2009-12-17 22:25 量子尺度下的時空可能有碎形特性
量子尺度下的時空可能有碎形特性
Spacetime May Have Fractal Properties on a Quantum Scale
As scale decreases, the number of dimensions of k-Minkowski spacetime (red line), which is an example of a space with quantum group symmetry, decreases from four to three. In contrast, classical Minkowski spacetime (blue line) is four-dimensional on all scales. This finding suggests that quantum groups are a valid candidate for the description of a quantum spacetime, and may have connections with a theory of quantum gravity. Image credit: Dario Benedetti.
(PhysOrg.com) -- 通常，我們認為時空（spacetime）是四維的，有三維的空間與一維的時間。然而，這種歐幾里德（Euclidean）的觀點只是多維度時空版本的眾多可能之一。例如，弦論（string theory）預測了額外維度的存在 -- 6 維、7 維甚至是 20 維或以上。如同物理學家常解釋的，不可能看見這些額外的維度；它們的存在主要是為了滿足數學等式。
(PhysOrg.com) -- Usually, we think of spacetime as being four-dimensional, with three dimensions of space and one dimension of time. However, this Euclidean perspective is just one of many possible multi-dimensional varieties of spacetime. For instance, string theory predicts the existence of extra dimensions - six, seven, even 20 or more. As physicists often explain, it’s impossible to visualize these extra dimensions; they exist primarily to satisfy mathematical equations.
好像額外的維度還不夠奇怪似的，新研究已探索到一種使人更難以理解的可能性：時空的維度會依照尺度（scale）而改變，而且維度在小尺度上可能具有碎形（fractal）的特性。在一項最近的研究中，Dario Benedetti，（加拿大）Ontario省 Waterloo 市 Perimeter Institute for Theoretical Physics（PI 理論物理學研究所）的物理學家，以「在短尺度上脫離古典值的、尺度依存的（scale-dependent）維度」研究了二種可能的時空例子。不僅止於一種有趣的點子，這種現象也許為相對論的量子理論（quantum theory of relativity）提供了洞見，那也已被指出具有尺度依存的維度。Benedetti 的研究發表在最近一期的 Physical Review Letters 中。
In his study, Benedetti explains that a spacetime with quantum group symmetry has in general a scale-dependent dimension. Unlike classical groups, which act on commutative spaces, quantum groups act on nocommutative spaces (e.g. where xy doesn’t equal yx), which emerges through their unique curvature and quantum uncertainty. Here, Benedetti considers two types of spacetime with quantum group symmetry - a quantum sphere and k-Minkowski spacetime - and calculates their dimensions. In both spaces, the dimensions have fractal properties at small scales, and only reach classical values at large scales.
"簡單的說，量子群與非交換幾何之間的關係如下，" 他解釋。"古典上，我們知道某些空間在某些古典群的作用下是不變的；例如，歐幾里德空間在旋轉與變換（translations）下不變。量子群是一給定古典群的某種變形（deformation）。不變空間（invariant space）也必須是一古典空間的變形，此一變形使得它不具交換性。這全都已知與碎形無關，但是在我的研究中，我發現它們確實有一種共通的特性，即某種非整數維度（在某些尺度下）。"
Compared to a Euclidean sphere, a quantum sphere’s curvature and uncertainty make it a noncommutative space. When calculating the spectral dimension of the quantum sphere, Benedetti found that it closely resembles a standard sphere on large scales; however, as the scale decreases, the dimensions of the quantum sphere deviate and go down to zero. He describes this phenomenon as a signature of the fuzziness, or uncertainty, of the quantum sphere, and also as resulting from fractal behavior at small scales.
在第二種空間（k-Minkowski 時空）中，維度也自古典 Minkowski 時空的恆常行為（constant behavior）背離。雖然後者總共有四維、尺度的獨立性，不過在量子版本中卻減少到三維，並成為尺度的一種函數。在 k-Minkowski 時空與量子球體中，維度變成非整數的，那是碎形幾何的一種典型特徵。
In the second kind of space, k-Minkowski spacetime, the dimensions also deviate from the constant behavior of classical Minkowski spacetime. While the latter always has four dimensions, independent of the scale, the number of dimensions in the quantum version decreases to three as a function of the scale. In both k-Minkowski spacetime and the quantum sphere, the dimensionality becomes non-integral, which is a typical signature of fractal geometry.
