Foundations of Physics

, Volume 18, Issue 4, pp 449–460

Empirical two-point correlation functions

  • Lawrence J. Landau

DOI: 10.1007/BF00732549

Cite this article as:
Landau, L.J. Found Phys (1988) 18: 449. doi:10.1007/BF00732549


Let A1, A2, A3 A4 be four observables, the compatible observables among them being (A1, A3), (A1, A4), (A2, A3), (A2, A4). In order that the empirical data be reproducible by a quantum or a classical theory, the two-point correlation functions
$$\{ C_{ij} = \left\langle {A_i A_j } \right\rangle :i,j a compatible pair\} $$
must necessarily satisfy
$$|X_{13} X_{14} - X_{23} X_{24} | \leqslant \left( {1 - X_{13} ^2 } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \left( {1 - X_{14} ^2 } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + \left( {1 - X_{23} ^2 } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \left( {1 - X_{24} ^2 } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} (*)$$
where Xij=CijCii1/2Cjj1/2. In the case ofGaussian data, this inequality is alsosufficient; If (*) holds, there is a Gaussian joint distribution for A1, A2, A3, A4 which reproduces the Gaussian data for compatible pairs. It follows that Bell's inequality is satisfied by all true-false propositions about the Gaussian data. A further consequence of the analysis is thatquantum Gaussian fields satisfy Bell's inequality for all true-false propositions aboutfield measurements.

The maximum violation of (*) corresponds to Rastall's example in the case of two-valued observables.

Copyright information

© Plenum Publishing Corporation 1988

Authors and Affiliations

  • Lawrence J. Landau
    • 1
  1. 1.Department of Mathematics, King's CollegeUniversity of LondonLondonEngland