Rock Brentwood
Jun28-08, 05:00 AM
In renormalization theory, we starting out with an action
S = integral L
describing a given field law. In the usual process of going through
and writing out the perturbation expansion, we find a need to
regularize otherwise ill-defined products. This can be regarded in the
following way: S is split into a purely kinetic part S_0 and extra
"perturbation" part S_1, in the interaction picture. Free fields are
constructed using S_0, and a perturbation matrix is defined formally
as the generating function T[exp(-i S_I/h-bar)].
The T[] operator is generally ill-defined when taken in the sense of
the Wick expansion. In retrospect the flaw is realised to be that Wick
only clusters 2-at-a-time, while divergent terms represent clusters of
3-or-more that Wick forgot to properly handle. Hence, one redefines
T[] to a new operator TR[] where everything becomes well-defined.
That's the regularization process.
Assuming TR[] gives you multi-point distributions
TR[A1(x1) A2(x2) ... An(xn)]
that are well-defined on all clusters possibly except the total
cluster x1=x2=...=xn, one can extend this to a form that's well-
defined on the total diagonal x1=x2=...=xn, hence effecting an n-point
cluster. The result, generally, will however only be determined up to
an undetermined finite order polynomial (when written in momentum
space) of degree equal to the degree of the singularity associated
with the n-cluster.
A formula that actually accomplishes this all in one fell swoop is the
cumulant expansion!
T[log TR[exp(A)]] = A
subject to the condition that
T[TR[exp(A)]] = TR[exp(A)],
for a generating function exp(A) in a linear combination of the field
variables A = A_1 phi^1 + ... + A_n phi^n.
In effect, the final result is equivalent to taking TR[exp(-i S_I/h-
bar)] = T[exp(-i S'_I/h-bar)], where S'_I has a set of counter terms
added onto it.
A theory is deemed renormalizable if the counter-terms assume only one
of a finite number of different forms. Hence, one can then expand the
"seed" Lagrangian to a finite polynoomial that accommodates all the
extra terms that arise. The coefficients then become what are known as
"scale dependent".
A stronger condition is that there only be a finite number of
divergent clusters.
Otherwise, if neither condition holds, the theory is deemed non-
renormalizable, since an infinite number of effective terms (and scale-
dependent) coefficient would arise.
But WAIT A MINUTE! By "scale-dependent", what we're really talking
about are coefficient that assume different effective values at
different scale of resolution. The powers of resolution are those seen
in a scattering process off a scattering center; so what this
translates to is a radial (and possibly angular) dependence on the
source point.
The coefficients that are, for all intents and purposes, functions of
position; possibly through their dependence on the field variables.
The most elementary example is electromagnetism. If one starts out
only assuming the Lagrangian is Lorentz invariant, then L = L(I, J) is
a function of the Lorentz invariants I = (E^2 - B^2 c^2)/2, J = E.B;
and the coefficients are epsilon = dL/dI, theta = dL/dJ, with theta an
axial coefficient that tends asymptotically to 0.
Counter terms, it turns out, can all be locked away into I and J for a
linear electrodynamic theory, where L = epsilon I + theta J, for
constant epsilon and theta.
But, more generally, for ANY Lagrangian, even though there may be an
infinite number of ways of writing down Lorentz invariants (I, J, I^2,
IJ, J^2, etc.) there are still only a *finite* number of counter terms
when only regarding the *variational* of the Lagrangian:
delta L = epsilon delta I + theta delta J.
Thus, regardless of whether we had found electromagnetism to be
renormalizable or not, there is still only 2 coefficients -- provided
we generalize "scale dependent" to not only mean "position dependent"
but "position dependent, possibly through a functional dependence on
the fields, themselves". Hence, for the effective Lagrangian of
(linear) electrodynamics, one finds
epsilon = epsilon_0 + 7 A I, theta = 4 A J
up to O(h-bar^2) for some constant coefficient A (and actually
epsilon_0 also shifts, when doing the effective Lagrangian).
Hence, there is a SECOND degree of "renormalizable" that's
intermediate between what we normally refer to as "renormalizable",
and what is presently referred to as "non-renormalizable". These are
the theories whose Lagrangians produce an effective number of counter-
terms, but whose Lagrangian *variationals* produce only a finite
number of counter-terms.
For a gravitational dynamics, one has as the configuration variables
the orthonormal frame field (10 components), the Lorentz connection (6
x 4 components), the torsion 2-form (6 x 4 components) and the
curvature 2-form (6 x 6 components).
The most general Lagrangian that can be constructed from these is a
function of these 4 quantities. If one requires at the outset that the
dynamics be given by a Lagrangian locally symmetric with respect to
the Lorentz gauge group, then the Lagrangian may be a function only of
the Lorentz invariants constructed out of these components. A Lorentz-
invariant Lagrangian will be independent of the connection, thus
resulting in a function of 10+24+36 = 70 components. No more than 70
independent Lorentz invariants I_1, I_2, ..., I_n (n <= 70) can be
constructed from these. The variational of the Lagrangian will
therefore reduce to one involving at most a finite set of counter-
terms delta L = sum (epsilon^i I_i: i = 1,2,...,n) ... assuming there
are no Lorentz-violating anomalies.
By this account, this makes the prospective gravitational theory
renormalizable in the newly designated sense. There will only be a
finite set of counter terms and a finite set of coefficients
(epsilon^1, ..., epsilon^n) each posessing a (newly generalized sense
of) scale-dependence.
