Abstract
The concept of event provides the essential bridge from the realm of virtuality of the quantum state to real phenomena in space and time. We ask how much we can gather from existing theory about the localization of an event and point out that decoherence and coarse graining—though important—do not suffice for a consistent interpretation without the additional principle of random realization.
Similar content being viewed by others
References
Bohr, N.: In: Schilpp, P.A., Einstein, A. (eds.) Philosopher-Scientist. Tudor, New York (1951)
Bohr, N.: Atomic Theory and the Description of Nature. Cambridge University Press, Cambridge (1934)
Bohm, D.: A suggested interpretation of the quantum theory in terms of hidden variables I. Phys. Rev. 85, 166–179 (1952)
Bopp, F.: Würfelbrettspiele, deren Steine sich annähernd quantenmechanisch bewegen. Z. Naturforsch. 10a, 783–789 (1955)
Heisenberg, W.: Die Physikalischen Prinzipien der Quantentheorie. Hirzl, Leipzig (1930)
Griffiths, R.B.: Consistent Quantum Theory. Cambridge University Press, Cambridge (2002)
Bell, J.: On the Einstein-Podolsky-Rosen paradox. Physics 1, 195–200 (1964)
Clauser, J.F., Horne, M.A., Shimony, A., Holt, R.A.: Proposed experiment to test local hidden variable theories. Phys. Rev. Lett. 49, 1804–1807 (1969)
Kochen, S., Specker, E.P.: The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59–87 (1967)
Maxwell, N.: Is the quantum world composed of propensitons. In: Suarez, M. (ed.) Probabilities, Causes and Propensities in Physics, pp. 221–243. Springer, Dodrecht (2011)
Haag, R.: Quantum physics and reality. In: Conference Proceedings Gargnano 2000, Zeitschrift für Naturforschung 56a, pp. 76–82 (2001)
Acknowledgements
I want to thank Heide Narnhofer for many discussions and encouragement during several years. I am greatly indebted to Detlev Buchholz and Erhard Seiler for essential criticism and vital assistance during the last stages of this work.
Author information
Authors and Affiliations
Corresponding author
Appendix: Non-uniqueness of Decomposition of a General State—Effective Coherence Length
Appendix: Non-uniqueness of Decomposition of a General State—Effective Coherence Length
Let us start from a pure state with almost sharp momentum with mean value \(\bar{p}\), mean position \(\bar{x}\) and momentum uncertainty γ −1/2 described by the wave function in momentum space (we omit normalization factors)
or in x space
We consider a mixture of such states corresponding to an ignorance of the precise values \(\bar{p}\) and \(\bar{x}\) expressed by the weight function \(\mathrm{e}^{-\frac{\beta}{2}(\bar{p} -\hat{p})^{2}-\frac{1}{2\alpha }{\bar{x}}^{2}}\). The statistical matrix in x space is \(\langle x'|\rho_{1} |x\rangle = \int\mathrm{d}\bar{x} \mathrm{d}\bar{p} \mathrm{e}^{-K_{x}}\) with
Integration over \(\bar{p}\) gives \(\langle x'|\rho|x\rangle= \int\mathrm{d}\bar{x} \mathrm {e}^{-K_{1}}\) with
with
The same statistical matrix is obtained if we start from a pure state given by the wave function in x space
and consider the mixture given with a weight factor \(\exp(-\frac{\bar{x}^{2}}{2\alpha'})\). It leads at first sight to the following expression for the statistical matrix
with
After integration over \(\bar{x}\) we see that ρ 1=ρ 2 provided
If γ≫β, in the first version we then have a very large coherent extension γ 1/2 of the pure components. In the second version the effective coherence length β′1/2 is much smaller corresponding to the much larger subjective ignorance of the momentum.
Rights and permissions
About this article
Cite this article
Haag, R. On the Sharpness of Localization of Individual Events in Space and Time. Found Phys 43, 1295–1313 (2013). https://doi.org/10.1007/s10701-013-9747-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10701-013-9747-z