The scheme of a simultaneous multiple pulse focusing on one spot naturally arises from the structural features of projected new laser systems, such as the Extreme Light Infrastructure (ELI) and High Power laser Energy Research (HiPER). It is shown that the multiple pulse configuration is beneficial for observing e+ e- pair production from a vacuum under the action of sufficiently strong electromagnetic fields. The field of focused pulses is described using a realistic three-dimensional model based on an exact solution of the Maxwell equations. The e+ e- pair production threshold in terms of electromagnetic field energy can be substantially lowered if, instead of one or even two colliding pulses, multiple pulses are focused on one spot. The multiple pulse interaction geometry gives rise to subwavelength field features in the focal region. These features result in the production of extremely short e+ e- bunches.
A simple method is developed for computing the eigenvalues of the invariant operators (the so-called Casimir operators) C of arbitrary order ρ for semisimple Lie groups. The resulting formulas (52) and (55) are applicable for the case that among the irreducible representations of the given group there is at least one representation with a simple spectrum -in particular, for all the classical groups, as well as the groups G 2 , E & , and E 7 . An expression is found (see (75)) for the generating function of the Casimir operators in the case of the classical groups.
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