Planck units (uselessness of).
Wednesday 2007 May 16
Planck units are determined by setting the speed of light c, Planck’s constant h, and the gravitational constant all equal to 1. The speed of light and Planck’s constant (give or take a factor of two pi) seem fundamental, but it is not clear why the gravitational constant G should be 1.
A minor problem is that it is probably missing a factor of 4π or maybe 8π as this is what appears in the Lagrangian. This is not serious but is a little worrying: no-one would suggest using 2c or c/2 as a fundamental unit of velocity, so an ambiguity of 2 is not a good sign for a supposedly fundamental unit.
A much more serious problem is that G appears as a non-renormalizable term in the Lagrangian of the standard model with gravitation; in fact, the only non-renormalizable term that has been measured to be non-zero. According to Wilson’s view of the renormalization group flow, this Lagrangian should be thought of as a low energy effective Lagrangian of some unknown theory. His theory predicts that there should be lots of non-zero non-renormalizable terms, which should all be extremely small. The constant G by a fluke happens to be detectable, because gravity just happens to be cumulative and is not masked by renormalizable interactions. So this suggests that G is just one of an infinite number of terms in the Lagrangian that could be used to determine a set of units.
Another problem is that according to Wilson’s theory all the coupling constants (presumably including G) change under the renormalization group flow so are not really constant. This suggests that there is nothing particularly significant about the Planck mass, or length, or energy, or whatever: all we know is that classical general relativity breaks down by Planck densities or lengths. By analogy, the Fermi constant for weak interactions could be used to produce a fundamental set of units, where the fundamental energy is about 300 GeV, but nothing special happens at this energy; it is just an order of magnitude estimate for the point where the old Fermi theory of the weak interaction breaks down (and for the masses of the vector bosons of the electroweak interaction).
Sunday 2007 May 20 at 8:09 pm
This is interesting. Naively, I would still think that even if
is part of an infinite set of non-renormalizable terms, the fact that it is not masked by renormalizable terms de facto sets it appart.
Then, at least according to this, since the RG flow is a semi-group, something like
should make sense. So the planck mass and so on as defined in the wikipedia page you mentionned would be meaningless, but replacing those 
by 
(perhaps mixed with some 
or 
factor indeed) would.
P.s. is it true this is professor Borcherds’ blog? (if so then perhaps a link from your webpage would help authentification
)
Monday 2007 May 21 at 2:51 am
Fantastic first post! Welcome to the physics blogosphere.
Monday 2007 May 21 at 10:15 am
Quite an otiose point.
Nobody give precise meaning to the Planck scale but the order of magnitude of where gravitational effects will mix with the quantum ones and one actually needs a consistent description of both. Yes, for the time being, c and h seem to be fundamental while G not and yes, when we will reach a proper theory for both the quantum and the gravity we will see what is the right choice for dimensionless units…
Monday 2007 May 21 at 2:12 pm
Interesting subject! The Planck scale is purely the result of dimensional analysis, and Planck’s claim that the Planck length was the smallest length of physical significance is vacuous because the black hole event horizon radius for the electron mass, R = 2GM/c^2 = 1.35*10^{-57} m, which is over 22 orders of magnitude smaller than the Planck length, R = square root (h bar * G/c^3) = 1.6*10^{-35} m.
Why, physically, should this Planck scale formula hold, other than the fact that it has the correct units (length)? Far more natural to use R = 2GM/c^2 for the ultimate small distance unit, where M is electron mass.
If there is a natural ultimate ‘grain size’ to the vacuum to explain, as Wilson does, simply why there are no infinite momenta problems with pair-production/annihilation loops beyond the UV cutoff (i.e. smaller distances than the grain size of the ‘Dirac sea’), it might make more physical sense to use the event horizon radius of a black hole of fundamental particle mass, than to use the Planck length.
All the Planck scale has to defend it is a century of obfuscating orthodoxy.
Monday 2007 May 21 at 7:54 pm
Welcome. The Planck units can be very misleading, since they assume that G, c and h are all constant.
