Readings on scale,
conformal and Weyl invariance
This is a subject that has given rise
to much misunderstanding, because the same words are often
used by different authors to mean subtly different things.
This page contains a list of papers on scale, conformal and
Weyl invariance, with comments. By reviewing these papers I
have tried to clarify for myself the source of many
misunderstandings and establish a set of conventions that is
ambiguity-free. I hope it may be useful to others too.
Preliminary mathematical
notions.
Two metric tensors g and g' are said to be conformally
related if g'=Ω2
g for some nowhere vanishing function Omega. This is
equivalent to the condition that the angles between any two
vectors measured with the metrics g and g' are the same.
A diffeomorphism of a manifold is said to be an isometry
if the pullback of the metric g is equal
to g, and a conformal transformation
if the pullback of g is conformally
related to g.
The isometries of a Riemannian manifold
form a Lie group (the isometry group) whose Lie algebra
consists of the Killing vectorfields, i.e. the vectorfields
along which the Lie derivative of the metric is zero. The
conformal transformations of a Riemannian manifold form a
group called the conformal
group, whose Lie algebra consists
of the conformal Killing vectorfields, i.e. the vectorfields
along which the Lie derivative of the metric is proportional
to the metric.
The isometry group of Minkowski space is the Poincare' group
(the semidirect product of translations and SO(1,3)) and the
conformal isometries form the group SO(2,4). Most Riemannian
manifolds do not have any conformal Killing vectors, so their
conformal group is trivial (a fortiori their isometry group is
trivial).
The term "conformal group" is sometimes also used in another
sense, namely as the infinite dimensional abelian group of all
conformal rescalings of a metric. This is not a group of
transformations acting on spacetime; it is instead akin to the
infinite dimensional group of gauge transformations in a gauge
theory.
-----------------------------------------
E. Cunningham,
Proc, London Math. Soc. 8, 77 (1909)
H. Bateman, Proc,
London
Math. Soc. 8, 223 (1910)
Any review of this subject must start from Weyl's classic paper. Its main focus is the unification of gravity and electromagnetism, which I have briefly discussed elsewhere, but here we are more interested in his critique of Einstein's theory. Weyl observes that in Riemannian geometry parallel transport is generally not integrable, so that two vectors at two different points cannot be compared. However, Riemannian parallel transport preserves the length of vectors, so the length of two vectors at two different points can be compared. Why should the direction of a vector have no absolute meaning, but its length have one? This, he says, "has no factual basis, and only seems to be due to the derivation of the theory from the flat one". For Weyl, this fact goes against the idea of an "infinitesimal geometry", where only infinitesimally close vectors can be compared. In modern geometrical language, Weyl advocates dropping the assumption that the connection is metric (i.e. the covariant derivative of the metric vanishes). The theory must then be invariant not just under coordinate transformations, but also under conformal rescalings of the metric.
In an appendix to the paper, Einstein points out that if the metric properties of a spacetime could only be determined by use of light rays, then indeed the metric could only be determined up to an unknown function. But in practice we have at our disposal (infinitesimal) rigid rods and clocks, which can be used to define the line element ds. This definition of ds would only be illusory if the notions of unit length and unit time depended on the previous history of the rods and clocks. This seems not to be the case, so, he concludes, the basic assumption of Weyl's theory is false. There are counterarguments by Weyl, but it is clear that Einstein has a more profound grasp of empirical reality. Weyl admits that he may be on the wrong track (he uses the wonderful expression "auf dem Holzwege sein", which stems from the fact that paths used to carry logs out of a forest do not lead anywhere). In a note added in 1955 he admits that his attempt was misplaced and that the proper setting for the gauge invariance of electromagnetism is the phase of the quantum wave function of charged particles and not the scale.
It is important to realize that while the physical
interpretation of Weyl's theory was certainly wrong, this does
not rule it out: it is sufficient to reinterpret Weyl's
potential as a different gauge field. Then, Einstein's
objection would not apply: the fact that atoms all have the
same size and energy levels just tells us that in the
spacetime region where atoms existed, the curvature of this
connection was zero. An extension of this argument to protons
would push the statement back to before baryogenesis. In most
conceivable models (see below) the Weyl gauge field would
probably have a Planck-size mass and this restriction could be
easily accommodated.
