The appearance of tracks, close to classical orbits, left by charged quantum particles propagating inside a detector, such as a cavity periodically illuminated by light pulses, is studied for a family of idealized models. In the semi-classical regime, which is reached when one considers highly energetic particles, we present a detailed, mathematically rigorous analysis of this phenomenon. If the Hamiltonian of the particles is quadratic in position- and momentum operators, as in the examples of a freely moving particle or a particle in a homogeneous external magnetic field, we show how symmetries, such as spherical symmetry, of the initial state of a particle are broken by tracks consisting of infinitely many approximately measured particle positions and how, in the classical limit, the initial position and velocity of a classical particle trajectory can be reconstructed from the observed particle track.
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Note that, in general, an abstract random variable is denoted by a capital letter, while its values are denoted by the corresponding lower-case letter. For example, the approximate position of a particle is a random variable denoted by Q and its measured values are denoted by q.
More precisely, that for any C > 0, there exists a finite such that for any n ≥ n0, Σn > C.
