We discuss interferometers in Bohmian quantum mechanics. It is shown that, with the correct configuration space, Bohm trajectories in a which way interferometer are not surrealistic, but behaves exactly as common sense suggests. Some remarks about a way to generalize Bohmian mechanics to treat density matrix are also made.
PACS: 03.65.Bz, 03.75.Dg
Key words: Bohm Trajectories, Which Way Interferometers, ESSW
Let us begin with a response to the summary of the Comment by D. Durr, W. Fusseder, S. Goldstein, and N. Zanghi [1]. We are accused to not stay inside orthodox quantum but to employ a naive, largely incoherent operationalism instead. We think that this charge is not justified, because we did apply standard quantum theory correctly and also followed the rules of Bohmian Mechanics (BM) carefully. In the following we react to the more essential points of the Comment in some detail. The Comment suggests that we have conceded many virtues to BM and that we have admitted that standard quantum theory is plagued by notorious conceptual difficulties. This is not the case. On the contrary, we maintain that BM is not needed to have the Schrodinger equation embedded into a physical theory. Standard quantum theory has already clarified the significance of Schrodinger's wave function as a tool used by theoreticians to arrive at probabalistic predictions. It is quite unnecessary, and indeed dangerous, to attribute any additional meaning to the (/^-function. The semantic difference between inconsistent and surrealistic is not the issue. It is the purpose of our paper to show clearly that the interpretation of the Bohm trajectory as the real retrodicted history of the atom observed on the screen is implausible, because this trajectory can be macroscopically at variance with the detected, actual way through the interferometer. And yes, we do have a framework to talk about path detection; it is based upon the local interaction of
de Broglie-Bohm theory (dBBT), treating quantum particles as point objects moving along well defined (Bohmian) trajectories, offers an appealing solution of the measurement problem in quantum mechanics; it has, however, problems relating to spin, relativity and lack of proper integration with the Hilbert space based framework. In this work, we present a consistent formalism which has the traditional state-observable framework integrated with the desirable features of dBBT. We adopt ensemble interpretation for the Schrodinger wave function
ψ. Given a Schrodinger wave function
ψ, we use its value
ψ0 at some fixed time (say,
t=0) to define the probability measure
∣ψ0∣2dx on the system configuration space
M (
=Rn). On the resulting probability space
M0, we introduce a stochastic process
ξ(t) corresponding to the Heisenberg position operator
XH(t) such that, in the Heisenberg state
∣ψh⟩ corresponding to
ψ0, the expectation value of
XH(t) equals that of
ξ(t) in
M0. This condition leads to the de Broglie-Bohm guidance equation for the sample paths of the process
ξ(t) which are, therefore, Bohmian trajectories supposedly representing time-evolutions of individual members of the
ψ0-ensemble. Stochastic processes and Bohmian trajectories corresponding to observables with discrete eigenvalues (in particular spin) are treated by extending the configuration space to the spectral space of the commutative algebra obtained by adding appropriate discrete observables to the position observables. Pauli's equation is treated as an example. A straightforward derivation of von Neumann's projection rule employing the Schrodinger-Bohm evolution of individual systems along their Bohmian trajectories is given. Some comments on the potential application of the formalism developed here to quantum mechanics of the universe are included.