Renormalization Grounded in Classical Field Theory -- Mass Shift vs.
Rock Brentwood
the long delay in approving this posting, which was originally dated
Tue, 10 Nov 2009 13:23:30 -0800 (PST). The delay was due to a mixup
on my part in saving the incoming message to the wrong mailbox.
-- jt]]
This is another in a series of articles I've recently posted on this
topic. Here, I'm going to lay out the general framework and apply it
to scalar fields to (a) show how mass renormalization and field
strength renormalization arise in CLASSICAL field theory and (b)
recover the relation (well-known in quantum field theory [1]) between
the two. Then I'll briefly discuss the classical origin behind the
relation between the vertex and gauge field strength renormalizations
(and how these, too, are grounded in classical field theory).
Contents:
(0) Background -- Maxwell's Renormalization Theory
(1) "Scale Dependence" Without Scale Dependence: The Classical
Interpretation
(2) Constitutive Coefficients, Symmetry and Renormalization
Coefficients
(2.1) The General Form of a Lagrangian Field Theory
(2.2) Constitutive Coefficients and Symmetry
(2.3) Beta Coefficients and Compatibility
(2.4) Renormalizability and Renormalization Group
(3) The General Scalar Field
(3.1) General Lorentz-Invariant Lagrangian
(3.2) The Mass, Metric and Axial Coefficients
(3.3) Singular Field Metrics and Landau Poles
(4) Mass Shift and Field Strength Renormalization
(4.1) Solving for Fixed z -- The Classical Renormalization Group
(4.2) Solving for Fixed M -- "Physical Mass"
(5) Vertex and Charge Renormalization vs. Gauge Group Metric
(5.1) Gauge Group Metric as Generalizations of Permittivity and
Permeability
(5.2) The Classical Version of Gauge Fixing
(6) Notes
(0) Background -- Maxwell's Renormalization Theory
The key insight is this: the mathematics of Sturm-Liouville systems
already contains the essence of renormalization theory -- but at the
CLASSICAL level in classical field theory; thus showing that this
issue does not pertain to quantum field theory, but is firmly grounded
at the classical level in field theories of the same type that were
dealt with in the 19th century.
In retrospect, this should not be a surprise: this study ultimately
originates from a close reading of Maxwell's treatise, where in
chapter 1 we find what amounts to the elements of modern
renormalization theory (e.g. bare vs. dressed charges, vacuum
polarization, etc.) cast in almost unrecognizable form in 19th century
language [2], but still discernible to those in the present era.
However, all of it takes place at the classical level. The key insight
relates directly to the nature of the relevant field equations as
Sturm-Liouville equations: the constitutive coefficients in a field
theory are not constant, but field-dependent. Thus, for instance,
Maxwell's theory if rendered in modern language in the setting of
Lagrangian field theory would take the form of the following action
principles:
(1) Lagrangian: S = integral L(I,J,K) d^4 x
(2) Routhian: S = integral (H.curl A - R(I,J',K')) d^4 x
involving the rotation invariant field combinations
I = E^2/2, J = E.B, K = B^2/2, J' = E.H, K' = H^2/2.
>From these we would then recover the constitutive coefficients as the
derivatives of the Lagrangian (or Routhian) with respect to the
invariants
Permittivity: epsilon = dL/dI
Dielectric Coefficient: kappa = dR/dI
Permeability; mu = dR/dK' = -1/(dL/dK)
Axial Permittivity: theta = dL/dJ
Axial Dielectric Coefficient: lambda = dR/dJ
with the variationals
delta(L) = epsilon delta(I) + theta delta(J) - (1/mu) delta(K)
delta(R) = kappa delta(I) + lambda delta(J') + mu delta(K')
Thus, with the variationals defined by
delta(L) = D.delta(E) - H.delta(B), delta(R) = D.delta(E) + B.delta
(H)
this leads to the constitutive laws
D = epsilon E + theta B, H = B/mu - theta E
D = kappa E + lambda H, B = mu H + lambda E
and the relations
epsilon = kappa + mu theta^2, lambda = mu theta,
epsilon mu = kappa mu - lambda^2.
Here, we will explore the general issue of the classical grounding of
renormalization theory by recapturing both mass shift and field
strength renormalization for the scalar field at the classical level.
