Bosonization in the path integral formulation

Abstract

We establish the direct $d=2$ on-shell bosonization ${\ensuremath{\psi}}_{L}({x}_{+})={e}^{i\ensuremath{\xi}({x}_{+})}$ and ${\ensuremath{\psi}}_{R}^{\ifmmode\dagger\else\textdagger\fi{}}({x}_{\ensuremath{-}})={e}^{i\ensuremath{\xi}({x}_{\ensuremath{-}})}$ in path integral formulation by deriving the off-shell relations ${\ensuremath{\psi}}_{L}(x){\ensuremath{\psi}}_{R}^{\ifmmode\dagger\else\textdagger\fi{}}(x)=\mathrm{exp}[i\ensuremath{\xi}(x)]$ and ${\ensuremath{\psi}}_{R}(x){\ensuremath{\psi}}_{L}^{\ifmmode\dagger\else\textdagger\fi{}}(x)=\phantom{\rule{0ex}{0ex}}\mathrm{exp}[\ensuremath{-}i\ensuremath{\xi}(x)]$. Similarly, the on-shell bosonization of the bosonic commuting spinor, ${\ensuremath{\phi}}_{L}({x}_{+})=\phantom{\rule{0ex}{0ex}}i{e}^{\ensuremath{-}i\ensuremath{\xi}({x}_{+})}{\ensuremath{\partial}}^{+}{e}^{\ensuremath{-}i\ensuremath{\chi}({x}_{+})}$, ${\ensuremath{\phi}}_{R}^{\ifmmode\dagger\else\textdagger\fi{}}({x}_{\ensuremath{-}})={e}^{\ensuremath{-}i\ensuremath{\xi}({x}_{\ensuremath{-}})\ensuremath{-}i\ensuremath{\chi}({x}_{\ensuremath{-}})}$ and ${\ensuremath{\phi}}_{R}({x}_{\ensuremath{-}})=i{e}^{i\ensuremath{\xi}({x}_{\ensuremath{-}})}{\ensuremath{\partial}}^{\ensuremath{-}}{e}^{+i\ensuremath{\chi}({x}_{\ensuremath{-}})}$, ${\ensuremath{\phi}}_{L}^{\ifmmode\dagger\else\textdagger\fi{}}({x}_{+})={e}^{i\ensuremath{\xi}({x}_{+})+i\ensuremath{\chi}({x}_{+})}$, is established in path integral formulation by deriving the off-shell relations ${\ensuremath{\phi}}_{L}(x){\ensuremath{\phi}}_{R}^{\ifmmode\dagger\else\textdagger\fi{}}(x)=i{e}^{\ensuremath{-}i\ensuremath{\xi}(x)}{\ensuremath{\partial}}^{+}{e}^{\ensuremath{-}i\ensuremath{\chi}(x)}$ and ${\ensuremath{\phi}}_{R}(x){\ensuremath{\phi}}_{L}^{\ifmmode\dagger\else\textdagger\fi{}}(x)=i{e}^{i\ensuremath{\xi}(x)}{\ensuremath{\partial}}^{\ensuremath{-}}{e}^{i\ensuremath{\chi}(x)}$.

Cite this paper

@article{fujikawa2015bosonization,
  title = {{Bosonization in the path integral formulation}},
  author = {Fujikawa, Kazuo and Suzuki, Hiroshi},
  journal = {Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D, Particles, fields, gravitation, and cosmology},
  volume = {91},
  number = {6},
  year = {2015},
  doi = {10.1103/physrevd.91.065010}
}

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