Zero Dimensional Field Theory of Tachyon Matter

D. D. Dimitrijevic* and G. S. Djordjevic

Department of Physics, University of Nis, P.O.Box 224, Nis, Serbia, [email protected] *Institute of Physics, Faculty of Sciences, Nis, Serbia, [email protected]

Abstract. The first issue about the object (now) called tachyons was published almost one century ago. Even though there is no experimental evidence of tachyons there are several reasons why tachyons are still of interest today, in fact interest in tachyons is increasing. Many string theories have tachyons occurring as some of the particles in the theory. In this paper we consider the zero dimensional version of the field theory of tachyon matter proposed by A. Sen. Using perturbation theory and ideas of S. Kar, we demonstrate how this tachyon field theory can be connected with a classical mechanical system, such as a massive particle moving in a constant field with quadratic friction. The corresponding Feynman path integral form is proposed using a perturbative method. A few promising lines for further applications and investigations are noted. Keywords: Tachyon, Feynman path integral, String Theory. PACS: 03.65-w, 10., 11.10-z.

INTRODUCTION ZERO DIMENSIONAL CASE In string theory, when physicists calculate mass of the particles, in some cases, their mass2 turned out to The corresponding zero dimensional analogue of a be negative. Such particles are called tachyons. For tachyon field can be obtained by the correspondence such a theory vacuum state is generally unstable. A. [3]: xi → t , T → x , V (T) → V (x) . The action Sen proposed a field theory of tachyon matter few years ago [1,2]. The action is given as: reads: 2 S = −∫ dtV (x) 1− x& , (3) S = − d n+1 xV (T ) 1+η ij ∂ T∂ T , (1) ∫ i j Corresponding equation of motion, including V (x) = e−αx , is: whereη00 = −1, η µν = δ µν , µ,ν = 1,...,n , T(x) 2 is the scalar tachyon field and V (T ) is the tachyon &x&+ αx& = α , (4) potential, which unusually appears in the action as a and coincide with the equation for the system under multiplicative factor and has (from string field theory gravity in the presence of quadratic damping: arguments) exponential dependence with respect to the tachyon field: 2 m&y& + βy& = mg . (5) V (T) = e−αT (x) . (2) This equation can be derived from the action:

It is very useful, at least from the pedagogical βy − reason, to understand and to investigate lower β 2 S = − dte m 1− y& . (6) dimensional analogs of this tachyon field theory [3]. ∫ mg

The solution can be found perturbatively:

y(t) = y0 (t) + y1 (t) , (7) One can go back to Eq. (6) and for very small β , it leads to the new form of action (6): where y0 (t) is solution of Eq. (5) for β = 0 , and β 2 β y (t) is obtained from the same equation after S → S′ = − dt[ y& + y −1], (13) 1 β →0 ∫ 2mg m inserting and neglecting all non linear terms: y0 (t) This action is quadratic with respect to velocity, and 2 standard procedure can be engaged for the path &y&1 + aty&1 = −bt , (8) integral.

2βg βg 2 where a = , b = . For y (0) = 0 and CONCLUSION m m 0 y (0) = 0 the final solution is given by: Sen`s proposal [1,2] and similar conjectures (see, & 0 e.g., [6]) have attracted important interests among physicists. Our understanding of tachyon matter, g b 3 at 2 especially its quantum aspects is still quite pure. y(t) = t 2 + t 2 ( F [1,1; ,2;− ] −1), (9) 2 2a 2 2 2 2 Perturbative solutions for classical particles analogous to the tachyons offer many possibilities for further investigations and toy models in quantum mechanics, 3 at 2 where F [1,1; ,2;− ] is hypergeometric quantum and string field theory and cosmology on 2 2 2 2 archimedean and nonarchimedean spaces [5]. function. For small t it gets quite simple form: ACKNOWLEDGMENTS g βg 2 y(t) = t 2 − t 4 . (10) The research of both authors is supported by the 2 12m Serbian Ministry of Science and Technology Projects No. 144014 and No. 141016. The financial support of This solution with nontrivial boundary conditions can the UNESCO-ROSTE under the Project be useful to get simpler-quadratic action for tachyons. ``Southeastern European Network in Mathematical Details will be presented elsewhere. and Theoretical Physics`` (SEENET-MTP) No. 8759145 is also kindly acknowledged. We would like FEYNMAN PATH INTEGRAL to thank S. Kar, J. Jeknic and Lj. Nesic for many fruitful discussions. According to Feynman’s idea [4], dynamical evolution of the system is completely described by the REFERENCES kernel K(y′′,T; y′,0) of the unitary evolution ′′ ′ 1. A. Sen, JHEP 0204, 048, 2002. operator U (0,T) , where y , y are initial and final 2 A. Sen, Tachyon Dynamics in Open String Theory, hep- positions and T is ``total`` time: th/0410103. T 2π i 3. S. Kar, A simple mechanical analog of the field theory of Ldt h ∫ tachyon matter, hep-th/0210108. K(y′′,T; y′,0) = Dye 0 . (11) 4. R.P. Feynman, Rev. Mod. Phys. 20, 367, 1948. ∫ 5. G.S.Djordjevic and Lj. Nesic, Real and p-adic aspect of quantization of tachyons, Proc. of the BW2003 There is very useful semi-classical expression for workshop, World Scientific, Singapore, 2005, pp. 197- the kernel if the classical action S (y′′,T; y′,0) is 207. 6. M. R. Garousi1, Tachyon couplings on non-BPS D-branes polynomial quadratic in y′ and y′′ (which holds for and Dirac-Born-Infeld action, hep-th/0003122. both real and p-adic number fields [5]):

1/ 2 2π i 2 S ( y′′,T;y′,0) ⎛ i ∂ S ⎞ h K(y′′,T; y′,0) = ⎜ ⎟ e . (12) ⎝ h ∂y′∂y′′ ⎠