there are some points about electrodynamics you can rather easily see from the field tensor.
in 1+1 dim, the field tensor is a 2X2 matrix with only one independent component.this component is the 01 component:which is one space and one time.so this is an electric field.note that E is a polar vector and so is the 01 component; hence the electric field is a scalar in 1+1 dimension.
all components exhausted, there is no B in 1+1 dimension!!
this also follows from Biot-savart law; the force on a current due to another current acts in a direction prependicular ot the line joining them, which is not possible in 1 space.
now lets come to 2 +1 dim.the electric field has 2 components.
what about the magnetic field ?
recall biot-savart law.force is given as a cross pdt, it has the structure of Fj=εjkvkB so it turns out that it is a scalar.also since F and v are polar vectors,B must ont change sign under parity and hence it is a true vector.
Bianchi identity: suppose u have an antisymmetric object and u have an operator that acts linearly on the tensor , then the sum of all cyclic permutations of the tensor add up to zero.
Fij=DjAi – DiAj (1)
define tijk=DkFij (2)
then the bianchi identity is
Tijk=tijk+tjki+tkij=0 (3)
as can be shown by substituting 1 & 2 in 3.
Next,using the anti-symmetric properties of Tijk in 3 we see that only 4 non-trivial equations result.knowing that this involves no source terms, we suspect these to be the source free maxwell equations.there are the right number of them!so the bianchi identity are the source free maxwell equations
why do you need a four-vector potential to characterize the field in the action formulation?well,this is because you have E from a scalar potential and B from a vector potential.
note that the convention E= -div (phi) comes from the convention that a positive charge moves in the direction of lower field.
the sturcture of the bianchi identity, ie, of some linear operator acting on an antisymetric structure is the same as for maxwell equations.
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