在這篇論文中，我們先描述費曼是如何只用牛頓運動方程式和兩個對易關係式導出Lorentz力和兩個同質的Maxwell方程式。在我們的敘述中，我們會用泊松括號而不用對易關係式去使用費曼方法因為所推導的方程式是古典的而非量子。我們也注意到使用泊松括號避免了量子算符順序的模稜兩可。
在介紹了使用泊松括號的費曼方法之後，我們考慮到對於向量場和純量場的非對易同位旋結構，這可把電磁學的阿貝爾結構推廣成非阿貝爾結構。在這推廣的費曼方法之中，泊松括號必須要修改以適應這個新結構。我們注意到在場方程中協變微分的推導和對時間全微分的計算都需要王方程式。王方程式描述了同位旋的動力學而且可以藉由使用修改過的泊松括號套入哈密頓力學來得到。
另一個費曼方法的推廣就是取代費曼的其中一個假設：${x^i,{dot x}^j}=delta^{ij}$。我們把位置和運動動量的對易關係寫成${x^i,{dot x}^j}=g^{ij}$，這也可以藉由對易空間的假設而得到。在這個假設中，我們擁有時空的度規張量和為了在對易空間中的粒子之Lagrangian的存在而生的四個Helmholtz條件。向量純量場可以被相對地擴展為$g_{i0}c$和$-1/2g_{00}c$。粒子的運動方程式可被視為在非相對論極限中沒有其它外力的側地線方程。度規張量也可以被視為推廣的電磁場，這給出了推廣的高斯定律和法拉第定律。法拉第定律可以藉由計算Helmholtz的其中一個條件來獲得，而且磁場的高斯定律和其推廣形式可以藉由Jacobi恆等式導出來。推廣的磁場高斯定律給出了黎曼張量的第一Bianchi恆等式。
最後，我們建議了一個可能的方法去建立一個符合規範和Lorentz不變性的Lagrangian，從這個Lagrangian我們可以得到兩條有電量來源的Maxwell方程式。電磁重力學也會被提到因為在電磁學和相對論重力當中存在相似性。對推廣場方程更進一步的研究與應用可能對於在物理中建立重要角色有幫助。

In the thesis, we first describe how Feynman proved the Lorentz force and the two homogeneous Maxwell equations using only Newton's equations of motion and two commutation relations. In our description, we will use the Feynman approach with the Poisson bracket instead of the commutation relation because the equations to be derived are classical not quantum. We also note that using the Poisson bracket avoids the quantum operator ordering ambiguity.
After introducing the Feynman approach with the Poisson bracket, we consider a non-commutative isospin structure for the vector field and scalar field, generalizing the Abelian structure of electromagnetism to a non-Abelian case. In this generalization of the Feynman approach, the Poisson bracket has to be modified to adopt the new structure. We note that the derivation of the covariant derivatives in the field equations and the calculation of the total time derivative require Wong's equations, which describe the dynamics of the isospin and could be derived by Hamiltonian mechanics using the modified Poisson bracket.
Another generalization of the Feynman approach is to replace one of Feynman's assumptions: ${x^i,{dot x}^j}=delta^{ij}$. We write the commutation relation of the position and the kinematic momentum as ${x^i,{dot x}^j}=g^{ij}$, which can also be obtained by the commutative space assumption. In this assumption, we have the metric tensor of space and time and four Helmholtz conditions for the existence of a Lagrangian of a particle in the commutative space. The vector and scalar fields can be extended as $g_{i0}c$ and $-1/2g_{00}c$, respectively. The equations of motion of the particle can be considered as the geodesic equations without other external forces in the non-relativistic limit. The metric tensor can also be treated as the generalized electromagnetic field to give the generalized Gauss' law and Faraday's law. Faraday's law can be obtained by calculating one of the Helmholtz conditions and Gauss' law for magnetism and its generalization form can be derived by the Jacobi identity. The generalized Gauss' law for magnetism gives the first Bianchi identity of Riemann curvature.
Finally, we suggest a possible method to construct a gauge- and Lorentz-invariant Lagrangian, from which we can obtain the two Maxwell equations with sources. Gravitoelectromagnetism will also be mentioned because there exist similarities between electromagnetism and relativistic gravitation. Further studies and applications of the generalized field equations may help to establish their significant roles in physics.

[1] F.J. Dyson, Feynman's proof of the Maxwell equations, Am. J. Phys. 58, 209 (1990).
[2] C.R. Lee, The Feynman-Dyson proof of the gauge eld equations, Phys. Lett. A 148, 146 (1990).
[3] I.E. Farquhar, Comment on "The Feynman-Dyson proof of the gauge eld equations", Phys. Lett. A 151, 203 (1990).
[4] I.E. Farquhar, Comment on "Feynman's proof of the Maxwell equations" by Freeman J. Dyson, Am. J. Phys. 59, 87 (1991).
[5] I.E. Farquhar, Comment on "The Feynman-Dyson proof of the gauge field equations", Phys. Lett. A 151, 203 (1990).
[6] M. Montesinos and A. Perez-Lorenzana, Minimal coupling and Feynman's proof, Int. J. of Theor. Phys. 38, 901 (1999).
