint (63). On the solution of the equation (60)

xµ(t) = iD−αt Σnkµnδ(t− tn)

=i

Γ(α)Σnkµn|t− tn|α−1, (64)

the action (52) takes value

S = − 1Γ(α)

Σn

In the limit, α→ 1, for p-adic case we obtain

xµ(t) = −ip− 1p lnp

Σnkµnln|t− tn|,

S[x(t)] =p− 1p lnp

Σn

Fractal Calculus in Quantum Field theory Applications

Matrix calculus in QFT perturbation theory [Isaev, 2003], can be interpreted as operator Fractalcalculus. Indeed, with the following definitions

[x̂n, p̂m] = iδn,m, x̂n|x >= xn|x >, p̂m|p >= pm|p >, < x|y >= δD(x− y)

< x|p >= 1(2π)D/2

exp(ipx),∫dDp|p >< p| =

∫dDx|x >< x| = 1, (68)

we have

G(x, y) =< x|p̂−2α|y >= A(α)(x− y)−2β ,

β =D

2− α, A(α) = Γ(β)

22απD/2Γ(α), (69)

G(x, y) =1

Γ(α)

∫ ∞0

dttα−1 < x|e−tp̂2 |y >

=1

Γ(α)

∫ ∞0

dttα−1∫dDp < x|p >< p|y > exp(−tp2)

=Γ(D2 − α)

Γ(α)22απD/2(x− y)−2(D/2−α) (70)

In coordinate representation, p̂n = −i∂/∂xn, we have D-dimensional fractal calculus. As anexample, consider Coulomb potential, the solution of the equation for potential of point source

∆φ = gδD(x) (71)

Note, that, ∆ = −p̂2,

ϕ(x) = −g < 0| 1p̂2|x >= −g

Γ(D2 − 1)4πD/2

1|x|D−2

(72)

As another example, we take the following useful integral

G3(x, y) =∫dDz(x− z)−2α(z − y)−2γ

= A−1(β)A−1(δ)∫dDz < x|p̂−2β|z >< z|p̂−2δ|y >

=A(β + δ)A(β)A(δ)

(x− y)−2η,

β =D

2− α, δ = D

2− γ, η = D

2− (β + δ) = α+ γ − D

2(73)

Fractal Supersymmetry

In the simplest form of superspace, (t, θ) we have super path

xm(t, θ) = xm(t) + θψm(t), (74)

of a fermion particle and supersymmetry transformations,

δθ = �, δt = θ�, (75)

corresponding generator Q and covariant derivative D

�Q = δθ∂θ + δt∂t = �(∂θ − θ∂t),D = ∂θ + θ∂t, {D,Q} = 0{∂θ, θ} = 1, θ2 = ∂2θ = 0,D2 = ∂t, Q2 = −∂t, (76)

θ is Grassmann variable, � is Grassmann constant.

The action of the free fermion motion is

S =∫dtdθ∂txDx =

∫dt(ẋẋ−ΨΨ̇) (77)

We have also the following matrix (operator) representation

D = a+ a+∂t, Q = a− a+∂t

a =

0 01 0

, a+ = 0 1

0 0

, {a, a+} = 1. (78)By fractal calculus we can define fractal superdynamics with

D = a+ a+∂tα, Q = a− a+∂tα, (79)

D2f(t, θ) = ∂αt f, α ∈ R,

S =∫dtdθ∂t

αxDx =∫dt((∂tαx)2 + ∂tαΨΨ) (80)

Superdynamics

The Hamiltonian mechanics (HM) is in the ground of mathematical description of the phys-ical theories [Faddeev, Takhtajan, 1987]. But HM is in a sense blind, e.g., it does not makedifference between two opposites: the ergodic Hamiltonian systems (with just one integral ofmotion) and integrable Hamiltonian systems (with maximal number of the integrals of motion).

By our proposal Nambu’s mechanics (NM) [Nambu, 1973] is proper generalization of the HM,which makes difference between dynamical systems with different numbers of integrals of mo-tion explicit.

