The Snyder space-time quantization, q-deformations, and ultraviolet divergences

The Snyder idea [Phys. Rev. 71 (1947) 38; 72 (1947) 68] to introduce the quantized space-time coordinates or, respectively, the curved momentum space is the oldest example of using the non-commutative geometry in physics. The goal of this letter is to draw attention to the connection between the geometry of momentum space and the character of singularities of the principal field-theoretical quantities. The well-known fact is that the structure of these singularites defines the character of ultraviolet divergences. In the Snyder theory, the problem of multiplication of singular field-theoretic functions is modified in a major way as compared with the usual Quantum Field Theory (QFT), and as a consequence, no ultraviolet divergences occur. It is shown that the modification of singularities is connected with the finite-difference (deformed) realizations of the Poincaré algebra.