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The work presented in this book is relevant to the interaction of quantum fields with the classical background manifested by a strong external field. In other words, the background field is adequately described by a classical theory and does not need to be quantized. A significant simplification comes from considering quantized scalar field interacts only with the gravitational field. The Renormalization using zeta function method is the essential requirement of this study for calculated the expectation values of the energy-momentum tensor, in which the quantum effects in a gravitational field induce corrections to the energy-momentum tensor of a matter field, this formulation express the vacuum polarization effects, and the relevant theory is called the Semi-classical theory of gravity.