 where \(M^k\) projects onto the entire (macroscopic) subspace in which the apparatus pointer is in position \(k\). Since any two situations which are macroscopically distinct correspond to mutually orthogonal subspaces of the quantum Hilbert space, the \(M^k\) for different \(k\) will be orthogonal to each other. In addition one can include a special projector, say \(\hat M\), in the sum to represent any additional possibilities (e.g., the pointer has fallen off the apparatus), to make the sum add up to \(I\). The \({|m^j\rangle }\) of Sec. [sbct7.1] are related to the \(M^k\) through