0 Quantization of the Relativistic Particle

We revise the problem of the quantization of relativistic particle, presenting a modified consistent canonical scheme, which allows one not only to include arbitrary backgrounds in the consideration but to get in course of the quanti-zation a consistent relativistic quantum mechanics, which reproduces literally the behavior of the one-particle sector of the corresponding quantum field. At the same time this construction presents a possible solution of the well-known old problem how to construct a consistent quantum mechanics on the base of a relativistic wave equation. 1 Already for a long time there exists a definite interest in quantization of classical and pseudoclassical models of relativistic particles (RP). This problem meets such difficulties as zero-Hamiltonian phenomenon and time definition problem. Consideration in arbitrary electromagnetic and gravitational backgrounds creates additional difficulties. The usual aim of the quantization is to arrive in a way to a corresponding relativistic wave equation without any attempt to demonstrate that a consistent quantum mechanics is constructed, since there is a common opinion that the construction of a such a mechanics on the base of relativistic wave equations is not possible due to existence of infinite number of negative energy levels, and due to existence of negative vector norms (in scalar case), and these difficulties may be only solved in QFT [1]. One of possible approach to the canonical quantization of RP models was presented in [2] on the base of a special gauge, which fixes reparametrization gauge freedom. However, the difficulties with inclusion of arbitrary backgrounds were not overcome and the consistent quantum mechanics was not constructed. It turns out that the whole scheme of quantization, which was used in that papers and repeated then in numerous works, has to be changed essentially to make it possible to solve the above problems and to construct a quantum mechanics which is consistent to the same extent to which a one-particle description is possible in the frame of the corresponding QFT. One of the main point of the modification is related to a principally new realization of the Hilbert space. At the same time this construction gives a solution of the above mentioned old problem how to construct a consistent quantum mechanics on the base of a relativistic wave equation Below we present a demonstration for a spinless particle case. The spinning particle case and all long technical details may be found by a reader in [3]. We start …