Unravelling the Non-Perturbative Structure of Gauge Theory

一个世纪前，马克斯·普朗克（Max Planck）假设能量不是一个连续的量，而是以称为量子的离散单位出现的。这些影响是如此之小，以至于我们在日常生活中通常看不到它们的影响。尽管如此，它们从根本上改变了理论的性质。这种能量和其他量的离散化，在经典理论中被称为量子化。它已经被用于电磁相互作用，以及核力，统称为规范理论。这些量子规范理论所做的预测已经与实验相匹配，达到了惊人的精确度。例如，核力规范理论预测质子和中子是由被称为夸克的极小粒子组成的，夸克是后来在实验中发现的。夸克不同于其他粒子，如电子或质子，因为它们不是单独出现的。它们之间的相互作用非常强烈，以至于单个夸克很快就会吸引其他夸克，形成可以观察到的粒子，比如质子或中子。这种性质被称为约束，是由于核相互作用的强或非摄动性质而产生的。目前我们还没有办法从核相互作用的规范理论推导出约束是如何发生的。这是当今理论物理学面临的一大挑战。在普朗克发现万有引力二十年后，爱因斯坦在一个独立的发展中提出了一个被称为广义相对论的万有引力理论，它推广了牛顿的万有引力定律。在广义相对论中，时空被物质弯曲，比如地球，正是这种弯曲使物体在引力作用下“下落”。GR也在许多实验中得到了验证。然而，GR是一个经典理论，能量是一个连续的量。鉴于量子化规范理论的成功，物理学家试图量子化广义相对论。结果证明，通常的量子化过程不能应用于广义相对论！但在物理学中，能量是一个普遍的量；物理世界的某些部分，如原子，是用能量量子化的理论来描述的，而另一些部分，如行星和恒星，是用能量连续的理论来描述的，这是不可能的！引力的量子化问题已经成为物理学的中心理论问题之一。另一种将引力量子化的方法是使用弦理论。在这种方法中，自然界的基本粒子（如电子、夸克或引力子）根本不是粒子，而是弦。到目前为止，我们还没有观察到这些弦，因为它们确实很小。到目前为止，弦理论的接受来自于一个理论上的事实，即它们给出了一个一致的量子规范和引力理论。最近，量子化引力和解释约束这两个显然截然不同的问题，通过规范/弦对应关系相互联系起来。这个令人难以置信的结果，大约30年前由t Hooft预测，最近由Maldacena提出，表明引力理论可以用类核相互作用理论来描述！这种对应关系是现代理论物理学中两个最具挑战性问题之间的一座迷人的桥梁。我相信这种对应关系一方面可以教给我们很多关于规范理论中约束的本质的知识，另一方面也可以教给我们很多关于引力量子化的知识。在我的工作中，我打算使用规范/弦对应关系来学习规范理论现象，如约束。特别地，我想找出一个引力理论如何重新排列成规范相互作用的理论。在此过程中，我将特别关注引力理论的“弦”性质。最初，我的工作将集中于规范理论，它比核相互作用理论更对称。尽管这些理论更加对称，但它们与现实世界中的理论有许多相似之处。由于我提出的理解这种重新排列的程序并不明确地依赖于规范理论中存在的额外对称性，因此对于这些理论，我已经期望学习大量关于规范理论行为的知识。一旦理解了这种将弦理论重新排列成规范理论的方法，对于更对称的理论，我打算将其应用于不对称的规范理论，以便更多地了解核相互作用的规范理论。我相信，找出“弦”引力理论是如何重新组织成这些现实的规范理论的，将教会我们关于非微扰规范理论动力学的知识，比如约束。一个世纪前，马克斯·普朗克（Max Planck）假设能量不是一个连续的量，而是以称为量子的离散单位出现的。这些影响是如此之小，以至于我们在日常生活中通常看不到它们的影响。尽管如此，它们从根本上改变了理论的性质。这种能量和其他量的离散化，在经典理论中被称为量子化。它已经被用于电磁相互作用，以及核力，统称为规范理论。这些量子规范理论所做的预测已经与实验相匹配，达到了惊人的精确度。例如，核力规范理论预测质子和中子是由被称为夸克的极小粒子组成的，夸克是后来在实验中发现的。夸克不同于其他粒子，如电子或质子，因为它们不是单独出现的。它们之间的相互作用非常强烈，以至于单个夸克很快就会吸引其他夸克，形成可以观察到的粒子，比如质子或中子。这种性质被称为约束，是由于核相互作用的强或非摄动性质而产生的。目前我们还没有办法从核相互作用的规范理论推导出约束是如何发生的。这是当今理论物理学面临的一大挑战。在普朗克发现万有引力二十年后，爱因斯坦在一个独立的发展中提出了一个被称为广义相对论的万有引力理论，它推广了牛顿的万有引力定律。在广义相对论中，时空被物质弯曲，比如地球，正是这种弯曲使物体在引力作用下“下落”。GR也在许多实验中得到了验证。然而，GR是一个经典理论，能量是一个连续的量。鉴于量子化规范理论的成功，物理学家试图量子化广义相对论。结果证明，通常的量子化过程不能应用于广义相对论！但在物理学中，能量是一个普遍的量；物理世界的某些部分，如原子，是用能量量子化的理论来描述的，而另一些部分，如行星和恒星，是用能量连续的理论来描述的，这是不可能的！引力的量子化问题已经成为物理学的中心理论问题之一。另一种将引力量子化的方法是使用弦理论。在这种方法中，自然界的基本粒子（如电子、夸克或引力子）根本不是粒子，而是弦。到目前为止，我们还没有观察到这些弦，因为它们确实很小。到目前为止，弦理论的接受来自于一个理论上的事实，即它们给出了一个一致的量子规范和引力理论。最近，量子化引力和解释约束这两个显然截然不同的问题，通过规范/弦对应关系相互联系起来。这个令人难以置信的结果，大约30年前由t Hooft预测，最近由Maldacena提出，表明引力理论可以用类核相互作用理论来描述！这种通信是两个最重要的人之间的一座迷人的桥梁

A century ago Max Planck postulated that energy is not a continuous quantity, but rather that it comes in discrete units called quanta. These are so small that we do not normally see their effect in our day-to-day life. Nonetheless they fundamentally alter the properties of a theory. This discretisation of the energy, and other quantities, in a classical theory, is known as quantisation. It has been carried out for electromagnetic interactions, as well as for nuclear forces, known collectively as gauge theories. The predictions made by these quantum gauge theories have been matched with experiments to a spectacular degree of precision.For example, the gauge theory of nuclear forces predicts that protons and neutrons are composed of extremely small particles called quarks, which have since been found experimentally. Quarks are different from other particles such as electrons or protons, in that they do not occur on their own. They interact so strongly with one another that a single quark very quickly attracts