Covariant formulation of the generalized uncertainty principle

We present a formulation of the generalized uncertainty principle based on a commutator $[{\widehat{x}}^i,{\widehat{p}}_j]$ between position and momentum operators defined in a covariant manner using normal coordinates. We show how any such commutator can acquire corrections if the momentum space is curved. The correction is completely determined by the extrinsic curvature of the surface $p^2=\text{constant}$ in the momentum space, and results in noncommutativity of normal position coordinates $[{\widehat{x}}^i,{\widehat{x}}^j]\ne 0$. We then provide a construction for the momentum space geometry as a suitable four dimensional extension of a geometry conformal to the three dimensional relativistic velocity space—the Lobachevsky space—whose curvature is determined by the dispersion relation $F(p^2)=-m^2$, with $F(x)=x$ yielding the standard Heisenberg algebra.

Figure 1

Normal coordinates defined in the tangent space $T_{{\mathcal{P}}_0}(M)$. Their variations are completely determined by the geometry of the equigeodesic surfaces anchored at ${\mathcal{P}}_0$ (dashed lines), with $\lambda$ the length of the geodesic connecting $\mathcal{P}$ to ${\mathcal{P}}_0$. The passive version of these variations correspond to shifting the base point ${\mathcal{P}}_0$, in which case the change in normal coordinates is determined by extrinsic geometry anchored at $\mathcal{P}$.

Figure 2

The behavior of functions $G_1(p^2)$ and $G_2(p^2)$ for three different dispersion relations. These functions are defined by: $[{\mathsf{x}}^a,{\mathsf{p}}_b]=i\hslash (G_1(p^2){\delta }_b^a+G_2(p^2){\mathsf{p}}^a{\mathsf{p}}_b)$. The three curves correspond to $F(x)$: (i) $x+x[\exp (x/\mathrm{\Lambda }{)}^2-1]$ (Red), (ii) $x\exp (x/\mathrm{\Lambda })$ (Blue), (iii) $x[1+(x/\mathrm{\Lambda }{)}^2]$ (Orange). Here, $\mathrm{\Lambda }$ is a momentum scale and $\mathrm{\Lambda }\to \infty$ corresponds to $F(x)=x$.

Figure 3

The choice of reference point in the momentum space can be obtained from the vector $u^{\mathsf{a}}$ giving normal coordinates in position space.