In the ADM formulation, spacetime is foliated by spacelike hypersurfaces, allowing spacetime to be divided into space an‎d time. In this framework, the evolution of fields can be separated into evolution on these hypersurfaces an‎d evolution from one hypersurface to the next. In this thesis the algebra of generators of deformations of constant-time hypersurfaces will be studied. Then, by exploring concepts such as path independence an‎d parameterized systems, solutions for a scalar density field theory are investigated. These solutions restrict the dependence on field derivatives in the time evolution generator to specific terms. This theory is incompatible with Teitelboimʹs assumptions, such as the independence of the matter time evolution generator from metric derivatives. Inspired by second quantization an‎d by Using the Dirac algebra in three-dimensional spacetime $R×S^2$, we expan‎d the matter field in the basis of spherical harmonics on $S^2$ an‎d, following the second quantization approach, promote the coefficients of this expansion to creation an‎d annihilation operators. We show that the algebra of evolution generators determines the Hamiltonian an‎d diffeomorphism generators on $S^2$. This quantization method systematically determines the ordering of creation an‎d annihilation operators in the evolution generators.

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سروش شاكري , احمد شيرزاد