Monday 2007 May 21 at 8:25 pm
Welcome Richard! I hope this is the first of many interesting posts.
Do you have the same concerns about the privileged status of the Planck charge in electromagnetism? There is an additional parameter here, of course (the fine structure constant) but I was not aware of any proposal that this would imply that the Planck charge is an artefact of some more fundamental unit of charge. But I suppose QED is much more renormalisable and so these objections do not apply. (Nevertheless, we don’t expect any really striking transitional behaviour at the Planck charge level, and perhaps this charge should indeed be viewed as simply a convenient mathematical normalisation.)
Tuesday 2007 May 22 at 4:35 am
It would seem that c, h or G could be set equal to some identity value.
One [1] might be used for this identity value with c, h or G – but perhaps not at the same time since the identity values may not be equal simultaneously?
Tuesday 2007 May 22 at 4:36 pm
(Reply to Tenece Tao’s question): I have no idea if the Planck charge is of any significance, or what the relation of the Planck charge to the electron charge is. One can speculate that they might be related by some sort of renomalization group flow. But this does not seem to work in QED, where it is suspected that the electric charge becomes infinite under renormalization group flow at some extremely large energy (Landau pole).
Tuesday 2007 May 22 at 6:24 pm
Welcome to the physics blog metaverse. I’ll be back for more later.
Wednesday 2007 May 23 at 4:31 pm
Actually, neutrino masses come from a nonrenormalizable term in the standard model.
Friday 2007 May 25 at 2:21 pm
You know, I’ve always thought that G was the odd-man-out in that whole system.
The speed of light gives a nice classical limit as
, and Planck’s constant gives a nice classical limit as 
.
So the difference between special relativity and Galilean relativity is the difference between the speed of light being infinite or not. Once it’s not, we can pick units to make it 1. Similarly the difference between quantum mechanics and classical mechanics is the difference between Planck’s constant being infinitesimal or not. Once it’s not, we can pick units to make it 1.
So what different (simpler?) theories does taking a limit of G give us? I honestly don’t know of any.
Friday 2007 May 25 at 3:03 pm
G = 0 gives special relativity (if done carefully). I would say there is enough simplification involved to consider that a “classical” limit.
Friday 2007 May 25 at 7:37 pm
Well, I just quickly thought of this and it would seem that G (the gravitational constant) is pretty close to 1/(k*c), where k would be 50, so would it be possible that you could derive G from speed of light?
Just for a quick thought, that would give acceleration due to gravity as:
a = m / (50*c*r^2)
And mass:
m = 50*c*a*r^2
Radius:
r = sqrt(m / (50*c*a))
And speed of light:
c = m / (50*a*r^2)
(all just rearranging the same thing)
ok, just some random friday-thoughts, but anyways, thanks for your interesting posts.
Saturday 2007 May 26 at 7:32 am
Something just doesn’t sit right with me saying that GR goes to SR as
. Doesn’t that put implicit restrictions on the background topology? Otherwise there’s a much more violent change going on in the limit than deforming the coupling between space and time away to zero (as 
. It may be correct, but still something just feels much more disturbing to me there than the other two cases.
Saturday 2007 May 26 at 9:27 pm
Dear Ari,
G and 1/c have different units (in SI, G has units
while 1/c has units 
), so the quantity k = 1/(Gc) is not dimensionless, it has units 
. If you decide to move from metres to feet, or from kilograms to pounds, the nominal value of k will change. So the fact that it is somewhat close to 1 in human units is probably just a coincidence (and given the number of important constants in physics, probably explainable by the birthday paradox).
Dear John,
It all depends on what sense one takes a limit, and what else is kept fixed. For instance, if one takes a manifold with non-trivial topology, say a 2-torus, and dilates it around a fixed point on the torus, then the manifold will converge locally to a flat 2-space; the topology has been “pushed out to infinity”, and thus disappears in the limit as measured in this “weak” sense. For similar reasons, any Einsteinian spacetime will after rescaling converge locally to Minkowskian spacetime, and if one rescales one’s unit of mass the right way, then one will get non-trivial but finite amounts of gravitation-less matter, with G vanishing in the limit.
