From the Measure Problem of Quantum Mechanics to Renormalization Theory of Quantum Field

The measure problem appears in Quantum mechanics when the experiment shows the duality wave-corpuscle of the light, and so this problem has been expressed by uncertainty principle of Heisenberg, this book treats this problem pursuant Bridgman's dimensional analyses. Thus, one can express similar problem appeared in NCQFT as non integrable Feynman integrals at small inner momentum that is known as UV/IR-mixing problem. One can consider that there is a certain perturbation or vibrational mode of space-time curvature causes the non-integrality, thus a certain correction on the action of QFT becomes necessarily. This correction is realized by Gross-Wulknhaar by an additional term expressed the oscillation of the curved background space in which is manifested as a harmonic oscillator potential term. This book presents the renormalization models at one-loop for gauge field in non-commutative space in which the requirements of constructing these models is that the non-commutative field theory described these models acts as quantum field theory in curved pace-time based essentially on the replacement of commutative algebra of Euclidean space by noncommutative algebra in which the theory simplified to non-commutative field theories in Minkowski space known as ‘θ- deformed space’ Rθ4 . The truncated Heisenberg algebra yields an interpretation of the relevant oscillation term as the coupling of the scalar field to the curvature. Since the action including the gravity effects as an oscillator term one expects that there is a certain mass, thus we obtain a Lagrangian containing the Higgs potential. Showing therefore in the end of this book, some characteristics of Higgs field and consequences.