99-10E

# 99-10E

Weyl-type quantization rules and N-particle canonical realization of the Poincar\'e algebra in the two-dimensional space-time

The quantization of canonical realization of Poincar\' e algebra corresponding to $N$-particle interacting system in the two-dimensional space-time ${\mathbb M}_2$ in the front form of dynamics is considered. Unitary realizations of the group ${\cal P}(1,1)$ are obtained by means of a set of Weyl-type quantization rules. We demonstrate that the requirement of preservation of Lie algebra of this group restricts the set of quantization rules but does not by itself remove the ambiguity of quantization procedure. The set of quantization rules fall apart into equivalence classes. The quantization rules from the same equivalence class give the same mass spectrum, evolution of the quantized system and lead to equivalent unitary representations of the group ${\cal P}(1,1)$. The quantizations which belong to different classes lead to non-equivalent unitary representations and may result in different values for observable quantities.