Puzzles with tachyon in SSFT and Cosmological Applications
I. Aref'eva Steklov Mathematical Institute, Moscow
String Field Theory 2010
Kyoto, Japan 18-22 October, 2010
Outlook
• Historical remarks (about non-locality)• Non-locality in SFT or SFT inspired models• Schemes of calculations • Applications of SFT non-locality to cosmology
Time dependent solutions in cubic (super)string field theories
Key word ----- non-locality
Historical remarks. Non-locality in SFT
• Hikedi Yukawa (1949)
2
2
( ) ( , ) 0
( ) ( , ) 0
m U x r
r r U x r
...)()()]([
...)()()(),(
1
xAxx
rrxBrxAxrxU
Yukawa’s field as a reduced SFT field
Phys.Rev., 76(1949) 300
Historical remarks. H.Yukawa (1949)
Refs.:9) Non-local and non-linear field theories. D.I. Blokhintsev, (Dubna, JINR) Fortsch.Phys.6:246-269,1958, Usp.Fiz.Nauk 61:137-159,195710) Nonlocal quantum field theory. D.A. Kirzhnits, (Lebedev Inst.) Usp.Fiz.Nauk 90:129-142,1966.154) Relativistic Wave Equations with Inner Degrees of Freedom and Partons. V.L. Ginzburg, V.I. Manko, (Lebedev Inst.) .Sov.J.Part.Nucl.7:1,1976.
Historical remarks. Non-locality in QFT
A.Pais, G.Uhlenbeck, 1950
Field theories with non-localized action.
Motivation: eliminate divergences
Strings
One of the main goals: eliminate divergences
• Veneziano amplitudes?• Nambu-Goto string --- extended object
(we expected “strings” from QCQ,
QCD string -- Wilson criteria – lattice calculations)• Polyakov’s approach? • Light-cone Kaku-Kikkawa SFT?• Covariant midpoint SFT : cubic/nonpolynomial,
Witten, AMZ-PTY, Zwiebach, Berkovich
• Covariant light-cone-like Hata-Itoh-Kugo-Kunitomo-Ogawa SFT
Historical remarks. Non-locality in strings
Question: can we see non-locality in
Non-locality in String Field Theory
Time–dependent solutions:Rolling solutions vs wild oscillations Moeller and Zwiebach, hep-th/0207710; Fujita and Hata, hep-th/0304163
Non-flat metricLevel truncation method vs analytical solutions
• In fundamental setting locality and causality non-locality and noncommutative geometry (Witten)
• In practical setting (cosmological applications)
Hata and Oda, Causality in Covariant SFT, hep-th/9608128 Erler and Gross, Locality, Causality, and an Initial Value Formulation for Open SFT, hep-th/0406199
• Cosmological constant
why it is now so small• Dynamical DE
w < - 1
periodic crossing the w=-1 barrier
Non-local Cosmology from String Field Theory
Questions we want to address:
• Physics after Big Bang • Inflation; Non- Gausianity • Primordial Black Holes
Non-local Cosmology from String Field Theory
• String Field Theory is the UV-complete theory
• Non-locality is the key point of UV completion
Motivation
Nonlocal Models in Cosmology
• Nonlocality in Matter (mainly string motivated)
• Nonlocality in Gravity
Nonlocal Models in Cosmology
24
422
( )1 1( ( ))2 2
g
s
pFM
d x g V Rg M
I. Nonlocality in Matter (mainly string motivated)
• Later cosmology w<-1 • Inflation steep potential, non-gaussianity
• Bouncing
solutions
• I.A., astro-ph/0410443 I.A., L.Joukovkaya, JHEP,05109 (2005) 087 I.A., A.Koshelev, JHEP, 07022 (2007) 041 L.Joukovskaya, PR D76(2007) 105007; JHEP (2009 ) G. Calcagni, M.Montobbio,G.Nardelli,0705.3043; 0712.2237; Calcagni, Nardelli, 0904.4245;
• I.A., L.Joukovskaya, S.Vernov, JHEP 0707 (2007) 087
• Nunes, Mulryne, 0810.5471; N. Barnaby, T. Biswas, J.M. Cline, hep-th/0612230, J.Lidsey, hep-th/0703007; IA, N.Bulatov, L.Joukovskaya, S.Vernov, 0911.5105; PR(2009)
Nonlocal Models in Cosmology
II. Nonlocality in Gravity
24
2 24
1 1( ( ))2
( ,2
...)
