Abstract

Let A 1 , A 2 , A 3 A 4 be four observables, the compatible observables among them being (A 1 , A 3 ), (A 1 , A 4 ), (A 2 , A 3 ), (A 2 , A 4 ). 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=CijC ii −1/2 C jj −1/2 . In the case ofGaussian data, this inequality is alsosufficient; If (*) holds, there is a Gaussian joint distribution for A 1 , A 2 , A 3 , A 4 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