Nonextensive statistics in stringy space-time foam models and entangled meson states

The possibility of generation of nonextensive statistics, in the sense of Tsallis, due to space-time foam is discussed within the context of a particular kind of foam in string/brane theory, the D-particle foam model. The latter involves pointlike brane defects (D-particles), which provide the topologically nontrivial foamy structures of space-time. A stochastic Langevin equation for the velocity recoil of D-particles can be derived from the pinched approximation for a sum over genera in the calculation of the partition function of a bosonic string in the presence of heavy D-particles. The string coupling in standard perturbation theory is related to the exponential of the expectation of the dilaton. Inclusion of fluctuations of the dilaton itself and uncertainties in the string background will then necessitate fluctuations in $g_s$. The fluctuation in the string coupling in the sum over genera typically leads to a generic structure of the Langevin equation where the coefficient of the noise term fluctuates owing to dependence on the string coupling $g_s$. The positivity of $g_s$ leads naturally to a stochastic modeling of its distribution with a $\chi$ distribution. This then rigorously implies a Tsallis-type nonextensive or, more generally, a superstatistics distribution for the recoil velocity of D-particles. As a concrete and physically interesting application, we provide a rigorous estimate of an $\omega$-like effect, pertinent to $CPT$ violating modifications of the Einstein-Podolsky-Rosen correlators in entangled states of neutral kaons. In the case of D-particle foam fluctuations, which respect the Lorentz symmetry of the vacuum on average, we find that the $\omega$ effect may be within the range of sensitivity of future meson factories.

Figure 1

Upper picture: A fluctuating D-particle is described by open strings attached to it. As a result of conservation of string fluxes [2, 5, 6] that accompany the D-branes, an isolated D-particle cannot occur, but it has to be connected to a D-brane world through flux strings. Lower picture: World sheet diagrams with annulus topologies, describing the fluctuations of D-particles as a result of the open string states ending on them. Conformal invariance implies that pinched surfaces, with infinitely long thin strips, have to be taken into account. In bosonic string theory, such surfaces can be resummed [21].

Figure 2

Schematic representation of a D-foam. The figure also indicates the capture/recoil process of a string state by a D-particle defect for closed (left panel) and open (right panel) string states, in the presence of D-brane world. The presence of a D-brane is essential due to gauge flux conservation, since an isolated D-particle cannot exist. The intermediate composite state at $t=0$, which has a lifetime within the stringy uncertainty time interval $\delta t$, of the order of the string length, and is described by world sheet logarithmic conformal field theory, is responsible for the distortion of the surrounding space-time during the scattering, and subsequently leads to induced metrics depending on both coordinates and momenta of the string state. This results on modified dispersion relations for the open string propagation in such a situation [4], leading to nontrivial “optics” for this space-time.

Figure 3

(a) World sheet annulus diagram for the leading quantum correction to the propagation of a string state $V$ in a D-brane background, and (b) the pinched annulus configuration which is the dominant divergent contribution to the quantum recoil.