other quarks to form observed particles such as protons or neutrons. This property, known as confinement, occurs because of the strong, or non-perturbative, nature of nuclear interactions. At present we have no way of deriving from the gauge theory of nuclear interactions how confinement happens. It is one of the big challenges of theoretical physics today.In an independent development, twenty years after Planck's discovery, Einstein formulated a theory of gravity known as General Relativity (GR), which generalised Newton's law of gravity. In GR spacetime is curved by matter, such as the earth, and it is this curvature that makes objects 'fall' under gravity. GR too has been verified in many experiments. However, GR is a classical theory, with energy being a continuous quantity.Given the success of quantising gauge theories, physicists tried to quantise GR. It turns out that the usual quantisation procedure cannot be applied to GR! But energy is a universal quantity in physics; it cannot be that some parts of the physical world, such as the atom are described by theories in which energy is quantised, while others, describing planets and stars are described by theories in which energy in continuous! The problem of quantising gravity has become one of the central theoretical problems in physics.An alternative way to quantise gravity has been to use string theory. In this approach, fundamental particles of nature (such as electrons, quarks or gravitons) are not particles at all but rather strings. We have not observed these strings to date because they are very small indeed. So far, the acceptance of string theory comes from the theoretical fact that they give a consistent quantum gauge and gravity theory.Recently, the two, apparently very different, problems of quantising gravity and explaining confinement have been related to one another via the gauge/string correspondence. This incredible result, predicted some 30 years ago by 't Hooft and recently presented by Maldacena, shows that a theory of gravity can be described by a theory of nuclear-like interactions! This correspondence is a fascinating bridge between two of the most challenging problems in modern theoretical physics. I believe that this correspondence can teach us a great deal about the nature of confinement in gauge theory on the one hand, and about the quantisation of gravity on the other.In my work I intend to use the gauge/string correspondence to learn about gauge theory phenomena such as confinement. In particular, I intend to find out how a theory of gravity can re-arrange itself into a theory of gauge interactions. In doing this I will be paying particular attention to the 'stringy' nature of the gravitational theory. Initially, my work will focus on gauge theories which are more symmetric than the theory of nuclear interactions. Despite being more symmetric, such theories possess many similarities with those in the real world. Since the procedure I propose for understanding this re-arrangement does not rely explicitly on the extra symmetries present in the gauge theory, already for such theories I expect to learn a great deal about gauge theory behaviour.Once an understanding of this re-arrangement of string theory into a gauge theory is understood for the more symmetric theories, I intend to apply it to gauge theories with less symmetry, in order to learn more about the gauge