理论物理:拓扑场论和二维量子引力【转载】
全文转载自知乎:拓扑场论和二维量子引力 – 知乎 (zhihu.com)
1. 引言
过去几年见证了量子场论技术在研究各种数学问题上丰硕的应用。这个主要归功于Edward Witten的项目,已经找到了量子场论和拓扑与几何量间意想不到的关联。这点的精彩范例是,三维规范理论和二维共形场论中出现了像Jones多项式这样的纽结不变量[1]。直到最近,这些发展也主要是带给我们,要么是新的数学上的不变量,要么是对从量子场论的直觉得到的结果的更好理解。用在这些构造中的量子场论,虽然是更物理的场论的近亲,但没有直接的物理应用。
随着对矩阵模型和一般的二维量子引力的研究,意想不到的转机出现了。一个称为拓扑引力[2]的特殊场论,最初被构造来处理Riemann面模空间的问题,被证明与二维量子引力有密不可分的关联。可将它看成引力的另一个更简单的相,其中关联函数更容易计算。借助矩阵模型在二维引力中取得重大突破[3]后不久,Witten指出量子引力可能不过是拓扑引力的简单微扰[4]。这点现在已经有了坚实的证据,提供了从拓扑引力开始得到所有矩阵模型结果的相当直接的方法[4][5][6][7][8][9]。拓扑引力有个推广,称为拓扑弦论,其中引力与各种物质系统耦合[4]。拓扑弦论可用于描述和解与引力耦合的所有 极小模型,以及很多其它理论[10][11][12][13]。
这份讲义旨在对一般的拓扑场论,特别是拓扑引力进行初步介绍。我们的最终目标,是建立起与矩阵模型结果的关联,特别是可积的KdV型层次结构[14][15]的出现。不过,最后一节才会涉及这点。沿途我们将讨论几何,代数和量子场论间诸多优美的关联。
这份讲义中,描述与E. 和H. Verlinde,以及与E. Witten合作的工作的部分,将基本照着同一主题其它已出版的讲义[16],但有些地方可能更加友好。这份讲义会和在冬季学校讲的四节课高度一致,也是依此来组织的。在第2节,我们讨论拓扑场论的一些一般性质。我们强调因子化的概念,并应用于二维情形。在第3节,我们考虑拓扑共形场论,它们与 超对称模型密切相关,并为上节的抽象讨论提供一些例子。拓扑引力放在第4节,那里我们还将用递归关系讨论它的解。最后,在第5节,我们讨论与矩阵模型和可积KdV层次结构的关联。
TED演讲视频:The death of the universe — and what it means for life
物理学:图形化演示视频 Waves: Light, Sound, and the nature of Reality
物理学:知乎上找到的一个系列文章——《Halbach array 的构建与模拟》
知乎上找到了一个系列文章——《Halbach array 的构建与模拟》,一共写了三篇,链接分别如下:
- Halbach array 的构建与模拟(一) – 知乎 (zhihu.com)
- Halbach array 的构建与模拟(二) – 知乎 (zhihu.com)
- Halbach array 的构建与模拟(终极版) – 知乎 (zhihu.com)
以下是全文转载。
飞行:电磁悬浮四轴飞行(器)
Electromagnetic Levitation Quadcopter – YouTube(英文原版)
Obviously this “quadcopter” is a demonstration device, showing how moving magnets over a conducting surface can generate levitation. It has not been optimized to minimize losses or be an efficient mode of transport. I still think it’s pretty cool. I’m used to seeing light things levitated by induced currents but not a 100+ lb machine.
这个飞行方式承载能力很强。
中文字幕版:四旋軸電磁懸浮「直升機」 – Electromagnetic Levitation Quadcopter – YouTube
https://www.youtube.com/watch?v=Tuat0uwdz_A
物理学+数学:旋量简介
知乎的回答:
Clifford代数
和Clifford代数联系紧密的是四元数 ,它是 上的4维线性空间,在四元数的乘法下成为代数。取一组基 ,那么四元数的乘法按 定义。四元数和 群有紧密联系。通常,由Pauli矩阵 , 和 ,可以取 为Lie代数 的一组基。从而 中的元素都可以写成 ,其中 , , 且 。易验证 是单位阵。应用指数映射, 的单参数子群为 ,由于矩阵Lie群可以展开为Taylor级数,所以有 ,从而 的矩阵均形如 ,其中 , 。易看出 ,所以实际上这四个参数构成了球面 ,即 。四元数有忠实表示 。令 , ,那么四元数 表示为 。
令 是代数,不妨将其上的二元运算看成乘法。称子代数 是左理想,若 和 使得 。若是右理想,则 。若同时是左理想和右理想,则称它是双边理想。显然,交换环上定义的交换代数本身就是交换环,所以在给定双边理想后,可以按环论中的做法,引入商环。给定交换环上的环 和其双边理想 ,定义等价关系 若 。这样可以直接记等价类为 。所有等价类的集合再次构成环,其加法按 定义,乘法按 定义。称这个环是商环,记作 。若 是代数,那么是交换环,并且商环 也是交换环,进而是代数。称代数 是因子代数。我们要指出,外代数实际上就是张量代数的因子代数。对于有限维线性空间 ,令 是只有下标的张量代数, 是所有形如 的和构成的双边理想,其中 , 。容易验证 是双边理想,并且 。为了说明这一点,注意到商环的定义,它实际上是让理想消灭了,那么我们理应让对称形式消灭,只留下交错形式。据定义,取 ,那么 ,从而 。对于 ,有 ,若其中两个分量相同,则 ,这就表明确实只剩下了交错形式。注意到,虽然代数 和理想 都是无穷维的,但因子代数却是有限维的。
物理学:An Epic Journey to a Black Hole to Give You Goosebumps
物理学:黑洞信息悖论进展——The Black Hole Information Paradox Comes to an End【转载】
原文来自:The Black Hole Information Paradox Comes to an End | Quanta Magazine
The Most Famous Paradox in Physics Nears Its End
In a landmark series of calculations, physicists have proved that black holes can shed information, which seems impossible by definition. The work appears to resolve a paradox that Stephen Hawking first described five decades ago.