S = integral L
describing a given field law. In the usual process of going through
and writing out the perturbation expansion, we find a need to
regularize otherwise ill-defined products. This can be regarded in the
following way: S is split into a purely kinetic part S_0 and extra
"perturbation" part S_1, in the interaction picture. Free fields are
constructed using S_0, and a perturbation matrix is defined formally
as the generating function T[exp(-i S_I/h-bar)].
The T[] operator is generally ill-defined when taken in the sense of
the Wick expansion. In retrospect the flaw is realised to be that Wick
only clusters 2-at-a-time, while divergent terms represent clusters of
3-or-more that Wick forgot to properly handle. Hence, one redefines
T[] to a new operator TR[] where everything becomes well-defined.
That's the regularization process.
Assuming TR[] gives you multi-point distributions
TR[A1(x1) A2(x2) ... An(xn)]
that are well-defined on all clusters possibly except the total
cluster x1=x2=...=xn, one can extend this to a form that's well-
defined on the total diagonal x1=x2=...=xn, hence effecting an n-point
cluster. The result, generally, will however only be determined up to
an undetermined finite order polynomial (when written in momentum
space) of degree equal to the degree of the singularity associated
with the n-cluster.
A formula that actually accomplishes this all in one fell swoop is the
cumulant expansion!
T[log TR[exp(A)]] = A
subject to the condition that
T[TR[exp(A)]] = TR[exp(A)],
for a generating function exp(A) in a linear combination of the field
variables A = A_1 phi^1 + ... + A_n phi^n.
In effect, the final result is equivalent to taking TR[exp(-i S_I/h-
bar)] = T[exp(-i S'_I/h-bar)], where S'_I has a set of counter terms
added onto it.
A theory is deemed renormalizable if the counter-terms assume only one
of a finite number of different forms. Hence, one can then expand the
"seed" Lagrangian to a finite polynoomial that accommodates all the
extra terms that arise. The coefficients then become what are known as
"scale dependent".
A stronger condition is that there only be a finite number of
divergent clusters.
Otherwise, if neither condition holds, the theory is deemed non-
renormalizable, since an infinite number of effective terms (and scale-
dependent) coefficient would arise.
But WAIT A MINUTE! By "scale-dependent", what we're really talking
about are coefficient that assume different effective values at
different scale of resolution. The powers of resolution are those seen
in a scattering process off a scattering center; so what this
translates to is a radial (and possibly angular) dependence on the
source point.
The coefficients that are, for all intents and purposes, functions of
position; possibly through their dependence on the field variables.
The most elementary example is electromagnetism. If one starts out
only assuming the Lagrangian is Lorentz invariant, then L = L(I, J) is
a function of the Lorentz invariants I = (E^2 - B^2 c^2)/2, J = E.B;
and the coefficients are epsilon = dL/dI, theta = dL/dJ, with theta an
axial coefficient that tends asymptotically to 0.
Counter terms, it turns out, can all be locked away into I and J for a
linear electrodynamic theory, where L = epsilon I + theta J, for
constant epsilon and theta.
But, more generally, for ANY Lagrangian, even though there may be an
infinite number of ways of writing down Lorentz invariants (I, J, I^2,
IJ, J^2, etc.) there are still only a *finite* number of counter terms
when only regarding the *variational* of the Lagrangian:
delta L = epsilon delta I + theta delta J.
Thus, regardless of whether we had found electromagnetism to be
renormalizable or not, there is still only 2 coefficients -- provided
we generalize "scale dependent" to not only mean "position dependent"
but "position dependent, possibly through a functional dependence on
the fields, themselves". Hence, for the effective Lagrangian of
(linear) electrodynamics, one finds
epsilon = epsilon_0 + 7 A I, theta = 4 A J
up to O(h-bar^2) for some constant coefficient A (and actually
epsilon_0 also shifts, when doing the effective Lagrangian).
Hence, there is a SECOND degree of "renormalizable" that's
intermediate between what we normally refer to as "renormalizable",
and what is presently referred to as "non-renormalizable". These are
the theories whose Lagrangians produce an effective number of counter-
terms, but whose Lagrangian *variationals* produce only a finite
number of counter-terms.
For a gravitational dynamics, one has as the configuration variables
the orthonormal frame field (10 components), the Lorentz connection (6
x 4 components), the torsion 2-form (6 x 4 components) and the
curvature 2-form (6 x 6 components).
The most general Lagrangian that can be constructed from these is a
function of these 4 quantities. If one requires at the outset that the
dynamics be given by a Lagrangian locally symmetric with respect to
the Lorentz gauge group, then the Lagrangian may be a function only of
the Lorentz invariants constructed out of these components. A Lorentz-
invariant Lagrangian will be independent of the connection, thus
resulting in a function of 10+24+36 = 70 components. No more than 70
independent Lorentz invariants I_1, I_2, ..., I_n (n <= 70) can be
constructed from these. The variational of the Lagrangian will
therefore reduce to one involving at most a finite set of counter-
terms delta L = sum (epsilon^i I_i: i = 1,2,...,n) ... assuming there
are no Lorentz-violating anomalies.
By this account, this makes the prospective gravitational theory
renormalizable in the newly designated sense. There will only be a
finite set of counter terms and a finite set of coefficients
(epsilon^1, ..., epsilon^n) each posessing a (newly generalized sense
of) scale-dependence.