We now fast forward by more than forty years and consider the
following famous work:
Brans
and R.H. Dicke,
Mach's principle and a relativistic theory of
gravitation
R.H. Dicke,
Mach's principle and invariance under transformations of
units
Phys.
Rev.
126, 2163 (1962)
These two papers were motivated by the
desire to formulate a theory of gravity that incorporates
Mach's ideas on the origin of inertia. This motivation has
been largely forgotten and the work is important because it is
one of the first instances of what we now call scalar-tensor
theories of gravity. They are also admirable examples of clear
physical thinking. The authors are careful to define the
theory in ways that have operational meaning, and in so doing
they find themselves echoing Weyl's call for a "truly
infinitesimal geometry". In the first paper they say that
"there is no direct way in which the mass of a particle...can
be compared with that of another at a different space-time
point". And in the second Dicke adds: "It is evident that two
rods side by side, stationary with respect to each other, can
be intercompared....this cannot be done for....rods with
either a space- or time-like separation". There follows that a
statement such as "a hydrogen atom on Sirius has the same
diameter as one on the Earth...is either a definition or else
meaningless". The resulting local arbitrariness in the choice
of units reflects itself in the definition of the metric:
different choices of units lead to metrics that are
conformally related. This does not mean that Riemannian
geometry is useless or incorrect, just that each Riemannian
geometry is a particular representation of the theory. The
physical content of the theory should be contained in the
invariants of position dependent transformations of units and
coordinate transformations. In general relativity the
representation is one in which units are chosen so that atoms
are described as having physical properties independent of
location. That such a representation exists is in itself a
nontrivial assumption.
In Brans and Dicke's theory, it is assumed that all mass
ratios of elementary particles are constant. In the first
paper a system of units is chosen such that the masses are
themselves constant throughout spacetime. Then, Newton's
constant is allowed to be position dependent and is treated as
a field. The Lagrangian of Brans-Dicke theory in these units
consists of a kinetic term for the scalar, coupled
nonminimally to the curvature scalar. The matter Lagrangian is
assumed to depend on the metric but not on the scalar, in such
a way that the equations of motion of matter in a given
background geometry are the same as in general relativity. In
the second paper the same theory is discussed in the system of
units where Newton's constant is assumed to be constant and
the particle masses are allowed to depend on position. This
reformulation differs from the previous one by a conformal
transformation. In the new "conformal frame" the equations for
the metric are the same as in Einstein's theory, but matter
now couples to the scalar, which can be seen as another matter
field. It is important to note that the Brans-Dicke action is
not Weyl-invariant (except for the special choice of parameter
-3/2).
P.A.M.
Dirac,
Long range forces and broken symmetries
Proc. Roy. Soc. 333, 403 (1973)
Dirac, like many others, was fascinated
by Weyl's theory, which he describes as "the outstanding
[unified theory], unrivalled by its simplicity and beauty",
Yet, it clashes with quantum theory: "quantum phenomena
provide an absolute standard of length...[and] there is no
need for the arbitrary metric standards of Weyl's theory". In
this late paper Dirac tries to resuscitate Weyl's theory. He
starts from the "large number hypothesis" that he had
formulated in 1938. It is a speculation that all the very
large dimensionless numbers that one can form are related by
some as yet unknown law, and it leads to the conclusion that
Newton's constant has to decrease in proportion to the age of
the universe. Assuming that this hypothesis is correct, the
question is how to reconcile it with Einstein's theory of
gravitation, which requires that G shall be constant. Dirac's
suggestion is that "the Einstein equations refer to an
interval dsE connecting two
neighboring points that is not the same as the interval dsA
measured by atomic apparatus. By taking the ratio of dsE
to dsA to vary with the epoch we get G varying with
the epoch".
"With the introduction of the two metrices dsE
and dsA we see that the objection
to Weyl's theory...falls away. One can apply Weyl's geometry
to dsE, supposing it is nonintegrable when
transported by parallel displacement, so that we must refer to
an arbitrary metric gauge to get a definite value for it...On
the other hand dsA is referred to atomic units and
is not affected by a [conformal] transformation..."
