arXiv subjects

Achim Kempf

Publications and source records attributed to Achim Kempf.

108 records · Page 6Linked to original sources

arXiv2007

Generalization of Quantum Error Correction via the Heisenberg Picture

We show that the theory of operator quantum error correction can be naturally generalized by allowing constraints not only on states but also on observables. The resulting theory describes the correction of algebras of observables (and may therefore suitably be called ``operator algebra quantum error correction''). In particular, the approach provides a framework for the correction of hybrid quantum-classical information and it does not require the state to be entirely in one of the corresponding subspaces or subsystems. We discuss applications to quantum teleportation and to the study of information flows in quantum interactions.

quant-ph↗

arXiv2006

Exact solution of inflationary model with minimum length

Within the inflationary scenario, Planck scale physics should have affected the comoving modes' initial conditions and early evolution, thereby potentially affecting the inflationary predictions for the cosmic microwave background (CMB). This issue has been studied extensively on the basis of various models for how quantum field theory (QFT) is modified and finally breaks down towards the Planck scale. In one model, in particular, an ultraviolet cutoff was implemented into QFT through generalized uncertainty relations which have been motivated from general quantum gravity arguments and from string theory. Here, we improve upon prior numerical and semi-analytical results by presenting the exact mode solutions for both de Sitter and power-law inflation in this model. This provides an explicit map from the modes' initial conditions, which are presumably set by quantum gravity, to the modes' amplitudes at horizon crossing and thus to the inflationary predictions for the CMB. The solutions' particular behaviour close to the cutoff scale suggests unexpected possibilities for how the degrees of freedom of QFT emerge from the Planck scale.

gr-qc↗

arXiv2006

Could the Casimir Effect explain the Energetics of High-Temperature Superconductors?

It is proposed that the Casimir energy of the electromagnetic field in between the layers of cuprate superconductors could significantly contribute to the energy balance in these materials. Since the Casimir energy is strongly dependent on the distance between the layers a corresponding dependence is predicted for the superconducting transition temperatures. Comparison with the experimental data on the transition temperatures of epitaxial superlattices of alternating layers of YBCO and PrBCO of varying thickness shows that these are well reproduced.

cond-mat.supr-con↗

arXiv2004

On Fields with Finite Information Density

The existence of a natural ultraviolet cutoff at the Planck scale is widely expected. In a previous Letter, it has been proposed to model this cutoff as an information density bound by utilizing suitably generalized methods from the mathematical theory of communication. Here, we prove the mathematical conjectures that were made in this Letter.

hep-th↗

arXiv2004

Short-Distance Cutoffs in Curved Space

It is shown that space-time may possess the differentiability properties of manifolds as well as the ultraviolet finiteness properties of lattices. Namely, if a field's amplitudes are given on any sufficiently dense set of discrete points this could already determine the field's amplitudes at all other points of the manifold. The criterion for when samples are sufficiently densely spaced could be that they are apart on average not more than at a Planck distance. The underlying mathematics is that of classes of functions that can be reconstructed completely from discrete samples. The discipline is called sampling theory and is at the heart of information theory. Sampling theory establishes the link between continuous and discrete forms of information and is used in ubiquitous applications from scientific data taking to digital audio.

gr-qc↗

arXiv2003

A Covariant Information-Density Cutoff in Curved Space-Time

In information theory, the link between continuous information and discrete information is established through well-known sampling theorems. Sampling theory explains, for example, how frequency-filtered music signals are reconstructible perfectly from discrete samples. In this Letter, sampling theory is generalized to pseudo-Riemannian manifolds. This provides a new set of mathematical tools for the study of space-time at the Planck scale: theories formulated on a differentiable space-time manifold can be completely equivalent to lattice theories. There is a close connection to generalized uncertainty relations which have appeared in string theory and other studies of quantum gravity.

gr-qc↗

arXiv2003

Aspects of Information Theory in Curved Space

The often-asked question whether space-time is discrete or continuous may not be the right question to ask: Mathematically, it is possible that space-time possesses the differentiability properties of manifolds as well as the ultraviolet finiteness properties of lattices. Namely, physical fields in space-time could possess a finite density of degrees of freedom in the following sense: if a field's amplitudes are given on a sufficiently dense set of discrete points then the field's amplitudes at all other points of the manifold are fully determined and calculable. Which lattice of sampling points is chosen should not matter, as long as the lattices' spacings are tight enough, for example, not exceeding the Planck distance. This type of mathematical structure is known within information theory, as sampling theory, and it plays a central role in all of digital signal processing.

gr-qc↗

arXiv2003

Unusual Properties of Superoscillating Particles

It has been found that functions can oscillate locally much faster than their Fourier transform would suggest is possible - a phenomenon called superoscillation. Here, we consider the case of superoscillating wave functions in quantum mechanics. We find that they possess rather unusual properties which raise measurement theoretic, thermodynamic and information theoretic issues. We explicitly determine the wave functions with the most pronounced superoscillations, together with their scaling behavior. We also address the question how superoscillating wave functions could be produced.

quant-ph↗

arXiv2002

On the Vacuum Energy in Expanding Space-Times

If there is a shortest length in nature, for example at the Planck scale of 10^-35m, then the cosmic expansion should continually create new comoving modes. A priori, each of the new modes comes with its own vacuum energy, which could contribute to the cosmological constant. I discuss possible mathematical models for a shortest length and an explicit model for a corresponding mode-generating mechanism.

