Int J Theor Phys (2008) 47: 1459–1470 DOI 10.1007/s10773-007-9588-7 Stochastically and Intrinsically Extended Non-relativistic Quantum Particles A. Smida · M. Hachemane · A.-H. Hamici · Y. Oualili Received: 13 May 2007 / Accepted: 5 October 2007 / Published online: 23 October 2007 © Springer Science+Business Media, LLC 2007 Abstract Stochastically and intrinsically extended non relativistic quantum particles are described by combining the ideas of a stochastic quantum theory and a quantum functional theory. The former relates the extension to imperfect real measurements while the latter considers it as intrinsic. Physical states, Positive-Operator-Valued measures connected to measurement, and propagators are given and discussed. The stochastic theory is sufficient when the bilocal field describing the particle has a product form. Keywords Positive-Operator-Valued measures · Extended particles · Stochastic theory · Quantum mechanics 1 Introduction Conventional quantum mechanics is based on a pointlike conception of elementary particles. This conception extrapolates to the relativistic regime and to conventional quantum field the- ory although it has been contested from the early days of quantum mechanics. De Broglie tried to conceive the pointlike image as a singularity in a physical wave u which represents an extended body [5]. Thereafter, Destouches proposed a generalized version of this idea in his functional theory [6, 7]. Its main feature is that the analysis of the concept of physical system with respect to the remaining part of the Universe leads to the influence of the lat- ter on the intrinsic characteristics of the former. As a consequence, an elementary particle may be represented by a function u describing these characteristics and, as such, it must be conceived as a nonrigid extended body. This replacement of the pointlike conception x ∈ R 3 by a functional conception u entails a replacement of the conventional quantum mechanical wave function ˆ ψ t (x) = ˆ ψ(t, x) by a quantum functional wave X t [u]= X(t,u). The state X may not belong to a Hilbert space (but to a space which contains one) and the physical wave u may be handled by associating it to a (realistic) model. Of course, abiding by a re- alistic standpoint is comforting but we prefer taking a detached point of view by replacing A. Smida · M. Hachemane ( ) · A.-H. Hamici · Y. Oualili Faculté de Physique, USTHB, B.P. 32 El-Alia Bab-Ezzouar, Alger 16111, Algeria e-mail: hachemane@wissal.dz