Graded gauge theory. (English) Zbl 0536.53069

The author deals with two things: first he defines a class of graded principal fibre bundles, secondly he introduces there some differential operators and studies a Lagrangian of the supergauge field theory. For the purpose of his graded bundles he presents a realization of a super Lie group in the case when a linear representation of its algebra is given. The total space of the supergroup is provided with a semigroup structure. Then the author constructs a graded fibre bundle. The resulting bundle contains both Lorentz spinors (at the base) and G- spinors (at the fibre). Note that there exist also non-trivial bundles of this type; an example is outlined by the reviewer in [Prepr., Inst. Math., Pol. Acad. Sci., Warsz. 277, 88 p. (1983; Zbl 0515.58013)].
In the second part the author tries to describe gauge fields in the above bundle and their Lagrangians. Commutation relations in some sectors of values of the fields are discussed. Physical phenomena like conformal symmetry breaking and mass in related fields are studied. Three coupling scale constants of length-dimension are pointed out. The number of variables is big as the author is unable to give an explicit form for the Lagrangian. On the other hand some physical fields cannot be described in this model. Note that the whole machinery of the first part is not used in the second one. For needs of the Lagrangian local coordinates of the bundle are quite sufficient. Formula 1.18 is not proved. 4.6 seems to be not true. Several notational mistakes can be noticed.
Reviewer: J.Czyz


53C80 Applications of global differential geometry to the sciences
58H99 Pseudogroups, differentiable groupoids and general structures on manifolds
55R99 Fiber spaces and bundles in algebraic topology
81T08 Constructive quantum field theory
17B70 Graded Lie (super)algebras


Zbl 0515.58013
Full Text: DOI


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