Hamiltonization of the general dynamical systems

Let us consider a general dynamical system described by the following system of the ordinarydifferential equations [Arnold, 1978]

ẋn = fn(x), 1 ≤ n ≤ N, (81)

and ẋn stands for the total derivative with respect to the parameter t.

When the number of the degrees of freedom is even, 1 ≤ n,m ≤ 2M , and

fn(x) = εnm∂H0∂xm

, 1 ≤ n,m ≤ 2M, (82)

the system (81) is Hamiltonian one and can be put in the form

ẋn = {xn,H0}0, (83)

where the Poisson bracket is defined as

{A,B}0 = εnm∂A

∂xn

∂B

∂xm= A

←∂

∂xnεnm

→∂

∂xmB, (84)

and summation rule under repeated indices has been used.

Let us consider the following Lagrangian

L = (ẋn − fn(x))ψn (85)

and the corresponding equations of motion

ẋn = fn(x),

ψ̇n = −∂fm∂xn

ψm. (86)

The system (86) extends the general system (81) by linear equation for the variables ψ. Theextended system can be put in the Hamiltonian form[Birkhoff, 1927, Makhaldiani, Voskresenskaya, 1997]

ẋn = {xn,H1}1,ψ̇n = {ψn,H1}1, (87)

where first level (order) Hamiltonian is

H1 = fn(x)ψn (88)

and (first level) bracket is defined as

{A,B}1 = A(←∂

∂xn

→∂

∂ψn−

←∂

∂ψn

→∂

∂xn)B. (89)

Note that when the Grassmann grading [Berezin, 1987] of the conjugated variables xn and ψn

{xn, ψm}1 = δnm (90)

are different, the bracket (89) is known as Buttin’s bracket[Buttin, 1969].

Geodesic motion of the point particles and integrals of motion

Geodesic motion of the particles maybe described by the following action functional

S =

2∫1

L(|ẋ|)ds, (91)

where

|ẋ|2 = gabẋaẋb (92)

and gab is metric tensor. The corresponding Euler-Lagrange equation

d

dt(∂L

∂ẋa)− ∂L

∂xa= 0 (93)

gives the extremal trajectories of the variation of the action (91)

δS =

2∫1

ds(∂L

∂xa− dds

(∂L

∂ẋa))δxa + (

∂L

∂ẋaδxa)21, (94)

with fixed ends, δxa(1) = δxa(2) = 0, and have the form

ẍa + Γabcẋbẋc = 0, (95)

where

ẋa =dxa

ds(96)

is the proper time derivative,

ds2 = gabdxadxb, (97)

gives the geodesic interval and

Γabc =12gad(gdb,c + gdc,b − gbc,d) (98)

is the Chistoffel’s symbols.

Usually considered forms of the Lagrangian are L = |ẋ| or 12 |ẋ|2. The first one gives the

reparametrization invariant action, the second one is easy for Hamiltonian formulation[DeWitt, 1965]. In the following we restrict ourselves by the last form of the Lagrangian

L =12gabẋ

aẋb. (99)

Corresponding Hamiltonian

H = paẋa − L (100)

is

H =12gabpapb, (101)

where the momentum is

pa =∂L

∂ẋa= gabẋb (102)

and gab is the inverse metric tensor,

gacgcb = δab . (103)

The Hamilton’s equations of motion are

ẋa = {xa,H}0 = gabpb,

ṗa = {pa,H}0 = −12∂gbc

∂xapbpc, (104)

where the Poisson bracket is

{A,B}0 =∂A

∂xa∂B

∂pa− ∂A∂pa

∂B

∂xa= A(

←∂ xa

→∂ pa −

←∂ pa

→∂ xa)B (105)

= A←∂ zn εnm

→∂ zm B, (106)

and with the unifying variables zn

zn = xn, zn+N = pn, 1 ≤ n ≤ N (107)

the Hamilton’s equations of motion (104) takes the form (81).

