NATURAL LAGRANGIANS FOR QUANTUM STRUCTURES OVER 4-DIMENSIONAL SPACETIME JOSEF JANY ˇ SKA Abstract. The natural quantum Lagrangians which appear in Einstein general relativistic quantum mechanics are classified by using methods of gauge-natural bundles and natural operators. It is proved that all natural quantum Lagrangians for scalar particles have a functional base formed by four Lagrangians. The in- variant description of these Lagrangians is given. Introduction In [1, 3, 4] the authors have proposed a new geometric formulation of quantum mechanics of a classical charged particle, with given gravitational and electromag- netic classical fields, in the framework of a general relativistic Galilei spacetime. An important role in this theory is played by the distinguished quantum Lagrangian constructed naturally from the metric tensor, a potential of the electromagnetic field and a section of the quantum bundle. In [5] all natural quantum Lagrangians in the Galilei approach are classified by using the theory of natural operators, [8, 9]. Recent papers [6, 7] have proposed the Einstein analogue to some results of [1, 3, 4]. The aim of this paper is to introduce the Einstein analogue of the distin- guished quantum Lagrangian in the Galilei approach and to classify all natural quan- tum Lagrangians for scalar particles in the Einstein approach. We prove that the set of natural quantum Lagrangians has a functional base formed by four Lagrangians and the only Lagrangian in the base involving both gravitational and electromag- netic structure of the spacetime is the distinguished quantum Lagrangian which is constructed by using a quantum connection. For the natural basic Lagrangians we give the corresponding Euler-Lagrange operators and generalized Euler-Lagrange equations. We assume the following fundamental unit spaces [1]: (1) the oriented 1-dimensional vector space T over IR of time intervals , (2) the positive 1-dimensional semi-vector space L over IR + of lengths , (3) the positive 1-dimensional semi-vector space M over IR + of masses . 1991 Mathematics Subject Classification. 53C05, 53C50, 58A20, 58F05, 70H35, 83C10. Key words and phrases. Jet, jet-contact structure, non-linear connection, quantum bundle, nat- ural differential operator, Lagrangian, Euler-Lagrange operator. This research has been supported by grant No. 201/96/0079 of GA CR and GNFM of CNR. 1