## Appendix E Analogy to Electrodynamics
Lightning is an example of arc discharge in nature. Under some conditions, however, the above mentioned positive feedback mechanism can stop when the density of free charges is relatively low. In this case, an arc-discharge like phenomenon can occur in a local region of a gas medium for a short duration, but it does not lead to electrical breakdown of the entire volume. It disappears and the gas medium regains the less conductive state. This type of discharge is called the spark discharge. Any mechanism that reduces the number of free charges (electrons) can prevent such a local arc-like discharge from growing to the global scale that leads to electric breakdown of the gas medium. The analogy between the present field Equation (6) and Ampere law (13) is not a formulaic coincidence. Compare the conduction current expression (14) and the longitudinal force expression for plasticity (9). Both represent longitudinal energy damping effects, and we find the following correspondence. \[\sigma_{0}\rho(\
abla\cdot\mathbf{v})\mathbf{v}\leftrightarrow\sigma_{e}\mathbf{E} \tag{15}\]  
Comparison of the present field equations and Maxwell equations indicates that the electric field \(\mathbf{E}\) and magnetic field \(\mathbf{B}\) correspond respectively to particle velocity \(\mathbf{v}\) and rotation \(\mathbf{\omega}\). Based on this correspondence, we can interpret that (15) indicates that the damping effect in the deformation field corresponds to a quantity proportional to \((\
abla\cdot\mathbf{E})\). According to [PERSON] law, \((\
abla\cdot\mathbf{E})=\rho_{e}\) (charge density). In other words, the energy dissipative dynamics in the deformation field is equivalent to Ohmic loss in electrodynamics. Above we discussed that when a shear band appears intermittently in association of serration the phenomenon can be interpreted as recoverable fracture. In electrodynamics, this corresponds to a spark discharge which eventually develops to an arc discharge followed by electrical breakdown. This transition corresponds to a transition from recoverable fracture to final (unrecoverable) fracture. At a more profound level of physics, the above similarity can be argued as follow. The present gauge field makes [PERSON]'s law locally symmetric. The vector potential \(\mathbf{A}\) aligns all the deformation structural elements (DSE) so that the deformation in the plastic regime can be expressed at the global level based on [PERSON]'s law. The electromagnetic field is a gauge field that makes Shrodinger equation locally symmetric [11]. Consider that an electron orbiting an atom changes its orbit for some reason (such as a positive ion attracts it). Then the question arises; \"Do all other electrons know this orbital change?\" The answer is \"Yes, they know because they feel a change in the electric potential due to this electron.\" We can also answer the question saying that the distance from this electron changes for all the other electrons. Those electrons for which the orbital change makes the distance from this electron shorter (longer) feel a stronger (weaker) Coulomb force. The important point is that Shrodinger equation cannot describe this electron-electron interactions. Electrodynamics does. In this dynamics, the electric charge is the conserved quantity, the charge of symmetry. Its temporal component is the electric charge density and spatial component is the electric current density. In the present case, \(j^{0}\) (the deformation charge density) corresponds to the electric charge density and \(j\) to the current density. All deformation charges know each others' behavior through vector potential \(A\). The interactions between deformation charges somewhat resembles dislocation-dislocation interaction. Charges of the same (opposite) sign repel (attract) each other as dislocations of the same (opposite) sign repel (attract) each other. More detailed discussions can be found in ref. [9].