Quantum mechanics onSO(3)via noncommutative dual variables

Abstract

We formulate quantum mechanics on the group SO(3) using a non-commutative dual space representation for the quantum states, inspired by recent work in quantum gravity. The new non-commutative variables have a clear connection to the corresponding classical variables, and our analysis confirms them as the natural phase space variables, both mathematically and physically. In particular, we derive the first order (Hamiltonian) path integral in terms of the non-commutative variables, as a formulation of the transition amplitudes alternative to that based on harmonic analysis. We find that the non-trivial phase space structure gives naturally rise to quantum corrections to the action for which we find a closed expression. We then study both the semi-classical approximation of the first order path integral and the example of a free particle on SO(3). On the basis of these results, we comment on the relevance of similar structures and methods for more complicated theories with group-based configuration spaces, such as Loop Quantum Gravity and Spin Foam models.

Bounding bubbles: The vertex representation of 3d group field theory and the suppression of pseudomanifolds

“…We use this labeling in (23), where upper indices correspond to colors of the triangles whereas lower ones correspond to that of the vertices (see Fig. 5).…”

“…This relation has been clarified and strengthened by the recent noncommutative metric representation of group field theories [15], based on the so-called group Fourier transform [16][17][18], a very natural construction on the type of phase space used in loop quantum gravity [6,19,20], Chern-Simons theory [21,22], and discrete BF theories [15,23,24]. In this representation, group field theories are written as noncommutative field theories on Lie algebras and the corresponding Feynman amplitudes take explicitly the form of simplicial gravity path integrals, which proves an exact duality between such path integrals and spin foam models.…”

Noncommutative Fourier transform for the Lorentz group via the Duflo map

“…The path-integral formulation of quantum mechanics in terms of non-commutative momenta was described in [43] for the case of SO(3), and the corresponding propagator for a free particle on a sphere was derived. In dealing with the homogeneous space H 3 ∼ = SL(2, C)/SU(2) (or SO(3,1)/SO(3)), we will refer to [43] for the details on the general characterization of quantum mechanics in non-commutative momentum basis.…”

“…Following the construction of [43], we define a set of states | P , R | P , R ∈ R 6 ⋆ in the non-commutative momentum basis, by their inner product with the group basis…”

“…This is to be expected, since harmonic functions are only eigenstates of the quadratic Casimir, while the states 120 are generalised eigenstates of all momentum operators. Finally, we can ascribe the correction terms in the exponential to the choice of (Duflo) quantization map (see the discussion of a similar term in the compact case in [43]).…”