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系統識別號 U0026-0707201914335000
論文名稱(中文) 費曼方法中的場方程
論文名稱(英文) The Field Equations in the Feynman Approach
校院名稱 成功大學
系所名稱(中) 物理學系
系所名稱(英) Department of Physics
學年度 107
學期 2
出版年 108
研究生(中文) 陳慕義
研究生(英文) Mu-I Chen
電子信箱 chenmy1989@gmail.com
學號 L28031067
學位類別 博士
語文別 英文
論文頁數 52頁
口試委員 指導教授-楊緒濃
口試委員-許祖斌
口試委員-游輝樟
口試委員-楊義清
口試委員-陳耀煌
中文關鍵字 費曼方法  泊松括號  李代數  對易空間  Helmholtz條件  推廣場方程 
英文關鍵字 Feynman approach  Poisson bracket  Lie algebra  commutative space  Helmholtz conditions  generalized field equations 
學科別分類
中文摘要 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。
英文摘要 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.
論文目次 1 Introduction . . . . . . . . . . . . . . . . . . . . . . 3
2 The Feynman Approach . . . . . . . . . . . . . . . . . . 7
2.1 The Equations of Motion . . . . . . . . . . . . . . . .8
2.2 The Field Equations . . . . . . . . . . . . . . . . . 9
3 The Feynman approach for Non-Abelian Gauge Field . . . .14
3.1 The Non-Abelian Structure . . . . . . . . . . . . . . 15
3.2 The Equations of Motion . . . . . . . . . . . . . . . 17
3.3 The Field Equations . . . . . . . . . . . . . . . . . 20
4 The Feynman approach for a Commutative Space . . . . . .25
4.1 The Helmholtz Conditions . . . . . . . . . . . . . . .27
4.2 The Equations of Motion . . . . . . . . . . . . . . . 30
4.3 The Field Equations . . . . . . . . . . . . . . . . . 33
4.3.1 The Helmholtz Conditions . . . . . . . . . . . . . .34
4.3.2 The Jacobi Identity . . . . . . . . . . . . . . . . 37
5 Discussion and Conclusion . . . . . . . . . . . . . . . 40
A The Time Derivative of F_{ij} . . . . . . . . . . . . . 43
B Derivation of the Fourth Helmholtz condition [30] . . . 45
C The Fourth Helmholtz Condition in O({dot x}^2) . . . . .47
Bibliography . . . . . . . . . . . . . . . . . . . . . . .49
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