Cosmological horizons, uncertainty principle, and maximum length quantum mechanics
Abstract
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 Δ x Δ p ≥ℏ2/1/1 α Δ x^{2} where α =l_{max}^{2}≃(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/√{α }, this form of GUP implies also the existence of a minimum measurable momentum p_{min}=3/√{3 } 4 ℏ√{α }. Using appropriate representation of the position and momentum quantum operators we show that the spectrum of the onedimensional harmonic oscillator becomes E_{¯n}=2 n +1 +λ_{n}α ¯ where E_{¯n}≡2 E_{n}/ℏω is the dimensionless properly normalized n th energy level, α ¯ is a dimensionless parameter with α ¯≡α ℏ/m ω and λ_{n}∼n^{2} for n ≫1 (we show the full form of λ_{n} in the text). For a typical vibrating diatomic molecule and l_{max}=c /H_{0} we find α ¯∼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.
 Publication:

Physical Review D
 Pub Date:
 May 2017
 DOI:
 10.1103/PhysRevD.95.103523
 arXiv:
 arXiv:1704.05681
 Bibcode:
 2017PhRvD..95j3523P
 Keywords:

 General Relativity and Quantum Cosmology;
 Astrophysics  Cosmology and Nongalactic Astrophysics;
 High Energy Physics  Phenomenology;
 High Energy Physics  Theory;
 Quantum Physics
 EPrint:
 11 pages, 7 Figures. The Mathematica file that was used for the production of the Figures may be downloaded from http://leandros.physics.uoi.gr/maxlenqm/