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

20

|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

R E F E R E N C E S

[1] I.V. Volovich, p-Adic space-time and string theory, Theor.Math. Phys. 71 (1987) 340; see also p-Adic string, Class.Quantum Grav. 4 (1987) L83 - L87.

[2] l. Brekke and P.G.O. Freund, p-Adic numbers in physics,Phys. Rep. 233 (1993) 1.

[3] V.S. Vladimirov, I.V. Volovich and E.I. Zelenov, p-Adicanalysis and mathematical physics, World Scientific, Singapore,1994.

[4] L. Brekke, P.G.O. Freund, M. Olson and E. Witten, Non-archimedean string dynamics, Nucl. Phys. B 302 (1988) 365.

[5] P.H. Frampton and Y. Okada, Effective scalar field theoryof p-adic string, Phys. Rev. D 37 (1988) 3077; The p-adicstring N-point function, Phys. Rev. Lett. 60 (1988) 484.

[6] D. Ghoshal and A. Sen, Tachyon condensation and branedescent relations in p-adic string theory, Nucl. Phys. B 584(2000) 300, hep-th/0003278.

[7] J.A. Minahan, Mode interactions of the tachyon conden-sate in p-adic string theory, JHEP 0103 (2001) 028, hep-th/0102071; A. Sen, Time evolution in open string theory,JHEP 0210 (2002) 003, hep-th/0207105; N. Moeller and B.Zwiebach, Dynamics with infinitely many time derivatives androlling tachyons, JHEP 0210 (2002) 034, hep-th/0207107;H. Yang, Stress tensors in p-adic string theory and trun-cated OSFT, JHEP 0211 (2002) 007; I.Ya. Aref’eva, L.V.Joukovskaya and A.S. Koshelev, Time evolution in super-string field theory on non BPS-brane. I. Rolling tachyon andenergy-momentum conservation, JHEP 0309 (2003) 012, hep-th/0301137.

25

[8] D. Ghoshal and T. Kawano, Towards p-adic string inconstant B-field, Nucl. Phys. B 710 (2005) 577, hep-th/0409311; P. Grange, Deformation of p-adic string ampli-tudes in a magnetic field, Phys. Lett. B 616 (2005) 135,hep-th/0409305.

[9] B. Dragovich and I.V. Volovich, p-Adic strings and non-commutativity, in ’Noncommutative Structures in Mathemat-ics and Physics’, eds. S. Duplij and J. Wess, Kluwer Acad.Publishers (2001); D. Ghoshal, Exact noncommutative solitonsin p-adic strings and BSFT, JHEP 0009 (2004) 041, hep-th/0406259.

[10] V.S. Vladimirov and Ya.I. Volovich, On the nonlinear dy-namical equation in the p-adic string theory, Theor. Math.Phys. 138 (2004) 297, math-ph/0306018.

[11] I.Ya. Aref’eva, Nonlocal string tachyon as a model forcosmological dark energy, AIP Conf. Proc. 826 (2006) 301,astro-ph/0410443; N. Barnaby, T. Biswas and J.M. Cline, p-Adic inflation, hep-th/0612230.

[12] D. Ghoshal, p-Adic string theories provide lattice dis-cretization to the ordinary string worldsheet, Phys. Rev. Lett.97 (2006) 151601.

[13] B. Dragovich, Adelic model of harmonic oscillator, Theor.Math. Phys. 101 (1994) 1404; Adelic harmonic oscillator, Int.J. Mod. Phys. A 10 (1995) 2349, hep-th/0404160.

[14] I.Ya Aref’eva and I.V. Volovich, Quantization of the Rie-mann zeta-function and cosmology, hep-th/0701284.

[15] A. Gerasimov and S. Shatashvilli, On exact tachyon po-tential in open string field theory, JHEP 10 (2000) 034, hep-th/0009103.

[16] N. Moeller and M. Schnabl, Tachyon condensation in

open-closed p-adic string theory, hep-th/0304213.

[17] B. Dragovich, Zeta-nonlocal Scalar Fields,

Theor. Math. Phys. 157 (3) (2008) 1671-1677;

arXiv:0804.4114v1 [hep-th].

[18] B. Dragovich, Towards effective lagrangians for

adelic strings, Fortschritte der Physik, to appear

(2009); arXiv:0902.0295v1 [hep-th].