Integrals of motion H fulfil the following equations

d

dsH(x, ẋ) = (ẋa

∂

∂xa+ ẍa

∂

∂ẋa)H

= (ẋa∂

∂xa− Γabcẋbẋc

∂

∂ẋa)H

= ẋa∇aH = 0,

d

dsH(x, p) = (gabpb

∂

∂xa− 1

2∂gbc

∂xapbpc

∂

∂pa)H

= pb∇bH = 0, (108)

whered

ds= ẋa∇a = pb∇b,

∇a =∂

∂xa− Γbacẋc

∂

∂ẋb,

∇b = gba∇a = gba∂

∂xa− 1

2∂gbc

∂xapc

∂

∂pa. (109)

For the linear in ẋ integrals

H1 = Ka(x)ẋa = Kapa (110)

we have

Ḣ1 = ẋa∇aH1 =∂Kb∂xa

ẋaẋb −KcΓcabẋaẋb

= (Ka,b −KcΓcab)ẋaẋb

= Ka;bẋaẋb =12(Ka;b +Kb;a)ẋaẋb

= K(a;b)ẋaẋb = 0. (111)

So, from the expression (111), we see one-to-one correspondence between the expression of thefirst order integrals of the motion (110) and the nontrivial solutions of the following equationfor the so-called Killing vector Ka

K(a;b) = 0. (112)

For quadratic in ẋ integrals

H2 = Kab(x)ẋaẋb (113)

we have

Ḣ2 = (Kab,c −KdbΓdac −KadΓdbc)ẋaẋbẋc

= Kab;cẋaẋbẋc =13(Kab;c +Kbc;a +Kca;b)ẋaẋbẋc

= K(ab;c)ẋaẋbẋc = 0. (114)

So, we have one-to-one correspondence between the existence of the second order integrals ofmotion (113) and the nontrivial solutions of the following equation for the tensor Kab

K(ab;c) = 0. (115)

Now we prove the following:

Theorem 1. A necessary and sufficient condition that the following polynomials

Hn = Ka1a2...an(x)ẋa1 ẋa2 ...ẋan = Ka1a2...an(x)pa1pa2 ...pan (116)

are integrals of the geodesic motion, (95, 104) is that the symmetric tensors Ka1a2...an , fulfilthe equation

K(a1a2...an;a) = 0. (117)

In fact,

Ḣn = ẋa∇aHn = K(a1a2...an;a)ẋa1 ẋa2 ...ẋan ẋa = 0,

= pa∇aHn = K(a1a2...an;a)pa1pa2 ...panpa = 0, (118)

which proves the theorem, see [Sommers, 1973].

The symmetric tensors, which fulfil the equation (117), are known as Killing tensors.

Note that, as the metric tensor is covariantly constant, gab;c = 0, there is always the secondorder Killing tensor

Kab = gab (119)

and the corresponding integral of motion, Hamiltonian, H0,

2H0 = gabẋaẋb. (120)

Let us define an interesting algebra on Killing tensors.

Theorem 2. The following symmetrized product of the Killing tensors Kn and Km

K(a1a2...anKan+1an+2...an+m) = Ka1a2...an+m , (121)

is (reducible) Killing tensor.

In fact, let us multiply the corresponding integrals of motion

HnHm = K(a1a2...anKan+1an+2...an+m)pa1pa2 ...pan+m == K(a1a2...anKan+1an+2...an+m)ẋ

a1 ẋa2 ...ẋan+m = Hn+m, (122)

which, using the Theorem 1, proves this theorem.

We have the following bracket algebra of the integrals of motion

{Hn,Hm}0 = Hn+m−1. (123)

This algebra gives another method of the construction of the Killing tensors. As an example

let us calculate the bracket for the integrals H1 = Kapa and H2 = Kabpapb

{H1,H2}0 = Ka{pa,Kbc}pbpc +Kbc{Ka, pbpc}pa= (KabKc,a +KacKb,a−Kbc,aKa)pbpc= K̃abpapb. (124)

Let us consider another, tensor, generalization of the scalar integral of motion (110)

Ha1a2...am−1 = Aa1a2...am(x)ẋam ,

Ha1a2...am−1 = Aa1a2...am(x)pam , (125)

where the tensors Aa1a2...am(x) and Aa1a2...am(x) are skew-symmetric. We have the following:

