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.
