p-adic Strings: Thermal Duality & the Cosmological Constant

Tirthabir Biswas

Loyola University, New Orleans

PRL 104, 021601 (2010) JHEP 1010:048, (2010)PRD 82:085028, (2010) with J. Kapusta and J .A. R. Cembranos.

1

2

2

2 1

1exp

2

1 pD

p

Ds

pMxd

g

mS

'2

1

ln

2

1

11 22

22

22

s

s

op

mp

mM

p

p

gg

The N-point tree amplitudes of the open string can begenerated from a non-local Lagrangian of a single scalarfield.

Volovich, Brekke, Freund, Olson, Witten, Frampton, Okada, late 80’s

open string coupling prime number string tension

p-adic Action

Generalizations Nonlocal Infinite derivative Actions

p-adic theories Strings on Random lattice (Douglas&Shenker,Gross&Migdal, 1990)

Regge trajectories (TB, Siegel, Grisaru 2004)

String Field Theories

Related cousins appear in Noncommutative Field theory Theory of unparticles

)( VFxdS D O

)exp()()( 2 OOO mF

What can we gain? Insights into string theory Hagedorn physics Brane Physics

Applications to Cosmology Novel kinetic energy dominated non-slow-roll inflationary

mechanisms (TB, Barnaby, Cline, 2006), large nongaussianities (Barnaby,Cline, 2007)

Dark Energy (Arefeva et.al.)

Thermal Cosmology in the Early Universe

Applications to Particle Physics [Moffat et.al.]

Interesting Properties (?) Usually higher derivative theories are plagued by ghosts:

But padic type theories have no extra states! Initial value problem may be well defined: Studied by Russian mathematicians, 2 degrees of freedom for every

pole, rigorously established for free theories. [Barnaby, Kamran] Diffusion equation formulation in one higher dimension suggest

finite number of IC’s. [Calcagni, Nardelli...]

No perturbative states, free theory is trivial, quantum contributions only arise when interactions are present.

Field Equations can be recast as integral equations and hence numerical progress/tests can be made.

• Rescale the fields to put the action in the form

1

2

2223 /

exp2

1 p

M

dxddS

1

21

1

p

s

p

m

g

p

with dimensionful coupling constant

and non-local propagator nT

MkkD

n

nn

2

/exp),( 2220

Thermal Field Theory

2

03

3

2 ),()2(

)3(ln

nn kD

kdTVZ

M

TN

N

MkD

kd

nn

N

2

2),(

)2(

3

03

3

p=3

xxex

n

xn 22

)(

2-loop & Thermal Duality

Compute Feynman diagrams

Partition Function

Analogous to t-duality

This was conjectured using “real string theory” arguments (momentum modes and winding modes)

It was also conjectured that non-perturbative corrections will violate the duality In the p-adic case, this happens at higher loops

2 )( 0

20

11

MT

T

TZTZ

242

1

22

43

M

TN

M

TMP

xxex

n

xn 22

)(

22 /21)( xex

x

2

21)( xex

Thermodynamics

• Low Temperature:

Behaves as pressureless dust (Deo et. al.,Vafa,Tseythlin, Brandenberger)

• High Temperature (Atick & Witten):

Behaves as stiff fluid

02

exp1

2exp

2

2

22

2

T

M

Tand

T

MP

12 TP

Vacuum Energy Vacuum energy appears only at 2-loops It is -ve Ghost appears due to self-energy

Adding counter-term makes the vacuum energy positive and hierarchically suppressed

122

2

1

4!!)1( with term-counter theAdd

pM

pp

.0at on contributienergy -self thecancel to T

2/)1(

03

3

1 ),()2(

!!)1(

p

nn kD

kdTpp

04

!!)112

(2

1vac

pM

pp

Planck Mass & Cosmological Constant

21926

862 GeV 1022.1

21

o

s

NP g

mV

GM

3

)(

p

P

s

M

mpc hierarchical suppression

known dimensionless function of p

volume of extra-dimensional compactified space

p=7

PeV 385m7p

TeV 1820m5p

MeV 550m3p

then )meV 3.2( If

s

s

s4

vac

Necklace diagrams

and sunset diagrams

Low T High T (Atick & Witten) OK

.2o

s

g

mT

Higher Loops

)1(2

)1(2 ~ln~ln

l

s

sl

lsl m

TgZgZ

Solitons at Finite temperature

maxmin fff

01 & 1

:1 Around

1 & 1 :0 Around

1 & 10 :1 Around

maxmin

maxmin

maxmin

ff

ff

ff

ppp ,7,3

p

M

2

222 /exp Classical Equation:

0at Gaussian T

cosine

2ln

for 1pM

TTf c

even solution p=3

95.0,85.0,5.0,3.0,1.0,01.0/ cTT

odd solution p=3

50.1,0.1,5.0,1.0,01.0/ cTT

0at Gaussians T

T as amplitude

increasing with sine

ESeVZ solitonln )()(

2TI

g

paS

oE

Even solitons are important for high temperatures

22 /in lexponentia MT

TTc /

Future Applications

Thermal fluctuations may dominate the early phase of inflation…any signatures (?)

Thermal solitons suggest existence of branes in extra compact directions

Exponential cut-off acts like a regularization parameter – can it be made physical?

What about bound states?What about Gravity? (TB,Mazumdar,Siegel’05)

In mathematics, and chiefly number theory, the p-adic number systemfor any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational numbersystem to the real and complex number systems. The extension isachieved by an alternative interpretation of the concept of absolute value.

Wikipedia

String Theories over p-adic Fields*

• Quantum fields valued in the field of complex numbers• Space-time coordinates valued in the field of real numbers• World-sheet coordinates valued in the field of p-adic numbers

*Freund & Olson (1987), Freund & Witten (1987)

2

12222

4!!

2/)1(12

1

p

M

T

M

T

M

T

M

TMpP

pp

04

!!)112

(2

1vac

pM

pp

04

!! 2/)1(

4/)1(32

1

p

p

TM

pP

vac2

2122

2

11 2

exp4

!!1

T

MMppP

p

vacuum energy:

low T:

high T:

no particle degreesof freedom

Solitons at Finite Temperature

p

M

2

222 /exp :motion ofequation classical

pxMppxf

xfxffxx

ps

nssn

4/)1(exp)(

)()()(),...,(

22)1(2/1

11

. period with periodic bemust )(function The f

.1,0 are solutions Trivial

Soliton solutions in Euclidean space at zero temperaturewere found by Brekke, Freund, Olson & Witten (1988).

Key Results

• There are no particle degrees of freedom so there is no one-loop contribution to the partition function

• The lowest order contribution arises from interactions• A counter-term must be added to avoid the appearance

of a ghost in a loop expansion which has the consequence that …

• The vacuum energy is positive and hierarchically suppressed

• Perturbation theory breaks down at a temperature of order

• Soliton solutions exist at all temperatures and become important when

2/~ osc gmT

2/ os gmT