Time

In this paper we wish to explore the possibility of the existence of a quantity that can be regarded as the complement of the Hamiltonian for a quantum system with discrete energy levels, recognizing that if this quantity is represented by an operator the appropriate commutator must be a generalized form of Eq. 2. Although this quantity will have dimensions of time, it will not be appropriate to refer to such an operator as a time operator. As an operator it would represent an observable of the quantum system and not time in the abstract or coordinate sense or as a reading on an external clock. Its eigenstates would represent a state of the quantum system and some measurement on the system, at least in principle, should tell us about the quantity involved. Although this quantity is not time, we would hope that for a particular ideal system its expectation value may change linearly with time, for example, so that a measurement of the quantity would also give a measure of time. In this case, we would be using the system as a clock. Before seeking an operator for the complement of the Hamiltonian we first represent this quantity by a nonorthogonal probability-operator measure, which is a more general concept 6.