Undecidability of Macroscopically Distinguishable States in Quantum Field Theory

A heuristic discussion is presented regarding quantum field theory as a synthesis of the complementary theories of classical mechanics and quantum mechanics. If the states of quantum field theory are partitioned in equivalence classes accordingly as their occupation numbers differ in a finite or an infinite number of places, it is suggested that we define states to be macroscopically distinguishable if they belong to different equivalence classes. It is then proven that there is, in general, no effective procedure for determining whether or not two arbitrarily given states of a quantum system having an infinite number of degrees of freedom are macroscopically distinguishable.