The Generalized Moyal Nahm and Continuous Moyal Toda Equations

We present in detail a class of solutions to the $4D SU(\infty)$ Moyal Anti Self Dual Yang Mills equations that are related to $reductions$ of the generalized Moyal Nahm quations using the Ivanova-Popov ansatz. The former yields solutions to the ASDYM/SDYM equations for arbitary gauge groups. A further dimensional reduction yields solutions to the Moyal Anti Self Dual Gravitational equations. The Self Dual Yang Mills /Self Dual Gravity case requires a separate study. SU(2) and $SU(\infty)$ (continuous) Moyal Toda equations are derived and solutions to the latter equations in $implicit$ form are proposed via the Lax-Brockett double commutator formalism . An explicit map taking the Moyal heavenly form (after a rotational Killing symmetry reduction) into the SU(2) Moyal Toda field is found. Finally, the generalized Moyal Nahm equations are conjectured that contain the continuous $SU(\infty)$ Moyal Toda equation after a suitable reduction. Three different embeddings of the three different types of Moyal Toda equations into the Moyal Nahm equations are discussed.Comment: Revised TEX file. 31 pages. The Legendre transform between the Moyal heavenly form and the Moyal Toda field is solve

Scalar field inflation driven by a modification of the Heisenberg algebra

We study the modifications induced on scalar field inflation produced by considering a general modification of the Heisenberg algebra. We proceed by modifying the Poisson brackets on the classical theory whenever the corresponding quantum commutator is modified. We do not restrict ourselves to a specific form for such modification, instead we constrain the functions involved by the cosmological behaviour of interest. We present whenever possible the way in which inflation can be realized approximately via three slow roll Hubble parameters that depend on the standard slow roll parameters in a very different form than in the usual case and that can be less restrictive. Furthermore we find a general analytical solution describing an expanding universe with constant Hubble parameter that generalizes the standard cosmological constant case by restricting the form of the modification of the Heisenberg algebra. It is found that even if such modification can be neglected in some limit and the cosmological constant is set to zero in that limit, the exponential expansion is present when the modification is important. Thus an appropriate modification of the Heisenberg algebra is sufficient to produce an exponentially expanding universe without the need of any other source.Comment: 34 pages, 1 figur

Lorentzian Vacuum Transitions in Ho\v{r}ava-Lifshitz Gravity

The vacuum transition probabilities for a Friedmann-Lema\^itre-Robertson-Walker universe with positive curvature in Ho\v{r}ava-Lifshitz gravity in the presence of a scalar field potential in the Wentzel-Kramers-Brillouin approximation are studied. We use a general procedure to compute such transition probabilities using a Hamiltonian approach to the Wheeler-DeWitt equation presented in a previous work. We consider two situations of scalar fields, one in which the scalar field depends on all the spacetime variables and other in which the scalar field depends only on the time variable. In both cases analytic expressions for the vacuum transition probabilities are obtained and the infrared and ultraviolet limits are discussed for comparison with the result obtained by using general relativity. For the case in which the scalar field depends on all spacetime variables we obtain that in the infrared limit it is possible to obtain a similar behavior as in general relativity, however in the ultraviolet limit the behavior found is completely opposite. Some few comments about possible phenomenological implications of our results are given. One of them is a plausible resolution of the initial singularity. On the other hand for the case in which the scalar field depends only on the time variable, the behavior coincides with that of general relativity in both limits, although in the intermediate region the probability is slightly altered.Comment: 26 pages, 2 figures. Some references adde