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Within Weber’s approach, we study the classical and quantum scalar field theory in the Schwarzschild metric. By employing the localized coordinates and the Bateman – Caldirola – Kanai method, we find a Lagrangian density of the scalar fields from the deformed motion equation of them. Also, we show that approximated theory has an exact established solution and Green’s function by replacement the ordinary field’s mass with the u-mass. We know that the new fields do not obey the second quantization relation. So the classical and quantum scalar fields have different treatments in Weber’s method. Thus, we must admit a new background field for quantum cases. Locally homeomorphism between pseudo Riemann's manifold and Minkowski one is considered.
 

Keywords: Quantum Field Theory, Riemann’s Coordinate, Weber’s Method, BCK Model

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