許多物理系統表現為約束系統。這包括具有局部規範不變性的系統。在狄拉克的開創性著作中研究了約束系統及其分類。引入適當的輔助條件，可使一級系統變為二級系統；益川和中島證明了該過程可產生定義明確的縮減相空間，是規範相空間。
因此，哈密頓演化由狄拉克括號而不是泊松括號控制。在此背景下，論文調查了一些重要案例；對這種系統的相空間分解其物理和非物理（規範）部分。我們引入了一組輔助條件可，原則上，用於分解受約束的系統。然後可以使用縮減規範函數積分執行量化，其中法捷耶夫-波波夫行列式（和相關的鬼場）自然出現。我們將這些想法應用於一些基礎理論和例子。其中包括電磁學、楊-米爾斯理論、愛因斯坦-希爾伯特引力及其擴展，以及克勞德模型。此外，在此框架內，可克服“格里博夫模糊＂的方法得到體現。

Many physical systems manifest themselves as constrained systems. These include systems with local gauge invariance. Constrained systems and their classifications were investigated in Dirac’s seminal works. Introduction of appropriate auxiliary conditions can turn first class systems into second class systems; and Maskawa and Nakajima demonstrated that this procedure has the advantage of yielding well-defined resultant phase spaces which are truly canonical. The Hamiltonian evolution is then governed by Dirac, rather than Poisson, brackets. Within this context, the thesis investigates a number of important cases; the study of
the phase spaces of such system reveals its decomposition into its physical and unphysical (gauge) parts. We introduce a set of auxiliary conditions that can, in principle, be applied quite generically to decompose a constrained system in this manner. Quantization can then be performed with reduced canonical functional integral, wherein Faddeev-Popov determinants (and associated ghost fields) emerge naturally. We apply these ideas to some fundamental theories and examples. These include Electromagnetism, Yang-Mills theory, Einstein-Hilbert gravity and its extensions, and also the Klauder model which is solved explicitly. In addition,
within this framework, a method to overcome the “Gribov ambiguity” is manifested.

[1] P. A. M. Dirac, Generalized hamiltonian dynamics. Canadian Journal of Mathematics, 2:129–148, (1950);
P. A. M. Dirac, Lectures on Quantum Mechanics (Dover Books on Physics. Dover Publications, Mineola, NY, 2001).
[2] V.N. Gribov, Quantization of non-abelian gauge theories. Nuclear Physics B, 139(1):1–19, 1978.
[3] Eyo Eyo Ita, Chopin Soo, and Abraham Tan, Quantization of constrained systems as Dirac first class versus second class: a toy model and its implications. Symmetry,15(5):1117, (2023);
Eyo Eyo Ita III, Chopin Soo, and Hoi-Lai Yu, Cosmic time and reduced phase space of general relativity. Physical Review D, 97(10):104021, (2018);
Chopin Soo and Hoi-Lai Yu, Intrinsic Time Geometrodynamics: At One With The Universe (World Scientific, 2023).
[4] Toshihide Maskawa and Hideo Nakajima, Singular lagrangian and the dirac-faddeev method: Existence theorem of constraints in’standard form’. Progress of Theoretical Physics, 56(4):1295–1309, (1976).
[5] Viktor Nikolaevich Popov, Functional integrals in quantum field theory and statistical physics (Springer Science & Business Media, 2001).
[6] Bruno Zumino, Normal forms of complex matrices. Journal of Mathematical Physics, 3(5):1055–1057, (1962).