It's not correct that the field can't be resolved at visible wavelengths, whether in principle or in practice (especially these days) and the issues aren't limited to resolution at those wavelengths.

(a)The intensity is the invariant quantity, that's constructed from the fields (electric and magnetic) that are invariant under rotations and Lorentz transformations. The field is just a means to obtaining the intensity, not the other way around. That's why it doesn't make sense to claim a measurement of the field, same as for any quantity that isn't invariant.

(b) For the reason mentioned above. The field isn't invariant under the symmetries of Maxwell's equations. It's the means for obtaining the invariant quantities, among them the intensity.

yes Dr Stam Nicolis, in quantum theory or semi-empirical theory, we use the probability of intensity and the field is a tool to obtain the intensity, field isn't invariant.

thank you Dr. Nicolis and Dr. Alagealy for the replies. I just read that the electric field can mesasured using attosecond pulses.

Regarding part (2) of my question about the detector, can you please provide a physical explanation? For me I cannot fully understand the physics behind your replies as I am very new into this field.

Τhe electric (or magnetic) field isn't an invariant quantity-it depends on the frame. It's one of the building blocks of invariant quantities. The invariant quantity is the intensity, the spatial average of the energy density of the electromagnetic field.

thank you. The intensity is given by the Poynting vector. It's defined as the vector product between the electric and magnetic fields. But if it's a vector then why is it invariant?