Bosonization in the path integral formulation

Abstract

We establish the direct $d=2$ on-shell bosonization ${\psi }_L(x_{+})=e^{i\xi (x_{+})}$ and ${\psi }_R^{†}(x_{-})=e^{i\xi (x_{-})}$ in path integral formulation by deriving the off-shell relations ${\psi }_L(x){\psi }_R^{†}(x)=\exp [i\xi (x)]$ and ${\psi }_R(x){\psi }_L^{†}(x)=\exp [-i\xi (x)]$. Similarly, the on-shell bosonization of the bosonic commuting spinor, ${\varphi }_L(x_{+})=i{e}^{-i\xi (x_{+})}{\partial }^{+}{e}^{-i\chi (x_{+})}$, ${\varphi }_R^{†}(x_{-})=e^{-i\xi (x_{-})-i\chi (x_{-})}$ and ${\varphi }_R(x_{-})=i{e}^{i\xi (x_{-})}{\partial }^{-}{e}^{+i\chi (x_{-})}$, ${\varphi }_L^{†}(x_{+})=e^{i\xi (x_{+})+i\chi (x_{+})}$, is established in path integral formulation by deriving the off-shell relations ${\varphi }_L(x){\varphi }_R^{†}(x)=i{e}^{-i\xi (x)}{\partial }^{+}{e}^{-i\chi (x)}$ and ${\varphi }_R(x){\varphi }_L^{†}(x)=i{e}^{i\xi (x)}{\partial }^{-}{e}^{i\chi (x)}$.

Comments: 18 pages, the final version to appear in Phys. Rev. D