It is here that we will see, in a simple and clear way, removed from
the (red herring) association with perturbation theory and removed
from the thickets of Feynman diagrams, what ACTUALLY lies behind the
relation between the two and what the explicit form of the relation
between the mass shift versus field strength renormalization actually
is.
(1) "Scale Dependence" Without Scale Dependence: The Classical
Interpretation
A quantum field theorist is taught to regard the coupling coefficients
in a Lagrangian field theory as "scale-dependent"; so that the "bare"
coefficients that arise in a Lagrangian serve as little more than
place-holders. Here, in contrast, we take a different point of view,
more firmly grounded in classical theory and in more sensible
interpretation: we simply stop treating the coefficients as constants
and regard them, instead, as field-dependent.
Thus, a "linear" field theory (one with constant constitutive
coefficients) is replaced by a "non-linear" field theory (one, like
Born-Infeld, in which the coefficients are variable).
Scale-dependency arises naturally in the following way. In a
scattering process, we probe a concentrated source to an energy
resolution q. Associated with this is a length scale r ~ 1/q. For a
given field whose field strength has radial dependence f(r), if a
constitutive coefficient k is a function k(f) of the field strength,
then it inherits the radial dependence from the field, thus leading to
a relation of the form k = k(f(r)) for this particular source.
Finally, from this we get scale dependence:
k(q) = k(f(1/q)).
One can then think of the setting of a scale as the "osculating linear
field theory" of a given non-linear field theory, where the
coefficients are expanded about a mean value; that mean value being
the value of the coefficients at a certain scale q_0 as k_0 = k(f(1/
q_0)). Then it expands as a series k = k_0 + k_1 (q - q_0) + k_2 (q -
q_0)^2/2 + ...
(2) Constitutive Coefficients, Symmetry and Renormalization
Coefficients
(2.1) The General Form of a Lagrangian Field Theory
What are the coefficients? In fact, they arise naturally when we step
back and stop trying to force the classical Lagrangian to be something
it cannot be. A linear field theory leads to the very constant
constitutive coefficients that Maxwell took pains to argue away from
(for the electromagnetic field) and which lead to the very self-force
and self-energy infinities in classical field theory that are
inherited by quantum field theory and become the divergences seen in
the loop integrals.
So, instead, we make NO assumptions about the Lagrangian, other than
it respect certain symmetry principles. As seen in the example above,
just from this assumption alone the structure of the field theory can
be clamped down on significantly.
A Lagrangian field theory starts with a Lagrangian L = L(x^m, q^A,
v^A_m) and forms from it an action principle
S = integral L(x^m, q^A(x), d_m(q^A)(x)) dx.
d_m = partial derivative with respect to x^m
The Lagrangian is a function of the KINEMATIC field variables
q^A = field configuration variables
v^A_m = field gradients
and, itself, serves as a generating function to relate the DYNAMIC
field variables
F_A = "force" density
P^m_A = "momentum" density
to the kinematic variables. The relations generated
F_A = dL/dq^A, P^m_A = dL/d(v^A_m)
give us the CONSTITUTIVE LAWS in their most general form. The two sets
of equations that come out of the Lagrangian field theory are:
Kinematic Law: v^A_m = d_m(q^A)
Dynamics Law: F_A = d_m(P^m_A)
(2.2) Constitutive Coefficients and Symmetry
So, what happens when we impose symmetry requirements on the
Lagrangian, imposing the condition that L respect a certain set of
symmetries (e.g. the rotation symmetry in the Maxwell field, in the
example considered at the top of the article) is that the Lagrangian
reduces to a function
L = L(I^1, I^2, ..., I^k)
of all the independent symmetry-invariant combinations
I^1(x,q,v), I^2(x,q,v), ..., I^k(x,q,v)
that can be formed from the fields. The CONSTITUTIVE COEFFICIENTS are
then just the respective derivatives
e_i = dL/dI^i, i = 1, 2, ..., k.
[In here and the following, the summation convention will be used on
repeated indices.]
The result of this reduction is that the constitutive law now
expresses the dynamic variables in terms of a FIXED set of field
combinations, with all the information about the original Lagrangian
locked up in a relatively small and fixed set of constitutive
coefficients. These combinations are derived as follows from the
symmetry invariants.
delta(I^i) = F^i_A delta(q^A) + P^{im}_A delta(v^A_m), i = 1, ..., k
The constitutive law reads
F_A = e_i F^i_A, P^m_A = e_i P^{im}_A.