Theorem 3a. A necessary and sufficient condition that the tensors (125) are (covariantly)constant (parallel) along any geodesic xa(s) is that the covariant derivative of the skew-symmetric tensor Aa1a2...am(x) is also skew-symmetric

Aa1a2...am;am+1 +Aa1a2...am+1;am = 0. (126)

aThis theorem is slight modification of the corresponding theorem from [Yano, 1952]

In fact, as xa(s) is geodesic, we have

Dẋa

Ds= ẍa + Γabcẋ

bẋc = 0,

DpaDs

= ṗa +12∂gbc

∂xapbpc = gab

Dẋb

Ds= 0 (127)

and

D

Ds(Aa1a2...am(x)ẋ

am) = Aa1a2...am;am+1 ẋam ẋam+1

=12(Aa1a2...am;am+1 +Aa1a2...am+1;am)ẋ

am ẋam+1 = 0,

D

Ds(Aa1a2...am(x)pam) = A

a1a2...am;am+1pampam+1

=12(Aa1a2...am;am+1 +Aa1a2...am+1;am)pampam+1 = 0, (128)

which proves the theorem. From the tensor integrals (125) we can construct the second rankKilling tensor.

Theorem 4. The following (symmetric) product of the tensors An and Bn gives a second rankKilling tensors

Aa1a2...an(aBa1a2...anb) = Kab. (129)

In fact, if we multiply the integrals

An = Aa1a2...an = Aa1a2...anapa,Bn = Ba1a2...an = Ba1a2...an

bpb, (130)

we obtain again integral

AnBn = Aa1a2...an(aBa1a2...anb)papb

= Kanbmpanpbm = H2, (131)

and using the Theorem 1, we prove the theorem 4.

So, if we have a nontrivial solution of the equations (126) , we can construct second integralof motion H2

H2 = Kabpapb, (132)

and with original Hamiltonian (101)

H1 = 2H = gabpapb, (133)

we will have bi-Hamiltonian system. So, we can apply the general method of the integra-tion of the bi-Hamiltonian systems [Magri, 1978, Okubo,Das,1988, Makhaldiani]. Also we can

construct the Nambu-Poisson formulation [Baleanu, Makhaldiani, 1998] of this system

ẋn = {xn,H1,H2}

= ωnmk(x)∂H1∂xm

∂H2∂xk

, (134)

where the Nambu-Poisson structure tensor ωnmk we identify [Makhaldiani] by comparision ofthe system (134) with the original system (104).

Note, that the skew-symmetric tensors, Aa1a2...an , which have the property, that its covariantderivative is also skew-symmetric, were considered by Bochner [Bochner, 1948] (see [Yano, 1952])and/but in literature are known as the Killing-Yano tensors [Gibbons, Rietdijk, van Holten, 1993].

Modified Bochner-Killing-Yano (MBKY) structures

Now we return to our extended system (86) and formulate conditions for the integrals of motionH(x, ψ)

H = H0(x) +H1 + ...+HN , (135)

where

Hn = Ak1k2...kn(x)ψk1ψk2 ...ψkN , 1 ≤ n ≤ N, (136)

we are assuming Grassmann valued ψn and the tensor Ak1k2...kn is skew-symmetric. For integrals(135) we have

Ḣ = {N∑n=0

Hn,H1} =N∑n=0

{Hn,H1} =N∑n=0

Ḣn = 0. (137)

Now we see, that each term in the sum (135) must be conserved separately.In particular for Hamiltonian systems (82), zeroth, H0 and first level H1, (88), Hamiltoniansare integrals of motion. For n = 0

Ḣ0 = H0,kfk = 0, (138)

which reduce to the condition (108), in the case of the geodesic motion of the particle (104)and defines corresponding modifications of the polynomial integrals of motion (116).

