Pictures Paths Particles Processes

November 7, 2013 13
of more fields, and ultimately to that of an infinity of fields. We find that the
nature of the two-point interactions between different fields can, under suitable
circumstances, be reinterpreted, or visualised, so that we are suddenly not
talking about infinitely many fields at a single point, but a chain of fields positioned
along an infinite line : this is the invention of space. To this end we
need to introduce a ‘length scale’, but we shall take care to arrange matters in
such a way that the length scale can be taken to be infinitesimal : this is the
continuum limit. We take the Feynman rules through this sequence of steps,
and find the rules for a one-dimensional continuum theory. Similar arguments
apply to derive higher-dimensional theories. We do the same for the action as
well, without however insisting that the Feynman rules must necessarily come
from that action. We shall also see that the classical field equations can be
derived from the action by a number of formal manipulations, called functional
differentiation, that lead to Euler-Lagrange equations. Throughout, however,
the Feynman rules have the primacy.
The next step, which in Chapter 3 takes us into our familiar Minkowski
space, is to assign a special rôle, that of time, to one of the dimensions. Doing
this requires a rather drastic assumption of admissibility : it goes under
the name of the Euclidean postulate. This is the point at which quantum field
theory and statistical physics part to go their separate ways. Having taken
this hurdle we can find the form, both of the Feynman rules, and of the action
in Minkowski space, and then we are ready to confront our theory with
a number of basic facts about our own world. It is seen that the so-called iɛprescription,
that we have to introduce to keep the Minkowski formulation of
our theory at least moderately well-defined, is closely related to the possibility
of encountering unstable particles, and in a deeper sense tells us the direction of
time. We also see that the collection of connected Green’s functions is related
to the wave function that determines the probability density to find particles
at a given space-time point. A simple example is a quick derivation of the
Yukawa potential, a Coulomb-type law for static sources. A more demanding
but also more rewarding calculation provides us with Newton’s first law since we
see that a localized source can emit particles that move with constant velocity
along straight lines, as long as there are no interactions : in fact, it is this that
justifies our statement that our fields describe particles in the first place ! A
closer investigation, and some elementary bookkeeping, shows that the fields
describe in fact not only particles, but antiparticles as well. We thus find the
prediction of antimatter as well as the CPT transformation that relates free
matter and antimatter 6 . As a by-product we obtain a natural prescription for
the density of states of free particles, that is, a rule for counting quantum states.
In Chapter 4 we take yet another step towards phenomenology, by discussing
how the knowledge gathered so far can be used to obtain cross sections and de-
6 This is not to be confused with the deeper property of CPT invariance of interacting
theories.