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I consider general interacting systems of quantum particles in one spatialdimension. These consist of bosons or fermions, which can have any number ofcomponents, arbitrary spin or a combination thereof, featuring low-energy two-and multiparticle interactions. The single-particle dispersion can be Galilean(non-relativistic), relativistic, or have any other form that may be relevantfor the continuum limit of lattice theories. Using an algebra of generalizedfunctions, statistical transmutation operators that are genuinely unitary areobtained, putting bosons and fermions in a one-to-one correspondence withoutthe need for a short-distance hard core. In the non-relativistic case,low-energy interactions for bosons yield, order by order, fermionic dualinteractions that correspond to the standard low-energy expansion for fermions.In this way, interacting fermions and bosons are fully equivalent to each otherat low energies. While the Bose-Fermi mappings do not depend on microscopicdetails, the resulting statistical interactions heavily depend on the kineticenergy structure of the respective Hamiltonians. These statistical interactionsare obtained explicitly for a variety of models, and regularized andrenormalized in the momentum representation, which allows for theoretically andcomputationally feasible implementations of the dual theories. The mapping isrewritten as a gauge interaction, and one-dimensional anyons are alsoconsidered.

Bose-Fermi dualities for arbitrary one-dimensional quantum systems in the universal low energy regime