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NELOKALNA TEORIJA POLJA
(NONLOCAL FIELD THEORIES)
Branko Dragovich
http://www.phy.bg.ac.yu/˜ dragovich
Institute of Physics, Belgrade
Seminar: Geometrija, vizuelizacija i obrazovanje saprimenama
Beograd, 19. i 26. februara 2009(Faculty of Mathematics)
1
1. Introduction
2. p-Adic and Adelic Strings
3. Lagrangian for p-Adic Strings
4. Lagrangians with Riemann Zeta function
5. Concluding Remarks
2
1. Introduction
polje = ?
ϕ(x) = ϕ(−→x , t)
fizicka polja: elektromagnetno polje, gravitaciono
polje, Klajn-Gordonovo polje, Dirakovo polje, ...
(lokalna) teorija polja = ?
L(x) = L(ϕi(x),∂ϕi(x)
∂xµ) S =
∫L(x)d4x
∂L∂ϕi
− ∂
∂xµ
∂L∂∂µϕi
= 0 ∂µ ≡ ∂
∂xµ
(nelokalna) teorija polja = ?
L(x) = L(ϕi, ∂µϕi, ..., ∂nµϕi, ...)
L(x) = L(ϕ, eα¤ϕ) ¤ = ∂µ∂µ
3
lagranzijan i jedn. kretanja za p-adicnu strunu:
L =mD
g2
p2
p− 1[− 1
2ϕp
− ¤2m2 ϕ +
1
p + 1ϕp+1]
p− ¤
2m2 ϕ = ϕp
lagranzijan i jedn. kretanja sa
Rimanovom ζ-funkcijom:
L = −mD
g2 [1
2φ ζ(
¤2m2)φ + φ +
1
2ln(1− φ)2]
ζ(¤
2m2)φ =1
(2π)D
∫
RDeixk ζ(− k2
2m2) φ(k)dk =φ
1− φ
4
p-Adic numbers:
• discovered by K. Hansel in 1897.
• many applications in mathematics, e.g. representa-tion theory, algebraic geometry and modern numbertheory.
• many applications in mathematical physics since1987, e.g. string theory, QFT, quantum mechanics,dynamical systems, ...
• Any p-adic number (x ∈ Qp ) has a unique canonicalrepresentation
x = pν(x)+∞∑n=0
xn pn , ν(x) ∈ Z , xn ∈ {0, 1, · · · , p− 1}.
• Real and p-adic numbers unify by adeles. An adele αis an infinite sequence
α = (α∞, α2, α3, · · · , αp , · · ·) , α∞ ∈ R , αp ∈ Qp
where for all but a finite set P of primes p one hasthat αp ∈ Zp = {x ∈ Qp : |x|p ≤ 1}.
5
2. p-Adic and Adelic Strings
Volovich, Freund, Witten, Vladimirov, Dragovich, ...
String amplitudes:
• standard crossing symmetric Veneziano amplitude
B∞(a,b) =
∫
R|x|a−1
∞ |1−x|b−1∞ d∞x =
ζ(1− a)
ζ(a)
ζ(1− b)
ζ(b)
ζ(1− c)
ζ(c)
• p-adic crossing symmetric Veneziano amplitude
Bp(a,b) =
∫
Qp
|x|a−1p |1−x|b−1
p dpx =1− pa−1
1− p−a
1− pb−1
1− p−b
1− pc−1
1− p−c
where a, b, c ∈ C with condition a + b + c = 1 and ζ(a)is the Riemann zeta function.
• product formula for adelic strings
B(a,b) = B∞(a,b)∏p
Bp(a,b) = 1
6
3. Effective Lagrangian for p-Adic Strings
• One of the greatest achievements in p-adic string the-ory is an effective field description of scalar open andclosed p-adic strings. The corresponding Lagrangiansare very simple and exact. They describe not onlyfour-point scattering amplitudes but also all higher(Koba-Nielsen) ones at the tree-level.
• The exact tree-level Lagrangian for effective scalarfield ϕ which describes open p-adic string tachyon is
Lp =mD
p
g2p
p2
p− 1[− 1
2ϕp
− ¤2m2
p ϕ +1
p + 1ϕp+1] ,
where p is any prime number, ¤ = −∂2t +∇2 is the D-
dimensional d’Alembertian and metric with signature(− + ...+) (Freund, Witten, Frampton, Okada, ...) .
• An infinite number of spacetime derivatives followsfrom
p− ¤
2mp = exp (− ln p
2mp¤) =
∑
k≥0
(− ln p
2mp)k 1
k!¤k .
