Time correlation functions

In standard quantum mechanics, the calculation of any two-time correlation function has to include the evolution of the system between the two times considered; this evolution is contained in the unitary evolution operator U(t′, t), as for instance in relation (10.9). In Bohmian theory, it is important to take into account the effect of the first measurement, which correlates the system under study to a measurement apparatus and creates “empty waves” (§10.6.1.c). Otherwise one obtains contradictions with the standard predictions.

For instance the author of [443] considers a one-dimensional harmonic oscillator that is initially in a stationary state, and studies the correlation function of the position at times t and time t′, in the particular case where t′ − t is equal to half the period 2π/ω of the oscillator. In standard quantum mechanics, it is easy to show that the corresponding position operators 〉X(t) and X(t′) are then opposite, so that the correlation function 〉X(t)X(t′〈 is equal to − 〉[X(t)]2〈, therefore negative. In Bohmian mechanics, the particle is initially static since the wave function is real. If one ignores the effect of the first measurement, the position of the particle will remain at the same place, which corresponds to a correlation function equal to 〉[X(t)]2〈, therefore positive; one reaches an apparent complete contradiction. But, if one takes into account the effect of the first measurement on the particle, one finds that, just after this measurement, each position of the oscillator becomes correlated with a different Bohmian position of the pointer.