In standard approaches to quantum field theory one tends to think of particles as some kind of small perturbations in a field. Normally for calculations these perturbations are on their own taken to be plane waves of definite frequency, and indeed in many ways they are direct analogs of waves in classical field theories like those of electromagnetism or fluid mechanics. To investigate collisions between particles, one thus looks at what happens with multiple waves. In a system described by linear equations, there is always a simple superposition principle, and waves just pass through each other unchanged. But what in effect leads to non-trivial interactions between particles is the presence of nonlinearities. If these are small enough then it makes sense to do a perturbation expansion in which one approximates field configurations in terms of a succession of arrangements of ordinary waves - as in Feynman diagrams. But just as one cannot expect to capture fully turbulent fluid flow in terms of a few simple waves, so in general as soon as there is substantial nonlinearity it will no longer be sufficient just to do perturbation expansions. And indeed for example in QCD there are presumably many cases in which it is necessary to look at something closer to actual complete field configurations - and correlations in them. .

The way the path integral for a quantum field theory works, each possible configuration of the field is in effect taken to make a contribution Exp[I s/h], where s is the so-called action for the field configuration (given by the integral of the Lagrangian density - essentially a modified energy density), and h is a basic scale factor for quantum effects (Planck's constant divided by 2 Pi). In most places in the space of all possible field configurations, the value of s will vary quite quickly between nearby configurations. And assuming this variation is somehow random, the contributions of these nearby configurations will tend to cancel out. But inevitably there will be some places in the space where s is stationary (has zero variational derivative) with respect to changes in fields. And in some approximation the field configurations in these places can be expected to dominate the path integral. But it turns out that these field configurations are exactly the ones that satisfy the partial differential equations for the classical version of the field theory. (This is analogous to what happens for example in classical diffraction theory, where there is an analog of the path integral - with h replaced by inverse frequency - whose stationary points correspond through the so-called eikonal approximation to rays in geometrical optics.) In cases like QED and QCD the most obvious solutions to the classical equations are ones in which all fields are zero. And indeed standard perturbation theory is based on starting from these and then looking at the expansion of Exp[I s/h] in powers of the coupling constant. But while this works for QED, it is only adequate for QCD in situations where the effective coupling is small. And indeed in other situations it seems likely that there will be all sorts of other solutions to the classical equations that become important. But apart from a few special cases with high symmetry, remarkably little is known about solutions to the classical equations even for pure gluon fields. No doubt the analog of turbulence can occur, and certainly there is sensitive dependence on initial conditions (even non-Abelian plane waves involve iterated maps that show this). Presumably much like in fluids there are various coherent structures such as color flux tubes and glueballs. But I doubt that states involving organized arrangements of these are common. And in general when there is strong coupling the path integral will potentially be dominated by large numbers of configurations not close to classical solutions.

In studying quantum field theories it has been common to consider effectively replacing time coordinates t by I t to go from ordinary Minkowski space to Euclidean space (see page 1047). But while there is no problem in doing this at a formal mathematical level - and indeed the expressions one gets from Feynman diagrams can always be analytically continued in this way - what general correspondence there is for actual physical processes is far from clear. Formally continuing to Euclidean space makes path integrals easier to define with traditional mathematics, and gives them weights of the form Exp[-? s] - analogous to constant temperature systems in statistical mechanics. Discretizing yields lattice gauge theories with energy functions involving for example Cos[Subscript[?,i]-Subscript[?,j]] for color directions at adjacent sites. And Monte Carlo studies of such theories suggest all sorts of complex behavior, often similar in outline from what appears to occur in the corresponding classical field theories. (It seems conceivable that asymptotic freedom could lead to an analog of damping at small scales roughly like viscosity in turbulent fluids.)

One of the apparent implications of QCD is the confinement of quarks and gluons inside color-neutral hadrons. And at some level this is presumably a reflection of the fact that QCD forces get stronger rather than weaker with increasing distance. The beginnings of this are visible in perturbation theory in the increase of the effective coupling with distance associated with asymptotic freedom. (In QED effective couplings decrease slightly with distance because fields get screened by virtual electron-positron pairs. The same happens with virtual quarks in QCD, but a larger effect is virtual gluon pairs whose color magnetic moments line up with a color field and serve to increase it.) At larger distances something like color flux tubes that act like elastic strings may form. But no detailed way to get confinement with purely classical gluon fields is known. In the quantum case, a sign of confinement would be exponential decrease with spacetime area of the average phase of color flux through so-called Wilson loops - and this is achieved if there is in a sense maximal randomness in field configurations. (Note that it is not inconceivable that the formal problem of whether quarks and gluons can ever escape to infinity starting from some given class of field configurations may in general be undecidable.)

Quantum field theory was originally thought to be simply the quantum theory of fields. That is, when quantum mechanics was developed physicists already knew about various classical fields, notably the electromagnetic field, so what else would they do but quantize the electromagnetic field in the same way that they quantized the theory of single particles?

