joe kapusta- thermodynamics and cosmological constant of non-local field theories from p-adic...

8/3/2019 Joe Kapusta- Thermodynamics and Cosmological Constant of Non-Local Field Theories from p-Adic Strings

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Thermodynamics and

Cosmological Constant ofNon-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.


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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 number

system to the real and complex number systems. The extension is

achieved by an alternative interpretation of the concept of absolute value.

Wikipedia

String Theories overp-adic Fields*

Quantum fields valued in the field of complex numbersSpace-time coordinates valued in the field of real numbersWorld-sheet coordinates valued in the field ofp-adic numbers*Freund & Olson (1987), Freund & Witten (1987)


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The N-point tree amplitudes of the open string can be

generated 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


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Key Results

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

The lowest order contribution arises from interactions

A counter-term must be added to avoid the appearanceof a ghost in a loop expansion which has theconsequence that

The vacuum energy is positive and hierarchicallysuppressed

Perturbation theory breaks down at a temperature oforder

Soliton solutions exist at all temperatures and becomeimportant when


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Rescale the fields to put the action in the form

with dimensionful coupling constant

and non-local propagator


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p=3 p=7


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Thermal Duality:

Analogous to dualities found in stringy calculationsDoes not hold at higher orderLeads to peculiar thermodynamicsGhost appears due to self-energy


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vacuum energy:

low T:

high T:

no particle degrees

of freedom


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Necklace diagrams

and sunset diagrams

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


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Planck Mass & Cosmological Constant

hierarchical suppression

known dimensionless function ofp

volume of extra-dimensional compactified space


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p=7


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Solitons at Finite Temperature

Soliton solutions in Euclidean space at zero temperature

were found by Brekke, Freund, Olson & Witten (1988).


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Variety of equivalent integral equations


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Three types of solutions in the inverted potential


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even solution

p=3


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odd solution

p=3


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Even solitons are important for high temperatures

quantum fluctuations


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Conclusions

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

We studied a nonlocal field theory arising fromstring theory at finite temperature.The theory has no particle degrees of freedom.Pertubation theory is accurate up to and beyondthe string scale when the string coupling is weak.

There is a positive cosmological constant andpower-law behavior at high temperature.

Soliton solutions were found at all temperatures.

joe kapusta- thermodynamics and cosmological constant of non-local field theories from p-adic...

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