On the mathematical structure of quantum measurement theory

Geoffrey Sewell, Queen Mary, University of London

We show that the key problems of quantum measurement theory, namely the reduction of the wave-packet of a microsystem and the specification of its state by a macroscopic measuring instrument, may be resolved within the traditional framework of the quantum mechanics of finite systems. The argument is centred on the generic model of a microsystem, S, coupled to a finite macroscopic measuring instrument, I, which itself is an N-particle quantum system. The pointer positions of I correspond to the macrostates of this instrument, as represented by orthogonal subspaces of the Hilbert space of its pure states. These subspaces, or "phase cells", are the simultaneous eigenspaces of a set of coarse grained macroscopic observables and, crucially, are of astronomically large dimensionalities, which increase exponentially with N. We formulate conditions on the conservative dynamics of the composite (S+I) under which it yields both a reduction of the wave packet of the state of S and a one-to-one correspondence, following a measurement, between the pointer position of I and the resultant eigenstate of S, and we show that these conditions are fulfilled, up to utterly negligible corrections that decrease exponentially with N, by the finite version of the Coleman-Hepp model.