7
• The equation of motion is
p− ¤
2m2p ϕ = ϕp ,
and its properties have been studied by many au-thors (see e.g. Vladimirov, Barnaby and referencestherein). It has trivial solutions ϕ = 0 and ϕ = 1.There are also inhomogeneous solutions resemblingsolitons. Equation separates in arguments and forany spatial direction xi one has
ϕ(xi) = p1
2(p−1) exp (− p− 1
2m2p p ln p
(xi)2) .
• Prime number p can be replaced by natural numbern ≥ 2 and such expression also makes sense. More-over, when p = 1 + ε → 1 there is the limit which isrelated to the ordinary bosonic string in the boundarystring field theory (Gerasimov-Shatashvili):
L =mD
g2[1
2ϕ
¤m2
ϕ +ϕ2
2(lnϕ2 − 1)] .
8
• This p-adic string theory has been significantly pushedforward when was shown (Ghoshal-Sen) that it de-scribes tachyon condensation and brane descent re-lations simpler than by ordinary bosonic strings.
• After that success, many aspects of p-adic string dy-namics have been investigated and compared withdynamics of ordinary strings (see, e.g. Minahan,Zwiebach, ...).
• Noncommutative deformation by the Moyal starproduct and of p-adic string world-sheet with a con-stant B-field was investigated (Ghoshal, ... ).
• A systematic mathematical study of spatially homo-geneous solutions of the nonlinear equation of motion(Vladimirov).
• Some possible cosmological implications of p-adicstring theory have been also investigated (Arefeva,Barnaby, Joukovskaya, ...).
• It was recently proposed (Ghoshal) that p-adic stringtheories provide lattice discretization to the world-sheet of ordinary strings.
• As a result of these developments it follows thatmany nontrivial features of ordinary strings are similarto p-adic ones and are related to the p-adic effectiveaction.
9
4. Lagrangians with Riemann Zeta Function
A. Additive approach
B.D.: hep-th/0703008v1, 0804.4114v1[hep-th],
0805.0403v1[hep-th], 0809.1601[hep-th]
• Now we want to introduce a model which incorpo-rates all the above n-adic string Lagrangians. Wetake the sum of the Lagrangians Ln in the form
L =∑
n≥1
CnLn
Ln =∑
n≥1
mDn
g2n
n2
n− 1[− 1
2φ
∑
n≥1
n− ¤
2m2n φ +
∑
n≥1
1
n + 1φn+1]
which depends on the choice of coefficients Cn,masses mn and coupling constants g2
n.
• There is a few simple and interesting cases (mn =m, g2
n = g2):
Cn =n− 1
n2+h
Cn =n2 − 1
n2
Cn = µ(n)n− 1
n2
where µ(n) is the Mobius function.
10
• Let us consider the first case Cn = n−1n2+h :
Lh =1
g2[− 1
2φ
∑
n≥1
n−¤
2m2−h φ +∑
n≥1
n−h
n + 1φn+1] .
• According to the famous Euler product formula onecan write ∑
n≥1
n−¤
2m2 =∏p
1
1− p−¤
2m2
.
• Recall that the Riemann zeta function is defined as
ζ(s) =∑
n≥1
1
ns=
∏p
1
1− p−s, s = σ + iτ , σ > 1 .
• We can rewrite Lagrangian in the form
Lh = − 1
g2[1
2φ ζ(
¤2m2
+ h)φ +AC+∞∑n=1
n−h
n + 1φn+1] .
We shall consider this Lagrangian with analytic con-tinuation of zeta functions and power series.
11
• ζ( ¤2m2) acts as a pseudodifferential operator in the
following way:
ζ(¤
2m2)φ(x) =
1
(2π)D
∫
RD
eixk ζ(− k2
2m2) φ(k)dk ,
where φ(k) =∫
e(−ikx) φ(x) dx is the Fourier transformof φ(x).
• Nonlocal dynamics of this field φ is encoded in thepseudo-differential form of the Riemann zeta function(the d’Alembertian is an argument of the Riemannzeta function).
• Potential of the above scalar field is equal to −Lh at¤ = 0, i.e.
Vh(φ) =1
g2(φ2
2ζ(h)−AC
+∞∑n=1
n−h
n + 1φn+1)
where h 6= 1 since ζ(1) = ∞. The term with ζ-function vanishes at h = −2,−4,−6, · · ·.
• The equation of motion in differential and integralform is
ζ(¤
2m2+ h)φ = AC
+∞∑n=1
n−h φn ,
1
(2π)D
∫
RD
eixk ζ(− k2
2m2+ h) φ(k) dk = AC
+∞∑n=1
n−h φn ,
respectively. It is clear that φ = 0 is a trivial solutionfor any real h. When h > 1 we have another constanttrivial solution φ = 1.