Sunday 2007 May 27 at 8:07 am
First, about limits of spacetimes: I’d just like quickly to point out an old but very helpful paper by Robert Geroch, a general relativity theorist who, with Hawking and Penrose, proved some of the first singularity theorems.
Geroch’s very helpful paper
http://tinyurl.com/yr9ans
clarifies, defines, and explores the notion of the limit of a 1-parameter family of (general-relativistic) spacetimes.
In a nutshell, the idea is that a 1-parameter family of spacetimes is actually a sort of fibre bundle over the (positive) real line, and that a limit spacetime for this 1-parameter family is a (4-dimensional) boundary affixed to the (5-dimensional) total space of this bundle. In general, various boundaries may be affixed in various ways, and Geroch gives a number of explicit examples that do much to clarify the whole idea of limiting processes in general relativity. Definitely worth a look.
Oh, and second, about the Planck units: it seems only fair to note that h-bar is, in a sense, just as suspect as G, for similar reasons.
Of course it’s true that the Einstein scalar curvature Lagrangian is non-renormalizable, which suggests that general relativity cannot be “quantized.” And yes, this is usually taken to mean that general relativity is merely a low-energy “effective” theory, whose fundamental variables should no more be “quantized” than should, say, the entropy or temperature variables assigned by classical thermodynamics to the equilibrium states of a dilute gas. And if this analysis is correct, then G is hardly fundamental; it’s much more like Boltzmann’s constant than, say, the speed of light in vacuum.
But it’s equally true that quantum theory cannot be “general-relativized.” Despite clear experimental evidence that quantum phenomena are just as useless as classical phenomena for drawing local distinctions between the effects of gravity and the effects of uniform acceleration, quantum theory still requires for its formulation an (unobservable) globally flat spacetime structure — which is to say, a sort of ether.
So, just as G appears suspect because general relativity fails to conform to the universal requirements of quantum theory, so does h-bar appear suspect because quantum theory fails to conform to the universal requirements of general relativity theory.
The key point, it seems to me, is that both theories impose non-trivial structural requirements on all other physical theories — and that neither theory can meet the requirements imposed by the other. In such a situation, with two exceedingly well confirmed and persuasive physical theories in direct conflict, it seems rash to maintain that we can say with any confidence which of these constants, G or h-bar, is “fundamental” (or, indeed, whether either of them is.)
I think this helps to explain why, even now, more than a hundred years after their advent, the Planck units remain so obscure.
Sunday 2007 May 27 at 12:44 pm
Thanks for the post. I believe that G is a conventional unit and an artifact of the unit systems. I tried to explain this here: http://www.alphysics.com/Slides/UnitStates_files/frame.htm
Any comments would be welcome.
There is also a similar discussion about constants/units in progress at angryphysicist: http://angryphysicist.wordpress.com/2007/05/16/anger-is-freedomor-something/
And, where do I find the “Wilson’s paper” referenced in the post?
Thanks.
Sunday 2007 June 3 at 5:49 pm
For a list of the papers of Nobel laureate Kenneth G. Wilson on renormalization problems, try http://scholar.google.com/scholar?q=Kenneth+G.+Wilson+&hl=en&lr=
The best introductory paper however is probably his 1982 Nobel lecture, The Renormalization Group and Critical Phenomena, http://nobelprize.org/nobel_prizes/physics/laureates/1982/wilson-lecture.pdf
Monday 2007 June 18 at 3:28 pm
Dimensionful units are somewhat less ad hoc in string models. There one has a limiting acceleration and temperature with alpha’ replacing G as the main dimensionful parameter. It is natural to define acceleration and temperature as fractions of these maximum values in a similar manner to v/c in special relativity.
Tuesday 2007 August 28 at 6:42 am
If we write G into form of its concepts by using dimensional analysis, we get G = 1/(P x t^2) where P = density and t = time.
