Snyder space revisited

Abstract

We examine basis functions on momentum space for the three dimensional Euclidean Snyder algebra. We argue that the momentum space is isomorphic to the SO(3) group manifold, and that the basis functions span either one of two Hilbert spaces. This implies the existence of two distinct lattice structures of space, on which continuous rotations and translations are unitarily implementable.Comment: 22 page

Quantum Mechanics on a Curved Snyder Space

“…In particular, it was shown that space is discretized [9] and deformed Heisenberg uncertainty relations hold, implying a lower bound on measurable length [8]. Both classical and quantum dynamics are modified with respect to the standard results, with deviations of order 2 2 , being the energy of the system [8].…”

“…The properties of the Snyder space and its dynamics, both in the nonrelativistic and relativistic version, have been investigated in several papers and in various contexts, both classical and quantum [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. In particular, it was shown that space is discretized [9] and deformed Heisenberg uncertainty relations hold, implying a lower bound on measurable length [8].…”

Nonassociative Snyder ϕ4 quantum field theory

“…The Snyder model has been studied in a series of papers [30][31][32][33][34][35][36] and the associated Hopf algebra investigated in [30] and [36], where the model has been generalized and the star product, coproducts and antipodes have been calculated using the method of realizations. A different approach was used in [35], where the Snyder model was considered in a geometrical perspective as a coset in momentum space, and the results are equivalent to those of Refs.…”

Misleading inferences from discretization of empty spacetime: Snyder-noncommutativity case study

“…[12], Snyder's spacetime leads to a picture of space in cartesian coordinates as a (quantum) cubical lattice of lattice spacing λ (for λ real number with dimensions of length, as we are also here assuming), while time remains a continuous variable. These results have been established by seeking formal Hilbert-space representations of the Snyder algebra, but the question remained so far concerning the physical interpretation of these results.…”