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Thirty-one years ago , Dick Feynamn told me about his 'sum over histories' version of Quantum Mechanics.'The electron does anything it likes',he said, 'It goes in any direction at any speed,forward or backward in time,however it likes, and then add up the amplitudes and it gives the wave-function'. 'I said to him,'You're crazy'. But he was't.                                                             - F.J Dyson

Four Vector : $$x^{\mu}=\{ct,\overrightarrow{\mathbf{r}}\}$$$$x_{\mu} = \{ct,-\overrightarrow{\mathbf{r}}\}$$ Four Momentum : $$p^{\mu}=\{\frac{E}{c},\overrightarrow{\mathbf{p}}\}$$$$p_{\mu}=\{{\frac{E}{c}},-\overrightarrow{\mathbf{p}}\}$$ Four Potential : $$A^{\mu}=\{\phi,\overrightarrow{\mathbf{A}}\}$$$$A_{\mu}=\{{\phi},-\overrightarrow{\mathbf{A}}\}$$ Four Current : $$j^{\mu}=\{c\rho,\overrightarrow{\mathbf{J}}\}$$$$j_{\mu}=\{c\rho,-\overrightarrow{\mathbf{J}}\}$$ More Details click  here [PDF]

In physics a Gauge theory is a type of field theory in which the Lagrangian is invariant under certain 'Lie' group of local Transformation.Gauge Transformationis the english translation of German word 'Eichinvarianz' ,Where Eich means 'Gauge'. Classical Gauge Transformation in Electrodynamics:   The Maxwell's equations in free space :   $$ (1) \nabla.\overrightarrow{E} = \frac{\rho}{\epsilon_0} \text{  (Gauess's Law)}$$   $$(2) \nabla\text{.}\overrightarrow{B} = 0 \text{  (No Name i.e no existance of magnetic monopole)} $$   $$(3) \nabla\text{x}\overrightarrow{E}=\frac{\partial\overrightarrow{B}}{\partial{t}} \text{  (Farady's Law)} $$   $$ (4) \nabla\text{x}\overrightarrow{B}  = \mu_0\overrightarrow{J}+\mu_0\epsilon_0\frac{\partial{\overrightarrow{E}}}{\partial {t}} \text{(Ampears Law with Maxwell's corrections)}$$ Where $$\mu_0$$ and $$\epsilon_0$$ are free space permitivity and free space permeability respectively. Suppose we h...