Cosmological Horizons, Uncertainty Principle and Maximum Length Quantum Mechanics

Understand

The cosmological particle horizon is the maximum measurable length in the Universe.

• The existence of such a maximum observable length scale implies a modification of the quantum uncertainty principle.
• Thus due to non-locality of quantum mechanics, the global properties of the Universe could produce a signature on the behaviour of local quantum systems.
• A Generalized Uncertainty Principle (GUP) that is consistent with the existence of such a maximum observable length scale $l_{max}$ is $\Delta x \Delta p \geq \frac{\hbar}{2}\;\frac{1}{1-\alpha \Delta x^2}$ where $\alpha = l_{max}^{-2}\simeq (H_0/c)^2$ ($H_0$ is the Hubble parameter and $c$ is the speed of light).