Gauge Theory Amplitudes and String Theory in Twistor Space

现代理论物理学的最终目标之一是发现万物理论，这是对我们在自然界中观察到的四种明显不同的力的统一描述：电、弱、强，最后是引力。在20世纪70年代，提出了一个理论框架，完成了将电磁学和弱、强相互作用纳入统一理论的任务，即所谓的基本相互作用标准模型。这一理论已经通过了许多重要的实验测试，并得到了非常详细的研究，这要归功于强大的对撞机，在对撞机中，粒子以非常高的能量散射以研究它们的相互作用。在自然界的四种基本力中，重力是我们最熟悉的一种。然而，正是引力在很长一段时间内逃脱了与其他引力统一的所有尝试。此外，也许令人惊讶的是，在所有耦合常数中——粗略地说，耦合常数量化了力的强度——牛顿常数（与引力强度有关）是迄今为止最小的常数，也是实验精度最低的常数。目前，只有一种理论将引力与标准模型的力结合起来，这就是弦理论。一个音乐上的类比在这里是合适的：当我们敲击钢琴的琴弦时，我们不仅产生了基本的声音，而且还产生了无限的（不易察觉的）泛音或谐波。在弦理论中，每个谐波对应一个不同的粒子。不同的谐波频率当然会增加，量子力学告诉我们频率与能量成正比。通过爱因斯坦著名的公式E=mc^2，能量也与质量有关。因此，这个无限大的塔中的所有粒子的质量都在增加。实际上弦理论是在一个稍微不同的统一尝试中发现的。在20世纪60年代，人们在粒子对撞机上发现了大量粒子的扩散。物理学家试图为这个神秘的事实找到一个解释，并认为弦理论及其无限的粒子塔可能是解决这个难题的办法。将粒子表示为弦（具有固有长度的物体）的激发，与我们自德谟克利特时代以来习惯的将粒子视为点状物体的传统概念相比，是一种非常不同的图景。所以对于同一个物体，我们似乎有两种截然不同的描述。物理学家在过去30年左右的时间里认识到，粒子相互作用的许多方面也可以通过点粒子的概念或替代弦的描述来描述。两个可选描述的存在被称为它们之间的对偶性。有人可能会问，为什么我们需要对同一现象进行多种描述。我们是否不满足于其中一种，可能是传统的点粒子的方法，它似乎工作得很好？对于这个反对意见，有很多可能的答案。首先，对同一现象有几种不同的描述不可避免地会导致对现象本身的更深层次的理解。其次，这些不同的描述可以是互补的，从某种意义上说，一个可能在另一个失败的地方起作用。另一个与这个研究项目直接相关的答案是，点粒子的传统方法（或用数学语言称为场论）并不总是能解释某些散射振幅的惊人和意想不到的简单性——物理学家计算的数量，可以在粒子对撞机上进行实验测量。描述物理可观测物的数学公式中无法解释的美，往往暗示着一些更深层次的数学结构有待揭示；弦理论经常扮演更深层次结构的角色。通过识别已知现象的新的、更简单的描述，我们也获得了很大的计算能力。如果我们希望发现新的物理学，这是至关重要的，这需要我们以极高的实验精度从标准模型物理学的过程中分离出真正的新现象。爱德华·威滕最近发现了一个对偶的例子，这个例子有望与未来的实验直接相关。它已经极大地提高了我们计算现象学上有趣的量的能力。在场论和弦论方面，对这种迷人的新二象性的研究，是我研究计划的主题。

One of the ultimate goals of modern theoretical physics is to discover the Theory of Everything, that is a unified description for the four apparently different forces we observe in nature: electromagnetic, weak, strong, and finally gravitational. In the 1970s, a theoretical framework was proposed which accomplishes the task of incorporating electromagnetism and the weak and strong force into a unified theory, the so-called Standard Model of fundamental interactions. This theory has passed many remarkable experimental tests, and has been studied in great detail thanks to powerful colliders, where particles are scattered at very high energy to study their interactions.Of the four fundamental forces of nature, gravity is the one we have the most experience of. It is however gravity which has escaped all attempts to be unified with the others for the longest time. Furthermore, and perhaps surprisingly, of all coupling constants - the number which, roughly speaking, quantifies the strength of the force - Newton's constant (related to the strength of gravity) is by far the smallest and also the one which is known with the smallest experimental precision. At present, there is only one theory which incorporates gravity with the forces of the Standard Model: this is String Theory. A musical analogue is appropriate here: When we hit a piano string, we not only produce the fundamental sound but also an infinite series of (less perceivable) overtones, or harmonics. In string theory, each harmonic corresponds to a different particle. Different harmonics have of course increasing frequency, and Quantum Mechanics taught us that frequency is proportional to energy. Energy is also related to mass, through Einstein's famous formula E=mc^2. Therefore all the particles in this infinite tower have an increasing mass.Actually string theory was discovered in a slightly different unification attempt. In the 1960s, a vast proliferation of particles had been discovered at particle colliders. Physicists tried to find an explanation for this mysterious fact, and thought that a theory of strings, with its infinite tower of particles, could be the solution to the puzzle. Representing a particle as the excitation of a string - an object with an intrinsic length - is a very different picture compared to the traditional concept of a particle as a pointlike object that we have been accustomed to since the time of Democritus. So it seems that we have two very different descriptions for the same object. What physicists realised in the last 30 years or so, is that many aspects of the interactions of particles can also be described by either resorting to the concept of point particle or to the alternative string description. The existence of two alternative descriptions is referred to as a duality between them.One could then ask why we need many descriptions of the same phenomena. Are we not satisfied with one, possibly the conventional one in terms of point particles, which seems to work pretty well? There are many possible answers to this objection. Firstly, having several different description of the same phenomenon leads inevitably to a deeper understanding of the phenomenon itself. Secondly, these different descriptions can be complementary, in the sense that one might work where the other fails. Another answer, directly relevant to this research project, is that the conventional approach of point particles (or Field Theory, in mathematical language) does not always account for the marvellous and unexpected simplicity of certain scattering amplitudes - quantities that physicists compute and can be measured experimentally at particle colliders. Unexplained beauty in mathematical formulae describing physical observables is often the hint of some deeper mathematical structure to be uncovered; string theory often plays the role of that deeper structure. By identifying new, simpler descriptions of known phenomena, we also gain a great deal in terms of computational power. This is crucial if we wish to discover new physics, which requires us to disentangle genuinely new phenomena from the processes due to Standard Model physics with great experimental precision. Edward Witten has recently discovered an example of duality which promises to be directly relevant for future experiments. It has already made it possible to dramatically increase our ability to compute phenomenologically interesting quantities. The study of this fascinating new duality, both on the field theory and on the string theory side, is the subject of my research proposal.