Cosmological Horizons, Uncertainty Principle and Maximum Length Quantum Mechanics

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). In addition to the existence of a maximum measurable length $l_{max}=\frac{1}{\sqrt \alpha}$, this form of GUP implies also the existence of a minimum measurable momentum $p_{min}=\frac{3 \sqrt{3}}{4}\hbar \sqrt{\alpha}$. Using appropriate representation of the position and momentum quantum operators we show that the spectrum of the one dimensional harmonic oscillator becomes $\bar{\mathcal{E}}_n=2n+1+\lambda_n \bar{\alpha}$ where $\bar{\mathcal{E}}_n\equiv 2E_n/\hbar \omega$ is the dimensionless properly normalized $n^{th}$ energy level, $\bar{\alpha}$ is a dimensionless parameter with $\bar{\alpha}\equiv \alpha \hbar/m \omega$ and $\lambda_n\sim n^2$ for $n\gg 1$ (we show the full form of $\lambda_n$ in the text). For a typical vibrating diatomic molecule and $l_{max}=c/H_0$ we find $\bar{\alpha}\sim 10^{-77}$ and therefore for such a system, this effect is beyond reach of current experiments. However, this effect could be more important in the early universe and could produce signatures in the primordial perturbation spectrum induced by quantum fluctuations of the inflaton field.