This is the first in a pair of articles that classify the configuration space and kinematic symmetry groups for
N identical particles in one-dimensional traps experiencing Galilean-invariant two-body interactions. These symmetries explain degeneracies in the few-body spectrum and demonstrate how tuning the trap shape and the particle interactions can manipulate these degeneracies. The additional symmetries that emerge in the non-interacting limit and in the unitary limit of an infinitely strong contact interaction are sufficient to algebraically solve for the spectrum and degeneracy in terms of the one-particle observables. Symmetry also determines the degree to which the algebraic expressions for energy level shifts by weak interactions or nearly-unitary interactions are universal, i.e. independent of trap shape and details of the interaction. Identical fermions and bosons with and without spin are considered. This article sequentially analyzes the symmetries of one, two and three particles in asymmetric, symmetric, and harmonic traps; the sequel article treats the
N particle case.
With the motivation of overcoming difficulties in studying systems of several one dimensional Tomonaga Luttinger liquid wires connected locally at a junction, we construct quadratic lattice field theories for the single non-chiral/chiral wire, the three wire fork and the chiral box junction. For a fork consisting of three identical non-chiral wires, we find that the permutation symmetries of the system, together with the requirement of charge and current continuity, determine the terms in the action governing the dynamics of the fields adjacent to the junction. These results are generalised to the case of N ≥ 3 identical (as well as different) TLL wires. A study of the chiral box junction circuit model reveals that junctions of several chiral wires can be formed by relating the TLL interaction parameters of their constituents.