Benedetti 的結果與量子重力的先前途徑相符，那也指出一種具有碎形特性的基底尺度（ground-scale）浮現。總的來說，這些研究也許幫助科學家了解時空的獨特普朗克尺度特性，也許結合了某種重力的量子理論（譯註：量子論的一個未解問題是，無法將重力場量子化，如果可行，那麼愛因斯坦企求的統一場論 (unified field theory，注意與萬有理論 ，Theory of everything，有點不同)，才算成真）。例如，如同 Benedetti 的解釋，量子時空的碎形本質也許讓重力藉由維度縮減（dimensional reduction）糾正它自己的「太超過行為（ultraviolet behavior）」。
“The main problem with gravity is that apparently it cannot be quantized as other field theories; in jargon it is said to be non-renormalizable,” he said. “This problem is specific to four-dimensional spacetime. If spacetime had only two dimensions, then quantum gravity would be much simpler and treatable. The problem with a two-dimensional theory is that it is unphysical, as we see four dimensions at our scales. Things can be solved combining four and two dimensions at different scales. That is, if gravity itself provides a mechanism by which the dimension of spacetime depends on the scale at which we probe it (four at our and larger scales and two at very short scales), then we could have a physical theory (compatible with observations) that is free of quantum (short scale) troubles.”
※ 相關報導：
＊ Fractal Properties of Quantum Spacetime
http://link.aps.org/doi/10.1103/PhysRevLett.102.111303
All rights reserved. This material may not be published, broadcast, rewritten or redistributed in whole or part without the express written permission of PhysOrg.com.
Spacetime May Have Fractal Properties on a Quantum Scale
March 25, 2009 By Lisa Zyga
As scale decreases, the number of dimensions of k-Minkowski spacetime (red line), which is an example of a space with quantum group symmetry, decreases from four to three. In contrast, classical Minkowski spacetime (blue line) is four-dimensional on all scales. This finding suggests that quantum groups are a valid candidate for the description of a quantum spacetime, and may have connections with a theory of quantum gravity. Image credit: Dario Benedetti.
(PhysOrg.com) -- 通常，我們認為時空（spacetime）是四維的，有三維的空間與一維的時間。然而，這種歐幾里德（Euclidean）的觀點只是多維度時空版本的眾多可能之一。例如，弦論（string theory）預測了額外維度的存在 -- 6 維、7 維甚至是 20 維或以上。如同物理學家常解釋的，不可能看見這些額外的維度；它們的存在主要是為了滿足數學等式。
(PhysOrg.com) -- Usually, we think of spacetime as being four-dimensional, with three dimensions of space and one dimension of time. However, this Euclidean perspective is just one of many possible multi-dimensional varieties of spacetime. For instance, string theory predicts the existence of extra dimensions - six, seven, even 20 or more. As physicists often explain, it’s impossible to visualize these extra dimensions; they exist primarily to satisfy mathematical equations.
好像額外的維度還不夠奇怪似的，新研究已探索到一種使人更難以理解的可能性：時空的維度會依照尺度（scale）而改變，而且維度在小尺度上可能具有碎形（fractal）的特性。在一項最近的研究中，Dario Benedetti，（加拿大）Ontario省 Waterloo 市 Perimeter Institute for Theoretical Physics（PI 理論物理學研究所）的物理學家，以「在短尺度上脫離古典值的、尺度依存的（scale-dependent）維度」研究了二種可能的時空例子。不僅止於一種有趣的點子，這種現象也許為相對論的量子理論（quantum theory of relativity）提供了洞見，那也已被指出具有尺度依存的維度。Benedetti 的研究發表在最近一期的 Physical Review Letters 中。
As if
extra dimensions weren’t strange enough, new research has probed an even more
mind-bending possibility: that spacetime has dimensions that change depending on
the scale, and the dimensions could have fractal
properties on small scales. In a recent study, Dario Benedetti, a physicist
at the Perimeter Institute for Theoretical Physics in Waterloo, Ontario, has
investigated two possible examples of spacetime with scale-dependent dimensions
deviating from classical values at short scales. More than being just an
interesting idea, this phenomenon might provide insight into a quantum theory of relativity, which also has been suggested to have
scale-dependent dimensions. Benedetti’s study is published in a recent issue of
Physical
Review Letters.
