Careful experiments with radiation, molecules, atoms and subatomic systems have convinced physicists over the years that the laws governing them (embodied in quantum mechanics) are quite different from those governing familiar objects of everyday experience (embodied in classical mechanics and electrodynamics). Quantum mechanics has turned out to be a very accurate and reliable theory though, even after more than seventy years of its birth, its interpretation continues to intrigue physicists and philosophers alike. Thanks to enormous technological advances over the last couple of decades, it has now become possible actually to perform some of the gedanken experiments that the pioneers had thought of to highlight the counterintuitive and bizarre consequences of quantum theory. So far, quantum mechanics has emerged unscathed in every case, and continues to defy all attempts at falsification. The spectacular success of its working rules has spurred physicists in recent times to grapple seriously with its foundational problems, leading to new theoretical and technological advances.

On the other hand, general relativity, the paradigm of classical field theory, continues to remain as accurate and reliable as quantum mechanics in its own domain of validity, namely, the large-scale universe. To wit, the agreement between the predictions of general relativity and observation of the energy loss due to gravitational waves emitted by binary pulsars is just as impressive as the agreement between the prediction of quantum electrodynamics and the measured value of the Lamb shift in atoms.

Introduction

Quantum theory predicts the striking and paradoxical result that when a system is continuously watched, it does not evolve! Although this effect was noticed much earlier by a number of people [146], [147], [148], [149], [150], it was first formally stated by Misra and Sudarshan [151] and given the appellation ‘Zeno's paradox’ because it evokes the famous paradox of Zeno denying the possibility of motion to a flying arrow. It is as startling as a pot of water on a heater that refuses to boil when continuously watched. This is why it is also called the ‘watched pot effect’. The effect is the result of repeated, frequent measurements on the system, each measurement projecting the system back to its initial state. In other words, the wave function of the system must repeatedly collapse. It is also necessary that the time interval between successive measurements must be much shorter than the critical time of coherent evolution of the system, called the Zeno time [152]. For decays this Zeno time is the time of coherent evolution before the irreversible exponential decay sets in, and is governed by the reciprocal of the range of energies accessible to the decay products. For most decays this is extremely short and hard to detect [153]. In the case of non-exponential time evolution, the relevant Zeno time can be much longer.

An Interpretive Introduction to Quantum Field Theory (Teller 1995; hereafter IIQFT) supersedes most of my prior work on quantum field theory. The gossip mill has described this book as a popularization of the most elementary parts of Bjorken and Drell (1965), which into the 1980s was the most widely used quantum field theory text. As with any good caricature, there is a great deal of truth in this comparison. Like Bjorken and Drell, who published in 1965, IIQFT presents the theory largely as it existed in the 1950s. But in order to see aspects of structure and interpretation more clearly, IIQFT presents the theory stripped of all the details needed for application. IIQFT also does not treat contemporary methods, such as the functional approach, and important recent developments, especially gauge theories and the renormalization group. Nonetheless it is hoped that by laying out the structure of the theory's original form in the 1950s, much of which survives in contemporary versions, and by developing a range of ways of thinking about that theory physically, one does essential ground work for a thorough understanding of what we have today.

Why do we use the term ‘quantum field theory’? A good fraction of the work done in IIQFT aims to clarify the appropriateness, accurate development, and limitations of the application of the epithet ‘field’, as well as examination of alternatives. While called a ‘field theory’, quantum field theory (QFT) is also taken to be our most basic theory of ‘elementary particles’.

During the almost 40 years that I have been following it, there has been an amazing (to me, at any rate) change in the tenor of the discussion about the relation between quantum field theory and general relativity. In 1957, I started graduate studies at Stevens Institute of Technology, then a world center of relativity research: there were actually three people there who worked on such problems! I soon started attending the famous informal Stevens relativity meetings, getting to know many of the leading figures in the field, and meeting most of the others at the 1959 Royaumont GRG meeting.

This was a time of high tension, of struggle between two rival imperialisms, one clearly much stronger than the other. I am referring, of course, to the dominant quantum field theory paradigm, which was stubbornly resisted by the much weaker unified field theory program. I call these two programs imperialisms because each had a universalist ideology used to justify an annexationist policy. Einstein's unified field program aimed to annex quantum phenomena by means of some generally covariant extension of general relativity that would include electromagnetism, somehow miraculously bypassing quantum mechanics. The structure of matter and radiation, including all quantum effects, would result from finding non-singular solutions to the right set of non-linear field equations. In spite of repeated failures over 30-40 years, Einstein persisted in working toward this goal, though not without increasing doubts, particularly towards the end of his life.

Quantum field theory offers physicists a tremendously wide range of application: it is a language with which a vast variety of physical processes can be discussed and also provides a model for fundamental physics, the so-called ‘standard model’, which thus far has passed every experimental test. No other framework exists in which one can calculate so many phenomena with such ease and accuracy. Nevertheless, today some physicists have doubts about quantum field theory, and here I want to examine these reservations. So let me first review the successes.

