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 nonlocality of quantum mechanics, the global properties of the Universe could produce a signature on the behavior 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 triangle x triangle p >= h/2 1/1-alpha triangle x(2) where alpha = l(max)(-2) similar or equal to (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) = 1/root alpha, this form of GUP implies also the existence of a minimum measurable momentum p(min) = 3 root 3/4 h root alpha Using appropriate representation of the position and momentum quantum operators we show that the spectrum of the one- dimensional harmonic oscillator becomes (epsilon) over bar (n) = 2n + 1 + lambda(n) (alpha) over bar where (epsilon) over bar (n) equivalent to 2En=/h omega is the dimensionless properly normalized nth energy level, alpha a is a dimensionless parameter with (alpha) over bar equivalent to alpha h/m omega and lambda(n) similar to n(2) for n >> 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 (alpha) over bar similar to 10(-77) and therefore for such a system, this effect is beyond the 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.