The uncertainty principle and electric field strength

[snoopies622]

As my knowledge of an electric field strength becomes more precise, what physical quantity becomes correspondingly less precisely known?

 

[dextercioby]

There's no (direct) analogue of the "uncertainy principle" in quantum field theory.

 

[tom.stoer]

There's no (direct) analogue of the "uncertainy principle" in quantum field theory.

One can start with the Maxwell Lagrangian and show that there are three pairs of canonical conjugate field operators

$$[A_i(x),E_k(y)] = i\,\delta_{ik}\,\delta^{(3)}(x-y)$$

The problem is that we have a gauge theory, so there are unphysical d.o.f. which we have to eliminate; a first choice is the A°=0 gauge b/c for A° there's no E° due to the antisymmetry of the field strength tensor F and therefore ∂°A° is not present in F. Even then not all three i=1..3 are physical, we have to eliminate a second one to arrive at two physical polarizations.

Using Fourier decomposition one can introduce creation and annihilation operators for the fields; in terms of these operators it becomes clear that the canonically conjugate pair x,p is replaced by something like the Fourier modes of A and E, therefore one can derive an uncertainty principle for A(k) and E(k) where k is representing momentum space.

 

[snoopies622]

Thank you both. Good stuff. : )

 

[andrien]

One can start with the Maxwell Lagrangian and show that there are three pairs of canonical conjugate field operators

$$[A_i(x),E_k(y)] = i\,\delta_{ik}\,\delta^{(3)}(x-y)$$

The problem is that we have a gauge theory, so there are unphysical d.o.f. which we have to eliminate; a first choice is the A°=0 gauge b/c for A° there's no E° due to the antisymmetry of the field strength tensor F and therefore ∂°A° is not present in F. Even then not all three i=1..3 are physical, we have to eliminate a second one to arrive at two physical polarizations.

Using Fourier decomposition one can introduce creation and annihilation operators for the fields; in terms of these operators it becomes clear that the canonically conjugate pair x,p is replaced by something like the Fourier modes of A and E, therefore one can derive an uncertainty principle for A(k) and E(k) where k is representing momentum space.

Here A are operators,similarly in quantum field theory these exist these kinds of relation for bosons and also for fermions but that is rather anticommutation.