We defined a non-commutative algebra representation for quantum systems whose phase space is the cotangent bundle of the Lorentz group, and the non-commutative Fourier transform ensuring the unitary equivalence with the standard group representation. Our construction is from first principles in the sense that all structures are derived from the choice of quantization map for the classical system, the Duflo quantization map.
I. INTRODUCTIONPhysical systems whose configuration or phase space is endowed with a curved geometry include for example point particles on curved spacetimes and rotor models in condensed matter theory, and they present specific mathematical challenges in addition to their physical interest. In particular, many important results have been obtained in the case in which their domain spaces can be identified with (non-abelian) group manifolds or their associated homogeneous spaces.Focusing on the case in which the non-trivial geometry is that of configuration space, identified with a Lie group, this is then reflected in the non-commutativity of the conjugate momentum space, whose basic variables correspond to the Lie derivatives acting on the configuration manifold, and that can be thus identified with the Lie algebra of the same Lie group. At the quantum level, this non-commutativity enters heavily in the treatment of the system. For instance, it prevents a standard L 2 representation of the Hilbert state space of the system in momentum picture, and may lead to question whether a representation in terms of the (non-commutative) momentum variables exists at all. Further, if such a representation could be defined, one would also need a generalised notion of (non-commutative) Fourier transform to establish the (unitary) equivalence between this new representation and the one based on L 2 functions on the group configuration space.The standard way of dealing with this issue is indeed to renounce to have a representation that makes direct use of the noncommutative Lie algebra variables, and to use instead a representation in terms of irreducible representations of the Lie group, i.e. to resort to spectral decomposition as the proper analogue of the traditional Fourier transform in flat space. In other words, instead of using some necessarily generalised eigenbasis of the non-commutative momentum (Lie algebra) variables, one uses a basis of eigenstates of a maximal set of commuting operators that are functions of the same. This strategy has of course many advantages, but it prevents the more geometrically transparent picture, more closely connected to the underlying classical system, that dealing directly with the non-commuting Lie algebra variables would provide.In recent times, most work on these issues has been motivated by quantum gravity, where they turned out to play an important role. This happened in two domains. The first is effective (field theory) models of quantum gravity based (or inspired) by non-commutative geometry [1]. Such models have attracted a considerable interest because they of...
