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Renormalization without infinities II
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In bare (divergent) QFT, infinities arise because integrals taken over
unbounded momenta don't exist; so doing it 'formally' leads to nonsense.
Instead, proper QFT takes regularized integrals, for example by
adding an explicit cutoff Lambda. This simply means that everything is
calculated with an action that depends on Lambda as an additional
parameter. Once this is done, everything is finite, but
Lambda-dependent.
The only problem with that is that the cutoff destroys Lorentz
covariance - apart from that it would be a completely respectable
field theory in itself. Now Lorentz invariance is violated only
at energies > O(Lambda); here O(...) is the Landau symbol that denotes
an unspecified factor that might itself depend on Lambda.
Hence to have the theory conform to physics that can be checked it
suffices to take Lambda large.
But for aesthetic reasons and/or since we believe that symmetries are
fundamental, we want to have fully invariant theories. This requires
that we let Lambda go to infinity.
But in order that the results have a finite limit we must at the
same time make the bare coupling constants g dependent on Lambda.
If this is done in a correct way (and the textbooks on QFT teach
one or more of the known correct ways under the heading of
'renormalization'), one encounters no infinities at all in the
whole process.
Thus renormalized quantities are (typically) never infinite.
The essentials of the renormalization process, namely the need for
Lambda-dependent coupling constants for sufficiently singular
Hamiltonians, can be understood nonperturbatively on the
nonrelativistic level.
What happens is that one has a family of Hamiltonians
H(Lambda,g) that depend on a scale parameter Lambda and and a coupling
constant g (or several). H(Lambda,g) has a good limit H(g) as Lambda
to inf, with g fixed, but the corresponding limit of the resolvent
G(Lambda,g) does not exist; hence if one tries to do calculations with
H(g) directly (the 1930 way of doing things, which was a dead end),
one gets infinities all over the place.
On the other hand, if one chooses a good parameterization g(Lambda,mu)
then, although H(Lambda,g(Lambda,mu)) has no longer a good limit as
Lambda to inf, its resolvent G(Lambda,g(Lambda,mu)) has a well-defined
limit G(mu). (This holds at least in quantum mechanics, which is simply
1D quantum field theory, and in 2D field theory, where this can be
proved in certain cases. In 3D and 4D, where things are not well
understood rigorously, one probably needs also a
Lambda-dependent inner product defining the Hilbert space
to ensure that one ends up in the right representation,
and Lambda-dependent wave functions to ensure that the limiting
renormalized wave functions remain bounded in the limiting
renormalized inner product.)
Since all dynamical information, including all scattering information,
is in the resolvent, G(mu) defines a good physical model for a
scattering process.
In some simple cases, renormalization can be done nonperturbatively.
For example, standard perturbation theory for a Hamiltonian
p^2/2m + g delta(x) produces infinities. The renormalization of this
particular example is treated nonperturbatively in hep-th/9305052.
Thus, infinities only appear if one takes the limit in a way it
cannot be taken consistently.
The relativistic case is a bit more involved and is at present
not understood nonperturbatively in the physically most important
case of 4 space-time dimensions, but there is no difference in
principle.
The local interaction of the formal Lorentz invariant action is
replaced by a nonlocal interaction depending on the UV cutoff Lambda.
Thus one has V(g,Lambda) in place of V(g), where g are the coupling
constants (including masses).
To do so, one writes the (Euclidean = Wick rotated) field as
Phi(x) = integral dp exp(-i p dot x) Phihat(p)
and substitutes it into the action. This gives an action in the
momentum representation. Then one regularizes the interaction term by
throwing away (or damping more gracefully) the momenta above some
cutoff Lambda.
Introducing the cutoff makes the interaction nonlocal, as one can see
by going from the momentum representation of the regularized
interaction term back to the position representation by substituting
Phihat(p) = const * integral dx exp(i p dot x) Phi(x).
Instead of the delta functions which would appear without the
cutoff there are now explicit nonlocal potential terms.
(Note that the Coulomb interaction in nonrelativistic QFT is nonlocal.
Thus nonlocalities are nothing particularly strange. See
H. Ekstein, Phys. Rev. 117, 1590-1595 (1960)
for more on nonlocal interactions and relations to the S-matrix.
But actually one does not need to care about locality or not,
since the regularized interaction in the momentum representation
is mathematically ok and one can do everything else in this
representation without interpreting it in space.)
More precisely, one starts with the smeared Lagrangian interaction
defined by the cutoff, uses the representation of the S-matrix as
a time-ordered exponential to work out the corresponding
Hamiltonian interaction in the interaction picture, and takes this
as definition of the regularized dynamics. (Note that Haag's theorem,
which asserts that a nontrivial Lorentz-invariant theory satisfying
microlocality cannot have an interaction picture, does not apply since
the theory with cutoff is neither Lorentz invariant nor microlocal.)
From here on, one can do standard perturbation theory without
encountering any infinity at all; one gets meaningful formulas
throughout the whole renormalization procedure, but everything
is dependent on the cutoff Lambda.
All contributions to the S-matrix elements of this regularized theory
are finite, and give (after analytic continuation back to real time)
the S-matrix of the regularized interaction.
The result is an asymptotic series S(g,Lambda) for the S-matrix of
the regularized interaction, with finite, computable coefficients.
This S-matrix is unitary and has all properties one would like to have,
except that, because of the cutoff, it is only approximately Lorentz
invariant.
Of course, for a nonlocal theory in position representation,
one gets more complicated Feynman rules than those traditionally
written down in the (nonexisting) formal limit Lambda-->inf.
In momentum space, the formulas become the standard formulas, but
with explicit cutoff included. Thus it is much simpler to always work
in the momentum representation.
To restore Lorentz invariance, one uses a running coupling constant
g=g(Lambda,mu) which, for fixed renormalization point mu (a vector of
the same dimension of g containing the free constants in the matching
of the renormalization conditions), is uniquely determined
(for any fixed renormalization scheme) as the solution of a
renormalization group equation whose coefficients are also defined
as a (presumably even convergent) asymptotic expansion.
Having this, one can take the limit
S(mu) = lim_{Lambda to inf} S(g(Lambda,mu),Lambda)
which is an asymptotic series in hbar with finite, computable
coefficients when the theory is renormalizable, and turns out to
be Lorentz invariant and microlocal.
Thus one gets the desired Lorentz invariant, microlocal theory
as a perturbatively well-defined limit of perturbatively well-defined
but not Lorentz invariant or microlocal theories.
At the very end one can pass to the limit, but not earlier.
The only infinity encountered is not worse than the infinity
encountered in defining Riemann integrals over the real line,
where one also gets a finite limit by letting a finite cutoff
go to infinity.
The real mathematical difficulties in QFT are not in the
renormalization procedure but in giving a nonperturbative construction
of the S-matrix S(mu), i.e., in proving that the above limits
exist not only order by order in the coupling constant g but
as a function defined for finite values of g.