"The measurements ordinarily made by
physicists in the laboratory use apparatus which is fixed by
the atomic properties of matter, so the measurements will
refer to the metric dsA. The metric dsE
cannot be measured directly, but it shows itself up through
the equations of motion.... The relation of the two metrics is
exemplified by radar observations of the planets. Here a
distance which is determined by equations of motion is
measured by atomic apparatus."
Dirac then goes on to review the formalism of Weyl geometry and proposes a Lagrangian which is essentially the same as the one of Brans and Dicke. Perhaps the most original part of the paper is the prediction that in this theory there will be C and T violating effects, with CT preserved. Unfortunately this conclusion only applies when the Weyl gauge field is interpreted as the electromagnetic potential. In the conclusions Dirac asks why one should believe in such a theory. Even if the variation of G was confirmed by experiment, it would not prove that Weyl's geometry is needed to explain the electromagnetic field. Dirac says that "it appears as one of the fundamental principles of Nature that the equations expressing basic laws should be invariant under the widest possible group of transformations... The passage to Weyl's geometry is a further step in the direction of widening the group of tranformations underlying physical laws."
I have listed this paper here because it makes a clear case
for the existence of different spacetime metrics, related to
different fundamental units. However, this notion had been
discussed earlier in the literature and Dirac's insistence on
Weyl's original interpretation is clearly misplaced.
Furthermore, some of its conclusions are incorrect, as pointed
out in the following paper.
The focus of this paper is the question whether the gravitational constant could be spacetime dependent. Starting from the observation that physical laws are invariant under global rescalings of the units of length, and the axiom that this invariance should be extended to local rescalings, it is observed that this would also imply that all Compton wavelength should be allowed to transform, and therefore all particle masses should become scalar fields. For the theory to be complete, it must prescribe a dynamics for each such mass field. The authors assume that all mass ratios are constant and therefore all masses are proportional to a single mass field. From a set of postulates, including Weyl invariance, the action of the mass field is derived to be that of a conformally coupled scalar, plus a quartic potential. This theory is known to be equivalent to Einstein's theory in the frame where the mass field is contant. But the authors point out that "this conclusion...is an understantement. The physical content of the theory is the same in any units. It is GR in any conformal frame." The ratio of Newton's constant to any particle mass is constant. The conclusion is that the "the simplest implementation of the principle of conformal invariance requires that gravitation be described by GR" with Newton's constant a constant when measured in particle masses.
There is then a sharp criticism of
Dirac's preceding paper, and other attempts to introduce a
variable Newton's constant. Regarding Dirac's paper they
observe that the proposed action for a pointlike particle is
the integral of the line element including a factor of the
universal mass field, and a constant dimensionless prefactor.
This action is Weyl invariant. Dirac defines Einstein units to
be those where the mass field is constant. He regards these as
distinct from atomic units. "Yet the particular theory at hand
fails to incorporate this distinction". In fact, in the
Einstein frame particle masses as defined above are also
constant. "Thus, despite his avowed intention to write a
theory which distingushes between Einstein and atomic units,
the theory at hand does not do this... it is evidently GR in a
conformally invariant garb.... The failure to incorporate a
distinction between the units can be traced to the use of a
single field... to describe the characteristic length
associated with gravitation, and that associated with rest
masses."
E.E.
Flanagan,
The conformal frame freedom in theories of gravitation
Class.
and Quantum Grav. 21, 3817 (2004)
This paper contains a clear discussion of the issue of the choice of conformal frame. The issue can be stated as follows: suppose we are given two Lagrangians for gravity and matter, which can be obtained from each other by means of a Weyl redefinition of the metric. Should the two theories be regarded as physically equivalent? The answer to this question depends on the physical interpretation of the metric in the two frames, and a lot of confusion has been generated in the literature by the fact that authors often fail to state such assumptions explicitly. If one assumes that the metric in both theories is the metric measured by means of atomic clocks, then indeed the two theories will be physically inequivalent. However, if one sticks to the interpretation mentioned above that conformal redefinitions of the metric simply amount to different choices of units, then the two theories must be physically equivalent. This interpretation is by far the most natural and attractive one, as it agrees fully with the notion that a choice of conformal frame is a choice of gauge, and is therefore in line with the terminology that has developed historically since Weyl's original paper.
The discussion is a little less
complete in the quantum case, which will be discussed below.
