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.

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We give the propagation of extended particles in a generic curved spacetime. The extended particle may be a scalar system of two quarks viewed as two quantum modes. Precisely, the first mode represents the global location of the extended particle in the curved spacetime and is quantized by the geometro-stochastic method which seems to be well suited for that purpose. The other mode, which represents a relative motion and is naturally confined in a de Sitter internal spacetime, is quantized by the method of induced representations. This corresponds to the relativistic harmonic oscillator in which the interaction has been replaced by a curvature of the internal space (relativistic rotator). States of the extended particle are then defined in a Hilbert bundle structure with the direct product of the external Poincaré and internal de Sitter symmetries playing the role of the structural group. Intertwining operators are used to define propagation in one fibre as a transition amplitude between the so-called local quantum frames. Parallel transport of these frames is used to define the total propagation of the extended particle as advocated by the geometro-stochastic theory.

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