Measurability of quantum fields and the energy–time uncertainty relation

Quantum restrictions on the measurability of an electromagnetic field strength and their relevance to the energy–time uncertainty relation are considered. The minimum errors in measuring electromagnetic field strengths, as they were estimated by the author (1988) in the framework of the phenomenological method of restricted path integral (RPI), are compared with the analogous estimates found by Landau and Peierls (1931) and by Bohr and Rosenfeld (1933) with the help of certain measurement setups. RPI-based restrictions, including those of Landau and Peierls as a special case, hold for any measuring schemes meeting the strict definition of measurement. Their fundamental nature is confirmed by the fact that their associated field detectability condition has the form of the energy–time uncertainty relation. The weaker restrictions suggested by Bohr and Rosenfeld rely on an extended definition of measurement. The energy–time uncertainty relation, which is the condition for the electromagnetic field to be detectable, is applied to the analysis of how the near-field scanning microscope works.