Thermodynamics and Cosmological Constant of Non-Local Field Theories from p-Adic Strings

1. 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.

2. Wikipedia 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=3 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.