for any scalar functionθ. The electric and magnetic fields are first order time and space-dervatives of
the electric potential and of the vector potential, and bot are gauge invariant. The most systematic
way to construct the electric and magnetic field is to precisely take advantage of these two properties.
Computing the general first order derivatives gives∂μAν, which behaves as follows under a gauge
transformation ofAμ,
∂μAν →∂μA′ν=∂μAν+∂μ∂νθ (21.51)Thus, the most general first order derivative ofAμis not gauge invariant. The gauge term∂μ∂νθ,
however, is always symmetric under the interchange ofμandν. Therefore, we are guaranteed that
anti-symmetricpart of the derivative will be gauge invariant. The correspondingfield strengthis
defined by
Fμν=∂μAν−∂νAμ (21.52)Counting the number of independent fields in the rank 2 anti-symmetric tensorFμν=−Fνμgives
4 × 3 /2 = 6, which is precisely the correct number to incorporate the 3 components ofEand
the three components ofB. DecomposingFμνaccording to its space and time indices, and using
anti-symmetry, we find the following parts,
F 0 i = ∂ 0 Ai−∂iA 0
Fij = ∂iAj−∂jAi i,j= 1, 2 , 3 (21.53)The standard electric and magnetic fields may now be identified as follows,
Fi 0 = EiFij =∑^3k=1εijkBk (21.54)As a matrix, the field strength tensor has the following entries,
Fμν=
0 −E 1 −E 2 −E 3
E 1 0 B 3 −B 2
E 2 −B 3 0 B 1
E 3 B 2 −B 1 0
μν(21.55)
21.8.2 Maxwell’s equations in Lorentz covariant form
Lorentz invariance of the equations dictates, to a large extent, the structure of the possible equations
we may have for the gauge fieldAμ, and its gauge invariant field strength tensorFμν. Maxwell’s
equations emerge in two groups; a first set independent of theexternal electric current densityjμ,
and a second set which does involvejμ.
The first group of Maxwell’s equations results directly fromthe fact thatFμνis a “curl” in the
4-dimensional sense. FromFμν=∂μAν−∂νAμ, it readily follows that
εμνρσ∂ρFμν= 2εμνρσ∂ρ∂μAν= 0 (21.56)