It is pointed out, however, that if the theory is treated as
an effective quantum field theory, then applying a standard
result, physical observables (more precisely: the S matrix) do
not change under field redefinitions, so also in this context
two theories that differ by a conformal redefinition of the
metric must be equivalent.
A.
Iorio, L. O'Raifeartaigh, I. Sachs and C.
Wiesendanger
Weyl gauging and conformal invariance
Nucl. Phys. B495, 433-450
(1997)
We now go over to the context of quantum field theory. The
first few papers listed below discuss mostly some results
regarding conformal invariance in flat space (i.e. invariance
under the conformal group SO(2,d)).
C.G.
Callan, S. Coleman and R. Jackiw,
A new improved energy-momentum tensor
S.
Deser,
Scale invariance and gravitational coupling
S.
Coleman and R. Jackiw,
Why dilatation generators do not generate
dilatations?
The motivation for this work comes from
the idea that at high energy masses become negligible and
therefore a theory should become approximately scale
invariant, in some suitable sense. The context of this work
are power counting renormalizable theories (i.e. theories with
dimensionless coupling constants). Then it appears that the
only possible source of breaking of scale invariance are mass
terms. If one applies standard canonical methods, one can
construct a current that generates dilatations and derive from
it theorems about the asymptotic behavior of Green functions.
These theorems are false in perturbation theory. The reason is
that the formal manipulations leading to them do not go
through when one introduces a cutoff. This is simple to
understand: a large cutoff is a large dimensionful scale in
the theory, and it leaves behind a residue called a scale anomaly.
The main results of the paper are: (1) in lowest order of
perturbation theory, the scale anomaly can be completely
described by a change in the scale dimension of the field, (2)
at higher orders, the anomaly cannot be explained simply by a
change of the scale dimension. The authors stress the
difference between the notion of scale dimension and the usual
notion of dimension as used in dimensional analysis. In both
cases the dimension is the power of the scaling parameter that
must multiply a field in order to preserve scale invariance.
But "the transformations of
dimensional analysis map the exact solutions of one field
theory into the exact solutions of a different field theory,
with different masses. Scale transformations... stay within
a given field theory, and do not map solutions into
solutions, unless the masses are zero". This is an
important difference between the notion of scale (or Weyl)
invariance as commonly used by particle physicists and general
relativists.
J.
Polchinski,
Scale and conformal invariance in quantum field
theory
R. Jackiw and S.-Y. Pi,
Tutorial
on scale and conformal symmetries in diverse
dimensions,
J. Phys. A44, 223001 (2011)
arXiv:1101.4886
[math-ph]
arXiv:1101:5385
[hep-th]
These three papers
are concerned with the following issue: under what conditions
does invariance under Poincare and scale transformations imply
invariance under special conformal transformations (and hence
under the whole SO(2,d))? (The converse statement, namely
Poincare and special conformal invariance implying scale
invariance, is always true just for Lie algebraic reasons.) It
is sometimes erroneously assumed that the implication is
always true. Polchinski gives several examples of theories
that are scale invariant and not conformally invariant, but
such examples are all somewhat pathological. He then proves
(completing an argument by Zamolodchikov) that in two
dimensions for every unitary theory having a discrete spectrum
of operator dimensions, scale invariance implies conformal
invariance. The question remained open whether the same
statement also holds in higher dimensions.
The other two papers provide a simple counterexample: Maxwell
theory in five dimensions.
Now we come to Weyl transformations and
we begin by discussing quantum fields in a background
geometry. The most important new phenomenon that we have to
take into account is the conformal anomaly. This can be
stated as follows. Suppose the classical action of the system
S(g,Φ) is invariant under
Weyl transformations: S(Ω2g,Ω-dΦ)=S(g,Φ),
where d is the length dimension of the field Φ.
(Note that even though we treat the background as fixed, we
allow coordinate transformations and Weyl transformations to
act on it. This means that we are not treating the metric g as
a fixed background, but rather its diffeomorphism equivalence
class and, in the conformal invariant case, its Weyl
equivalence class.) Next we compute the effective action Γ(g,Φ). Many different
calculations lead to the conclusion that even though the
classical action is invariant under Weyl transformations, the
effective action is not. Before discussing this in greater
detail let me first quote some classic papers on the subject: