We determine the energy eigenvalues and eigenfunctions of the harmonic oscillator where the coordinates and momenta are assumed to obey the modified commutation relations ͓x i ,p j ͔ϭiប͓(1ϩ p 2 )␦ i j ϩЈp i p j ͔. These commutation relations are motivated by the fact that they lead to the minimal length uncertainty relations which appear in perturbative string theory. Our solutions illustrate how certain features of string theory may manifest themselves in simple quantum mechanical systems through the modification of the canonical commutation relations. We discuss whether such effects are observable in precision measurements on electrons trapped in strong magnetic fields.
We continue our investigation of the phenomenological implications of the ''deformed'' commutation relations ͓x i ,p j ͔ϭiប͓(1ϩ p 2 )␦ i j ϩЈp i p j ͔. These commutation relations are motivated by the fact that they lead to the minimal length uncertainty relation which appears in perturbative string theory. In this paper, we consider the effects of the deformation on the classical orbits of particles in a central force potential. Comparison with observation places severe constraints on the value of the minimum length.
We investigate the effect of the minimal length uncertainty relation, motivated by perturbative string theory, on the density of states in momentum space. The relation is implemented through the modified commutation relation ͓x i ,p j ͔ϭiប͓(1ϩ p 2 )␦ i j ϩЈp i p j ͔. We point out that this relation, which is an example of a UV/IR relation, implies the finiteness of the cosmological constant. While our result does not solve the cosmological constant problem, it does shed new light on the relation between this outstanding problem and UV/IR correspondence. We also point out that the blackbody radiation spectrum will be modified at higher frequencies, but the effect is too small to be observed in the cosmic microwave background spectrum.
The energy spectrum of the Coulomb potential with minimal length commutation relations ͓X i , P j ͔ = iប͕␦ ij ͑1+P 2 ͒ + ЈP i P j ͖ is determined both numerically and perturbatively for arbitrary values of Ј /  and angular momenta ᐉ. The constraint on the minimal length scale from precision hydrogen spectroscopy data is of the order of a few GeV −1 , weaker than previously claimed. DOI: 10.1103/PhysRevA.72.012104 PACS number͑s͒: 03.65.Ge, 02.40.Gh, 31.15.Md, 32.10.Fn Quantum gravity incorporates Newton's constant as a dimensional parameter that could manifest itself as a minimal length in the system. Recent string theoretic considerations suggest that this length scale might imply an ultravioletinfrared ͑UV-IR͒ correspondence, contrary to the normal perceptions on momentum and spatial separations. Large momenta are now directly tied to large spatial dimensions, which then implies the existence of a minimal length. Earlier studies have focused upon its amelioration of ultraviolet divergences ͓1͔, but did not take into full account the UV-IR correspondence.There are various ways of implementing such an idea, but the simplest is to suppose that coordinates no longer commute in D-dimensional space. This, in turn, leads to a deformation of the canonical commutation relations. In our previous works, we adopted the equivalent hypothesis that the fundamental commutation relations between position and momentum are no longer constant multiples of the identity. In this paper, we report on constraints on the minimal length hypothesis from precision measurements on hydrogenic atoms. This system has a potential that is singular at the origin, and is therefore particularly sensitive to whether there is a fundamental minimal length. Considerations based upon higher-dimensional theories suggest that such lengths may be large ͓2͔.To set the context, we note that if in one dimension we havewhere  is a small parameter, then the resulting uncertainty relation ͑⌬X͒͑⌬P͒ ജ iប͕1+͑⌬P͒ 2 ͖ exhibits a form of the UV-IR correspondence, and gives as minimal length ⌬X ജប ͱ  ͓3͔.We had examined the harmonic oscillator system under this hypothesis in ͓4͔, but no real constraint can be obtained on the minimal length, presumably because of the softness of the potential at the origin. An interesting approach is to take the classical limit ប → 0 of the commutation relations; it yields an unbelievably strong bound, but its robustness might be questioned ͓5͔.We will work in arbitrary D Ͼ 1 dimensions, where ͑1͒ takes the tensorial formwhich, assuming that the momenta commute ͓P i , P j ͔ =0, leads via the Jacobi identity to the nontrivial position commutation relationsThe position and momentum operators can be represented bywhere the operators x i and p j satisfy the canonical commutation relations ͓x i , p j ͔ = iប␦ ij . The simplest representation is momentum diagonal,In this representation the eigenvalue equation for the distance squared operator R 2 = X i X i can be solved exactly. With z = ͑ + Ј͒p 2 − 1 ͑ + Ј͒p 2 + 1 , ͑5͒ the eig...
We present theories ofgravitation based, respectively, on the general linear group GL(n, R) and its inhomoIgeneous extension IGL(n, R) [SO(n-l, 1) and ISO(n -1, 1) for torsion-free manifolds]. Noting that the geometry of the conventional gauge theories can be described in terms of a fiber bundle, and that their action is a scalar in such a superspace, we construct principal fiber bundles based on the above gauge groups and propose to describe gravitation in terms of their corresponding scalar curvatures. To ensure that these manifolds do indeed have close ties with the space-time of general relativity, we make use of the notion of the parallel transport of vector fields in space-time to uniquely relate the connections in space-time to the gauge potentials in fiber bundles. The relations turn out to be similar to that suggested earlier by Yang. The actions we obtain are related to those of Einstein and Yang but are distinct from both and have an Einstein limit. The inclusion of internal symmetry leads to the analogs of Einstein-Yang-Mills equations. A number of variations and less attractive alternatives based on the above groups or their subgroups are also discussed,
In this paper, we study the phenomenology of right-handed neutrino isosinglets. We consider the general situation where the neutrino masses are not necessarily given by m 2 D /M , where m D and M are the Dirac and Majorana mass terms respectively. The consequent mixing between the light and heavy neutrinos is then not suppressed, and we treat it as an independent parameter in the analysis. It turns out that µ − e conversion is an important experiment in placing limits on the heavy mass scale (M ) and the mixing. Mixings among light neutrinos are constrained by neutrinoless double beta decay, as well as by solar and atmospheric neutrino experiments. Detailed one-loop calculations for lepton number violating vertices are provided.
Abstract. We discuss the properties of Galois Field Quantum Mechanics constructed on a vector space over the finite Galois field GF (q). In particular, we look at 2-level systems analogous to spin, and discuss how SO(3) rotations could be embodied in such a system. We also consider two-particle 'spin' correlations and show that the Clauser-Horne-Shimony-Holt (CHSH) inequality is nonetheless not violated in this model.
We review our work on the minimal length uncertainty relation as suggested by perturbative string theory. We discuss simple phenomenological implications of the minimal length uncertainty relation and then argue that the combination of the principles of quantum theory and general relativity allow for a dynamical energy-momentum space. We discuss the implication of this for the problem of vacuum energy and the foundations of non-perturbative string theory.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.