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Register a new userOptimal uncertainty relations in a modified Heisenberg algebra
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Various theories that aim at unifying gravity with quantum mechanics suggest modifications of the Heisenberg algebra for position and momentum. From the perspective of quantum mechanics, such modifications lead to new uncertainty relations which are thought (but not proven) to imply the existence of a minimal observable length. Here we prove this statement in a framework of sufficient physical and structural assumptions. Moreover, we present a general method that allows to formulate optimal and state-independent variance-based uncertainty relations. In addition, instead of variances, we make use of entropies as a measure of uncertainty and provide uncertainty relations in terms of min- and Shannon entropies. We compute the corresponding entropic minimal lengths and find that the minimal length in terms of min-entropy is exactly one bit.
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We analyze general uncertainty relations and we show that there can exist such pairs of non--commuting observables $A$ and $B$ and such vectors that the lower bound for the product of standard deviations $Delta A$ and $Delta B$ calculated for these vectors is zero: $Delta A,cdot,Delta B geq 0$. We show also that for some pairs of non--commuting observables the sets of vectors for which $Delta A,cdot,Delta B geq 0$ can be complete (total). The Heisenberg, $Delta t ,cdot, Delta E geq hbar/2$, and Mandelstam--Tamm (MT), $ tau_{A},cdot ,Delta E geq hbar/2$, time--energy uncertainty relations ($tau_{A}$ is the characteristic time for the observable $A$) are analyzed too. We show that the interpretation $tau_{A} = infty$ for eigenvectors of a Hamiltonian $H$ does not follow from the rigorous analysis of MT relation. We show also that contrary to the position--momentum uncertainty relation, the validity of the MT relation is limited: It does not hold on complete sets of eigenvectors of $A$ and $H$.
Analyzing general uncertainty relations one can find that there can exist such pairs of non-commuting observables $A$ and $B$ and such vectors that the lower bound for the product of standard deviations $Delta A$ and $Delta B$ calculated for these vectors is zero: $Delta A,cdot,Delta B geq 0$. Here we discuss examples of such cases and some other inconsistencies which can be found performing a rigorous analysis of the uncertainty relations in some special cases. As an illustration of such cases matrices $(2times 2)$ and $(3 times 3)$ and the position--momentum uncertainty relation for a quantum particle in the box are considered. The status of the uncertainty relation in $cal PT$--symmetric quantum theory and the problems associated with it are also studied.
We study perturbation theory for large-scale structure in the most general scalar-tensor theories propagating a single scalar degree of freedom, which include Horndeski theories and beyond. We model the parameter space using the effective field theory of dark energy. For Horndeski theories, the gravitational field and fluid equations are invariant under a combination of time-dependent transformations of the coordinates and fields. This symmetry allows one to construct a physical adiabatic mode which fixes the perturbation-theory kernels in the squeezed limit and ensures that the well-known consistency relations for large-scale structure, originally derived in general relativity, hold in modified gravity as well. For theories beyond Horndeski, instead, one generally cannot construct such an adiabatic mode. Because of this, the perturbation-theory kernels are modified in the squeezed limit and the consistency relations for large-scale structure do not hold. We show, however, that the modification of the squeezed limit depends only on the linear theory. We investigate the observational consequences of this violation by computing the matter bispectrum. In the squeezed limit, the largest effect is expected when considering the cross-correlation between different tracers. Moreover, the individual contributions to the 1-loop matter power spectrum do not cancel in the infrared limit of the momentum integral, modifying the power spectrum on non-linear scales.
Uncertainty relation is a core issue in quantum mechanics and quantum information theory. We introduce modified generalized Wigner-Yanase-Dyson (MGWYD) skew information and modified weighted generalizedWigner-Yanase-Dyson (MWGWYD) skew information, and establish new uncertainty relations in terms of the MGWYD skew information and MWGWYD skew information.
According to several quantum gravity theories there is an effective minimal length beyond which space cannot be probed, possibly the Planck length. It has been suggested that this fundamental limit implies a generalised uncertainty principle (GUP) and corresponding modification of the canonical position-momentum commutation relations. Some of these modified relations are also consistent with general principles that may be supposed of any physical theory. Here we study the noisy behaviour of an optomechanical system assuming a certain commonly studied modified commutator. From recent observations of radiation pressure noise in tabletop optomechanical experiments as well as the position noise spectrum of Advanced LIGO we derive bounds on the modified commutator, which may be used to constrain the minimal length. We find how such experiments can be adjusted to provide significant improvements in such bounds, potentially surpassing those from sub-atomic measurements.
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