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A Generalized Shannon Sampling Theorem, Fields at the Planck Scale as Bandlimited Signals
High Energy Physics - Theory · Submitted 16 May 1999
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We find that these fields are continuous - but possess only a finite density of degrees of freedom, similar to fields on lattices.
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The abstract, sentence by sentence
It has been shown that space-time coordinates can exhibit only very few types of short-distance structures, if described by linear operators: they can be continuous, discrete or "unsharp" in one of two ways. In the literature, various quantum gravity models of space-time at short distances point towards one of these two types of unsharpness. Here, we investigate the properties of fields over such unsharp coordinates. We find that these fields are continuous - but possess only a finite density of degrees of freedom, similar to fields on lattices. We observe that this type of unsharpness is technically the same as the aperture induced unsharpness of optical images. It is also of the same type as the unsharpness of the time-resolution of bandlimited electronic signals. Indeed, as a special case we recover the Shannon sampling theorem of information theory.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | It has been shown that space-time coordinates can exhibit only very few types of short-distance structures, if described by linear operators: they can be continuous, discrete or "unsharp" in one of two ways. | background 1.00 | 0.08 | 0.10 | asserted 0.86 |
| s1 | In the literature, various quantum gravity models of space-time at short distances point towards one of these two types of unsharpness. | background 1.00 | 0.04 | 0.03 | hedged possibility 0.68 |
| s2 | Here, we investigate the properties of fields over such unsharp coordinates. | contribution 1.00 | 0.53 | 0.03 | unclear 0.51 |
| s3 | We find that these fields are continuous - but possess only a finite density of degrees of freedom, similar to fields on lattices. | result 1.00 | 0.95 | 0.63 | asserted 1.00 |
| s4 | We observe that this type of unsharpness is technically the same as the aperture induced unsharpness of optical images. | interpretation 0.60 | 0.77 | 0.10 | asserted 1.00 |
| s5 | It is also of the same type as the unsharpness of the time-resolution of bandlimited electronic signals. | interpretation 0.73 | 0.52 | 0.08 | asserted 1.00 |
| s6 | Indeed, as a special case we recover the Shannon sampling theorem of information theory. | result 0.79 | 0.82 | 0.17 | asserted 0.98 |
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The paper
Original abstract
It has been shown that space-time coordinates can exhibit only very few types of short-distance structures, if described by linear operators: they can be continuous, discrete or "unsharp" in one of two ways. In the literature, various quantum gravity models of space-time at short distances point towards one of these two types of unsharpness. Here, we investigate the properties of fields over such unsharp coordinates. We find that these fields are continuous - but possess only a finite density of degrees of freedom, similar to fields on lattices. We observe that this type of unsharpness is technically the same as the aperture induced unsharpness of optical images. It is also of the same type as the unsharpness of the time-resolution of bandlimited electronic signals. Indeed, as a special case we recover the Shannon sampling theorem of information theory.
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References
How the literature uses it
5 arXiv papers cite this in 9 sentences
Papers citing it · 7 sentences without a confident label
Cites it as background 2 papers
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The kinematics of covariant natural ultraviolet cutoffs are discussed in detail in Refs. [ref] .
Covariant Predictions for Planck-Scale Features in Primordial Power Spectra · Covariant ultraviolet cutoffs · 0.96
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Here we give a short summary of some mathematical results [ref] relevant for the self-adjoint extension of the coordinate operator (c.f. also [ref] ).
Some consequences of GUP induced ultraviolet wavevector cutoff in one-dimensional Quantum Mechanics · Self-adjoint extensions of the coordinate operator · 0.97
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