For 1 ≤ n ≤ N we have

Ḣn = Ȧk1k2...knψk1ψk2 ...ψkN +Ak1k2...knψ̇k1ψk2 ...ψkN + ...+Ak1k2...knψk1ψk2 ...ψ̇kN= (Ak1k2...kn,kfk −Akk2...knfk1,k − ...−Ak1...kn−1kfkn,k)ψk1ψk2 ...ψkN = 0, (139)

and there is one-to-one correspondence between the existence of the integrals (136) and theexistence of the nontrivial solutions of the following equations

D

DtAk1k2...kn = {Ȧk1k2...kn − fk1,kAkk2...kn − ...− fkn,kAk1...kn−1k}

= {Ak1k2...kn,kfk −Akk2...knfk1,k − ...−Ak1...kn−1kfkn,k} = 0, (140)

where under the bracket operation, {Bk1,...,kN } = {B} we understand complete anti-symmetrizationwith respect to the free indexes.

For n = 1 the system (140) gives

Ak1,kfk −Akfk1,k = 0 (141)

and this equation has at list one solution, Ak = fk. If we have two (or more) independent firstorder integrals

H(1)1 = A

1kΨk; H

(2)1 = A

2kΨk, ... (142)

we can construct corresponding reducible second (or higher)order MBKY tensor(s)

H2 = H(1)1 H

(2)1 = A

1kA

2lΨkΨl = AklΨkΨl;

HM = H(1)1 ...H

(M)M = Ak1...kM Ψk1 ...ΨkM ,

Ak1...kM = {A(1)k1...A

(M)kM

}, 2 ≤M ≤ N (143)

The system (140) defines a Generalization of the Bochner-Killing-Yano structures (117, 126),of the geodesic motion of the point particle, for the case of the general (81) (and extended(86)) dynamical systems. Having AM , 2 ≤ M ≤ N independent MBKY structures, we canconstruct corresponding second order Killing tensors and Nambu-Poisson dynamics. In thesuperintegrable case, we have maximal number of the motion integrals, N-1.

The structures defined by the system (140) we will call the Modified Bochner-Killing-Yanostructures or MBKY structures for short, [Makhaldiani,1999].

Extended geodesic motion of the point particles andGrassmann valued integrals of motion

Let us take the following Lagrangian

L = (ẋa − gabpb)φa − (ṗa +12∂gbc

∂xapbpc)ψa

= (ẋa − gabpb)φa + (ψ̇a −12∂gab

∂xcψcpb)pa +

d

ds(ψapa)

= L1 +d

ds(ψapa). (144)

New momentum variables are

∂L1∂ẋa

= φa,∂L1

∂ψ̇a= pa, (145)

(fundamental) brackets are

{xa, φb}1 = δab ,{ψa, pb}1 = δba,{A,B}1 = A(

←∂ xa

→∂ φa +

←∂ψa

→∂ pa −

←∂ φa

→∂ xa −

←∂ pa

→∂ψa)B

= A←∂Zn εnm

→∂Zm B. (146)

The Hamiltonian is

H1 = gabφapb +12∂gbc

∂xaψapbpc, (147)

the equations of motion are

ẋa = gabpb,

ṗa = −12∂gbc

∂xapbpc,

φ̇a = −∂gbc

∂xapbφc −

12∂2gbc

∂xa∂xepbpcψ

e,

ψ̇a =∂gab

∂xcpbψ

c + gabφb. (148)

Note that the extended system (148) and Hamiltonian (147) can be obtained from the system(104) and Hamiltonian (101) by the following simple shift of the variables

xa ⇒ xa + θψa,pa ⇒ pa + θφa, (149)

where θ- Grassmann parameter, θ2 = 0.

In fact,

H0 =12gabpapb ⇒

12gabpapb + θ(

12gab,c papbψ

c + gabφapb)

= H0 + θH1. (150)

The Lagrangian L1 (144) can be obtained by the shift (149) from the following first orderLagrangian

L = −12gabpapb + ẋapa, (151)

which is equivalent to the Lagrangian (99).

