Defects in integrable classical and quantum field theory

可积场论是一个精细调谐的系统，在其中加入杂质通常会破坏可积性。顾名思义，添加缺陷意味着字段的某些属性将不连续。例如，在流体力学中，流体（液体或气体）的速度由取决于时间和空间位置的场（实际上是矢量场）表示;典型的缺陷是速度突然变化的“孔”或“冲击”。在可积系统的背景下，一个数学问题就是简单地问，在不破坏可积性的微妙性质的情况下，允许的可能的不连续性是什么。乍一看，这似乎是不可能安排的。然而，在2002年，Bowcock、科里根和Zambon指出，这可以通过一组合适的缺陷条件来实现，这些缺陷条件与缺陷两侧的场的导数有关。此外，这些特殊的条件必须与“冻结”在缺陷位置的贝克兰德变换有关。这一发现是令人兴奋的，因为贝克兰德变换首先出现在100多年前的微分几何背景下（在研究常负曲率曲面时），然后在60年代在产生孤子解时发挥了作用，但在这里，它们出人意料地出现在拉格朗日不连续公式中，回答了一个显然无害的问题。在sine-Gordon模型和其他可积模型中存在孤子，人们自然会问，当孤子遇到缺陷时，它会发生什么。这个问题的答案也是出乎意料的，有三种可能性：孤子可能被延迟，它可能被改变为反孤子，或者它可能被缺陷吸收。可以插入几个缺陷，每个缺陷的行为都独立于其他缺陷。事实上，它也被证明，缺陷可以移动和散射一致，在他们自己和孤立子。一旦它是明确的一个可能有缺陷的可积系统中的另一个数学问题是分类所有可能的类型。也有兴趣在量子场论与任何可积系统，因为它提供了一个理论实验室的想法和技术，希望适用于不可积场论模型，在凝聚态物质或粒子物理学中，可以尝试并随后进行调整。在这种情况下，询问缺陷的作用是很自然的，实际上这个问题在前面已经被问过了（在90年代中期由Delfino，Mussardo和Simonetti），并且事实证明，sine-Gordon模型中的可积缺陷完全符合早期的工作，并显着深化它。此外，现在很清楚，存在新的效应，包括与可积缺陷相关的具有有限寿命的不稳定状态（并且与缺陷必须“吃掉”孤子的奇怪能力密切相关）。同样，数学问题是对所有的可能性进行分类，在量子背景下，这与对“量子群”的无限维表示进行分类密切相关。最后，孤子通过缺陷或被缺陷吸收的奇怪行为表明，逻辑门可以用孤子和缺陷来构造。通过构造一个通用门（科里根和赞邦），已经证明了这是可行的，并且研究如何安排孤子和缺陷来执行所有标准逻辑功能将是有趣的。仍然存在的问题是，是否有可能构建一个真实的物理设备，可以显示的属性（至少近似）的可积缺陷。有很多真实的世界的情况下，几乎可集成的描述，并有过多的缺陷和杂质，但它仍然有待观察，这两个是否可以联系在一起，在一个真正有用的方式。

An integrable field theory is a finely-tuned system and adding an impurity to it will generally destroy integrability. As the name implies, adding a defect means some property of the field or fields will be discontinuous. For example, in fluid mechanics the velocity of a fluid (liquid or gas) is represented by a field (actually a vector field) depending on time and position in space; and a typical defect is a 'bore' or 'shock' through which the velocity changes abruptly. In the context of integrable systems a mathematical question is simply to ask what might the possible discontinuities be that are permitted without destroying the delicate property of integrability. At first sight it seems most unlikely that this can be arranged. However, it was pointed out in 2002 (by Bowcock, Corrigan and Zambon) that this can be achieved by a suitable set of defect conditions relating the derivatives of fields on either side of the defect. Moreover, these special conditions must be related to a Backlund transformation 'frozen' at the location of the defect. This discovery is exciting because Backlund transformations first arose more than 100 hundred years ago in the context of differential geometry (in the study of surfaces of constant negative curvature), and then played a role in the sixties in generating soliton solutions, yet here they arrive unexpectedly in a Lagrangian formulation of a discontinuity, in answer to an apparently innocuous question. Within the sine-Gordon and other integrable models there are solitons and it is natural to ask what might happen to a soliton when it meets a defect. The answer to this question is also unexpected and there are three possibilities: the soliton might be delayed, it might be changed to an anti-soliton, or it might be absorbed by the defect. It is possible to insert several defects and each will behave independently of the others. In fact it has also been shown that the defects may move and scatter consistently among themselves and with solitons.Once it is clear one may have defects within integrable systems another mathematical question is to classify all possible types.There is also interest in the quantum field theory associated to any integrable system because it provides a theoretical laboratory where ideas and techniques hopefully applicable to non-integrable field theory models, in condensed matter or particle physics, can be tried out and subsequently adapted. It is natural to ask about the role of defects in this context, and actually that question was asked earlier (in the mid-nineties by Delfino, Mussardo and Simonetti), and it turns out that the integrable defects within the sine-Gordon model fit perfectly to the earlier work, and significantly deepen it. Moreover, it is now clear there are new effects including unstable states with finite lifetimes associated with an integrable defect (and intimately related to the curious capacity a defect has to 'eat' a soliton). Again, the mathematical question is to classify all the possibilities and, in the quantum context, this is intimately related to classifying the infinite dimensional representations of 'quantum groups'.Finally, the curious behaviour of solitons passing a defect, or being absorbed by it, suggests that logical gates may be constructed using solitons and defects. It has alread been demonstrated, by constructing a universal gate (Corrigan and Zambon), that this is feasible, and it will be interesting to investigate how to arrange solitons and defects to perform all the standard logical functions. There remains the question of whether it is possible to construct a real physical device that can display the properties (at least approximately) of an integrable defect. There are plenty of real world situations with near integrable descriptions, and there is a plethora of defects and impurities but it remains to be seen whether the two can be linked together in a genuinely useful manner.