" 在量子重力（quantum
gravity）中那是種老想法：短尺度時空也許看起來像泡沫、碎形或類似的東西，" Benedetti
表示。"在我的研究中，我提出為了描述這種量子時空，量子群（quantum
groups）是一種夠格的候選者。此外，在計算這種幽靈維度（spectral
dimension）的過程中，我首度在量子群／非交換幾何（noncommutative geometries，亦稱 quantum
geometries）以及與量子重力顯然無關的嘗試間，例如因果動態三角化（Causal Dynamical
Triangulations，CDT）與精確重整化群（Exact Renormalization
Group，ERG），提供了某種關聯。而在不同的主題之間建立連結，通常是我們能夠理解這些主題的最佳方式之一。"
“It is an
old idea in quantum gravity that at short scales spacetime might appear foamy,
fuzzy, fractal or similar,” Benedetti told PhysOrg.com. “In my work, I suggest
that quantum groups are a valid candidate for the description of such a quantum
spacetime. Furthermore, computing the spectral dimension, I provide for the
first time a link between quantum groups/noncommutative geometries and
apparently unrelated approaches to quantum gravity, such as Causal Dynamical
Triangulations and Exact Renormalization Group. And establishing links between
different topics is often one of the best ways we have to understand such
topics.”
在他的研究中，Benedetti
解釋，一個具有量子群對稱性的時空一般有一個尺度依存的維度。不同於古典群，那作用在交換空間，量子群是作用在非交換空見（例如：xy 不同於
yx），那透過它們獨特的曲率（curvature）以及量子不確定性浮現出來。在此， Benedetti
考慮二類具有量子群對稱性的時空 -- 量子球體（quantum
sphere）與 k-Minkowski 時空
-- 並計算它們的維度。在這二種空間中，維度在小尺度下具有碎形的特性，並只有在大尺度下才達到古典值。In his study, Benedetti explains that a spacetime with quantum group symmetry has in general a scale-dependent dimension. Unlike classical groups, which act on commutative spaces, quantum groups act on nocommutative spaces (e.g. where xy doesn’t equal yx), which emerges through their unique curvature and quantum uncertainty. Here, Benedetti considers two types of spacetime with quantum group symmetry - a quantum sphere and k-Minkowski spacetime - and calculates their dimensions. In both spaces, the dimensions have fractal properties at small scales, and only reach classical values at large scales.
"簡單的說，量子群與非交換幾何之間的關係如下，" 他解釋。"古典上，我們知道某些空間在某些古典群的作用下是不變的；例如，歐幾里德空間在旋轉與變換（translations）下不變。量子群是一給定古典群的某種變形（deformation）。不變空間（invariant space）也必須是一古典空間的變形，此一變形使得它不具交換性。這全都已知與碎形無關，但是在我的研究中，我發現它們確實有一種共通的特性，即某種非整數維度（在某些尺度下）。"
“In
simple words, the relation between quantum groups and noncommutative geometry is
as follows,” he explained. “Classically, we know that certain spaces are
invariant under the action of some classical groups; for example, Euclidean
space is invariant under rotations and translations. A quantum group is a
deformation of a given classical group, and is such that no classical space can
have it as a symmetry group. The invariant space has to be as well a deformation
of a classical space, a deformation that makes it noncommutative. No relation of
all this to fractals is known, but in my work I've found that they do have a
common property, that is, a non-integer dimension (at some
scale).”
