We consider the energy levels of a hydrogen-like atom in the framework of θmodified, due to space noncommutativity, Dirac equation with Coulomb field. It is shown that on the noncommutative (NC) space the degeneracy of the levels 2S 1/2 , 2P 1/2 and 2P 3/2 is lifted completely, such that new transition channels are allowed.

In this paper, we present in detail consistent QED (and scalar QED) calculations of particle creation effects in external electromagnetic field that correspond to three most important exactly solvable cases of t-electric potential steps: Sauter-like electric field, T -constant electric field, and exponentially growing and decaying electric fields. In all these cases, we succeeded to obtain new results, such as calculations in modified configurations of the above mentioned steps and detailed considerations of new limiting cases in already studied before steps. As was recently discovered by us, the information derived from considerations of exactly solvable cases allows one to make some general conclusions about quantum effects in fields for which no closed form solutions of the Dirac (or Klein-Gordon) equation are known. In the present article we briefly represent such conclusions about an universal behavior of vacuum mean values in slowly varying strong electric fields.

In a U (1) ⋆ -noncommutative (NC) gauge field theory we extend the Seiberg-Witten (SW) map to include the (gauge-invariance-violating) external current and formulate -to the first order in the NC parameter -gauge-covariant classical field equations. We find solutions to these equations in the vacuum and in an external magnetic field, when the 4-current is a static electric charge of a finite size a, restricted from below by the elementary length. We impose extra boundary conditions, which we use to rule out all singularities, 1/r included, from the solutions. The static charge proves to be a magnetic dipole, with its magnetic moment being inversely proportional to its size a. The external magnetic field modifies the long-range Coulomb field and some electromagnetic form-factors.We also analyze the ambiguity in the SW map and show that at least to the order studied here it is equivalent to the ambiguity of adding a homogeneous solution to the current-conservation equation. * adorno@dfn.if.usp.br † gitman@dfn.if.usp.br ‡ shabad@lpi.ru §

We analyze the creation of fermions and bosons from the vacuum by the exponentially decreasing in time electric field in detail. In our calculations we use QED and follow in main the consideration of particle creation effect in a homogeneous electric field. To this end we find complete sets of exact solutions of the d-dimensional Dirac equation in the exponentially decreasing electric field and use them to calculate all the characteristics of the effect, in particular, the total number of created particles and the probability of a vacuum to remain a vacuum. It should be noted that the latter quantities were derived in the case under consideration for the first time. All possible asymptotic regimes are discussed in detail. In addition, switching on and switching off effects are studied.

The particle creation by the so-called peak electric field is considered. The latter field is a combination of two exponential parts, one exponentially increasing and another exponentially decreasing. We find exact solutions of the Dirac equation with the field under consideration with appropriate asymptotic conditions and calculate all the characteristics of particle creation effect, in particular, differential mean numbers of created particle, total number of created particles, and the probability for a vacuum to remain a vacuum. Characteristic asymptotic regimes are discussed in detail and a comparison with the pure asymptotically decaying field is considered.