One-dimensional quantum systems admit duality relations that put hard-core spinless bosons and fermions in one-to-one correspondence via Girardeau's mapping theorem. The simplest models of soft bosons interacting via zero-range potentials can also be mapped onto dual interacting fermions. However, a systematic approach to one-dimensional statistical transmutation for arbitrary low-energy interactions in the spinless and spinful or multicomponent cases has remained elusive. I develop a general theory of local unitary transformations between one-dimensional quantum systems of bosons and fermions with arbitrary spin or internal structure, single-particle dispersion—including nonrelativistic, relativistic, or otherwise—and low-energy interactions in the universal regime. These transformations generate families of duality relations and models that relate the strong- and weak-coupling limits of the respective dual theories.

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• Bose-Einstein condensates
• Cold atoms & matter waves
• Fermi gases