The algebraic structure of the generalized uncertainty principle

We show that a deformation of the Heisenberg algebra which depends on a dimensionful parameter κ is the algebraic structure which underlies the generalized uncertainty principle in quantum gravity. The deformed algebra and therefore the form of the generalized uncertainty principle are fixed uniquely by rather simple assumptions. The string theory result is reproduced expanding our result at first order in ∆p/MPL. We also briefly comment on possible implications for Lorentz invariance at the Planck scale. In recent years it has been suggested that measurements in quantum gravity are governed by a generalized uncertainty principle ∆x ≥ h̄ ∆p + const. G∆p (1) ( G is Newton’s constant). At energies much below the Planck mass MPL the extra term in eq. (1) is irrelevant and the Heisenberg relation is recovered. As we approach the Planck mass, this term becomes important and it is responsible for the existence of a minimal observable length on the order of the Planck length. The result (1) was first suggested in the context of string theory [1-4], in the kinematical region where 2GE is smaller than the string length (for a review, see [5]). However, heuristic arguments [6] suggest that this formula might have a more general validity in quantum gravity, and it is not necessarily related to strings. It is therefore natural to ask whether there is an algebraic structure which reproduces eq. (1) (or, more in general, which reproduces the existence of a minimal observable length), in the same way in which the Heisenberg uncertainty principle follows from the algebra [x, p] = ih̄. In this Letter we answer (in the affirmative) this question. Our strategy is as follows. Since it is relatively clear that no Lie algebra can reproduce eq. (1), we turn our attention to deformed algebras. A deformed algebra is an associative algebra where it is defined a commutator which is non-linear in the elements of the algebra; and there is a deformation parameter such that, in an appropriate limit, a Lie algebra is recovered. We therefore look for the most general deformed algebra which can be constructed from coordinates xi and momenta pi (i = 1, 2, 3). We restrict the range of possibilities making the following assumptions. 1) The threedimensional rotation group is not deformed; the angular momentum J satisfies the undeformed SU(2) commutation relations, and coordinate and momenta satisfy the undeformed commutation relations [Ji, xj ] = iǫijkxk, [Ji, pj] = iǫijkpk. 2) The momenta commutes between themselves: [pi, pj] = 0, so that also the translation group is not deformed. 3) The [x, x] and [x, p] commutators depend on a deformation parameter κ with dimensions of mass. In the limit κ → ∞ (that is, κ much larger than any energy), the canonical commutation relations are recovered. We will not require the existence of a coproduct and of an antipode, which would promote the deformed algebra to a quantum algebra, see below.