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S8b. Renormalization without infinities I
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Renormalization in QFT is often associated with the need to handle
infinities. This makes everything look like nonsense from a
mathematical point of view. But this is just the sloppiness of
physicists; it is not difficult to get a satisfying view of
renormalization without encountering any weird infinities.
The basic principles can be explained without knowing anything about
quantum mechanics, since renormalization is a much more general
phenomenon associated with idealizations in a theory and the
corresponding limits. As such it is also needed in various classical
situations (classical point electrons, turbulence, etc.)
My tutorial paper
Renormalization: An elementary tutorial
http://www.mat.univie.ac.at/~neum/papers/physpapers.html#ren
contains an elementary treatment of many of the issues.
The nice paper hep-th/0212049 by Delamotte also discusses most of
renormalization without ever mentioning fields (which come in quite
late).
In all cases, we want to describe a situation which is a limit of more
complex and often less symmetric situations. This limit is the only
problematic thing, and sometimes generates infinities if done in an
improper way. Just as when trying to compute
s_N = sum_{k=0:N} (-1)^k/(k+1)^s = u_N - v_N
by summing the even and odd contributions u_N and v_N separately.
The limit N to inf is well-defined for s>0, but can be obtained
by going to the limit in u_N and v_N separately only for s>1.
For s<=1, one gets instead (in physicists' parlance) the difference
of two infinities, indicating an invalid way of computing the limit
(which in this simple case is known to exist, as we have an alternating
series with coefficients whose absolute values converge to zero).
To remove the infinities, one proceeds similarly as wnen evaluating
limits of expressions that give naively inf-inf: One needs to use
some transformation that cancels the infinities analytically. Example:
lim sqrt(n^2+n)-sqrt(n^2+1)
= lim ((n^2+n)-(n^2+1))/(sqrt(n^2+n)+sqrt(n^2+1))
= lim (n-1)/(sqrt(n^2+n)+sqrt(n^2+1)) = 1/2.
In ordinary quantum mechanics (which is essentially 1+0-dimensional
field theory) renormalization is only needed for Hamiltonians that
are artificial, or singular limits of Hamiltonians describing real
sitiations. In 1+1-dimensional field theory, infinities occur already
in the naive treatment of _any_ local relativistic field theory, but
the situation can still be rectified without much difficulties.
Represent the Hamiltonian as a limit
H(g) = lim_{Lambda->inf} H_Lambda(g)
with a cutoff Lambda, where g is a vector of constants, one for each
integral of a field monomial occuring in the formal expression for H.
Since the naive limit is meaningless, one has to clarify which sort of
limit is intended and meaningful.
As a first step in mainkg sense of the limit, we note that the
Heisenberg representation shows that only the commutator
[H(g),A] = lim_{Lambda->inf} [H_Lambda(g),A]
of H with arbitrary operators A, must be definable in order to have
a well-defined quantum dynamics. We also note that this commutator
does not change when a constant is subtracted from H. We therefore write
H_Lambda(g) = :H_Lambda(T(Lambda)g): + const(Lambda), (*)
where :H: denotes what is called the normally ordered version of H.
It has the property that all annihilation operators stand to the right
of the creation operators, Under the standard assumptions on the form
of H(g), this is always possible with a matrix T(Lambda), which can be
expressed in terms of expectations of suitable operators. For example,
normal ordering can be achieved for any polynomial expression H by
repeated use of the canonical commutation rules. It can also be
achieved in more general cases; see, e.g., Section 10 of
R. M. Wilcox,
Exponential Operators and Parameter Differentiation in
Quantum Physics,
J. Math. Phys. 8, 962 (1967).
Now (*) leads to
[H(g),A] = lim_{Lambda->inf} [:H_Lambda(T(Lambda)g):,A].
In 1+1D field theory, one observes that upon replacing this
(still undefined) limit by
[H(g),A] = lim_{Lambda->inf} [:H_Lambda(T(g):,A], (**)
everything comes out finite, both perturbatively and nonperturbatively.
This is rigorously proved in the book by Glimm and Jaffe.
The only infinities that would show up when taking a limit Lambda->inf
would arise in T(Lambda), which has been eliminated by a simple linear
substitution.
In 4D field theory, a linear transformation is not sufficient, but a
more complex nonlinear one (now involving mass renormalization and
field renormalization) still does the job, at least perturbatively.