p
s
Md x g V G R
g MFM
Arkani-Hamed at al hep-th/0209227; Khoury, hep-th/0612052T.Biswas, A.Mazumdar, W.Siegel hep-th/0508194 ,
G.Dvali, S. Hofmann, J Khoury, hep-th/0703027,
S.Deser, R.Woodard, arXiv:0706.2151S. UV - completion
Non-local Cosmology from String Field Theory
• Rolling tachyon in flat background• Rolling tachyon in curved background• Mathematical aspects
(rolling vs wide oscillations )• Non-local vs local models
Non-locality as a key point
Non-locality in level truncation
4 2 31 1[ ]
2 2 3open bosontachyon
gS d x
( ) exp[ ] ( )x r x
r is a number
2 20 i
24
( ) ( ) ( ) ( ) ( ) ( ) ( )
1[ ]
2 2 3ijki
open i i i ij i j i j kscalars
S d x M
2 2( ) ( ) ( )( ) 0r
i ik ik k ijk j kM e 2 0i for auxiliary fields
Non-locality in level truncation
Sen’s conjecture : f=1/4, I.A, D.Belov,A.Koshelev,P.Medvedev, Nucl.Phys.(2000), Ohmori (2001);
42 4
24
1 1[ ]
2 2 4NS GSOtachyon
d xS f
g
646
2 24
1
s s
co
v M MMg g
u u
Assumption:
How to understand .
2 2( ) ( ) ( )( ) 0r
i ik ik k ijk j kM e
2 2 2 2( ) 0m e c
Non-locality in level truncation in SFT
( )F J
?
?How to solve .
Non-locality in level truncation.
There are several options:
with the Fourier transform; the Laplace transform.
F is an analytical function in a neighbourhood of z=0
In particular,
( ) ?F
Non-locality in level truncation.
with the Fourier transform .
( )F
where
This definition is natural is SFT, where all expressions came from calculations in the momentum space
2t t
( )F
Non-locality in level truncation
where
Definition depends on “c”
as a symbol with the Laplace transformation
( ) ?F
2 0t t
Non-locality in level truncation
With the Laplace transformation with a closed contour
where
N.Barnaby, N.Kamran to study cosmological perturbations, arXiv:0712.2237;arXiv:0809.4513 (previous works by R.Woodard and coauthors)
Suitable for the Cauchy problem
0 t
2 2 32( 1) e 3( ') ( ') ' ( )K t t t dt t
I.A., Joukovskaya, Koshelev; Vladimirov, Ya.Volovich; Prokhorenko
Non-locality in level truncation. Rolling tachyon
Solution: kink2 02 1 Later oscillations
Boundary problem 1)(
30 0
2e 30 0 0K
0( ) 1 2Je
BUT NO Cauchy problem for
K0 is heat kernel
12( ) Je We cannot write (on our call of functions)
32Je We can write
14
f
Rolling tachyon in curved background
New conjecture:
( )t
t Effective cosmological constant
4 42 24
21 1 1( ( ) )2 24
g
s
pMg fFg
Rd xM
( ) ( ) zF z z e 2 1
I.A., astro-ph/0410443
4 42 24
21 1 1( ( ) )2 24
g
s
pMg fFg
Rd xM
Numbers. Hubble Parameter
62 22
26
p o c
pss
M g Mm
v MM
1/ pM
1/ sM
,s PM stri ng scal e M Pl anck scal e
2 pmH
pc MM 9
p
sp M
MMH
10sM
ps MM 6.610 pMH 6010
Local Mode Decomposition of Nonlocal Models
( ) gS d x F
F - entire function of order N
{ . . . }( )( ) (1 )
N
Nn n
z z
Nm P z
n n
zF z z e e
Weierstrass (Hadamard) product
2( ) ( ) . .g
n n g n ns
F c cM
Linearization
Non-local <--> Local (Linear Approximation)
The non-local model
42 24
21
( )2
)2
g
s
pMg RS d x Fg M
4 ' 22
,24
1( )( )
2 2M i i i ii
piS d
Mg Rx F M M
g
is equivalent to the local model
Here are the roots of the characteristic equation ( ) 0iF M iM'( ) 0iF M We assume
This equivalence is background independent
Ostrogradski (1859), Pais, Uhlenbeck (1950), Volovich (1969), Nakamura, Hamamoto (1996),…
IA, A.Koshelev, hep-th/0605085
Nonlocal Stringy Models – Linear Approximation
22 2( )F a ce
( )( ) WzWz e z
n-s branch of Lambert nW
Quadruples of complex roots
IA, L. Joukovskaya, S.Vernov, hep-th/0701184
214 2 ( 2 ) 0, 1, . . .
2a
n na W ce n
Nonlocal Stringy Models. Removing complex masses -----> zeta-function
1
1( ) , , 1
zn
z z in
1( ) ( )
2F z iz
1( )2
S i
/ 2( 1)( ) ( ) ( )
2 2zz z z
z z
The zeros of the Riemann zeta-function become the masses of particles
I.A., I.Volovich, hep-th/0701284;B.Dragovich, hep-th/0703008
zeta-function ksi-function
1½
tachyons
1( ) (1 ) z
p
z p
Summary
Non-local interacting theories can define globalcosmological background
Cosmological perturbations of nonlocal theoriesare the same as in local models
with w<-1without singularity at t=0
Specifying class of functions in the problem we erase singularities/instabilities
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