22
GAUGE FIELDS AND STRINGS
the effective action will be finite in the given order. The result of the
above computation can be interpreted in two different ways. The most
naive interpretation has already been given: namely we compute the
effective action and see that the ultraviolet divergence in it can be
absorbed into a redefinition of the coupling constant. Another interpretation is the following. Let us suppose that we have a theory with the
cut-off A (which is of the order of the inverse lattice spacing). If we
integrate out the fields h(x) with the wave vectors A < p < A we shall
obtain the effective action which in the low energy limit has again the
form (2.1) but with a renormalized value of el, Cq(A):
A
el(A) = el(A) + ^C ,(G ) \og^ = el
(2.13)
This formula follows directly from (2.11), provided that we restrict the
p-integration by the condition A < |p| < A. As a result we conclude
that the physical theory, formulated with the cut-off A and the bare
charge el must be equivalent (for small momenta \p\ A) to the one
with the cut-off A and the specially chosen new bare charge el. This
statement is called renormalizability and the transformations from el to
el and from A to A the renormalizability group. The formula (2.13), as
is seen from its derivation, is correct provided that:
el log y ^ 1
(2.14)
Nothing prevents us from repeating the procedure and passing from A
to A < A etc.
The most important consequence of renormalizability is that it
controls the momentum dependence of different physical quantities. Let
us consider as an example the behaviour of the effective charge e^(p) for
the fluctuations with momentum p. This quantity can be defined in
different ways. One of the possibilities is to consider the four point
function with all momenta equal to p. There are many other options
and we shall comment on this ambiguity later.
Since our theory does not contain any dimensional parameter except
for A, we must have:
e\p) = e^(log(A/p), el)
(2.15)
Let us express e^(p) in terms of
where p is some fixed value of the
momentum. Inverting (2.15) we have:
«0 = eo(log(A/p), e \ p ) )
(2.16)
GAUGE FIELDS AND STRINGS
the effective action will be finite in the given order. The result of the
above computation can be interpreted in two different ways. The most
naive interpretation has already been given: namely we compute the
effective action and see that the ultraviolet divergence in it can be
absorbed into a redefinition of the coupling constant. Another interpretation is the following. Let us suppose that we have a theory with the
cut-off A (which is of the order of the inverse lattice spacing). If we
integrate out the fields h(x) with the wave vectors A < p < A we shall
obtain the effective action which in the low energy limit has again the
form (2.1) but with a renormalized value of el, Cq(A):
A
el(A) = el(A) + ^C ,(G ) \og^ = el
(2.13)
This formula follows directly from (2.11), provided that we restrict the
p-integration by the condition A < |p| < A. As a result we conclude
that the physical theory, formulated with the cut-off A and the bare
charge el must be equivalent (for small momenta \p\ A) to the one
with the cut-off A and the specially chosen new bare charge el. This
statement is called renormalizability and the transformations from el to
el and from A to A the renormalizability group. The formula (2.13), as
is seen from its derivation, is correct provided that:
el log y ^ 1
(2.14)
Nothing prevents us from repeating the procedure and passing from A
to A < A etc.
The most important consequence of renormalizability is that it
controls the momentum dependence of different physical quantities. Let
us consider as an example the behaviour of the effective charge e^(p) for
the fluctuations with momentum p. This quantity can be defined in
different ways. One of the possibilities is to consider the four point
function with all momenta equal to p. There are many other options
and we shall comment on this ambiguity later.
Since our theory does not contain any dimensional parameter except
for A, we must have:
e\p) = e^(log(A/p), el)
(2.15)
Let us express e^(p) in terms of
where p is some fixed value of the
momentum. Inverting (2.15) we have:
«0 = eo(log(A/p), e \ p ) )
(2.16)