Diffeomorphisms in Momentum Space: Physical Implications of Different Choices of Momentum Coordinates in the Galilean Snyder Model

Abstract

It has been pointed out that different choices of momenta can be associated to the same noncommutative spacetime model. The question of whether these momentum spaces, related by diffeomorphisms, produce the same physical predictions is still debated. In this work, we focus our attention on a few different momentum spaces that can be associated to the Galilean Snyder noncommutative spacetime model and show that they produce different predictions for the energy spectrum of the harmonic oscillator.

Quantum Mechanics of the Extended Snyder Model

“…These results are in accordance with (23) and can be compared with the spectrum of the standard Snyder oscillator [22][23][24], for which E n ∼ ω ∑ i n i + D 2 + o(β 2 mω). It turns out that the vacuum energy is different in the two cases, while the leading order correction, although different, are of the same order of magnitude.…”

“…Hence , while the canonical extended oscillator has a standard spectrum depending only on the quantum number n = ∑ µ<ν n µν , the Snyder extended oscillator has eigenvalues that depend on combinations of all the quantum numbers n µν . The order of magnitude of the corrections to the energy spectrum is the same as in the standard Snyder oscillator [22][23][24] for M = β 2 m/λ 2 , with m the mass of the vectorial degrees of freedom.…”

“…where n i , n ij are the occupation numbers for the vector and tensor degrees of freedom, respectively. The leading-order corrections due to the Snyder structure arising from (24) are instead…”

“…As we have mentioned, the energy spectrum depends on the realization [24]. For example, let us consider the classical realization, with commutation relations (3), ( 5).…”

“…In this paper, we shall attempt to investigate a quantum mechanical problem based on the Euclidean version of the model and inspired by an analogous one introduced in the context of Moyal space, where the objects of noncommutativity were promoted to antisymmetric operators [21]. In particular, we study the harmonic oscillator in this theory, with the aim of understanding in a simple case the physical implications of the addition of the tensorial degrees of freedom, comparing the results with those obtained in [22][23][24] for the standard Snyder model.…”