In the non-relativistic case the Green’s functionĜonly had one pole, atp 0 =
|p|^2 / 2 m. Two possible choices of how to extend integration overp 0 into the
complex plane avoiding the pole gave either a retarded Green’s function and
propagation from past to future, or an advanced Green’s function and propaga-
tion from future to past. In the Klein-Gordon case there are now two poles, at
p 0 =±ωp. Micro-causality requires that the pole at positive energy be treated
as a retarded Green’s function, with propagation of positive energy particles
from past to future, while the one at negative energy must be treated as an
advanced Green’s function with propagation of negative energy particles from
future to past.
43.6 Interacting scalar field theories: some com-
ments
Our discussion so far has dealt purely with a theory of non-interacting quanta,
so this theory is called a quantum theory of free fields. The field however can
be used to introduce interactions between these quanta, interactions which are
local in space. The simplest such theory is the one given by adding a quartic
term to the Hamiltonian, taking
Ĥ=
∫
R^3:
1
2
(Π(̂x)^2 + (∇Φ(̂x))^2 +m^2 Φ(̂x)^2 ) +λΦ(̂x)^4 :d^3 xThis interacting theory is vastly more complicated and much harder to under-
stand than the non-interacting theory. Among the difficult problems that arise
are:
- How should one make sense of the expression
:Φ(̂x)^4 :since it is a product not of operators but of operator-valued distributions?- How can one construct an appropriate state space on which the interact-
ing Hamiltonian operator will be well-defined, with a well-defined ground
state?
Quantum field theory textbooks explain how to construct a series expansion
in powers ofλabout the free field valueλ= 0, by a calculation whose terms
are labeled by Feynman diagrams. To get finite results, cutoffs must first be
introduced, and then some way found to get a sensible limit as the cutoff is
removed (this is the theory of “renormalization”). In this manner finite results
can be found for the terms in the series expansion, but the expansion is not
convergent, giving only an asymptotic series (for fixedλ, no matter how small,
the series will diverge at high enough order).
For known calculational methods not based on the series expansion, again
a cutoff must be introduced, making the number of degrees of freedom finite.