Because G is universal constant we may think that P and t can be associated with universal quantities. Now, if t is the age of the universe and P is the density of the universe at the time t we calculate P(t) = 1/(G x t^2) and we get (supposing t = 13.7 x 10 ^ 9 years) P(t) is about 10 ^ -27 kg/m^3.
*
Critical density for the expanding universe is: Pc = 3c^2H^2/(8PiG) and H is current value of Hubble parameter. If Hubble parameter range from 50 – 100 km/s/Mpc then Pc is approximately 10 ^ -27 kg/m^3.
*
Hence, the equation of G (dimensional analysis) gives the value of P which is very close to the value given by equation of Pc.
Now, numerator of equation of G is a pure number = 1. If this number is a relation of two quantities Pc/P(t) and this implies something like omega O – the universe is flat
we get G = O/(P(t) x T^2) and now T = is the age of the universe and t = characteristic expansion time so that P(t)^2 = Pc/(G x T^2).
Friday 2007 September 14 at 7:05 pm
[...] 14, 2007 Posted by A.J. Tolland in QFT, mathematical physics. trackback Richard Borcherds wrote a post a little while back in which he remarked that we shouldn’t take the Planck units very [...]
Sunday 2007 September 16 at 10:50 pm
About Planck charge, it remembers me that if you consider electromagnetism, then special relativity already gives you an angular momentum to be used instead of Planck constant: namely, the maximum angular momentum allowed for a closed orbit in the atom of hydrogen. Of course this is just $\alpha \hbar$, and then it suffers of the same renormalisation group criticism.
Amazingly these units are say (see Stoney Units in the wikipedia) to predate Planck… and relativity.
Wednesday 2008 February 20 at 9:40 pm
Pioneer, thanks for links
Thursday 2008 April 3 at 4:34 am
Integer multiples of 4π are ubiquitous in physics, simply because 4π appears in the formulae for the surface area and volume of the sphere.
“A minor problem is that it is probably missing a factor of 4π or maybe 8π…”
I warmly concur, simply because in general relativity, G nearly always appears as 8πG. Normalizing to 1 nicely streamlines the equations of general relativity, but introduces a factor of 8πG in the nondimensionalised version of Newton’s law. I can live with that.
Planck units should be modified so as to eliminate the stray instances of 4π in Maxwell’s equations, and in the curious gravitational analog of those equations:
http://en.wikipedia.org/wiki/Gravitoelectromagnetism
For my part, the purpose of Planck units is to purge from the algebra used to represent physical theories all constants that are, deep down, only conversion factors stenmming from arbitrary human choices of units of measurement. This suggest choosing units that normalize these conversion factors are 1 and hence eliminate them. The units that do this are precisely the Planck units.
A major problem with operationality of Planck units outside of pure theory, is that G has been experimentally determined to no better than 1 part in 10,000. This is pathetic, when compared to the accuracy of the electron & proton masses, the fine structure constant, the elementary charge, and so on. Moreover, G appears in the definition of nearly every Planck unit.
See
http://en.wikipedia.org/wiki/Natural_units
for several systems of natural units that do not normalize G, and so avoid many of problems raised in this interesting discussion. And here’s hoping that G will be known to within 1 part per million within the lifetimes of some of the readers of this blog.
Wednesday 2008 April 23 at 5:03 pm
In order to understand Planck’s “normal” constants, it might be helpful to know what Planck actually said about them. Planck’s theory was based on the atom as an electronic oscillator (“Planck’s Columbia Lectures”). These “oscillators” (atoms) exchange energy as a function of temperature, and Planck’s state equation describes the frequency spectrum of the normal distribution of energy. The discrete changes in energy are equal to hf, where h is Planck’s constant, and f is the frequency. The constants h and k (Boltzmann’s constant) allows “…the means of establishing units of length, mass, time and temperature, which are independent of special bodies or temperatures, which necessarily retain their significance for all environments, terrestrial and human or otherwise, and which may therefore be describes as “natural units.”.