theory of nuclear interactions. Finding out how the 'stringy' gravity theory re-organises itself into these realistic gauge theories, I believe, will teach us about non-perturbative gauge theory dynamics such as confinement.A century ago Max Planck postulated that energy is not a continuous quantity, but rather that it comes in discrete units called quanta. These are so small that we do not normally see their effect in our day-to-day life. Nonetheless they fundamentally alter the properties of a theory. This discretisation of the energy, and other quantities, in a classical theory, is known as quantisation. It has been carried out for electromagnetic interactions, as well as for nuclear forces, known collectively as gauge theories. The predictions made by these quantum gauge theories have been matched with experiments to a spectacular degree of precision.For example, the gauge theory of nuclear forces predicts that protons and neutrons are composed of extremely small particles called quarks, which have since been found experimentally. Quarks are different from other particles such as electrons or protons, in that they do not occur on their own. They interact so strongly with one another that a single quark very quickly attracts other quarks to form observed particles such as protons or neutrons. This property, known as confinement, occurs because of the strong, or non-perturbative, nature of nuclear interactions. At present we have no way of deriving from the gauge theory of nuclear interactions how confinement happens. It is one of the big challenges of theoretical physics today.In an independent development, twenty years after Planck's discovery, Einstein formulated a theory of gravity known as General Relativity (GR), which generalised Newton's law of gravity. In GR spacetime is curved by matter, such as the earth, and it is this curvature that makes objects 'fall' under gravity. GR too has been verified in many experiments. However, GR is a classical theory, with energy being a continuous quantity.Given the success of quantising gauge theories, physicists tried to quantise GR. It turns out that the usual quantisation procedure cannot be applied to GR! But energy is a universal quantity in physics; it cannot be that some parts of the physical world, such as the atom are described by theories in which energy is quantised, while others, describing planets and stars are described by theories in which energy in continuous! The problem of quantising gravity has become one of the central theoretical problems in physics.An alternative way to quantise gravity has been to use string theory. In this approach, fundamental particles of nature (such as electrons, quarks or gravitons) are not particles at all but rather strings. We have not observed these strings to date because they are very small indeed. So far, the acceptance of string theory comes from the theoretical fact that they give a consistent quantum gauge and gravity theory.Recently, the two, apparently very different, problems of quantising gravity and explaining confinement have been related to one another via the gauge/string correspondence. This incredible result, predicted some 30 years ago by 't Hooft and recently presented by Maldacena, shows that a theory of gravity can be described by a theory of nuclear-like interactions! This correspondence is a fascinating bridge between two of the most