Measuring quantities at a point

Bjorken and Drell, Relativistic Quantum Fields, 36:

We may make [a certain divergent result] less unpleasant with the observation that one cannot in fact measure the square of a field amplitude at a point. In order to probe a single isolated point of space-time, one needs infinitely large frequencies and infinitesimally short wavelengths — and these are not to be achieved at less than infinite energies.

Bohr and Rosenfeld, On the Question of the Measurability of Electromagnetic Field Quantities:

a field-theoretic determination of the electromagnetic energy in a given space-time domain would require knowledge of the values of the field components at each space-time point of a region, which are inaccessible to measurement. A physical measurement of the field energy can be carried out only by means of a suitable mechanical device that would make it possible to separate the electromagnetic fields in a given region from the rest of the field, so that the energy contained in a region could be measured subsequently by application of the conservation law. However, because of the interaction with the measuring mechanism, any such separation of the fields would be accompanied by an uncontrollable change in the field energy in the region in question, the consideration of which is essential for clarification of the well-known paradoxes that arise in the discussion of energy fluctuations in black-body radiation.

Landau and Peierls, Extension of the Uncertainty Principle to Relativistic Quantum Theory, in Collected Papers of L. D. Landau, pp. 49-50:

We have seen that no predictable measurements can exist for the fundamental quantities of wave mechanics (except when these quantities are constant in time, and then an infinitely long time is needed for an exactly predictable measurement). It cannot, of course, be formally demonstrated that there are not in nature some particularly complicated quantities for which predictable measurements are possible, but such a speculation need not be discussed. The assumptions of wave mechanics which have been shown to be necessary in section 2 are therefore not fulfilled in the relativistic range, and the application of wave mechanics methods to this range goes beyond their scope. It is therefore not surprising that the formalism leads to various infinities; it would be surprising if the formalism bore any resemblance to reality.
In the correct relativistic quantum theory (which does not yet exist), there will therefore be no physical quantities and no measurements in the sense of wave mechanics. One can, of course, cause the system to interact with some apparatus and ask what happens to the latter. The theory will give a probability for the result of this experiment, but this cannot be interpreted as the probability of a parameter of the system under investigation., since it can in no way be ensured that the probability of a given result is unity and that of all other results is zero. IN addition, it is in principle impossible to make the duration of such an experiment arbitrarily short.