Unpacking Space-Time Quanta: Snyder’s Algebra Meets Bekenstein’s Bound

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Abstract and Introduction

Space-time quanta and Becken Universal bound

Shape of space-time quanta

Symmetry of space-time quanta

Space-time quanta and Spectral mass gap

Phenomenological implications

Conclusion, Acknowledgments, and References

II. SPACE-TIME QUANTA AND BEKENSETIN UNIVERSAL BOUND

In this section, we investigate the physical properties of space-time quanta implied by Snyder’s algebra. It is clear that Eq. (1) only vanishes if there is no fundamental minimal/quantum length (i.e κℓP l = 0). This means non-commutative geometry would vanish if there is no minimal/quantum length. On the contrary, we find that the GUP commutation relation in Eq. (2) vanishes. The time-energy commutation relation of Eq. (2) vanishes when:

where E = p0 and Eκ represents the maximum bound on energy. The position-momentum commutation relation Eq. (2) vanishes when:

On another side, Bekenstein found a universal bound [35–37] that defines the maximal amount of information that is necessary to perfectly and completely describes a physical object up to the quantum level. Bekenstein universal bound is given by:

When we compare Eq. (3) with Eq. (6), we get:

that completely describes the quanta of space-time. Notice here that Hκ depends only on π and is independent of κ and nature constants.