A famous quote from Richard Feynman goes “I think it is safe to say that no one understands quantum mechanics.”[22]. In this book we’ll pursue one possible route to such an understanding, emphasizing the deep connections of quantum mechanics to fundamental ideas of modern mathematics. The strangeness inherent in quantum theory that Feynman was referring to has two rather different sources. One of them is the striking disjunction and incommensurability between the conceptual framework of the classical physics which governs our everyday experience of the physical world, and the very different framework which governs physical reality at the atomic scale. Familiarity with the powerful formalisms of classical mechanics and electromagnetism provides deep understanding of the world at the distance scales familiar to us. Supplementing these with the more modern (but still “classical” in the sense of “not quantum”) subjects of special and general relativity extends our understanding into other much less familiar regimes, while still leaving atomic physics a mystery.
Read in context though, Feynman was pointing to a second source of difficulty, contrasting the mathematical formalism of quantum mechanics with that of the theory of general relativity, a supposedly equally hard to understand subject. General relativity can be a difficult subject to master, but its mathematical and conceptual structure involves a fairly straightforward extension of structures that characterize 19th century physics. The fundamental physical laws (Einstein’s equations for general relativity) are expressed as partial differential equations, a familiar if difficult mathematical subject. The state of a system is determined by a set of fields satisfying these equations, and observable quantities are functionals of these fields. The mathematics is largely that of the usual calculus: differential equations and their real-valued solutions.
In quantum mechanics, the state of a system is best thought of as a different sort of mathematical object: a vector in a complex vector space with a Hermitian inner product, the so-called state space. Such a state space will sometimes be a space of functions known as wavefunctions. While these may, like classical fields, satisfy a differential equation, one non-classical feature is that wavefunctions are complex-valued. What’s completely different about quantum mechanics is the treatment of observable quantities, which correspond to self-adjoint linear operators on the state space. When such operators don’t commute, our intuitions about how physics should work are violated, as we can no longer simultaneously assign numerical values to the corresponding observables.
During the earliest days of quantum mechanics, the mathematician Hermann Weyl quickly recognized that the mathematical structures being used were ones he was quite familiar with from his work in the field of representation theory. From the point of view that takes representation theory as a central theme in mathematics, the framework of quantum mechanics looks perfectly natural. Weyl soon wrote a book expounding such ideas [101], but this got a mixed reaction from physicists unhappy with the penetration of unfamiliar mathematical structures into their subject (with some of them characterizing the situation as the “Gruppenpest”, the group theory plague). One goal of this book will be to try and make some of this mathematics as accessible as possible, boiling down part of Weyl’s exposition to its essentials while updating it in the light of many decades of progress towards better understanding of the subject.
Weyl’s insight that quantization of a classical system crucially involves understanding the Lie groups that act on the classical phase space and the unitary representations of these groups has been vindicated by later developments which dramatically expanded the scope of these ideas. The use of representation theory to exploit the symmetries of a problem has become a powerful tool that has found uses in many areas of science, not just quantum mechanics. I hope that readers whose main interest is physics will learn to appreciate some of such mathematical structures that lie behind the calculations of standard textbooks, helping them understand how to effectively exploit them in other contexts. Those whose main interest is mathematics will hopefully gain some understanding of fundamental physics, at the same time as seeing some crucial examples of groups and representations. These should provide a good grounding for appreciating more abstract presentations of the subject that are part of the standard mathematical curriculum. Anyone curious about the relation of fundamental physics to mathematics, and what Eugene Wigner described as “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”[102] should benefit from an exposure to this remarkable story at the intersection of the two subjects.
The following sections give an overview of the fundamental ideas behind much of the material to follow. In this sketchy and abstract form they will likely seem rather mystifying to those meeting them for the first time. As we work through basic examples in coming chapters, a better understanding of the overall picture described here should start to emerge.
来源与版本
正文：英文 · 原著转录
核对状态：AI 辅助转录核对，未作人工审阅
原书 PDF · 印刷页 1、2
来源版本：2025-10-20
来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837