The transactional interpretation, counterfactuals, and weak values in quantum theory

https://doi.org/10.1016/j.shpsb.2008.02.005Get rights and content

Abstract

In recent years, a number of authors have studied pre- and post-selected quantum systems and associated time symmetry considerations. In this context, numerous paradoxes have arisen which raise questions about what can be inferred about such a system based on theoretically calculated quantities such as the Aharonov–Bergmann–Lebowitz (ABL) probabilities; and, more recently, “weak values”—time-symmetric quantities applicable to pre- and post-selected systems. This paper applies to some of these problems a time-symmetric interpretation of quantum theory: the “transactional interpretation” (TI) of Cramer, first proposed in 1980. The TI picture supports the conclusion that weak values are properly interpreted as multiple-time amplitudes rather than as generalized expectation values. It also prompts a stricter constraint on the counterfactual usage of the ABL rule than the consistency of the associated family of histories, which has previously been regarded as sufficient.

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 |ψ 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 A,B 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 |a and |b is defined asOw=b|O|ab|a

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 Hˇ 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.

References (30)

  • R.E. Kastner

    Time-symmetrised quantum theory, counterfactuals and advanced action

    Studies in History and Philosophy of Modern Physics

    (1999)
  • R.E. Kastner

    Weak values and consistent histories in quantum theory

    Studies in History and Philosophy of Modern Physics

    (2004)
  • Aharonov, Y., Botero, A., Popescu, S., Reznik, B., & Tollaksen, J. (2002). Revisiting Hardy's Paradox: Counterfactual...
  • Y. Aharonov et al.

    Properties of a quantum system during the time interval between two measurements

    Physical Review A

    (1990)
  • Y. Aharonov et al.

    Complete description of a quantum system at a given time

    Journal of Physics A

    (1991)
  • J. Bub

    Interpreting the quantum world

    (1997)
  • L. Chiatti

    Path integral and transactional interpretation

    Foundations of Physics

    (1995)
  • O. Cohen

    Pre- and postselected quantum systems, counterfactual, measurements and consistent histories

    Physical Review A

    (1995)
  • J.G. Cramer

    Generalized absorber theory and the Einstein–Podolsky–Rosen paradox

    Physical Review D

    (1980)
  • J.G. Cramer

    The transactional interpretation of quantum mechanics

    Reviews of Modern Physics

    (1986)
  • J.G. Cramer

    An overview of the transactional interpretation of quantum mechanics

    International Journal of Theoretical Physics

    (1988)
  • Cramer, J. G. (1997). Quantum nonlocality and the possibility of superluminal effects. In Proceedings of the NASA...
  • J.G. Cramer

    A transactional interpretation of interaction free measurements

    AIP conference proceedings

    (2001)
  • R.B. Griffiths

    Consistent histories and quantum reasoning

    Physical Review A

    (1996)
  • R.B. Griffiths

    Consistent quantum counterfactuals

    Physical Review A

    (1999)
  • View full text