It's often a good strategy to study simple theories before you study complicated theories because it's easier to see how they work, but the purpose of physics is to find out why nature is the way it is, and simplicity by itself is I think never the answer. But renormalizability was a condition of simplicity which was being imposed for what seemed after Dyson's 1949 papers8 like a rational reason, and it explained not only why the electron has the magnetic moment it has, but also (together with gauge symmetries) all the detailed features of the standard model of weak, electromagnetic, and strong, interactions, aside from some numerical parameters.

The other good thing about quantum field theory during this period of tremendous optimism was that it offered a clear answer to the ancient question of what we mean by an elementary particle: it is simply a particle whose field appears in the Lagrangian. It doesn't matter if it's stable, unstable, heavy, light - if its field appears in the Lagrangian then it's elementary, otherwise it's composite. Now my point of view has changed. It has changed partly because of my experience in teaching quantum field theory. When you teach any branch of physics you must motivate the formalism - it isn't any good just to present the formalism and say that it agrees with experiment - you have to explain to the students why this the way the world is. After all, this is our aim in physics, not just to describe nature, but to explain nature. In the course of teaching quantum field theory, I developed a rationale for it, which very briefly is that it is the only way of satisfying the principles of Lorentz invariance plus quantum mechanics plus one other principle. Let us run through this argument very rapidly. The first point is to start with Wigner's definition of physical multi-particle states as representations of the inhomogeneous Lorentz group.9 You then define annihilation and creation operators a(~p, ?, n) and a†(~p, ?, n) that act on these states (where ~p is the three-momentum, ? is the spin z-component, and n is a species label). There's no physics in introducing such operators, for it is easy to see that any operator whatever can be expressed as a functional of them. The existence of a Hamiltonian follows from time-translation invariance, and much of physics is described by the S-matrix, which is given by the well known Feynman-Dyson series of integrals over time of time-ordered products of the interaction Hamiltonian HI(t) in the interaction picture;

The other principle that has to be added is the cluster decomposition principle, which requires that distant experiments give uncorrelated results.10 In order to have cluster decomposition, the Hamiltonian is written not just as any functional of creation and annihilation operators, but as a power series in these operators with coefficients that (aside from a single momentum-conservation delta function) are sufficiently smooth functions of the momenta carried by the operators. This condition is satisfied for an interaction Hamiltonian of the form

where H(x) is a power series (usually a polynomial) with terms that are local in annihilation fields, which are Fourier transforms of the annihilation operators:

together of course with their adjoints, the creation fields. So far this all applies to nonrelativistic as well as relativistic theories.† Now if you also want Lorentz invariance, then you have to face the fact that the time-ordering in the Feynman-Dyson series (1) for the S-matrix doesn't look very Lorentz invariant. The obvious way to make the S-matrix Lorentz invariant is to take the interaction Hamiltonian density H(x) to be a scalar, and also to require that these Hamiltonian densities

commute at spacelike separations in order to exploit the fact that time ordering is Lorentz invariant when the separation between spacetime points is timelike. In order to satisfy the requirement that the Hamiltonian density commute with itself at spacelike separations, it is constructed out of fields which satisfy the same requirement. These are given by sums of fields that annihilate particles plus fields that create the corresponding antiparticles

where n denotes the antiparticle of the particle of species n. For a field ?? that transforms according to an irreducible representation of the homogeneous Lorentz group, the form of the coefficients u? and v? is completely determined (up to a single over-all constant factor) by the Lorentz transformation properties of the fields and one-particle states, and by the condition that the fields commute at spacelike separations. Thus the whole formalism of fields, particles, and antiparticles seems to be an inevitable consequence of Lorentz invariance, quantum mechanics, and cluster decomposition, without any ancillary assumptions about locality or causality. This discussion has been extremely sketchy, and is subject to all sorts of qualifications. One of them is that for massless particles, the range of possible theories is slightly larger than I have indicated here. For example, in quantum electrodynamics, in a physical gauge like Coulomb gauge, the Hamiltonian is not of the form (2) - there is an additional term, the Coulomb potential, which is bilocal and serves to cancel a non-covariant term in the propagator. But relativistically invariant quantum theories always do turn out to be quantum field theories, more or less as I have described them here. One can go further, and ask why we should formulate our quantum field theories in terms of Lagrangians. .