For a fixed ultraviolet cutoff, corresponding physically to only allowing fields
with momentum components smaller than a given value, one can construct a
sensible theory with non-trivial interactions (e.g., scattering of one particle by
another, which does not happen in the free theory). This gives a sensible theory,
for momenta far below the cutoff. However, it appears that, for three or more
spatial dimensions, removal of the cutoff will always in the limit give back the
non-interacting free field theory. There is thus no known continuum relativistic
quantum field theory of scalar fields, other than free field theory, that can be
constructed in this way.
43.7 For further reading
Pretty much every quantum field theory textbook has a treatment of the rel-
ativistic scalar field with more detail than given here, and significantly more
physical motivation. A good example is [15] the lectures of Sidney Coleman,
one that has some detailed versions of the calculations discussed here is chapter
5 of [16]. Chapter 2 of [67] covers the same material, with more discussion of
the propagator and the causality question.
For a more detailed rigorous construction of the Klein-Gordon theory in
terms of Fock space, closely related to the outline given in this chapter, three
sources are
- Chapter X.7 of [73].
- Chapter 5.2 of [28]
- Chapter 8.2.2 of [17]
For a general axiomatic mathematical treatment of relativistic quantum
fields as distribution-valued operators, some standard references are [87] and
[12]. The second of these includes a rigorous construction of the Klein-Gordon
theory.
For a treatment of relativistic quantum field theory relatively close to ours,
not only for the Klein-Gordon theory of this chapter, but also for the spin-^12
theories of later chapters, see [33].