12
• In the weak field approximation (|φ(x)| ¿ 1) the aboveexpression becomes
∫
RD
eikx [ζ(− k2
2m2+ h)− 1] φ(k) dk = 0 ,
which has a solution φ(k) 6= 0 if equation
ζ(−k2
2m2+ h) = 1
is satisfied. According to the usual relativistic kine-
matic relation k2 = −k20 +
−→k
2= −M2, equation in the
form
ζ(M2
2m2+ h) = 1 ,
determines mass spectrum M2 = µh m2, where setof values of spectral function µh depends on h. Theabove equation gives infinitely many tachyon masssolutions.
13
EXAMPLE (h = 0)
• The related Lagrangian is
L0 = −mD
g2[1
2φ ζ(
¤2m2
)φ + φ +1
2ln(1− φ)2] .
• The corresponding potential is
V0(φ) =mD
g2[ζ(0)
2φ2 + φ +
1
2ln(1− φ)2 ] ,
where ζ(0) = −12. It has two local maxima: V0(0) = 0
and V0(3) ≈ 1.443mD
g2 . There are no stable points and
limφ→1 V0(φ) = −∞ , limφ→±∞ V0(φ) = −∞ .
• The equation of motion for φ is
ζ(¤
2m2)φ =
1
(2π)D
∫
RD
eixk ζ(− k2
2m2) φ(k)dk =
φ
1− φ.
It has two trivial solutions: φ = 0 and φ = 3. Thesolution φ = 0 is evident. The solution φ = 3 fol-lows from the Taylor expansion of the Riemann zetafunction operator
ζ(¤
2m2) = ζ(0) +
∑
n≥1
ζ(n)(0)
n!(
¤2m2
)n , ζ(0) = −1
2.
So far nontrivial solutions are unknown.
14
• In the weak field approximation
ζ(¤
2m2)φ =
1
(2π)D
∫
RD
eixk ζ(− k2
2m2) φ(k)dk = φ
on mass shell one obtains equation for the massspectrum
ζ(m2
2) = 1
which has infinitely many tachyon solutions.
15
Case Cn = n2−1n2
• In this case Lagrangian becomes
L =mD
g2[− 1
2φ
+∞∑n=1
(n−¤
2m2+1 + n−¤
2m2)φ ++∞∑n=1
φn+1]
and it yields
L =mD
g2[ − 1
2φ {ζ( ¤
2m2− 1) + ζ(
¤2m2
)}φ +φ2
1− φ] .
• The corresponding potential is
V(φ) = −mD
g2
31− 7φ
24 (1− φ)φ2 ,
which has the following properties: V (0) = V (31/7) =0 , V (1 ± 0) = ±∞ , V (±∞) = −∞. At φ = 0 potentialhas local maximum.
• The equation of motion is
[ζ(¤
2m2− 1) + ζ(
¤2m2
)]φ =φ((φ− 1)2 + 1)
(φ− 1)2,
which has only φ = 0 as a constant real solution.
16
Case Cn = µ(n)n−1n2
µ(n) =
0, n = p2m
(−1)k, n = p1p2 · · · pk, pi 6= pj1, n = 1, (k = 0) .
(1)
• In this case Lagrangian becomes
Lµ =mD
g2[− 1
2φ
+∞∑n=1
µ(n)
n¤
2m2
φ ++∞∑n=1
µ(n)
n + 1φn+1]
and it yields
Lµ =mD
g2[− 1
2φ
1
ζ( ¤2m2)
φ +
∫ φ
0
M(φ)dφ] ,
where M(φ) =∑+∞
n=1 µ(n)φn = φ−φ2−φ3−φ5+φ6−φ7+φ10 − φ11 − . . ..
• The corresponding potential is
Vµ(φ) = −Lµ(¤ = 0) = −mD
g2[φ2 +
∫ φ
0
M(φ)dφ] .
• The equation of motion is
1
ζ( ¤2m2)
φ−M(φ) = 0
which has φ = 0 as a constant real solution.
17
• Its weak field approximation implies condition on themass spectrum
1
ζ( M2
2m2)− 1 = 0.
18
B. Multiplicative approach
B.D.: arXiv:0902.0295
Let us now consider a new approach, which is not
based on a summation of p-adic Lagrangians, but
the Riemann zeta function will emerge through its
product form. Our starting point is again p-adic
Lagrangian with equal masses, i.e. m2p = m2 for
every p. It is useful to rewrite Lagrangian, first in
the form,
Lp =mD
g2p
p2
p2 − 1{− 1
2ϕ [p
− ¤2m2+1
+p− ¤
2m2 ]ϕ+ ϕp+1}
and then, by addition and substraction of ϕ2, as
Lp =mD
g2p
p2
p2 − 1{12
ϕ [(1− p− ¤
2m2+1)
+(1− p− ¤
2m2)]ϕ− ϕ2(1− ϕp−1)} .
