as computed in the toy model
is not a completely arbitrary functional of its argument
. For example,
is real and not complex, and it is local in the sense that (to any finite order
in
) it consists of a finite sum of powers of the field
and its derivatives, all evaluated at the same
point.
Why should this be so? Both of these turn out to be general features (so long as only massive degrees of freedom are integrated out) which are inherited from properties of the underlying physics at higher energies:
Since is constructed to reproduce
this time evolution of the full theory, it must be real in order to
give a Hermitian Hamiltonian as is required by unitary time
evolution1
.
That is, heavy particles may be produced so
long as they are then re-destroyed sufficiently quickly. Such
virtual production is possible because the Uncertainty Relations
permit energy not to be precisely conserved for states which do not
live indefinitely long. A virtual state whose production requires
energy non-conservation of order therefore
cannot live longer than
, and so its influence must
appear as being local in time when observed only with probes having
much smaller energy. Similar arguments imply locality in space for
momentum-conserving systems. (This is a heuristic explanation of
what goes under the name operator product
expansion [157
, 41
] in the quantum
field theory literature.)
Since it is the mass of the heavy
particle which sets the scale over which locality applies once it
is integrated out, it is
which appears with
derivatives of low-energy fields when
is written in a
derivative expansion.