This monograph is written on topics in the subject of Continuum Quantum Geometric Path Integrals applied to Yang-Mills Theory and variants (QCD, Chern-Simons Theory, Ising Models, etc.)- the called Random Geometry in Quantum Field Theory, which are hoped to be useful to graduate students of quantum physics and applied mathematics, with a focused weight towards to those interested in applying the concepts of continuum quantum geometry in other branches of modern physics, like superconductivity, nuclear physics, polymer theory, string theory, etc. The methodology used to in this monograph is the same exposed in previous work in random classical physics: "Methods of Bosonic Path Integrals Representations- Random Systems in Classical Physics - Nova Publishers, (2006) U.S.A.': Expositions and formulas should be chewed, swallowed and digested. This process of analysis should not be abandoned until it yields a comprehension of the overall pattern of the proposed ideas and math, so after this step, one is ready to make improvements, corrections or criticisms on the path integrals representations of this book.

Table of Contents:
Preface; Loop Space Path Integrals Representations for Euclidean Quantum Fields Path Integrals and the Covariant Path Integral; Path Integrals Evaluations in Bosonic Random Loop Geometry - Abelian Wilson Loops; The Triviality - Quantum Decoherence of Quantum Chromodynamics in the Presence of An External Strong White-Noise Eletromagnetic Field; The Confining Behaviour and Asymptotic Freedom for QCD(SU)- A Constant Gauge Field Path Integral Analysis; Triviality - Quantum Decoherence of Fermionic Quantum Chromodynamics SU (N_c) in the Presence of an External Strong U Flavored Constant Noise Field; Fermions on the Lattice by Means of Mandelstam-Wilson Phase Factors A Bosonic Lattice Path-Integral Framework; A Connection between Fermionic Strings and Quantum - Gravity States - A Loop Space Approach; A Fermionic Loop Wave Equation for Quantum Chromodynamics at N_c = +; String Wave Equations in Polyakov's Path Integral Framework; A Random Surface Menbrane Wave Equation for Bosonic Q.C.D. (SU); Covariant Functional Diffusion Equation for Polyakov's Bosonic String; Covariant Path Integral for Nambu-Goto String Theory; Topological fermionic string representation for Chern-Simons non-Abelian gauge theories; Fermionic String Representation for the Three-Dimensional Ising Model; A Polyakov Fermionic String as a Quantum State of Einstein theory of Gravitation; A Scattering Amplitude in the Quantum Geometry of Fermionic Strings; Path-Integral Bosonization for the Thirring Model on a Riemann Surface; A Path-Integral Approach for Bosonic effective theories for Fermion Fields in Four and Three Dimensions; Domains of Bosonic Functional Integrals and Some Applications to the Mathematical Physics of Path Integrals and String Theory; Non-linear diffusion in R D and in Hilbert Spaces, A Path Integral Study; Basics Integrals Representations in Mathematical Analysis of Euclidean Functional Integrals; Supplementary Appendixes; Index.