Schrödinger’s equation says that the Hamiltonian is the generator of time translations. This seems to imply that any reasonable definition of time operator must be conjugate to the Hamiltonian. Then both time and energy must have the same spectrum since conjugate operators are unitarily equivalent. Clearly this is not always true: normal Hamiltonians have lower bounded spectrum and often only have discrete eigenvalues, whereas we typically desire that time can take any real value. Pauli concluded that constructing a general a (...) time operator is impossible. Here we show how the Pauli argument fails when one uses an external system to track time, so that time arises as correlations between the system and the clock. In this case, the time operator is conjugate to the clock Hamiltonian and not to the system Hamiltonian, but its eigenvalues still satisfy the Schrödinger equation for arbitrary system Hamiltonians. (shrink)

Field propagation on fractal structures can generate a large scale symmetric space-time geometry. The significance of this fact and the nature of the resulting space-time are discussed.“Each contained all the others, but in this totality each was confused and comingled with all the others without order and system.”—Haïm Vital, 1543–1620 (Kabbalist of Safed); English translation taken from “Sabbatai Sevi” by G. Scholem (Routledge and Kegan Paul, London, 1973), p. 36.