BV quantization of ${\phi }^3$-theory on $\lambda$-Minkowski space: Tree-level correlation functions

We review the quantization of scalar field theory on $\lambda$-Minkowski space using the Batalin–Vilkovisky (BV) formalism. We consider ${\phi }^3$-theory in two different quantization schemes: standard and braided. While standard BV quantization is based on an ordinary $L_{\mathrm{\infty }}$-algebra, braided BV quantization is based on a braided $L_{\mathrm{\infty }}$-algebra. We compare the tree-level three-point and four-point correlation functions in the two approaches. For the four-point function, standard quantization leads to two inequivalent classes of diagrams with different noncommutative contributions, whereas braided quantization yields only a single class of diagrams with noncommutativity entering solely through an overall phase factor depending on the external momenta.