Natural Science Forum / Physics / Research / January 2005

I find it a bit hard to pin down exactly what Moffat is saying, but
he's saying what I think he is, then he's wrong.  Anyway, rather than
trying to read minds, I'll just state specifically what I think is
true.

  Consider two theories that are the same except for the values of
  various constants.  If all of the dimensionless constants are the
  same, then the two theories are equivalent (cannot be distinguished
  by any experiment), even if the two theories have different values
  of dimensionful constants.

I think that this statement is (a) true, and (b) what people generally
mean when they say that only variations in dimensionless constants can
be measured.  If Moffat is disagreeing with this statement, then I
think he's wrong.

I'll give some examples, in case things aren't clear.

Imagine a universe consisting of only electrons, positrons, and
photons interacting via quantum electrodynamics (no gravity, strong,
or weak force).  In such a universe, the only dimensionless constant
is the fine-structure constant

alpha = e^2 / (hbar c).

There's no way to distinguish in this universe between a 5% decrease
in the value of e (holding hbar and c fixed) and a 10% increase in the
value of c (holding e and hbar fixed), because both of these alter the
fine-structure constant in the same way.

In our actual universe, there are particles other than electrons,
positrons, and photons, and interactions other than QED.  That means
that it may be possible to distinguish between a 5% drop in e and
a 10% rise in c.  Whether it's possible or not depends on what's being
held fixed as you change these constants.

To keep things relatively simple, let's imagine a universe only
slightly more complicated than the one above.  Suppose there are still
only electrons, positrons, and photons, but now they interact via both
QED and gravity.  In such a theory, the dimensionful constants we can
imagine varying include e, c, hbar, m_e (electron mass), G (gravitation
constant).

There are only two independent dimensionless constants we can make
out of these:

e^2 / hbar c = fine-structure constant
m_e / (hbar c/G)^(1/2) = ratio of electron mass to Planck mass.

Here's the question: in this theory, can you distinguish between a 5%
decrease in e and a 10% increase in c?  If we hold m_e and G fixed,
then the answer is yes, because the second dimensionless constant will
change.  On the other hand, if we instead allow G and/or m_e to vary
along with c, and we do it in such a way that the ratio of masses
remains constant, then there's no experimental way to distinguish
between a varying-e universe and a varying-c universe.

This may seem like sophistry.  After all, surely it's obvious that
when we talk about varying c, we're not talking about varying G at the
same time, right?  Wrong.  We can imagine varying c while holding G
and hbar fixed, or we can imagine varying hbar while holding the
Planck mass and hbar fixed.  Which one seems more "natural" to us
depends on whether we regard G or the Planck mass as more fundamental.
But that's an artifact of how we were taught physics, not a property
of the universe.

Moffat lists a bunch of examples in his paper of situations in which
changes in dimensionful constants can be measured.  In every case, if
you think carefully about what's being held constant, I think you'll
find that some dimensionless constant is varying.

Some papers in the field, especially theory papers regarding the
interpretation of recent observations that suggest variations in the
fine-structure constant, are a bit confusing on this point.  They talk
about whether the observations are better fit by models in which c
varies or models in which hbar varies, for instance.  They don't
always say explicitly which other constants (e.g., G or Planck mass?)
are being fixed when c and hbar are varied.  

I talked to Joao Magueijo, one of the leaders in this field, about
this issue once, and, if I understood him correctly, he agreed with
the "conventional wisdom" as I've tried to describe it above.  When he
and others distinguish between varying-c theories and varying-hbar
theories, those theories involve fixing other dimensionful constants
in a certain way, so that there are dimensionless constants that
differ between the two cases.

-Ted

ebunn@lfa221051.richmond.edu wrote:

I raised the issue and dealt with the topic extensively here over the
past 5 years.  The papers sound like echoes of the comments I made
(and probably, in fact, came from them, postdating the articles
posted here), but apparently, still not getting as far as I did.

It's true that all you have access to are dimensionless values.  So
transformations in dimensionful values (coupled with transformations
in the actual physical quantities, like lengths) of the kind that
preserve everything that's dimensionless are physically irrelevant.

However, there is one place where the rubber hits the ground.  Calculus
is not invariant under this transformation.  In general, if T is a
time unit for which (T/second) is not constant, then
d(t/second)*second vs. d(t/T)*T
are different.

So, what it comes down to is this: if you accept the invisibility and
irrelevance of transformations that preserve dimensionless values,
then the derivative operator ceases to be meaningful!