In fact, under the shift (149) we have

L = −12gabpapb + ẋapa

⇒ −12(gab + gab, cθψc)(pa + θφa)(pb + θφb) + (ẋa + θψ̇a)(pa + θφa)

= L0 + θL1. (152)

Let us define (extending the zeroth level bracket (105)) the Grassmann even bracket [Berezin, 1987]

{xa, pb}0 = δab ,{ψa, φb}0 = δab ,

{A,B}0 =∂A

∂xa∂B

∂pa+

∂A

∂ψa∂B

∂φa− ∂A∂pa

∂B

∂xa− ∂A∂φa

∂B

∂ψa

= A(←∂ xa

→∂ pa +

←∂ψa

→∂ φa −

←∂ pa

→∂ xa −

←∂ φa

→∂ψa)B

= A←∂Zn εnm

→∂Zm B. (153)

An interesting problem is to construct an even Hamiltonian.

General form H(x, ψ, p, φ) of the integrals of motion of the extended system (148) fulfils thefollowing equation

Ḣ = (ẋa∂

∂xa+ ψ̇a

∂

∂ψa+ ṗa

∂

∂pa+ φ̇a

∂

∂φa)

= (pb∇b + φb∇b1 + ψc∇2c)H = 0, (154)

where

∇b = gba ∂∂xa

− 12∂gbc

∂xapc

∂

∂pa,

∇b1 = gba∂

∂ψa− ∂g

bc

∂xapc

∂

∂φa,

∇2c =∂gab

∂xcpb

∂

∂ψa− 1

2∂2gbe

∂xc∂xapbpe

∂

∂φa. (155)

Field theory and condensed state physics applications

In various applications of the renormdynamic (RD) method [Bogoliubov,1959, Wilson,1974],for fractal space-time with dimension d = n− 2ε, the RD β−function is

β(α, ε) = β(α)− εα. (156)

For any given α (and corresponding scale a), there is the value of ε (fractal dimension), with

β(α, ε) = 0, ε = β(α)/α, d = n− 2β(α)/α. (157)

In the region of scales with linear β(α) function, we have self similar fractal space-time, orcorresponding physical fields live on the fractal subspace of the space-time manifold.

One loop perturbative value of β(α) in quantum electrodynamics [Weinberg,1995] is

β(α) = β1α2, β1 =13π, (158)

so

ε =13π

1137

= 77.5× 10−5, d = 3.998 < 4. (159)

For one loop perturbative QCD,

β1 = −94π, (160)

and if we take, e.g., α=0.1,

d = 4.001 > 4. (161)

Fractal dimension of the subspace occupied by a physical field ϕ(x) can be measured bycorrelation function

< ϕ(x)ϕ(y) >∼ |x− y|2−d. (162)

For macroscopic phenomena (e.g. Ball Lightning) we can consider mathematical models of frac-tals in 4 dimensional space-time manifolds. For hadronic phenomena (like Fair Ball) we shouldconsider more than 4 dimensional manifolds. One of the possible mechanisms of the sonolu-minescence (see e.g. [Barber,1997]) can be the dimensional phase transition with decreasingscale of the babbles from the phase with d < 4, to the phase with d > 4.

Standard Model of Fundamental Interections and Beyond

After the unification of electro-magnetic and weak interactions in Gleshow-Weinberg-Salammodel with gauge group U(1)×SU(2), [Weinberg,1996] a next step is a unification of electro-magnetic, weak and strong interactions (with gauge group SU(3)) in a Grand UnificationTheory (GUT ), with a simple group G with one coupling constant g, [Weinberg,1996]. Abridge between the SM scale 100 GeV and grand unification scale 1016GeV (Planck scale1019GeV ) can be provided by supersymmetry [Weinberg,2000]. At the scale of unification M,the U(1), SU(2) and SU(3) coupling constants are equal to g. In Supersymmetric Generaliza-tion of the Standard Model of Fundamental Interactions [Kazakov,2004],

α−1u = 26.3± 1.9± 1.0 (163)

Note that, in this interval, the only prime number is 29. Our proposal is that the (ultravioletasymptotic) value α−1uv is 29.0.., which corresponds to the (infrared asymptotic) value of theelectro-magnetic fine structure constant α−1 = 137.0...

