WHAT TO LOOK FOR

In its simplest characterization, the quantum theory of scalar fields is nothing but the quantum mechanics of N canonical degrees of freedom in the limit that N → ∞. Of course, things are not quite that simple since sequences do not always have the virtue of converging, and even if they do converge they do not always converge to acceptable limits. However, in this chapter we take a simple, pragmatic point of view and assume that any needed limits converge and, in addition, that the resultant answers are acceptable. Partway through this chapter we introduce units in which ħ = 1.

Classical scalar fields

For purposes of the present section we let g(x) denote a real scalar field defined for x ∈ ℝn, where n ≥ 1 denotes the dimension of space-time. If n = 1 then that single dimension is the time dimension and so there is no space dimension – hence no space – and thus one is really back in the case of classical mechanics. We shall have essentially no occasion to study the case n = 1 in this chapter. For n ≥ 2 the number of space dimensions is s ≡ n − 1 ≥ 1. (We note that the symbol n has several uses in this chapter, often as a dummy index of summation, but the context is generally clear in each case.)