Then, either the underlying continuum used to model physical events
has to be foregone in place of something discrete, or the derivatives
have to be unit-gauged.

In either case, if you're using derivatives to model events (either
as an approximation to the underlying discrete reality or a direct
representation of it), then there is implicit a natural "horizontal"
direction in the unit-gauge space.

However -- the ultimate conclusions raised in my dealings with the
topics -- the gauging need not be trivializable!  That means, then,
that NO system of units may exist which provides the correct
"horizontal" direction for the derivatives.  For instance, you may
have one system of equations that work naturally with one set of
units (meter, second, kilogram) and another system of equations
that work naturally with another set of units (L, T, M) for which
L/meter, T/second, M/kilogram are not all constant.  Differentiation
is equivalent under transformation only between equivalent sets of
units (i.e. sets whose ratios are all non-zero constants).  A
case of this is where one system may be naturally formulated in
terms of the units (e,h,m_e), another in terms of (c,h,G), but where
the ratio e/(hc) need not be a constant!

Under such circumstances, you are forced to accept the relativity
of scaling, to reject universally equating of one set of
constants to 1, and to admit an extra gauge structure for dimensional
scale invariance.

Perhaps it's just a semantic disagreement. Wouldn't Moffat agree that if
you simultaneously changed several dimensionful constants in such a way
that *none* of the dimensionless constants were affected, there would be
no new physically measurable effects? If so, then the claim "c increased
by a factor of 4" could be seen as just a kind of shorthand for a
statement like "all dimensionless constants which have a factor of c^1/2
in them increased by a factor of 2, all dimensionless constants which
have a factor of c^-1 in them decreased by a factor of 4, all
dimensionless constants which have a factor of c^-5 in them decreased by
a factor of 1024, etc."Then the disagreement between Moffat and Duff
would just be about whether it is acceptable to use "c increased by a
factor of 4" as a shorthand for this detailed description of a change in
various dimensionless constants or not (an argument against this
shorthand would be if it were possible to reproduce the same changes in
dimensionless constants by varying one or more other dimensionful
constants besides c).

I'm curious though, is there any basic principle that you can appeal to
that would prove that changing various dimensionful constants in a way
that leaves all the dimensionless constants unaltered could not possibly
have any measurable physical effects? I ask because I know of a popular
creationist theory which tries to explain how we see light from distant
galaxies by postulating that the speed of light was much higher in the
past, and in order to avoid this leading to certain death for Adam & Eve
(say, because the energy released by nuclear reactions in the sun would
be much larger according to E=mc^2) he has to vary a bunch of other
dimensionful constants as well, and I suspect that the net result is
that all of the dimensionless constants are unchanged...if there was
some simple proof that this could have no measurable effects, it would
be useful in debunking this theory. The page at
http://www.phys.unsw.edu.au/~dzuba/varyc.html gives some physical
arguments as to why specific methods of trying to measure a change in c
wouldn't work, but I'm wondering if there's a more general and rigorous
argument.

Jesse

robert bristow-johnson Wrote:

I thought I new about such trivial things as units, but Duff educated
me!!! Duff is quite right. Duff's mistake is to publicly attack the
folklore around "variable constants". It has been a small industry to
publish on "variable constants" and Duff shows that much of it is
nonsense. Duff demonstrates different sets of units, showing that a
change in the fine structure constant would be blamed on the speed of
light in one set of units and on Planck's constant in another, etc. He
references some critics, but their argument is essentially: "We know
the folklore and Duff is wrong" Burning him at the stake makes about as
much sense.

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robert bristow-johnson Wrote:

Duff mixes up a few things and therefor weakens his argument:  

1)  A variation of a constant (in %) is always dimensionless. It does
in no way dependent on how we choose our unit of measurement.
An N% change in liters would be the same N% change in gallons
Duff should leave such arguments outside the discussion.  

The real problem is that we may not have our previous reference
measurement stick anymore when it has changed. So the
problem becomes how to determine the % change when we can
not directly compare the "before" and the "after" values.  

It is this problem that occurs only in dimensional but not in
dimension-
less constants. Dimensionless constant are always ratio's of values
with the same dimension. The "after" ratio can always be determined
without a need for a "before" measurement stick.  

2)  It's simply not true that we can not distinguish if a varying alpha

comes from a change in c or e, even though the consequences of any
of the two changing can be quite complex.  

In general: If this kind of impossibility of distinction was true in
any
case of two or more constants then it would only mean that we can
describe nature with less constants than we do.

Regards, Hans

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