[7] S. Tanimura, Relativistic generalization and extension to the non-Abelian gauge theory of Feynman's proof of the Maxwell equations, Ann. Phys. 220, 229 (1992).
[8] S. Tanimura, Relativistic generalization of Feynman's proof of the Maxwell equations, Vistas Astron. 37, 329 (1993).
[9] M.Y. Chen and S.L. Nyeo, Non-Abelian gauge theory from the Poisson bracket, Int. J. Mod. Phys. A 33, 1859182 (2018).
[10] M.Y. Chen and S.L. Nyeo, The field equations in a three-dimensional commutative space, Int. J. Mod. Phys. A 34, 1950078 (2019).
[11] M.C. Land, N. Shnerb, and L.P. Horwitz, On Feynman's approach to the foundations of gauge theory, J. Math. Phys. 36, 3263 (1995).
[12] A. Berard, Y. Grandati, and H. Mohrbach, Magnetic monopole in the Feynman's derivation of Maxwell equations, J. Math. Phys. 40 3732 (1999).
[13] Z.K. Silagadze, Feynman's derivation of Maxwell equations and extra dimensions, Ann. Fond. Louis de Broglie 27, 241 (2002).
[14] A. Boulahoual and M.B. Sedra, Noncommutative geometry framework and the Feynman's proof of Maxwell equations, J. Math. Phys. 44, 5888 (2003).
[15] A. Berard, H. Mohrbach, J. Lages, P. Gosselin, Y. Grandati, H. Boumrar, and F. M enas, From Feynman proof of Maxwell equations to noncommutative quantum mechanics, J. Phys.: Conf. Ser. 70, 012004 (2007).
[16] R.J. Hughes, On Feynman's proof of the Maxwell equations, Am. J. Phys. 60, 301 (1992).
[17] P. Bracken, Poisson brackets and the Feynman problem, Int. J. of Theor. Phys. 35, 2125 (1996).
[18] P. Bracken, Relativistic equations of motion from Poisson brackets, Int. J. of Theor. Phys. 37, 1625 (1998).
[19] P. Bracken, Determination of the electromagnetic Lagrangian from a system of Poisson brackets, Int. J. of Theor. Phys. 44, 127 (2005).
[20] K.H. Yang and J.O. Hirschfelder, Generalizations of classical Poisson brackets to include spin, Phys. Rev. A 22, 1814 (1980).
[21] M. Lakshmanan and M. Daniel, Comment on the classical models of electrons and nuclei and the generalizations of classical Poisson brackets to include spin, J. of Chem. Phys. 78, 7505 (1983).
[22] S.K. Soni, Classical spin dynamics and quantum algebras, J. Phys. A: Math. Gen. 25, L837 (1992).
[23] J.W. van Holten, Covariant hamiltonian dynamics, Phys. Rev. D 75, 025027 (2007).
[24] B. Aycock, A. Roe, J.L. Silverberg, and A. Widom, Classical Hamiltonian dynamics and Lie group algebras, arXiv:0807.4725v1 [physics.class-ph].
[25] S.K. Wong, Field and particle equations for the classical Yang-Mills eld and particles with isotopic spin, Il Nuovo Cimento 65A, 689 (1970).
[26] A. Stern and I. Yakushin, Deformed Wong particles, Phys. Rev. D 48, 4974 (1993).
[27] S. Hojman and L.F. Urrutia, On the inverse problem of the calculus of variations, J. Math. Phys. 22, 1896 (1981).
[28] W. Sarlet, The Helmholtz conditions revisited. A new approach to the inverse problem of Lagrangian dynamics, J. Phys. A: Math. Gen. 15, 1503 (1982).
[29] F. Pardo, The Helmholtz conditions in terms of constants of motion in classical mechanics, J. Math. Phys. 30, 2054 (1989).
[30] S.A. Hojman and L.C. Shepley, No Lagrangian? No quantization!, J. Math. Phys. 32, 142 (1991).
[31] M. Crampin, T. Mestdag, and W. Sarlet, On the generalized Helmholtz conditions for Lagrangian systems with dissipative forces, Z. Angew. Math. Mech. 90, 502
(2010).
[32] B. Mashhoon, "Gravitoelectromagnetism: a Brief Review." arXiv:gr-qc/0311030.
[33] A. Bakopoulos, "Gravitoelectromagnetism: Basic Principles, Novel Approaches and Their Application to Electromagnetism." Master's Thesis, University of Ioannia
(2016).
[34] J.M. Romero, J.A. Santiago, J.D. Vergara, Newton's second law in a non-commutative space, Phys. Lett. A 310, 9 (2003).
[35] M. Le Bellac and J.M. Levy-Leblond, Galilean electromagnetism, Il. Nuovo Cimento 14B, 217 (1973).
[36] A. Vaidya and C. Farina, Can Galilean mechanics and full Maxwell equations coexist peacefully?, Phys. Lett. A 153, 265 (1991).
[37] O.D. Je menko, On the Relativistic Invariance of Maxwell's Equation, Z. Naturforsch. 54a, 637 (1999).