Field theory has been applied over a remarkably broad energy range and whenever detailed calculations are feasible and justified, numerical agreement with experiment extends to many significant figures. Arising from a mathematical account of the propagation of fluids (both ‘ponderable’ and ‘imponderable’), field theory emerged over a hundred years ago in the description within classical physics of electromagnetism and gravity. [1] Thus its first use was at macroscopic energies and distances, with notable successes in explaining pre-existing data (relationship between electricity and magnetism, planetary perihelion precession) and predicting new effects (electromagnetic waves, gravitational bending of light). Schrödinger's wave mechanics became a bridge between classical and quantum field theory: the quantum mechanical wave function is also a local field, which when ‘second’ quantized gives rise to a true quantum field theory, albeit a non-relativistic one. This theory for atomic and chemical processes works phenomenally well at electron-volt energy scales or at distances of O(10−5 cm).

This two-tier conference signals a new phase of philosophers' interest in quantum field theory, which has been growing noticeably in the last few years. However, some prominent physicists have shown their deep suspicions against ignorant philosophers' intrusion into their profession, and have expressed their hostility quite openly. In the philosophy community, some prominent philosophers of physics have also expressed their suspicions against the rationale of moving away from the profound foundational problems raised by Einstein and Bohr, Bohm and Bell, such as those concerning the nature of space-time and measurement, possibility and implications of hidden variables and nonlocality, and stepping into the technical complexity of quantum field theory, which is only an application of quantum mechanics in general without intrinsically distinct philosophical questions to be explored. In order to dispel these suspicions, it is desirable to highlight certain aspects of quantum field theory which require philosophical reflections and deserve further investigations. This discussion intends to suggest that philosophers can learn many important lessons from quantum field theory, and may be of some help in clarifying its conceptual foundations. At this stage of crisis that quantum field theory is experiencing now, the clarification may contribute to the radical transformation of our basic concepts in theoretical physics, which is necessary for a happy resolution of the crisis and the emergence of a new promising fundamental physical theory.

Generally speaking, philosophers are interested in the metaphysical assumptions adopted by science and the world picture suggested by science.

Introduction

Feynman diagrams have long been a staple tool for theoretical physicists. The line drawings were originally developed in the late 1940s by the American physicist Richard Feynman for the perturbative study of quantum electrodynamics, and are taught today to students of quantum field theory in much the same manner as Feynman originally designed them. Yet for a time during the 1950s and 1960s, Feynman diagrams became an equally indispensable tool in the hands of quantum field theory's rivals - particle physicists following the lead of Lev Landau and Geoffrey Chew, who aimed to construct a fully autonomous S-matrix theory built around a scaffolding of Feynman diagrams, but conspicuously lacking all field-theoretic notions of quantum fields or Hamiltonians. By the early 1960s, Chew and his collaborators had commandeered the field-theoretic diagrams while denouncing quantum field theory as being both bankrupt and dead. In the process, Feynman diagrams were re-invested with distinct meanings by the S-matrix theorists, and were taken to represent a fundamentally different basis for studying the constitution and interactions of particles.

There was a series of steps by which S-matrix theorists crafted an autonomous diagrammatic method, some of which we consider here. First came Geoffrey Chew's 1958 particle-pole conjecture. This work was soon extended by Chew in collaboration with Francis Low in their 1959 work on scattering off of unstable targets.

One recognizes that there has been, and continues to be, a great deal of common ground between statistical mechanics and quantum field theory (QFT). Many of the effects and methods of statistical physics find parallels in QFT, particularly in the application of the latter to particle physics. One encounters spontaneous symmetry breaking, renormalization group, solitons, effective field theories, fractional charge, and many other shared phenomena.

Professor Fisher [1] has given us a wonderful overview of the discovery and role of the renormalization group (RG) in statistical physics. He also touched on some of the similarities and differences in the foundations of the RG in condensed matter and high-energy physics, which were amplified in the discussion. In the latter subject, in addition to the formulation requiring cutoff-independence, we have the very fruitful Callan-Symanzik equations. That is, in the process of renormalizing the divergences of QFT, arbitrary, finite mass-scales appear in the renormalized amplitudes. The Callan-Symanzik equations are the consequence of the requirement that the renormalized amplitudes in fact be independent of these arbitrary masses. This point of view is particularly useful in particle physics, although it does make its appearance in condensed matter physics as well.

The very beautiful subject of conformal field theory spans all three topics we are considering: critical phenomena, quantum field theory, and mathematics. The relationship between conformal field theory and two-dimensional critical phenomena has become particularly fruitful in recent years.

The triumph of quantum field theory

Although the title of this session is ‘The foundations of quantum field theory’, I shall talk, not of the foundations of quantum field theory (QFT), but of its triumphs and limitations. I am not sure it is necessary to formulate the foundations of QFT, or even to define precisely what it is. QFT is what quantum field theorists do. For a practising high energy physicist, nature is a surer guide as to what quantum field theory is as well to what might supersede it, than is the consistency of its axioms.

Quantum field theory is today at a pinnacle of success. It provides the framework for the standard model, a theory of all the observed forces of nature. This theory describes the forces of electromagnetism, the weak interaction responsible for radioactivity, and the strong nuclear force that governs the structure of nuclei, as consequences of local (gauge) symmetries. These forces act on the fundamental constituents of matter, which have been identified as pointlike quarks and leptons. The theory agrees astonishingly well with experiment to an accuracy of 10−6–10−10 for electrodynamics, of 10−1–10−4 for the weak interactions and of 1–10−2 for the strong interactions. It has been tested down to distances of 10−18 cm in some cases. We can see no reason why QFT should not be adequate down to distances of order the Planck length of 10−33 cm where gravity becomes important.