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We consider dynamics of a free relativists particle at very short distances treating space-time as Archimedean as well as no Archimedean one. Usual action for the relativistic particle is nonlinear. Meanwhile, in the real case, that system may be treated like a system with quadratic (Hamiltonian) constraint. We perform similar procedure in p-adic case, as the simplest example of a no Archimedean space. The existence of the simplest vacuum state is considered and corresponding Green function is calculated. Similarities and differences between obtained results on both spaces are examined and possible physical implications are discussed.

Propagator for the free relativistic particle on Archimedean and no Archimedean spaces