Thermodynamics and Cosmological Constant of

Non-Local Field Theories from p-Adic Strings

Joe Kapusta*

University of Minnesota

*Based on: PRL 104, 021601 (2010), arXiv:1005.0430 and arXiv:1006.4098 with T. Biswas and J .A. R. Cembranos.

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)

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 (the tachyon) which has a tachyon-free vacuum withno particles but with soliton solutions.

Brekke, Freund, Olson & Witten (1988); Frampton & Okada (1988)

open string coupling prime number string tension

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

1

2

2223 /

exp2

1 p

M

dxddS

1

21

1

p

s

p

m

g

p

Rescale the fields to put the action in the form

with dimensionful coupling constant

and non-local propagator 2220 /exp),( MkkD nn

2/)1(

03

3

1 ),()2(

)(!!ln

p

nn kD

kdTVpZ

M

TN

N

MkD

kd

nn

N

2

2),(

)2(

3

03

3

p=3 p=7

2/)1(12

1

22

4!!

pp

M

TN

M

TMpP

xxex

n

xn 22

)(

22 /21)( xex

x

2

21)( xex

2/)1(

03

3

1 ),()2(

!!)1(

p

nn kD

kdTpp

Thermal Duality:2

)( 0

20

11

MT

T

TZTZ

•Analogous to dualities found in stringy calculations•Does not hold at higher order•Leads to peculiar thermodynamics•Ghost appears due to self-energy

122

2

1

4!!)1( with term-counter theAdd

pM

pp

.0at on contributienergy -self thecancel to T

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

Necklace diagrams

and sunset diagrams

can be evaluated. They become comparable inmagnitude to lower order diagrams when .

2o

s

g

mT

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

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).

Variety of equivalent integral equations

)'cos()'()cos()( :solutioneven 2/

2/

/ 22

n

n

pn

M fdeTf n

)'sin()'()sin(2)( :solution odd1

2/

2/

/ 22

n

n

pn

M fdeTf n

224

12/

2/

'exp)'(2

)(

nMfdM

fn

p

224

1 'exp)'(2

)(

MfdM

f p

Three types of solutions in the inverted potential

maxmin fff

01 and 1 :1 around nsoscillatio

1 and 1 :0 around nsoscillatio

1 and 10 :1 around nsoscillatio

maxmin

maxmin

maxmin

ff

ff

ff

ppp ,7,3

0at Gaussian T

cosine

2

lnfor 1

pMTTf 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

ESeKVZ solitonln )()(

2TI

g

paS

oE

Even solitons are important for high temperatures

22 /in lexponentia MT

TTc /

quantum fluctuations

Conclusions

Supported by the U.S. Department of Energy under Grant No. DE-FG02-87ER40328.

•We studied a nonlocal field theory arising from string theory at finite temperature.•The theory has no particle degrees of freedom.•Pertubation theory is accurate up to and beyond the string scale when the string coupling is weak.•There is a positive cosmological constant and power-law behavior at high temperature.•Soliton solutions were found at all temperatures.