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Einstein’s GTR initiated a new programme for describing fundamental interactions, in which the dynamics was described in geometrical terms. After Einstein’s classic paper on GTR (1916c), the programme was carried out by a sequence of theories. This chapter is devoted to discussing the ontological commitments of the programme (Section 5.2) and to reviewing its evolution (Section 5.3), including some topics (singularities, horizons, and black holes) that began to stimulate a new understanding of GTR only after Einstein’s death (Section 5.4), with the exception of some recent attempts to incorporate the idea of quantization, which will be addressed in Chapter 11. Considering the enormous influence of Einstein’s work on the genesis and developments of the programme, it seems reasonable to start this chapter with an examination of Einstein’s views of spacetime and geometry (Section 5.1), which underlie his programme.
In comparison with STR, which is a static theory of the kinematic structures of Minkowskian spacetime, general theory of relativity (GTR) as a dynamical theory of the geometrical structures of spacetime is essentially a theory of gravitational fields. The first step in the transition from STR to GTR, as we discussed in Section 3.4, was the formulation of EP, through which the inertial structures of the relative spaces of the uniformly accelerated frames of reference can be represented by static homogeneous gravitational fields. The next step was to apply the idea of EP to uniformly rotating rigid systems. Then Einstein (1912a) found that the presence of the resulting stationary gravitational fields invalidated the Euclidean geometry. In a manner characteristic of his style of theorizing, Einstein (with Grossmann, 1913) immediately generalized this result and concluded that the presence of a gravitational field generally required a non-Euclidean geometry and that the gravitational field could be mathematically described by a four-dimensional Riemannian metric tensor gμv (Section 4.1). With the discovery of the generally covariant field equations satisfied by gμv, Einstein (1915a–d) completed his formulation of GTR.
This chapter is devoted to examining the mathematical, physical, and speculative roots of nonabelian gauge theory. The early attempts at applying this theoretical framework to various physical processes will be reviewed, and the reasons for their failures explained.
The historical study of 20th century field theories in the preceding chapters provides an adequate testing ground for models of how science develops. On this basis I shall argue in this chapter that one of the possible ways of achieving conceptual revolutions is what I shall call “ontological synthesis” (Section 12.4). This notion is based on, and gives a strong support to, a special version of scientific realism (Sections 12.3 and 12.5). It has also provided a firm ground for the rationality of scientific growth (Section 12.6).
For a gauge invariant system of quantum fields to be a self-consistent framework for describing various interactions, mechanisms responsible for short-range interactions must be found (Sections 10.1 and 10.2), and its renormalizability be proven (Section 10.3). In addition, nonabelian gauge theories exhibit some novel features, which have suggested certain interpretations concerning the structure of the vacuum state and the conditions for the quantization of physical parameters such as electric charge. Thus, a new question, which had not appeared in the investigations of Abelian-gauge-invariant QED or of other, nongauge-invariant, local field theories, has posed itself with some urgency, and in recent years become a favorite research topic among a sizable portion of mathematics-oriented physicists. This is the question of the global features of nonabelian gauge field theories (Section 10.4). Thus this chapter reviews the formation of the conceptual foundations of gauge theories, both as a theoretical framework and as a research programme, and points to some questions that remain to be addressed by future investigators.
The study of the interactions between electrically charged particles and electromagnetic fields within the framework of quantum field programme (QFP) is called quantum electrodynamics (QED). QED, and in particular its renormalized perturbative formulation, was modeled by various theories to describe other interactions, and thus became the starting point for a new research programme, the quantum field programme (QFP). The programme has been implemented by a series of theories, whose developments are strongly constrained by some of its characteristic features, which have been inherited from QED. For this reason, I shall start this review of the sinuous evolution of QFP with an outline of these features.
Although the developments that I plan to explore began with Einstein’s general theory of relativity (GTR), without a proper historical perspective, it would be very difficult to grasp the internal dynamics of GTR and subsequent developments as further stages of a field programme. Such a perspective can be suitably furnished with an adequate account of the rise of the field programme itself. The purpose of this chapter is to provide such an account, in which major motivations and underlying assumptions of the developments that led to the rise of the field programme are briefly outlined.1
Einstein, in his formative years (1895–1902), sensed a deep crisis in the foundations of physics. On the one hand, the mechanical view failed to explain electromagnetism, and this failure invited criticisms from the empiricist philosophers, such as Ernst Mach, and from the phenomenalist physicists, such as Wilhelm Ostwald and Georg Helm. These criticisms had a great influence on Einstein’s assessment of the foundations of physics. His conclusion was that the mechanical view was hopeless. On the other hand, following Max Planck and Ludwig Boltzmann, who were cautious about the alternative electromagnetic view and also opposed to energeticism, Einstein, unlike Mach and Ostwald, believed in the existence of discrete and unobservable atoms and molecules, and took them as the ontological basis for statistical physics. In particular, Planck’s investigations into black body radiation made Einstein recognize a second foundational crisis, a crisis in thermodynamics and electrodynamics, in addition to the one in the mechanical view. Thus it was “as if the ground had been pulled out from under one, with no firm foundation to be seen anywhere, upon which one could have built” (Einstein, 1949).
The origin of the relativity theories was closely bound up with the development of electromagnetic concepts, a development that approached a coherent field-theoretical formulation, according to which all actions may vary in a continuous manner. In contrast, quantum theory arose out of the development of atomic concepts, a development that was characterized by the acknowledgment of a fundamental limitation to classical physical ideas when applied to atomic phenomena. This restriction was expressed in the so-called quantum postulate, which attributed to any atomic process an essential discontinuity that was symbolized by Planck’s quantum of action and soon incarnated in quantization condition (commutation or anticommutation relations) and uncertainty relations.
Quantum field theory (QFT) can be analyzed in terms of its mathematical structure, its conceptual scheme, or its basic ontology. The analysis can be done logically or historically. In this chapter, only the genesis of the conceptual foundations of QFT relevant to its basic ontology will be treated carefully; no detailed discussion of its mathematical structures or its epistemological underpinnings will be given. Some conceptual problems, such as those related to probability and measurement, will be discussed, but only because of their relevance to the basic ontology of QFT, rather than their intrinsic philosophical interest.
The Utrecht proof of the renormalizability of gauge-invariant massive vector meson theories in 1971 (Section 10.3), as observed by influential contemporary physicists, “would change our way of thinking on gauge field theory in a most profound way” (Lee, 1972) and “caused a great stir, made unification into a central research theme” (Pais, 1986). Confidence quickly built up within the particle physics community that a system of quantum fields whose dynamics is fixed by the gauge principle was a self-consistent and powerful conceptual framework for describing fundamental interactions in a unified way.