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Modifications of Heisenbergs uncertainty relations have been proposed in the literature which imply a minimum position uncertainty. We study the low energy effects of the new physics responsible for this by examining the consequent change in the quantum mechanical commutation relations involving position and momenta. In particular, the modifications to the spectrum of the hydrogen atom can be naturally interpreted as a varying (with energy) fine structure constant. From the data on the energy levels we attempt to constrain the scale of the new physics and find that it must be close to or larger than the weak scale. Experiments in the near future are expected to change this bound by at least an additional order of magnitude.
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Studies in string theory and quantum gravity suggest the existence of a finite lower limit $Delta x_0$ to the possible resolution of distances, at the latest on the scale of the Planck length of $10^{-35}m$. Within the framework of the euclidean path integral we explicitly show ultraviolet regularisation in field theory through this short distance structure. Both rotation and translation invariance can be preserved. An example geometry is studied in detail.
In the framework of the generalized uncertainty principle, the position and momentum operators obey the modified commutation relation $[X,P]=ihbarleft(1+beta P^2right)$ where $beta$ is the deformation parameter. Since the validity of the uncertainty relation for the Shannon entropies proposed by Beckner, Bialynicki-Birula, and Mycieslki (BBM) depends on both the algebra and the used representation, we show that using the formally self-adjoint representation, i.e., $X=x$ and $P=tanleft(sqrt{beta}pright)/sqrt{beta}$ where $[x,p]=ihbar$, the BBM inequality is still valid in the form $S_x+S_pgeq1+lnpi$ as well as in ordinary quantum mechanics. We explicitly indicate this result for the harmonic oscillator in the presence of the minimal length.
In this paper, we study the thermodynamics of quantum harmonic oscillator in the Tsallis framework and in the presence of a minimal length uncertainty. The existence of the minimal length is motivated by various theories such as string theory, loop quantum gravity, and black-hole physics. We analytically obtain the partition function, probability function, internal energy, and the specific heat capacity of the vibrational quantum system for $1<q<frac{3}{2}$ and compare the results with those of Tsallis and Boltzmann-Gibbs statistics without the minimal length scale.
In quantum mechanics with minimal length uncertainty relations the Heisenberg-Weyl algebra of the one-dimensional harmonic oscillator is a deformed SU(1,1) algebra. The eigenvalues and eigenstates are constructed algebraically and they form the infinite-dimensional representation of the deformed SU(1,1) algebra. Our construction is independent of prior knowledge of the exact solution of the Schrodinger equation of the model. The approach can be generalized to the $D$-dimensional oscillator with non-commuting coordinates.
Based on quantum mechanical framework for the minimal length uncertainty, we demonstrate that the generalized uncertainty principle (GUP) parameter could be best constrained by recent gravitational waves observations on one hand. On other hand this suggests modified dispersion relations (MDRs) enabling an estimation for the difference between the group velocity of gravitons and that of photons. Utilizing features of the UV/IR correspondence and the obvious similarities between GUP (including non-gravitating and gravitating impacts on Heisenberg uncertainty principle) and the discrepancy between the theoretical and the observed cosmological constant (apparently manifesting gravitational influences on the vacuum energy density), we suggest a possible solution for the cosmological constant problem.
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