And the dynamic law assumes the form
d_m (e_i P^{im}_A) = e_i F^i_A
of a STURM-LIOUVILLE system of partial differential equations.
So, finally, it is at this point we can answer our original question
and tie all the threads together:
What are the constitutive coefficients? These are the classical
equivalents of the renormalization coefficients.
(2.3) Beta Coefficients and Compatibility
The coefficients themselves are not arbitrary. Since they arise as
derivatives of the Lagrangian, we have the following exact
differential
delta(L) = e_i delta(I^i),
which entails the following compatibility relation
Symmetry Condition: beta_{ij} = beta_{ji}
involving the derivatives of the coefficients
beta_{ij} = de_i/dI^j.
These are the classical analogues of the BETA coefficients. An
interesting exercise is to try and find the quantum field theoretic
analogue of the "symmetry condition".
(2.4) Renormalizability and Renormalization Group
Because we have the extra set of coefficients, and because they're no
longer forced to be constants, we have room to pull completely back to
the classical level an additional set of symmetries -- the RESCALING
SYMMETRIES.
This is the essence of what lies behind the very concept of
renormalization, itself.
A field theory is deemed to be "renormalizable" with respect to a
given transformation
q^A -> Q^A = Q^A(x, q, v)
if the Sturm-Liouville system transforms into an altered version of
itself
d_m (e_i P^{im}(x,q,v)) = e_i F^i_A(x,q,v)
-> d_m (E_i P^{im}(x,Q,V)) = E_i F^i_A(x,Q,V)
with modified coefficients e_i(x,q,v) -> E_i(x,Q,V), where
V^A_m = d_m(Q^A).
Certain cardinal points may then naturally arise. For instance, there
may exist a transformation that converts some or all of the
coefficients e_i into constants or converts the Sturm-Liouville system
into an ordinary wave equation.
We'll see this play out for the scalar field.
(3) The General Scalar Field
(3.1) General Lorentz-Invariant Lagrangian
So, let's start with a scalar field with components q = (q^A: A =
1, ..., N). The symmetry requirement we'll impose here is that the
Lagrangian be a Lorentz invariant; so that it reduces to a function
L = L(T^{AB}, U^{AB}, V^{ABCD})
of all the Lorentz invariants
T^{AB} = 1/2 root(-g) q^A q^B
U^{AB} = 1/2 root(-g) g^{mn} v^A_m v^B_n
V^{ABCD} = 1/24 epsilon^{mnrs} v^A_m v^B_n v^C_r v^D_s
where epsilon^{mnrs} is the anti-symmetric form, for which epsilon^
{0123} = 1.
(3.2) The Mass, Metric and Axial Coefficients
>From this, we have the following coefficients
m_{AB} = dL/dT^{AB}, k_{AB} = -dL/dU^{AB}, l_{ABCD} = dL/dV^{ABCD}
and the constitutive law
delta(L) = delta(q^A) F_A + delta(v^A_m) P^m_A
with
F_A = -root(-g) k_{AB} q^B
P^m_A = root(-g) m_{AB} g^{mn} v^B_n + 1/6 epsilon^{mnrs} l_{ABCD}
v^B_n v^C_r v^D_s.
Finally, from this we obtain the field law
0 = 1/root(-g) d_m(m_{AB} g^{mn} d_n(q^B)) + k_{AB} q^B
+ 1/6 epsilon^{mnrs} d_m l_{ABCD} d_n q^B d_r q^C d_s q^D = 0.
This is a Sturm-Liouville system with coefficients m_{AB} appearing
inside the "wave operator"
Delta = 1/root(-g) d_m(g^{mn} d_n(_))
The coefficients k_{AB} give us the square of the "mass matrix", while
m_{AB} serve as a metric for the configuration variables q^A. These
correspond, respectively, to the mass and field strength
renormalization coefficients.
Finally, the axial coefficients l_{ABCD} reflect parity non-symmetry
(since they're pseudo-scalar).
(3.3) Singular Field Metrics and Landau Poles
The metric m_{AB} is symmetric, by construction. However, it may be
singular as a matrix m = (m_{AB}: A, B = 1, ..., N). Singularities in
m_{AB} give us the classical analogue of the LANDAU POLE phenomenon.
