Lorentz-Invariant “Elements of Reality” and the Question of Joint Measurability of Commuting Observables

It is shown that the joint measurements of some physical variables corresponding to commuting operators performed on preand post-selected quantum systems invariably disturb each other. The significance of this result for recent proofs of the impossibility of realistic Lorentz invariant interpretation of quantum theory (without assumption of locality) is discussed. hep-th/9305162 , TAUP 2027-93, to be published in PRL. Recently, few authors 1−4 by taking very plausible definition of “elements of reality” claimed to show that Lorentz invariant realistic interpretation of quantum mechanics is not possible. Contrary to the Einstein-Podolsky-Rosen (EPR) argument, their proof have not based on the locality assumption. In this Letter we will show that contradiction disappear when we abandon a “product rule” of elements of reality, even for elements of reality corresponding to commuting operators. We will show that the product rule has to be abandoned because joint measurements of commuting operators in the considered situations invariably disturb each other. The plan of this Letter is as follows: We shall start with the discussion of the problem of measurement performed on a preand post-selected quantum system. We will show that even commuting operators can not be measured on such systems without disturbing each other. Then we shall briefly present the arguments against Lorentz invariant realistic interpretation of quantum mechanics due to Pitowsky, 1−2 and Hardy, 3−4 and we shall explain their usage of the “product rule”. We shall conclude with a brief discussion of the two-state vector approach in which the elements of reality are Lorentz invariant. In every textbook of quantum mechanics we can find a condition for simultaneous measurability of variables A and B: the corresponding operators must commute: