[Paper Review] Covariant formulation of Generalised Uncertainty Principle

Raghvendra Singh, Dawood Kothawala|arXiv (Cornell University)|Oct 29, 2021

Noncommutative and Quantum Gravity Theories2 references4 citations

TL;DR

This paper presents a covariant formulation of the Generalised Uncertainty Principle (GUP) by defining position and momentum operators via normal coordinates in a geodesically convex spacetime neighborhood, introducing momentum space geometry based on a four-dimensional extension of the Lobachevsky space. The key result is that non-commutativity of position operators $[\hat{x}^i, \hat{x}^j] \neq 0$ arises naturally from the extrinsic curvature of the $p^2 = \text{const}$ surface in curved momentum space, with the GUP correction $\Theta^i{}_j(\hat{p}_k)$ fully determined by this geometry.

ABSTRACT

We present a formulation of the generalised uncertainty principle based on commutator $\left[ {\hat x}^i, {\hat p}_j ight]$ 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 $p^2=$ constant in the momentum space, and results in non-commutativity of normal position coordinates $\left[ {\hat x}^i, {\hat x}^j ight] eq 0$. 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.

Motivation & Objective

• To develop a manifestly covariant formulation of the Heisenberg algebra in curved spacetime, overcoming the breakdown of general covariance in standard commutator definitions.
• To geometrically motivate the GUP correction $\Theta^i{}_j(\hat{p}_k)$ using a four-dimensional momentum space structure derived from relativistic velocity space.
• To show that non-commutativity of position operators arises from the extrinsic curvature of the $p^2 = \text{const}$ hypersurface in momentum space.
• To establish a consistent, Lorentz-invariant GUP framework that incorporates modified dispersion relations $F(p^2) = -m^2$ through a geometric construction of momentum space.

Proposed method

• Use of normal coordinates defined via the exponential map from a spacetime point $\mathcal{P}_0$, ensuring covariant dependence on spacetime curvature.
• Construction of a four-dimensional momentum space metric conformal to the Lobachevsky metric on relativistic velocity space, with curvature determined by the dispersion relation $F(p^2) = -m^2$.
• Extension of the momentum space geometry to ensure it reduces to $ (\Delta m)^2 = (m_2 - m_1)^2 $ for zero relative velocities.
• Derivation of the GUP correction $\Theta^i{}_j(\hat{p}_k)$ as a function of the extrinsic curvature $K^{\textsf{a}}{}_{\textsf{b}}$ of the $p^2 = \text{const}$ surface in momentum space.
• Application of the Jacobi identity to derive the non-vanishing commutator $[\hat{x}^i, \hat{x}^j]$, showing it depends on $\Theta^{-1}$ and its derivatives.
• Derivation of a symmetric momentum-space representation of the position operator $\hat{x}^a_{\text{sym}}$ using the metric determinant and connection coefficients to preserve unitarity.

Experimental results

Research questions

• RQ1 How can the Heisenberg algebra be generalized to curved spacetime in a manifestly covariant way, avoiding the explicit dependence on coordinate charts?
• RQ2 What is the geometric origin of the GUP correction $\Theta^i{}_j(\hat{p}_k)$ in a relativistic, curved momentum space?
• RQ3 How does the non-commutativity of position operators $[\hat{x}^i, \hat{x}^j] \neq 0$ emerge from the momentum space geometry?
• RQ4 What role does the extrinsic curvature of the $p^2 = \text{const}$ surface play in determining the structure of the GUP?
• RQ5 How can a consistent, symmetric momentum representation of the position operator be constructed in a curved momentum space?

Key findings

• The GUP correction $\Theta^i{}_j(\hat{p}_k)$ is fully determined by the extrinsic curvature of the $p^2 = \text{const}$ hypersurface in momentum space, providing a geometric origin for the modification.
• For the standard dispersion relation $F(p^2) = -m^2$, the momentum space is flat (Riemann tensor vanishes), and $\Theta^i{}_j = \delta^i_j$, recovering the standard Heisenberg algebra.
• When the dispersion relation is generalized to $F(p^2) = -m^2$ with $F \neq \text{id}$, the momentum space curvature becomes non-zero, leading to non-trivial $\Theta^i{}_j(\hat{p}_k)$ and non-commutative position operators.
• The commutator $[\hat{x}^i, \hat{x}^j]$ is derived as $i\hbar \{ \hat{x}^l, (\Theta^{-1})^m{}_l \Theta_n^{[a]} \Theta_m^{b],n} \}$, showing explicit dependence on the inverse metric and derivatives of $\Theta$.
• The momentum-space representation of the position operator is constructed as $\hat{x}^a_{\text{sym}} = i\hbar \left( \frac{1}{2} \Theta^{m a} \Gamma^i{}_{m i} + \frac{1}{2} \Theta_m^{a, m} + \Theta_m^a \partial^m \right)$, ensuring symmetry under the measure $d^4\textsf{p} \sqrt{-g}$.
• The formalism is non-local due to dependence on a base point in both position and momentum space, with the choice of origin in momentum space tied to the geodesic flow from a spacetime point, making it position-dependent.