It is shown that the stochastic quantization of a fermion introducing an anisotropy in the internal space so that this gives rise to two internal helicities corresponding to particle and antiparticle leads us to describe a fermion as a Skyrme soliton. The Skyrme term appears here as a consequence of this internal anisotropy and can be treated as a quantum effect. Some topological properties of this fermionization are then discussed.

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On the basis of earlier work on relativistic generalization of Nelson’s stochastic quantization procedure introducing an anisotropy in the internal space it is shown here that in the non-relativistic limit the equivalence of stochastic and hydrodynamical quantizations formulated respectively by Nelson and Takabayashi can be achieved. Some difficulties regarding interpretation in both the formalisms may possibly be removed from the geometry of internal space-time.

Quantum dissipative scalar and fermionic fields are studied here from classical stochastic fields. These classical stochastic fields are the outcome of the relativistic generalization of Nelson's stochastic mechanics based on a new microlocal geometry. Results show that the dissipation is the external classical phenomena whereas quantum nature comes from within (microlocal structure).

The relativistic generalization of stochastic quantization helps us to introduce a stochastic-phase-space formulation when a relativistic quantum particle appears as a stochastically extended one. The nonrelativistic quantum mechanics is obtained in the sharp point limit. This also helps us to introduce a gauge-theoretical extension of a relativistic quantum particle when for a fermion the group structure of the gauge field is SU(2). The sharp point limit is obtained when we have a minimal contribution of the residual gauge field retained in the limiting procedure. This is shown to be equivalent to the geometrical approach to the phase-space quantization introduced by Klauder if it is interpreted in terms of a universal magnetic field acting on a free particle moving in a higher-dimensional configuration space when quantization corresponds to freezing the particle to its first Landau level. The geometric quantization then appears as a natural consequence of these two formalisms, since the Hermitian line bundle introduced there finds a physical meaning in terms of the inherent gauge field in stochastic-phase-space formulation or in the interaction with the magnetic field in Klauder quantization.