Abstract: In this paper, we present an explicit construction of the unitary irreducible representations of the two-dimensional Euclidean and Poincaré groups, together with their Spin double covers, by means of Mackey's theory of induced representations for semidirect products. In dimension D = 2, the simplicity of the corresponding little groups allows a complete explicit treatment of momentum orbits, equivariant wavefunctions, and representation operators. For the Euclidean group, the matrix elements of the infinite-dimensional representations are expressed in terms of Bessel functions. For the Poincaré group, the richer Lorentzian orbit structure leads to matrix elements involving modified Bessel and Hankel functions and, in some cases, tempered distributions, requiring the use of Rigged Hilbert Spaces. This work illustrates the interplay among induced representations, harmonic analysis on Lie groups, spin geometry, and special functions in a fully explicit relativistic setting.

Abstract: We develop the complete theoretical framework for open quantum systems coupled to scale-invariant environments. Such environments, we show, are universally and uniquely described by unparticle baths [1] characterized by a single scaling dimension d(u), which is a property of the bath operator entering the coupling, not of the probe. Different probes coupled to the same bath operator O-E must yield the same d(u), providing a nontrivial cross-probe consistency test for a fixed coupling channel. This companion paper provides the full proof of the uniqueness theorem, the mathematical formalism of the resulting nonMarkovian dynamics, and worked applications to three physical realizations omitted from the shorter letter [2]. Starting from the uniqueness theorem, we derive the complete set of non-Markovian memory kernels, the exact noise kernel, including vacuum and thermal contributions via Matsubara summation, and the fractional generalization of the Caldeira-Leggett master equation for arbitrary d(u). The unparticle dimension acts as a control parameter governing a rich phase structure, including a thermalization transition at d(u) = 3/2, the Ohmic boundary at d(u) = 2, and a decoherence phase transition at d(u) = 5/2 in the thermal regime (d(u) = 2 in the vacuum regime), beyond which quantum coherence is protected at long times. These transitions are universal for free-field, weakly interacting, and engineered power-law baths; for strongly interacting CFT baths they describe intermediate-time scaling in the window 1/omega(UV) << t << 1/T. Three physical realizations are presented. For the quantum Ising model, d(u) is derived from first principles from the known CFT operator dimension. For inflationary cosmology, d(u) = 2 is obtained by matching to established results; for heavy-fermion materials, d(u) = 3/2 is inferred empirically from two-channel transport data. Coupling to the energy operator in (1 + 1) spacetime dimensions yields d(u) = 3/2, providing a field-theoretic derivation of 1/f noise; the (2 + 1)D case yields d(u) approximate to 1.413 from the conformal bootstrap [3]. For high-energy astrophysical neutrinos in the regime E>> T, the energy- and baselinedependent decoherence rate Gamma(decoh) alpha B(E; TU)L-u(5-2d) provides a direct observable imprint of the scaling dimension. A systematic comparison with the Caldeira-Leggett model, phenomenological Lindblad equations, and the non-Markovian literature establishes the precise relationship between these approaches and the unparticle framework. The regime of validity is analyzed for each physical system, including the crossover between vacuum and thermal regimes of the noise kernel. Experimental predictions and consistency tests are detailed for trapped-ion quantum simulators, neutrino telescopes, and superconducting qubits.