In a series of breakthrough papers, theoretical physicists have come tantalizingly close to resolving the black hole information paradox that has entranced and bedeviled them for nearly 50 years. Information, they now say with confidence, does escape a black hole. If you jump into one, you will not be gone for good. Particle by particle, the information needed to reconstitute your body will reemerge. Most physicists have long assumed it would; that was the upshot of string theory, their leading candidate for a unified theory of nature. But the new calculations, though inspired by string theory, stand on their own, with nary a string in sight. Information gets out through the workings of gravity itself — just ordinary gravity with a single layer of quantum effects.
This is a peculiar role reversal for gravity. According to Einstein’s general theory of relativity, the gravity of a black hole is so intense that nothing can escape it. The more sophisticated understanding of black holes developed by Stephen Hawking and his colleagues in the 1970s did not question this principle. Hawking and others sought to describe matter in and around black holes using quantum theory, but they continued to describe gravity using Einstein’s classical theory — a hybrid approach that physicists call “semiclassical.” Although the approach predicted new effects at the perimeter of the hole, the interior remained strictly sealed off. Physicists figured that Hawking had nailed the semiclassical calculation. Any further progress would have to treat gravity, too, as quantum.
That is what the authors of the new studies dispute. They have found additional semiclassical effects — new gravitational configurations that Einstein’s theory permits, but that Hawking did not include. Muted at first, these effects come to dominate when the black hole gets to be extremely old. The hole transforms from a hermit kingdom to a vigorously open system. Not only does information spill out, anything new that falls in is regurgitated almost immediately. The revised semiclassical theory has yet to explain how exactly the information gets out, but such has been the pace of discovery in the past two years that theorists already have hints of the escape mechanism.
“That is the most exciting thing that has happened in this subject, I think, since Hawking,” said one of the co-authors, Donald Marolf of the University of California, Santa Barbara.
“It’s a landmark calculation,” said Eva Silverstein of Stanford University, a leading theoretical physicist who was not directly involved.
You might expect the authors to celebrate, but they say they also feel let down. Had the calculation involved deep features of quantum gravity rather than a light dusting, it might have been even harder to pull off, but once that was accomplished, it would have illuminated those depths. So they worry they may have solved this one problem without achieving the broader closure they sought. “The hope was, if we could answer this question — if we could see the information coming out — in order to do that we would have had to learn about the microscopic theory,” said Geoff Penington of the University of California, Berkeley, alluding to a fully quantum theory of gravity.
What it all means is being intensely debated in Zoom calls and webinars. The work is highly mathematical and has a Rube Goldberg quality to it, stringing together one calculational trick after another in a way that is hard to interpret. Wormholes, the holographic principle, emergent space-time, quantum entanglement, quantum computers: Nearly every concept in fundamental physics these days makes an appearance, making the subject both captivating and confounding.
And not everyone is convinced. Some still think that Hawking got it right and that string theory or other novel physics has to come into play if information is to escape. “I’m very resistant to people who come in and say, ‘I’ve got a solution in just quantum mechanics and gravity,’” said Nick Warner of the University of Southern California. “Because it’s taken us around in circles before.”
But almost everyone appears to agree on one thing. In some way or other, space-time itself seems to fall apart at a black hole, implying that space-time is not the root level of reality, but an emergent structure from something deeper. Although Einstein conceived of gravity as the geometry of space-time, his theory also entails the dissolution of space-time, which is ultimately why information can escape its gravitational prison.
The Curve Becomes the Key
In 1992, Don Page and his family spent their Christmas vacation house-sitting in Pasadena, enjoying the swimming pool and watching the Rose Parade. Page, a physicist at the University of Alberta in Canada, also used the break to think about how paradoxical black holes really are. His first studies of black holes, when he was a graduate student in the ’70s, were key to his adviser Stephen Hawking’s realization that black holes emit radiation — the result of random quantum processes at the edge of the hole. Put simply, a black hole rots from the outside in.
物理学:Does Time Really Flow? New Clues Come From a Century-Old Approach to Math.
Strangely, although we feel as if we sweep through time on the knife-edge between the fixed past and the open future, that edge — the present — appears nowhere in the existing laws of physics.