In QED, the Landau pole arises as a singularity in the "running value"
of alpha = e^2/(4 pi epsilon h-bar c). Here, epsilon plays the
analogous role of the metric coefficients m_{AB}. So, given our
approach of equating the "running" and "scale-dependency" of the
coupling coefficients with actual variability in the constitutive
coefficients, what the Landau Pole corresponds to is the case where
epsilon -> 0; i.e., where the metric becomes singular.
In the following, we will assume the absence of such poles and regard
the metric m_{AB} as non-singular, with inverse m^{AB}.
(4) Mass Shift and Field Strength Renormalization
Here, we will take "renormalization" at its barest essence and require
the field dynamics to be invariant under the scaling transformation
q^A -> z^A_B Q^B,
where the coefficients z^A_B form a non-singular matrix.
Then the axial coefficients l_{ABCD} can no longer be present and the
field law reduces to the simpler form
1/root(-g) d_m(m_{AB} g^{mn} d_n(q^B)) + k_{AB} q^B = 0
Provided the metric coefficients m = (m_{AB}: A,B = 1,...,N) form a
non-singular matrix, there will be N independent Sturm-Liouville
equations in the field law, effectively giving us N wave equations for
the field components (q^1,...,q^N).
If we assume the transformation takes on the following form
1/root(-g) d_m(m_{AB} g^{mn} d_n(z^B_C Q^C)) + k_{AB} z^B_C Q^C
= Z^A_B (1/root(-g) d_m(M^{BC} g^{mn} d_n Q^C) + K_{BC} Q^C)
where Z^A_B also form a non-singular matrix; then the field equations
1/root(-g) d_m(M^{BC} g^{mn} d_n Q^C) + K_{BC} Q^C = 0
will be equivalent to the original field equations.
This transformation leads to the following requirements
Z^B_A M_{BC} = m_{ab} z^B_C
dZ^B_A M_{BC} = -m_{ab} dz^B_C
(or equivalently, Z^B_A dM_{BC} = -dm_{ab} z^B_C)
Z^B_A K_{BC} = k_{AB} z^B_C + 1/root(-g) d_m(g^{mn} root(-g) m_{AB}
d_n z^B_C)
or in more abbreviated form with obvious notation:
ZM = mz
dZ M = -m dz (and Z dM = -dm z)
ZK = kz + D(m dz)
What stands out most prominently is that a field strength
renormalization associated with Z MUST lead to a mass shift
renormalization k -> K. Hence, we recover at the classical level the
link between mass shift and field strength renormalization.
(4.1) Solving for Fixed z -- The Classical Renormalization Group
This system can be solved for fixed z as follows.
(a) find Z as a solution to dZ Z^{-1} m = -m dz z^{-1}.
This requires the metric m to be non-singular.
(b) Set M = Z^{-1} m Z
(c) Set K = Z^{-1} k Z + Z^{-1} D(m dz).
The extra term Z^{-1} D(m dz) is the "mass shift", amounting to the
very kind of shift delta(mass^2) seen in renormalization theory.
As a result, we find that transformations
q^A -> Z^A_B Q^B
by non-singular matrices Z^A_B form a GROUP.
This is the classical equivalent of the RENORMALIZATION GROUP. Only
here, it actually IS a group, not merely a semi-group.
(4.2) Solving for Fixed M -- "Physical Mass"
The analogue of Physical Mass arises from converting the scalar field
equations to bona fide wave equations. So, here we start out with a
fixed constant non-singular metric M = (M_{AB}). Having this on hand,
we may then pull out the coefficients from the transformed equations
to obtain
D(M dQ) + KQ = 0 ==> M DdQ + KQ = 0 ==> DdQ + M^{-1}K Q = 0
which gives us our wave equations.
Here, the process is as follows:
(a) Solve dz z^{-1} = 1/2 m^{-1} dm for z
(b) Set Z = m z M^{-1}
(c) Set K = Z^{-1} (kz + D(m dz)).
The matrix M^{-1} K gives us the square of the "physical mass" matrix
and reduces to
M^{-1} K = z^{-1} m^{-1} (kz + D(m dz)).