19
Now we introduce a Lagrangian for the entire p-adic
sector by taking products
∏p
g2p = C ,
∏p
1
1− p−2,
∏p(1− p
− ¤2m2+1
) ,
∏p(1− p
− ¤2m2)
∏p(1− ϕp−1)
at the corresponding places. Then this new La-
grangian becomes
L =mD
Cζ(2) {1
2φ[ζ−1(
¤2m2
− 1)
+ζ−1(¤
2m2)]φ− φ2∏p
(1− φp−1)} ,
where ζ−1(s) = 1/ζ(s) and new scalar field is de-
noted by φ. For the coupling constant gp there are
two interesting possibilities: (1) g2p = p2
p2−1, what
yields ζ(2)/C = 1 , and (2) gp = |r|p, where r
may be any non zero rational number and it gives
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|r|∞∏p |r|p = 1 (this possibility was already consid-
ered by Dragovich). Both these possibilities are con-
sistent with adelic product formula. For simplicity, in
the sequel we shall take C = ζ(2). It is worth noting
that having Lagrangian one can easily reproduce its
p-adic ingredient.
Let us rewrite the above Lagrangian in the simple
form
L =1
2φ [ζ−1(
¤2− 1) + ζ−1(
¤2)]φ− φ2 Φ(φ) ,
with m = 1 and Φ(φ) = AC∏p(1 − φp−1), where
AC denotes analytic continuation of infinite product∏p(1 − φp−1), which is convergent if |φ|∞ < 1. One
can easily see that Φ(0) = 1 and Φ(1) = Φ(−1) = 0.
The corresponding equation of motion is
[ζ−1(¤2− 1) + ζ−1(
¤2)]φ = 2φΦ(φ) + φ2 Φ′(φ) ,
and has φ = 0 as a trivial solution. In the weak-field
approximation (φ(x) ¿ 1), equation becomes
[ζ−1(¤2− 1) + ζ−1(
¤2)]φ = 2φ .
Note that the above operator-valued zeta function
can be regarded as a pseudodifferential operator.
21
Mass spectrum of M2 is determined by solutions of
equation
ζ−1(M2
2− 1) + ζ−1(
M2
2) = 2 .
There are infinitely many tachyon solutions, which
are below largest one M2 ≈ −3.5.
The potential follows from −L at ¤ = 0, i.e.
V(φ) = [7 + Φ(φ)]φ2 ,
since ζ(−1) = −1/12 and ζ(0) = −1/2. This
potential has local minimum V (0) = 0 and values
V (±1) = 7. To explore behavior of V (φ) for all
φ ∈ R one has first to investigate properties of the
function Φ(φ).
22
It is worth noting that a Lagrangian similar to the
above one can be obtained by an additive approach.
Namely, let us start from g2p = p2/(p2−1) and m = 1,
we have
L = −+∞∑
n=1
µ(n)Ln =1
2φ [
+∞∑
n=1
µ(n)p−¤2+1
++∞∑
n=1
µ(n)p−¤2 ]φ−
+∞∑
n=1
µ(n)φn+1 ,
where µ(n) is the above Mobius function. Intro-
ducing zeta function one can rewrite it in the form
L =1
2φ [ζ−1(
¤2− 1)
+ζ−1(¤2)]φ− φ2 F(φ) ,
where F (φ) =∑+∞
n=1 µ(n)φn−1. The difference be-
tween two Lagrangians is only in functions Φ(φ) and
F (φ). Since Φ(φ) = (1 − φ)(1 − φ2)(1 − φ4)... =
1−φ−φ2+φ3−φ4+... and F (φ) = 1−φ−φ2−φ4+..., it
follows that these functions have the same behavior
for |φ| ¿ 1. Hence, Lagrangians have the same mass
spectrum and in weak-field approximation describe
the same scalar field theory.
23
5. Concluding Remarks
• p-Adic strings are in many ways related to ordinary strings.It seems that ordinary and p-adic strings are different facesof an adelic string.
• p-Adic scalar tachyons at the tree level are described bysimple and exact nonlocal Lagrangian.
• All here constructed Lagrangians contain Riemann zetafunction nonlocality.
• These Lagrangians with ζ-function nonlocality are new andsignificant in their own right.
24
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