Stringy Nonlocal Theories
Miami 2012
Tirtho Biswas
My Collaborators
•
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•
•
•
•
•
•
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N. Barnaby (University of Minnesota)
R. Brandenberger (McGill)
J. Cembranos (Madrid)
• TB, J. Cembranos and J. Kapusta,
J. Cline (McGill)
PRL 104, 021601 (2010)
M. Grisaru (McGill)
[arXiv:0910.2274 [hep-th]]
J. Kapusta (U of M)
• TB, E. Gerwick, T. Koivisto and A. Mazumdar,
T. Koivisto (Utrecht)
PRL 108, 031101 (2012)
A. Kosheylev (Brussels)
[arXiv:1110.5249 [gr-qc]]
A. Mazumdar (Lancaster)
A. Reddy (U of M)
W. Siegel (Stony Brook)
S. Vernov (Moscow)
Outline
Nonlocal Scalar Field Theory



Stringy Motivations
Ghostfree higher derivative theories
Finite Loops & some results
Nonlocal Gravity



The problem of Ghosts
Nonsingular Black Holes?
Nonsingular Cosmology?
Nonlocal Actions in String Theory
String Field Theory Tachyons
1

 2
1
D
2
2
S  2  d x   (  m )e M   V ( )
go
 2

2
Open string coupling
[Witten, Kostelecky & Samuel, Sen]
string tension
Mass square has the wrong sign
p-adic string theory [Volovich, Brekke, Freund, Olson, Witten, Frampton]
2




msD
1

1
D
p 1
S  2  d x   exp   2   
 
gp
p 1
 M 
 2


An inifinte series of higher derivative kinetic operators, mildly nonlocal
Interesting Properties
Ghostfree S ~ d D x  1  ( 2  m 2 ) 2   ( 2  m 2 ) 2  0
 

2
( p 2 ) ~


1
1
1
~

p 2 ( p 2  m2 ) p 2 p 2  m2
But SFT/padic type theories have no extra states!
Quantum loops are finite

UV under better control, like usual HD theories

Linear Regge Trajectories [TB, Grisaru & Siegel]
exp(  p 2 M 2 )
( p ) 
p 2  m2
2
 ms2 

Z1 (T )  Z1 
T 

Thermal duality

Can there be any phenomenological implications for LHC?
al]
[TB, Cembranos & Kapusta, 2010 PRL]
[Moffat et
Applications
Insights into string theory



Brane Physics & Tachyon condensation
[Zwiebach & Moeller; Forini,
Gambini & Nardelli; Colleti, Sigalov & Taylor; Calcagni…]
Hagedorn physics [Blum; TB, Cembranos & Kapusta]
Spectrum [TB, Grisaru & Siegel, Minahan]
Applications to Cosmology



Novel kinetic energy dominated non-slow-roll inflationary
mechanisms [TB, Barnaby & Cline; Lidsey…]
Large nongaussianities [Barnaby & Cline]
Dark Energy [Arefeva, Joukovskaya, Dragovich, ...]
Applications to Particle Physics
[Moffat et.al.]
Nonlocal Gravity



Can Nonlocal higher derivative terms be
free from ghosts?
Can they address the singularity problems
in GR ?
What about quantum loops?

Stelle demonstrated 4th order gravity to be renormalizable (1977),
but it has ghosts
Ghosts
From Scalars to Gravity

The metric has 6 degrees (graviton, vector, and two
scalars)

Gauge symmetry is subtle, some ghosts are allowed

Several Classical (time dependent) backgrounds.
Linearized Gravity
Free from ghosts in Minkowski vacuum

Only interested in quadratic action
PRL]
[with Mazumdar, Koivisto, Gerwick, 2012
g      h  R ~ O(h)

 ' ' ' '
 S   d 4 x  g R   R  Oˆ 
R ' ' ' '


Only 6 linearly independent combinations using BI


S   d 4 x  g R  RF1 ( 2 ) R  R  F2 ( 2 ) R  R  F3 ( 2 ) R  ...

Covariant derivatives must be Minkowski, most general form
1
S    d 4 x  h  a( 2 ) 2 h  h b( 2 )  h   hc( 2 )   h 
2
2
1
f
(

)
2
2

 
 hd ( ) h  h
      h 
2
2



Covariant to Minkowski

We noticed rather curious relations
a( x)  1 
1
xF2 ( x)  2 xF3 ( x)
2
ab  0
cd 0
ca f 0



They in fact follow from Bianchi identity!
By inverting Field equations we obtain the propagators
Decouple the different multiplets using projection
operators: P 2 , P1 , Ps0 , Pw0 [van Nieuwenhuizen]
Precisely because of the above relations, the dangerous
w-scalar ghost and the Vector ghost vanishes 

General Covariance dictates the propagator is of the
form
Ps0
P2
P 2 1 Ps0
GR
( p ) 

 2 
2
2
2
2
2
a( p ) p [a( p )  3c( p )] p
p 2 p2
2





At low energies, p  0, we automatically recover GR
In GR a = c = 1, scalar ghost cancels the longitudinal mode
a has to be an entire function, otherwise Weyl ghosts
a-3c can have a single zero -> f(R)/Brans-Dicke theory
Exponential non-local Gravity,

a  c  exp  
2

M
2
0
2
2




P
P
1
p
2
s
 exp   2 
( p )   2 
2 
2 p 
 M 
p
Newtonian Potentials
ds  (1  2)dt  (1  2 )dx
2
2
(r )   (r ) ~  d p
3


e
2
and
 p 2 / M 2 ip.r
e
p2
 00

 m (r )
erf ( Mr )
~
r
Large r, reproduces gravity; small r, asymptotic freedom
No small mass black holes, no horizon and no
singularity!
M2
m
p
M
Gravity Waves

Similar arguments imply nonsingular Green’s functions
for quadrupole moments
Exact Solutions
Bouncing Solutions
 deSitter completions, a(t) ~ cosh(Mt)

S   d 4 x  g R  RF1 ( 2 ) R


Stable attractors, but there are singular attractors.

Can provide a geodesically complete models of inflation.

Perturbations can be studied numerically and
analytically, reproduces GR at late times… can provide
geodesic completion to inflation
Conclusions



Nonlocal gravity is a promising direction in QG
It can probably solve the classical singularities
How to constrain higher curvatures?



New symmetries
Look at ghost constraints on (A)dS – relevant for DE
Can we implement Stelle’s methods?
Emergent Cosmology

Space-time begins with pure vacuum
g  t
 
 
a(t )  1  et
and
 t
 0
 

a(t ) t
1
a(2 )  3c(2 )  0


You cannot find a consistent solution for GR
There must be a scalar degree of freedom
c  a(1  p 2 / m2 )
with a  exp( p 2 )
and
~m

t’ Hooft dual to string theory

Polyakov action:

Strings on Random lattice

Dual Field theory action
[Douglas,Shenker]
Motivation




Standard Models of Particle Physics & Cosmology
have been remarkably successful
Too successful, no experimental puzzles
Hints at new meV physics (Dark energy &
Neutrinos)
Fall back on theoretical prejudices


Hierarchy problem, Unification - GUT, SUSY, String
Theory
Nonsingularity – can we use this to guide us?