The transactional interpretation, counterfactuals, and weak values in quantum theory
Section snippets
Introduction and background
Cramer, 1980, Cramer, 1986, Cramer, 1988 presented his transactional interpretation (TI) in the 1980s. TI proposes that the usual quantum-mechanical state characterizes an “offer wave” (OW) emitted in the usual forward time direction from a source, and adds to this picture the idea that absorbers in the future light cone of such a source emit advanced or backward time-directed “confirmation waves” (CW) back to the source, upon receiving all or part of such an OW. The overlap of such offer
A specific example of the TI
To the above rough sketch we now fill in some details using a specific example, the famous (or perhaps infamous) three-state or three-box experiment.
Fig. 2 shows the basic setup for the three-state experiment. It usually involves three boxes or shutters labeled and C (essentially three possible locations at which a particle could be found). In this version, we use a “three-slit” arrangement, in which detectors might be placed at one or more of the slits. Particles are pre-selected in the
Weak values in the TI picture
“Weak values” are quantities introduced by Aharonov and Vaidman (1990) in the context of pre- and post-selection experiments (such as the three-state example discussed above).
The weak value of the operator O with respect to states and is defined as
Several authors have used weak values as indicators of properties in pre- and post-selected systems, and as answers to counterfactual questions about the properties of such systems between measurements. For example, in the
TI and the possession of properties
In the TI, the possession of properties corresponding to values of observables is underdetermined by such theoretical quantities as probabilities or weak values. Instead, the precise nature of the particular experiment must be specified, as the latter will determine what types of transactions are possible; and under the TI, property possession corresponds to a special type of transaction (to be described below).
For example, under TI it may the case that a system seems to be characterised by a
TI and consistent histories
In the consistent histories formulation pioneered by Griffiths, 1996, Griffiths, 1999, Griffiths, 2002, one can assign standard “classical” or Kolmogorov-type probabilities for different sequences of events, called ‘histories,” provided that the set of such histories fulfills a consistency criterion (see below) which ensures that probabilities for distinct histories are additive. A history F is a projector on the multiple-time Hilbert space of the system corresponding to the number n of
Conclusion
Cramer's transactional interpretation has been applied to several commonly discussed pre- and post-selection experiments. It has been argued that TI provides insight into the nature of time-symmetric weak values: namely, that they should be interpreted as multiple-time amplitudes, rather than as generalized expectation values. Weak values of projection operators—even when the pre- and post-selection states are the same—do not reflect Born probabilities, which under TI arise from the overlap of
Acknowledgment
I have benefited from helpful comments on an earlier draft from two anonymous referees.
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