General Collision Theory

Abstract

We now want to divorce the treatment of collision processes from the assumption of a field-particle duality. Specifically we want to cover situations where particles occur which are not related to one of the basic fields via an asymptotic condition (II.3.23) (“composite particles”) or where basic fields occur which have no counterparts among the species of observable particles (e.g. quark fields). The physical interpretation of the quantum fields will not be primarily attached to particles but to local operations. Specifically an operator Φ(ƒ) (a basic field smeared out with a test function ƒ) represents a physical operation performed on the system within the space-time region given by the support of ƒ.¹ Naively speaking, the argument x of a basic field has direct physical significance. It marks the point where Φ(x) applied to a state produces a change. This sounds obvious and in fact one may regard it as the reason for axiom E. However one should remember that the argument x in Φⁱⁿ(x) does not have this direct physical meaning though Φⁱⁿ being formally a free covariant field, also satisfies axiom E. Also it is not obvious at this stage that such a purely space-time-geometric interpretation of the basic fields will suffice to analyze the phenomenological consequences of the theory since it introduces no distinction between the different types of fields occurring. The suggested interpretation suffices indeed to construct the states |λ₁, ⋯〉ⁱⁿ and |λ₁, ⋯〉^out. This will be demonstrated under some simplifying assumptions in this section and discussed more fully in Chapter VI.