The situation resembles here the one also used for singular
Hamiltonians in the case of ordinary quantum mechanics (though the
details are much more complex in the 4D case), which is outlined below.
In quantum physics, the data (the Hamiltonian in QM, the action in QFT)
depends on some parameter vector v of dimension d, say, without direct
physical meaning. The components of v are called the bare parameters.
For example, v may consist of bare mass, bare charge, and bare coupling
constant.
Without the renormalization conditions we get a family of solutions
parameterized by v, from which we can compute measurable quantities
combined into a vector q=q_N(v) of some dimension e>=d.
where N is the parameter in which we want to take the limit.
(For example, N might be an energy cutoff at energies beyond
observability, and q observed cross sections or the observed particle
spectrum.)
Anything we can reliably measure must clearly be essentially independent
of N, once N is large enough. Therefore, to be physically reasonable,
the equation q=q_N(v) must define (generically) a d-dimensional
manifold in R^e whose limit as a set when N-->inf is also (generically)
a well-defined d-dimensional manifold. This is the manifold of interest,
since picking a particular finite value for N is usually subjective.
In a theory with finite renormalization, this limit manifold can still
be parameterized by v, since the limit
q(v)= lim_{N to inf} q_N(v) (*)
exists. Although v is unobservable it can be calculated from the
measurements by solving the equation q=q(v) in the least squares sense.
Rather than doing that (which would be numerically best in case the
measurements are inexact or q(v) is not exactly known) one proceeds
in theoretical work as if an s-dimensional vector mu of key physical
data and a corresponding subset of d equations were known exactly,
and could be solved exactly for v=v(mu).
Then one gets a renormalized parameterization
q=q_ren(mu), with q_ren(mu)= q(v(mu)), (**)
expressing everything in terms of the physical parameters mu.
When the limit (*) does not exist, the situation is more complicated.
Since there is no limiting q, one has to work at finite N. Proceeding
as before, one solves d of the equations in q=q_N(v) for v, getting
v=v_N(mu), but since the limit (*) does not exist, there will also be
no limit
v(mu) = lim_{N to inf} v_N(mu)
which would enable the use of (**). Instead, v_N(mu) diverges.
Loosely speaking, we get infinite bare masses and bare coupling
constants. But this limit will never be used, hence this nonexistence
causes no real problems. It is just the loose way of speaking that
creates the impression of weirdness. The 'infinities' are caused by
the nature of the interactions. If they are too singular for a
standard treatment then the limits needed for a finite renormalization
simple do not exist anymore.
But this does not mean that the theory becomes meaningless but only
that one has to be careful in performing the limit N-->inf only on
expressions where it exists. This requires a small change in our
procedure.
At finite N, we can still define a renormalized parameterization
q = q_{N,ren}(mu), with q_{N,ren}(mu)= q_N(v_N(mu)).
For a renormalizable theory, the limit
q_ren(mu) = lim_{N to inf} q_N,ren}(mu)
exists although neither q_N nor v_N converge.
Once this limit replaces the naive bare recipe (*)-(**) which is
ill-defined, everything behaves properly as it should.
The situation may be slightly more complex than indicated above.
Instead of working with directly measurable quantities one often
works with formally more tractable quantities q that are finitely
related to the key measurable quantities mu (such as observed mass
spectra). However, their definition, and hence their relation to mu,
depends on an additional scale parameter E that fixes the
renormalization conditions. (This parameter should not be confused
with the cutoff energy, which after renormalization is always infinite!)
Thus we actually have q=q_N(v,E), solve some of these equations for
v=v_N(mu,E), and get as a result
q = q_{N,ren}(mu,E), with q_{N,ren}(mu,E)= q_N(v_N(mu),E),
hence
q_ren(mu,E) = lim_{N to inf} q_{N,ren}(mu,E).
But since the scale E can be chosen arbitrarily, the final renormalized
result of physical predictions P(q,E) must be
independent of E. Thus,
d/dE P(q_ren(mu,E),E) = 0,
which is a form of the renormalization group equations.
To get a renormalized Hamiltonian, one also needs wave function
renormalization, which means using a cutoff-dependent inner
product in the space of wave functions (in the functional
Schr"odinger picture). The limiting Hamiltonian is perturbatively
well-defined in the physical Hilbert space obtained as limit of
renormalized Hilbert spaces at finite cutoff, as the cutoff goes
to infinity.