gr-qc↗

arXiv1997

Corrections to the Emergent Canonical Commutation Relations Arising in the Statistical Mechanics of Matrix Models

We study the leading corrections to the emergent canonical commutation relations arising in the statistical mechanics of matrix models, by deriving several related Ward identities, and give conditions for these corrections to be small. We show that emergent canonical commutators are possible only in matrix models in complex Hilbert space for which the numbers of fermionic and bosonic fundamental degrees of freedom are equal, suggesting that supersymmetry will play a crucial role. Our results simplify, and sharpen, those obtained earlier by Adler and Millard.

hep-th↗

arXiv1996

String/Quantum Gravity motivated Uncertainty Relations and Regularisation in Field Theory

The possibility of the existence of small correction terms to the canonical commutation relations and the uncertainty relations has recently found renewed interest. In particular, such correction terms could induce finite lower bounds $Δx_0, Δp_0$ to the resolution of distances and/or momenta. I review a general framework for the path integral formulation of quantum field theories on such generalised geometries, and focus then on the mechanisms by which $Δp_0>0$, and/or $Δx_0>0$ lead to IR and/or UV regularisation.

hep-th↗

arXiv1996

Nonpointlike Particles in Harmonic Oscillators

Quantum mechanics ordinarily describes particles as being pointlike, in the sense that the uncertainty $Δx$ can, in principle, be made arbitrarily small. It has been shown that suitable correction terms to the canonical commutation relations induce a finite lower bound to spatial localisation. Here, we perturbatively calculate the corrections to the energy levels of an in this sense nonpointlike particle in isotropic harmonic oscillators. Apart from a special case the degeneracy of the energy levels is removed.

hep-th↗

arXiv1996

On Path Integration on Noncommutative Geometries

We discuss a recent approach to quantum field theoretical path integration on noncommutative geometries which imply UV/IR regularising finite minimal uncertainties in positions and/or momenta. One class of such noncommutative geometries arise as `momentum spaces' over curved spaces, for which we can now give the full set of commutation relations in coordinate free form, based on the Synge world function.

hep-th↗

arXiv1996

On Noncommutative Geometric Regularisation

Studies in string theory and in quantum gravity suggest the existence of a finite lower bound to the possible resolution of lengths which, quantum theoretically, takes the form of a minimal uncertainty in positions $Δx_0$. A finite minimal uncertainty in momenta $Δp_0$ has been motivated from the absence of plane waves on generic curved spaces. Both effects can be described as small noncommutative geometric features of space-time. In a path integral approach to the formulation of field theories on noncommutative geometries, we can now generally prove IR regularisation for the case of noncommutative geometries which imply minimal uncertainties $Δp_0$ in momenta.

hep-th↗

arXiv1996

On Quantum Field Theory with Nonzero Minimal Uncertainties in Positions and Momenta

We continue studies on quantum field theories on noncommutative geometric spaces, focusing on classes of noncommutative geometries which imply ultraviolet and infrared modifications in the form of nonzero minimal uncertainties in positions and momenta. The case of the ultraviolet modified uncertainty relation which has appeared from string theory and quantum gravity is covered. The example of euclidean $ϕ^4$-theory is studied in detail and in this example we can now show ultraviolet and infrared regularisation of all graphs.

hep-th↗

arXiv1995

Maximal Localisation in the Presence of Minimal Uncertainties in Positions and Momenta

Small corrections to the uncertainty relations, with effects in the ultraviolet and/or infrared, have been discussed in the context of string theory and quantum gravity. Such corrections lead to small but finite minimal uncertainties in position and/or momentum measurements. It has been shown that these effects could indeed provide natural cutoffs in quantum field theory. The corresponding underlying quantum theoretical framework includes small `noncommutative geometric' corrections to the canonical commutation relations. In order to study the full implications on the concept of locality it is crucial to find the physical states of then maximal localisation. These states and their properties have been calculated for the case with minimal uncertainties in positions only. Here we extend this treatment, though still in one dimension, to the general situation with minimal uncertainties both in positions and in momenta.

hep-th↗

arXiv1992

Quantum Group Symmetric Bargmann Fock Construction

Usually in quantum mechanics the Heisenberg algebra is generated by operators of position and momentum. The algebra is then represented on an Hilbert space of square integrable functions. Alternatively one generates the Heisenberg algebra by the raising and lowering operators. It is then natural to represent it on the Bargmann Fock space of holomorphic functions. In the following I show that the Bargmann Fock construction can also be done in the quantum group symmetric case. This leads to a 'q- deformed quantum mechanics' in which the basic concepts, Hilbert space of states and unitarity of time evolution, are preserved.

hep-th↗

arXiv1992

Approach to Quantum Group deformed Quantum Field Theory

In quantum field theory the creation and annihilation operators that are located at the points in 3-momentum space have commutation relations that are conserved under the action of a $U({\infty})$ group. Here it is shown how to define an appropriate $U_q({\infty})$ which then leads to deformed commutation relations. A formalism that was previously defined for quantum group symmetric quantum mechanics can be applied by imposing simple adjustments. The simplest example of a free scalar $q$-bosonic field is discussed. The deformation occurs as a multi-particle effect, which introduces a kinetic energy (i.e. $({\vec{p}})^2$) - ordering of the involved particles.

hep-th↗