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The discussion made in this book is about investigating the spontaneous violation of the Lorentz symmetry at low energy systems, where the effects associated with the violation of the Lorentz symmetry can be viewed through quantum effects, such as geometric quantum phases, Landau quantization and quantum Hall effect. This line of research looks for a more fundamental theory that goes beyond the Standard Model by extending the mechanism of the spontaneous symmetry breaking through a background determined by a vector field or a tensor field.