What Are the Degrees of Freedom in Lorenz and Columb Gauges?

[mkbh_10]

My teacher told me that in columb gauge we have div A = 0 along with d/dt(Ao) = 0 where A is the vector potential and Ao is its time component , and time & space are not on equal footing , so these are two degrees of freedom and in lorenz gauge we have 4 dimensional divA = 0 where space & time are on equal footing so there are 3 degrees of freedom .

These are the constraints on the potential , so how come they are degree of freedom and what are the 3 degrees of freedom in lorenz gauge .

 

[rkrsnan]

The electromagnetic potential [tex](A^0, A^1, A^2, A^3)[/tex] represents four numbers assigned at every point in space and time. Therefore on the face of it, it seems like we have four degrees of freedom(four numbers to represent the system at every space time point). But the fact is that some degrees of freedom are redundant. I will give an example. You are given one liter of water which is 1000 cm^3. But I can say the volume as "10cmx10cmx10cm" or "100cmx10cmx1cm" or any other set of values that gives the same volume. So as far as the volume is concerned giving three numbers causes extra degrees of freedom which is of no relevance. For the electromagnetic potential only two degrees of freedom are relevant. Now you can remove the extra degrees of freedom by giving some constraints. For the volume case let me give the constraints as the length=100 cm and width = 10cm. Under this constraint giving the height alone gives the volume. So we have only one free degree of freedom.

In coulomb gauge as far as I know there is only one constraint div A = 0. Lorentz gauge has also only one constraint 4DDiv A =0. So they add one constraint between the four variables and even after putting that constraint the Maxwell equations remain the same. So we prove that one extra degree of freedom was there and we remove it by having the coulomb or lorentz gauge. Now the source part in the Maxwell equations (the charge and current densities) can be expressed in terms of the the electromagnetic potential. We have a continuity equation which is a constraint between the charge and the current densities. So this causes a second constraint for the electromagnetic potential. Thus only two degrees of freedom are relevant.(We fixed one degree of freedom by the choice of the gauge and we lost one degree of freedom because of the continuity equation constraint). It is like taking water in a box having its base area constrained to be 100cm^2. you can fix the length(this is gauge fixing), then the width is constrained to be 100/length(this is like continuity equation constraint), now the height is the only degree of freedom that determines the volume.

My apologies if my volume comparison is silly :)