In 1926, in one of the very first papers on quantum mechanics,1 Born, Heisenberg and Jordan presented the quantum theory of the electromagnetic field. For simplicity they left out the polarization of the photon, and took spacetime to have one space and one time dimension, but that didn't affect the main results.

Born et al. gave a formula for the electromagnetic field as a Fourier transform and used the canonical commutation relations to identify the coefficients in this Fourier transform as operators that destroy and create photons, so that when quantized this field theory became a theory of photons. Photons, of course, had been around (though not under that name) since Einstein's work on the photoelectric effect two decades earlier, but this paper showed that photons are an inevitable consequence of quantum mechanics as applied to electromagnetism.

The quantum theory of particles like electrons was being developed at the same time, and made relativistic by Dirac2 in 1928-1930. For quite a long time many physicists thought that the world consisted of both fields 1 and particles: the electron is a particle, described by a relativistically invariant version of the Schr?odinger wave equation, and the electromagnetic field is a field, even though it also behaves like particles. Dirac I think never really changed his mind about this, and I believe that this was Feynman's understanding when he first developed the path integral and worked out his rules for calculating in quantum electrodynamics. When I first learned about the path-integral formalism, it was in terms of electron trajectories (as it is also presented in the book by Feynman and Hibbs3). I already thought that wasn't the best way to look at electrons, so this gave me an distaste for the path integral formalism, which although unreasonable lasted until I learned of 't Hooft's work4 in 1971.

It was quite soon after the Born-Heisenberg-Jordan paper of 1926 that the idea came along that in fact one could use quantum field theory for everything, not just for electromagnetism. This was the work of many theorists during the period 1928-1934, including Jordan, Wigner, Heisenberg, Pauli, Weisskopf, Furry, and Oppenheimer.

Although this is often talked about as second quantization, I would like to urge that this description should be banned from physics, because a quantum field is not a quantized wave function. Certainly the Maxwell field is not the wave function of the photon, and for reasons that Dirac himself pointed out, the Klein-Gordon fields that we use for pions and Higgs bosons could not be the wave functions of the bosons. In its mature form, the idea of quantum field theory is that quantum fields are the basic ingredients of the universe, and particles are just bundles of energy and momentum of the fields. In a relativistic theory the wave function is a functional of these fields, not a function of particle coordinates.

Quantum field theory hence led to a more unified view of nature than the old dualistic interpretation in terms of both fields and particles. There is an irony in this. It is that although the battle is over, and the old dualism that treated photons in an entirely different way from electrons is I think safely dead and will never return, some calculations are actually easier in the old particle framework. When Euler, Heisenberg and Kockel5 in the mid-thirties calculated the effective action (often called the Euler-Heisenberg action) of a constant external electromagnetic field, they calculated to all orders in the field, although their result is usually presented 2 only to fourth order.

This calculation would probably have been impossible with the old fashioned perturbation theory techniques of the time, if they had not done it by first solving the Dirac equation in a constant external electromagnetic field and using those Dirac wave functions to figure out the effective action. These techniques of using particle trajectories rather than field histories in calculations have been revived in recent years. Under the stimulus of string theory, Bern and Kosower,6 in particular, have developed a useful formalism for doing calculations by following particle world lines rather than by thinking of fields evolving in time.

Although this approach was stimulated by string theory, it has been reformulated entirely within the scope of ordinary quantum field theory, and simply represents a more efficient way of doing certain calculations. One of the key elements in the triumph of quantum field theory was the development of renormalization theory. I'm sure this has been discussed often here, and so I won't dwell on it. The version of renormalization theory that had been developed in the late 1940s remained somewhat in the shade for a long time for two reasons: (1) for the weak interactions it did not seem possible to develop a renormalizable theory, and (2) for the strong interactions it was easy to write down renormalizable theories, but since perturbation theory was inapplicable it did not seem that there was anything that could be done with these theories.

Finally all these problems were resolved through the development of the standard model, which was triumphantly verified by experiments during the mid-1970s, and today the weak, electromagnetic and strong interactions are happily all described by a renormalizable quantum field theory. If you had asked me in the mid-1970s about the shape of future fundamental physical theories, I would have guessed that they would take the form of better, more all-embracing, less arbitrary, renormalizable quantum field theories. I gave a talk at the Harvard Science Center at around this time, called "The Renaissance of Quantum Field Theory," which shows you the mood I was in. There were two things that especially attracted me to the ideas of renormalization and quantum field theory. One of them was that the requirement that a physical theory be renormalizable is a precise and rational criterion of simplicity. In a sense, this requirement had been used long before the advent of renormalization theory.

When Dirac wrote down the Dirac equation in 1928 he could have added an extra 'Pauli' term7 which would have given the electron an arbitrary anomalous magnetic moment. Dirac could (and 3 perhaps did) say 'I won't add this term because it's ugly and complicated and there's no need for it.' I think that in physics this approach generally makes good strategies but bad rationales.