1. Joe Kapusta* University of Minnesota
Thermodynamics and Cosmological Constant of Non-Local Field Theories from p-Adic Strings
Joe Kapusta*
University of Minnesota
*Based on: PRL 104, (2010), arXiv: and arXiv:
with T. Biswas and J .A. R. Cembranos.

2. Wikipedia String Theories over p-adic Fields*
In mathematics, and chiefly number theory, the p-adic number system
for any prime number p extends the ordinary arithmetic of the rational
numbers in a way different from the extension of the rational number
system to the real and complex number systems. The extension is
achieved by an alternative interpretation of the concept of absolute value.
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)

3. The N-point tree amplitudes of the open string can be
generated from a non-local Lagrangian of a single scalar
field (the tachyon) which has a tachyon-free vacuum with
no particles but with soliton solutions.
open string coupling prime number string tension
Brekke, Freund, Olson & Witten (1988); Frampton & Okada (1988)

4. 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

5. Rescale the fields to put the action in the form
with dimensionful coupling constant
and non-local propagator

6. p= p=7

8. Thermal Duality: Analogous to dualities found in stringy calculations
Does not hold at higher order
Leads to peculiar thermodynamics
Ghost appears due to self-energy

9. vacuum energy:
low T:
no particle degrees
of freedom
high T:

10. Necklace diagrams
and sunset diagrams
can be evaluated. They become comparable in
magnitude to lower order diagrams when

11. Planck Mass & Cosmological Constant
volume of extra-dimensional compactified space
hierarchical suppression
known dimensionless function of p

12. p=7

13. Solitons at Finite Temperature
Soliton solutions in Euclidean space at zero temperature
were found by Brekke, Freund, Olson & Witten (1988).

14. Variety of equivalent integral equations

15. Three types of solutions in the inverted potential

16. even solution
p=3

17. odd solution
p=3

18. Even solitons are important for high temperatures
quantum fluctuations

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