(5) Vertex and Charge Renormalization vs. Gauge Group Metric
(5.1) Gauge Group Metric as Generalizations of Permittivity and
Permeability
Though I won't go into it in any depth here, a second issue arises
naturally when considering the couplihng of a charged scalar field to
a gauge field -- vertex or charge renormalization.
Here, too, we find another mysterious that occurs, when approaching
this problem from within the setting of quantum field theory, that
suddenly gets cleared up: the link between vertex renormalization and
gauge field strength renormalization.
The key insight lies in explicitly writing out the form of the fine
structure coefficient. It is NOT
alpha = e^2/(4 pi h-bar c)
but
alpha = e^2/(4 pi epsilon h-bar c) = (mu c/4pi) e^2/h-bar
Given that epsilon plays the role of the gauge group metric for a 1-
component gauge field
A = A^0, epsilon c = k_{00},
with mu being associated with the inverse metric
mu c = k^{00}
then the resulting generalization to a n-mode gauge field with charge
vector e = (e_a:a=1,...,n),
alpha -> (k^{ab}/4pi) e_a e_b/h-bar.
For a simple gauge group, if we assume the metric is adjoint-invariant
(as is the case for a Yang-Mills field, for instance) then the metric
reduces to the form
k_{ab} = kappa_{ab}/g^2
where g is then deemed the "coupling coefficient", and kappa_{ab} is
the (fixed, constant) Killing metric. Then we have
alpha(k,e) = kappa^{ab}/(4pi h-bar) (e_a/g) (e_b/g).
A quantum field theorist is trained to regard the vertex
renormalization as a "scale-dependent" on the charge -- except that
what's meant by "charge" is conventionally taken NOT as the charge,
itself, but the charge divided by the square root of the metric; that
is: e/g.
A scale-dependent e/g can be more simply explained as a variable g;
and, in turn (in the more general setting of gauge fields for NON-
simple gauge groups), as a variable gauge group metric k_{ab}.
These coefficients play the same role for gauge fields that the m_{AB}
coefficients do for the scalar field. Consequently, they undergo a
change under a rescaling of the gauge field.
Hence, the link between the vertex and gauge field strength
renormalization. The mystery is cleared up, as we recognize that
what's actually going on here is simply that the gauge group metric k_
{ab} is now variable.
(5.2) The Classical Version of Gauge Fixing
These coefficients arise from a Lorentz-invariant Lagrangian
L = L(I^{ab})
I^{ab} = -1/4 root(-g) g^{mr} g^{ns} A^a_m A^b_n
as the derivatives of the Lagrangian, itself,
epsilon_{ab} = dL/dI^{ab},
k_{ab} = epsilon_{ab} c.
However, the field law given by such a Lagrangian is NOT invariant
under the rescaling transformation
A^a_m -> z^a_b A'^b_n + dy^a_b
Extra gauge-dependent terms appear in the field law. Therefore, it is
necessary to add extra counter-terms that involve explicit dependence
on the gauge potentials.
This brings us directly head-to-head with the classical version of
what in quantum field theory is referred to as "gauge fixing".
(6) Notes:
[1] c.f. Section 10.1 in Gauge Theories in Particle Physics (Vol. I),
Aitchison & Hey (2003) IoP
[2] Article 62 of the treatise makes a distinction between what we now
call the bare vs. renormalized vacuum; while also indirectly
mentioning the phenomenon of vacuum polarization. It goes in to
mention the screening of the surrounding space (vacuum or not) in the
presence of strong sources. Article 55 discusses the breakdown of the
insulating capacity of the surrounding space (including the vacuum) in
the presence of strong fields, lays out a thought experiment showing
how a concentrated source will screen the surrounding space, leading
to a distinction between bare vs. renormalized source strength; and
explicitly expresses the idea that this mechanism serves as a basis
for canceling out the self-force infinity of field theory. Article 83
lays out a second thought experiment demonstrating the screening
effect in the presence of strong point-like sources and discusses the
relation between the bare vs. effective potential.
Peter
...
> Thus, with the variationals defined by
> � � � � delta(L) = D.delta(E) - H.delta(B), � � delta(R) = D.delta(E) + B.delta
> (H)
How does this comply with the view that E and H (D and B) are
intensive (extensive) field variables, ie, that, according to Maxwell
(Dynamical Theory; Treatise),
delta(field_energy) = E delta(D) + H delta (B) ?
Thank you,
Peter