In the beginning, when Dirac, Jordan, Heisenberg, and Pauli created the quantum theory of fields, it was not expected that it would provide a consistent description of Nature. After all, it was only a quantized version of the classical theory of Maxwell and Lorentz, a theory which was well known to be afflicted with diseases arising from the infinite electromagnetic inertia of point particles. Many physicists were of the opinion that any project to make the theory’s mathematical foundation more rigorous was probably ill-advised; first the classical foundation should be set right. Such alterations might so change the basis of...

Throughout this book, states will be described in the Heisenberg picture of quantum mechanics. The Schrodinger picture is much less convenient for the description of a relativistic theory because it treats the time coordinate on a very different footing from the space coordinates; as will be proved in Chapter 4, the other commonly used picture, the interaction picture, in general does not exist. In the Heisenberg picture, to each state of the system under consideration there corresponds a unit vector, sayΦ, in a Hilbert space ℋ. The vector does not change with time, whereas the observables, represented by hermitian...

The two mathematical notions in terms of which everything in the present chapter will be expressed are distribution and holomorphic function. These are discussed in the first four sections of the chapter. The last section is devoted to a few remarks on Hilbert space.

Distributionis a generalization of the notion offunction, which makes it possible to make precise various formal mathematical manipulations common among physicists. The Diracδ-function and its derivatives are examples of distributions; they are defined by the equations\[\begin{array}{l} \int{f(x)\delta (x)dx=f(0)} \\ \int{f(x){\delta }'(x)dx}=-{f}'(0) \\ \quad \vdots \\ {{\left. \int{f(x){{\delta }^{(n)}}(x)dx={{(-1)}^{n}}}\frac{{{d}^{n}}f}{d{{x}^{n}}} \right|}_{x=0}}, \\ \end{array} \caption {(2-1)}\]wheref(x) is some suitably smooth function on the real line. It is clear thatδ(x)...

The classical notion of field originated in attempts to avoid the idea of action at a distance in the description of electromagnetic and gravitational phenomena. In these important cases, the field turns out to have two basic properties: (1) it is observable, and (2) it is defined by a set of functions on space-time with a well-defined transformation law under the appropriate relativity group. Since in quantum mechanics observables are represented by hermitian operators which act on the Hilbert space of state vectors, one expects the analogue in relativistic quantum mechanics of a classical observable field to be a set...

In the preceding chapters, we have defined what is meant by a relativistic quantum field theory and assembled some tools to aid in the analysis of its structure. In the present chapter, these are used to establish a series of general properties of relativistic quantum field theories.

Local commutativity asserts the vanishing of commutators or anti-commutators, [φ(x),ψ(y)]_±, for all space-likex−y. An assumption which is apparently weaker is that this condition holds in some smaller region, say (x−y)²< −a < 0. Theorem 4–1 asserts that such apparently weaker assumptions are in fact not weaker, since...