# The Field Equations in the Feynman Approach

• 陳 慕義

Student thesis: Doctoral Thesis

### Abstract

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
Date of Award 2019 English Su-Long Nyeo (Supervisor)

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