Low energy unification

According to the resent (non-) perturbative calculations in QCD, strong coupling constant risefrom perturbative to the maximum value of order 2 and then decrease to the value of order1 at the scales reached in the lattice calculations. In QED with magnetic monopoles, we haveDirac quantization of the electric -e, and magnetic -g, charges, eg = 2π, so, at the self-dualpoint, e = g,

αe = αg = 0.5 (164)

We propose, that at the self-dual point, and corresponding energy scale, we have low energyunification of the strong and electro-magnetic coupling constants. It is very interesting to testthis possibility with nonperturbative calculations.

Besides that, nonperturbative β−functions define corresponding fractal dimensions accordingto equation (157).

Some observations on the coupling constants values

There is an indication, [Rañada,(1999)], [Ritus,(2003)], that ultra violet fixed value of the finestructure constant is

α−1uv = 4π = 12.57 ' 13 (165)

For this value we have

α−1LEU = 4π = 12.57 ' 13= 22 + 32 = | ± 2± 3i|2 = | ± 3± 2i|2 (166)

For the corresponding high energy unification value, we have

α−1GUT = 29= 22 + 32 + 42 = | ± 2i± 3j ± 4k|2 = ... (167)

where i, j, k are quaternion unit vectors.

For the electromagnetic fine structure coupling constant, we have

α−1 = 137.0... (168)

For the corresponding twin prime number, we have

139 = 22 + 32 + 42 + 52 + 62 + 72

= | ± 2i± 3j ± 4k ± 5l ± 6n± 7m|2 (169)

where i, j, k, l, n, m are octonion unit vectors,

137 = |4± i11|2 (170)

A field theory model for the coupling constants

We can consider simple O(N), N = 2; 3; 6 scalar field models with Lagrangian

L = (∂φ)2 − α(|φ|2 − α−1)2, (171)

which generates the coupling constant values by scale transformation of the fields, φ→ α−1/2φ,

L =1α

((∂φ)2 − (|φ|2 − 1)2) (172)

Indeed, in scalar gauge interaction terms,α−1A2φ2, we can compensate the α factor, by scaletransformation A →

√αA, which gives needed interaction constant between fundamental

fermions and gauge fields,√αΨ2A. If we define the scalar fields on the two, three, and six

dimensional lattice spaces, and take corresponding ground states, we will have the values ofthe coupling constants. So, we reconstructed clear geometric structures which lies under thevalues of the coupling constants.

p-adic deformations of classical theories

The electromagnetic fine structure constant

α =e2

~c(173)

contains (unifies) three fundamental quantities, electron charge e, Plank’s constant ~, and lightvelocity c. We usually take units as ~ = 1, c = 1, but we can take as well, e = 1, c = 1, ~ =137, 0... or e = 1, ~ = 1, c = 137.0... In this system of units, ~ = 137.0...(or c = 137.0...) andquantum perturbation theory may be p-adic convergent(, correspondingly, relativistic theory,considered as an expansion in powers of c = 137, is p-adic theory). In this mathematical sense,the quantum theory is p-adic (phase or part of the unified) theory. Real phase or part is usualclassical (relativistic) theory. The same consideration on the scale (level) of GUT , in units,g = 1, c = 1, gives a p-adic convergent quantum theory with ~ = 29. We can considerrelativistic and quantum theories as p-adic deformations of classical theory.

I would like to thank members of �NMπ for conversations.

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Fractal Calculus (H)and some Applications Introduction Real, p - adic and q - uantum fractal calculus Euler, ... Liouville, ... Holmgren, ... p - adic fractal calculus Fractal qalculusFractal finite - difference calculusHypergeometric functionsn-dimensional hypergeometric functionsLauricella Hypergeometric functions (LFs)Riemann - function

Field theory applications of FCString theory applicationsFractal Calculus in Quantum Field theory ApplicationsFractal Supersymmetry

Superdynamics Hamiltonization of the general dynamical systemsGeodesic motion of the point particles and integrals of motionModified Bochner-Killing-Yano (MBKY) structures Extended geodesic motion of the point particles and Grassmann valued integrals of motionField theory and condensed state physics applicationsStandard Model of Fundamental Interections and BeyondLow energy unificationSome observations on the coupling constants valuesA field theory model for the coupling constantsp-adic deformations of classical theories