相較於歐幾里德球體，一個量子球體的曲率與量子不確定性使得它成為一種非交換空間。當計算量子球體的幽靈維度時，Benedetti
發現它十分類似一個在大尺度下的標準球體；然而，當尺度減小時，量子球體的維度偏離並降到零。他將這種現象描述成量子球體的某種模糊（fuzziness）（或不確定）特徵，而且與小尺度下產生的碎形行為（fractal behavior）一樣。Compared to a Euclidean sphere, a quantum sphere’s curvature and uncertainty make it a noncommutative space. When calculating the spectral dimension of the quantum sphere, Benedetti found that it closely resembles a standard sphere on large scales; however, as the scale decreases, the dimensions of the quantum sphere deviate and go down to zero. He describes this phenomenon as a signature of the fuzziness, or uncertainty, of the quantum sphere, and also as resulting from fractal behavior at small scales.
在第二種空間（k-Minkowski 時空）中，維度也自古典 Minkowski 時空的恆常行為（constant behavior）背離。雖然後者總共有四維、尺度的獨立性，不過在量子版本中卻減少到三維，並成為尺度的一種函數。在 k-Minkowski 時空與量子球體中，維度變成非整數的，那是碎形幾何的一種典型特徵。
In the second kind of space, k-Minkowski spacetime, the dimensions also deviate from the constant behavior of classical Minkowski spacetime. While the latter always has four dimensions, independent of the scale, the number of dimensions in the quantum version decreases to three as a function of the scale. In both k-Minkowski spacetime and the quantum sphere, the dimensionality becomes non-integral, which is a typical signature of fractal geometry.
Benedetti 的結果與量子重力的先前途徑相符，那也指出一種具有碎形特性的基底尺度（ground-scale）浮現。總的來說，這些研究也許幫助科學家了解時空的獨特普朗克尺度特性，也許結合了某種重力的量子理論（譯註：量子論的一個未解問題是，無法將重力場量子化，如果可行，那麼愛因斯坦企求的統一場論 (unified field theory，注意與萬有理論 ，Theory of everything，有點不同)，才算成真）。例如，如同 Benedetti 的解釋，量子時空的碎形本質也許讓重力藉由維度縮減（dimensional reduction）糾正它自己的「太超過行為（ultraviolet behavior）」。
Benedetti’s results match previous approaches to
quantum gravity, which also point to the emergence of a ground-scale spacetime
with fractal properties. Together, these studies may help scientists understand
the unique Planck scale properties of spacetime, and possibly tie in to a quantum
theory of gravity. For instance, as Benedetti explains, the fractal nature
of quantum spacetime might enable gravity to cure its own ultraviolet behavior
by dimensional reduction.
"重力的主要問題是，它顯然無法同其他場論那樣被量子化；以行話來說，那是指不可重整化（non-renormalizable），"
他說。"這個問題為四維時空所特有。如果時空僅有二維，量子重力將更加簡單與能處理。不過二維理論的問題是，當我們在我們的尺度下看待四維時空時，它是非物理的。這些事能藉由在不同尺度下結合四維與二維而獲得解決。亦即，若重力本身提供了某種機制，時空的維度藉此依存在我們探索它的尺度下（在我們的或較大的尺度下是四維，在非常短的尺度下是二維），那麼我們就能擁有一種沒有量子（短尺度）麻煩的重力理論（能與觀察共存）。" “The main problem with gravity is that apparently it cannot be quantized as other field theories; in jargon it is said to be non-renormalizable,” he said. “This problem is specific to four-dimensional spacetime. If spacetime had only two dimensions, then quantum gravity would be much simpler and treatable. The problem with a two-dimensional theory is that it is unphysical, as we see four dimensions at our scales. Things can be solved combining four and two dimensions at different scales. That is, if gravity itself provides a mechanism by which the dimension of spacetime depends on the scale at which we probe it (four at our and larger scales and two at very short scales), then we could have a physical theory (compatible with observations) that is free of quantum (short scale) troubles.”
※ 相關報導：
＊ Fractal Properties of Quantum Spacetime
http://link.aps.org/doi/10.1103/PhysRevLett.102.111303
Dario Benedetti
Phys. Rev. Lett. 102, 111303 (2009) [4 pages]
doi: 10.1103/PhysRevLett.102.111303
More information: Benedetti, Dario.
“Fractal Properties of Quantum Spacetime.” Physical Review Letters 102,
111303 (2009).
Copyright
2009 PhysOrg.com. All rights reserved. This material may not be published, broadcast, rewritten or redistributed in whole or part without the express written permission of PhysOrg.com.
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