mjhjmtu

So I have the following curiosity: Consider for example, in QED, the quantity

$$
j^muequivpartial_nu (lambda(x) F^mu nu)
$$

where $lambda(x)$ is an arbitrary scalar function of spacetime, constructed from elements of the theory, e.g.

$$
lambda(x)=A_mu A^mu F_rho sigma F^rho sigma
$$

or anything else you can think of. Then, since $F^mu nu$ is antisymmetric, $partial_mu j^mu=0$ identically.

In fact this can be generalized to any theory; construct from various elements an antisymmetric two-rank tensor, and a scalar, multiply them together, and there is a corresponding conserved current for any choice you make. If these currents are not trivial (e.g. giving only vanishing charges) then it appears that all theories give an infinite landscape of conserved currents. Is this so? Am I missing something? How is this justified logically?

OP wrote (v2):

If these currents are not trivial [...]

In fact most$^dagger$ of them are trivial. The corresponding charges $$Q ~:=~ int_V d^3x~ j^0~=~int_V d^3x~sum_i=1^3d_i(lambda F^i0)~=~int_partial V d^2x~(ldots) ~=~0$$
vanish if the components $lambda F^i0$ fall off fast enough at spatial infinity $partial V$.

--

$^dagger$ The case $lambda=1$ corresponds to ordinary electric current, cf. Maxwell's eqs.

If we have $partial_nu F^munu=0$ on-shell, as happens e.g. in EM without a source current, $F^munuX_nu$ is conserved provided $partial_mu X_nu$ is symmetric, as happens e.g. with $X_nu=partial_nulambda$. Not all these conserved currents are trivial. See Sec. 2.1.2 of my thesis, which generalises this to curved spacetime.