Volume 108A, number 8 PHYSICS LETTERS 22 April 1985 ON THE COVARIANT DESCRIPTION OF WAVEFUNCTION COLLAPSE D. DIEKS Fysisch Laboratorium, Rijksuniversiteit Utrecht, Postbus 80.000, 3508 TA Utrecht, The Netherlands Received 20 February 1985; accepted for publication 28 February 1985 The instantaneous change of the wavefunction known as the "collapse" of the wavefunction leads to ambiguities in relativistic quantum mechanics. It is here shown that a relativistically satisfactory generalization follows from the treatment of the measurement as a physical interaction. 1. Introduction. According to the usual account of measurement in quantum mechanics an ideal mea- surement can be described by means of the instanta- neous collapse of the wavefunction (or, more general- ly, of the density matrix). As this procedure is not relativistically covariant, it leads to ambiguities in relativistic quantum theory [1-3]. For instance, if by some local measurement the position of a particle is measured, the wavefunction collapses to become a mixture at all points which are simultaneous with the measurement event. However, application of this prescription by different Lorentz observers results in different assignments of the equal time hyperplane along which the collapse takes place. It depends there- fore on which Lorentz frame is taken as the frame of reference whether a pure state or a mixture is attrib- uted to one and the same space-time point. Clearly, these candidates for the state description cannot be Lorentz transforms of one another. The lack of covariance of the usual prescription has induced various authors [2,3] to propose relativis- tically covariant generalizations of the collapse postu- late. It is the purpose of this note to show that the consistent treatment of the measurement as a quan- tum mechanical interaction unambiguously leads to the requ~ed generalization; in fact, we shall corrob- orate the solution proposed by Aharonov and Albert [3]. 2. Measurement and collapse. We take as the basis of our account the treatment of the measurement as 0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) a physical interaction. Let Iq~)= ~,kCk [~k ) be the statevector of the system under consideration before the interaction; I~k) is an eigenvector of the operator corresponding to the physical quantity which is about to be measured. Similarly, I~0) is the statevector of the measuring device before the interaction. An ideal measurement interaction effects the following change of state: I xI') I~0) ~ k ~ Ck I~kk)I~k). (1) Here the I~k ) are the apparatus states which become correlated, through the interaction, with the object states [¢Jk ). The criterion for a "good" measurement interaction, which discriminates between the various values of the physical quantity which is measured, is that the Iek) are (practically) orthogonal: (~kl~k') = 8kk' • (2) It is important to note that by virtue of relation (2) the expectation value of any operator pertaining to the object system alone, calculated in the final state of (1), can be written as (0) = ~ [Ckl2 (~klOl~k). (3) k That is, the interaction with the measurement device "decouples" the various Iffk) so that there is no effect of interference between them. (We have assumed in the foregoing that there is only one "pointer basis" {lek)}, see refs. [4,5] .) 379