Dynamics with Infinitely Many Time Derivatives and Rolling Tachyons

Abstract

Both in string field theory and in p-adic string theory the equations of motion involve infinite number of time derivatives. We argue that the initial value problem is qualitatively different from that obtained in the limit of many time derivatives in that the space of initial conditions becomes strongly constrained. We calculate the energy-momentum tensor and study in detail time dependent solutions representing tachyons rolling on the p-adic string theory potentials. For even potentials we find surprising small oscillations at the tachyon vacuum. These are not conventional physical states but rather anharmonic oscillations with a nontrivial frequency--amplitude relation. When the potentials are not even, small oscillatory solutions around the bottom must grow in amplitude without a bound. Open string field theory resembles this latter case, the tachyon rolls to the bottom and ever growing oscillations ensue. We discuss the significance of these results for the issues of emerging closed strings and tachyon matter.

Figures (14)

• Figure 1: The p-adic string potentials. For odd $p$ the potential is even and has two unstable maxima. For even $p$ there is only one unstable maximum.
• Figure 2: A smooth function that is identically one for $|t|<a$, and identically zero when $|t|>b$. For such a function the differential form of the equation of motion does not coincide with the convolution form.
• Figure 3: A monotonically decreasing field configuration going from the top of the potential all the way to the tachyon vacuum. Such field configuration cannot satisfy the equation of motion.
• Figure 4: A bounded field configuration attaining a maximum value of $b$ and a minimum value of $a$. The field equation gives constraints on the possible values of $a$ and $b$.
• Figure 5: A monotonic lump with endpoint values $b$ and unique minimum value of $a$. This kind of lump solution cannot exist for even $p$. The dashed line represents the gaussian convolution of the field configuration.