With the additional value of the velocity of propagation of light in a vacuum, c, and that of the constant of gravitation, f (BigG), the means of determining the four units of length, mass, time and temperature are given. The “Planck length”, as it is now referred to is 3.99 x 10-35 meter, the “Planck mass”
is 5.37 x 10^-8 kg, the Planck time” is 1.33 x 10^-43 sec, and the “Planck temperature” is 3.60 x 10^32 deg. Note that most all temperatures we would ever see would be less than Planck’s natural unit, whereas for length, some believe that Planck’s natural unit is the minimum length that can be attributed to any natural process.
In the second edition of his book, he does not state that these natural units are either maximums or minimums. That conclusion appears to have come later in time by others.
The present methods of measurement were made “… essentiallwith reference to the special needs of our terrestrial civilization. Thus the units of length and time were derived from the present dimensions and motion of our planet, and the units of mass and temperature from the density and the most important temperature points of water, as being the liquid which plays the most important part on the surface of the earth…”.
He makes an important qualifying statement: “These quantities retain their natural significance as long as the law of gravitation and that of the propagation of light in a vacuum and the two principles of thermodynamics remain valid; they therefore must be found always the same, when measured by the most widely differing intelligences according to the most widely differing methods. After over 100 years, the constants, h, f, and c have not changed. The gravitational constant may be in various amounts of error, depending on which reference you wish to cite.
Some authors prefer to divide h by 2-pi (h-bar), as a matter of convenience, but this constant appears in many equations in many circumstances in many applications, which is an indication of its great significance.
It is beneficial to research what the great scientists actually have stated about their theories. I wonder how many physicists realize that Planck’s theory is based on the energy states as determined by chemical measurements and laws, thermodynamics, radiation theory and electronic oscillators.
Saturday 2008 May 10 at 7:31 pm
I’ve arrived at this post rather late but I’ve a few observations about this. It must be kept in mind that the Planck mass has context at or near the quantum gravity regime. The exception to this in the semi classical regime is where the inverse square of the Planck mass is the weak gravity condition of the normal Newton’s constant G. However, there is good reason to consider that G runs as you go up the scale toward the Planck energy and that it possibly hits a unity value. Again, emphasize that Planck mass only means something near the Planck density (in the compact space). After the semi classical gravity there would be quantum corrections to GR (as it is very close but not yet in the quantum gravity era). At the stage near, at Planck densities dimensionless numbers have a tendency to go beyond astronomically large (such as 10^120, 10^123 or the more or less arbitrary 10^500 vacua of the landscape though I doubt this number). The accounting for 8pi is still required and necessary (and difficult) where there are quantum corrections to GR in 4D and the only way I’ve been able to determine placement is by doing volumes of calculations. The dimensionless Planck units are the natural language and result of quantum gravity.
Monday 2008 May 26 at 1:17 pm
It is not necessary to include the gravitational constant in order to manipulate Planck’s natural units (see http://www.science-site.net/plancksunits.htm). In fact, the gravitational constant is the most questionable natural unit. Planck’s approach was to work forwards and backwards with the four constants, setting them all at unity. The same thing can be done with three constants, and they all work out the exact values of Planck’s units. This eliminates the radical in all of the equations, and the limitations of the fuzziness of the gravitational constant. Therefore, it is not necessary to even consider gravity in the derivation of Planck’s constants, although he made that work out too. Try substituting modern constants, and you will find that they do not work out exactly, which was exactly what Planck sought to avoid. As he put it, “…these quantities retain their natural significance as long as the law of gravitation, and that of the propagation of light in a vacuum and the two principles of thermodynamics remain valid…”.
Saturday 2008 October 4 at 10:49 am
[...] Plank units (uselessness of) [...]
Thursday 2008 November 20 at 1:49 pm
Cool post, maybe you dream fof writters?
___________________________________
Sry, hehe))
Friday 2008 December 5 at 2:46 pm
According to FH theory the Planck Mass is the mass equivalent of the vacuum energy of the volume described by the planck length cubed.
1/G is the mass of 1 cubic meter of vacuum space.
Work it out yourself and you will see that this is true..
The derivation of this can either be done by using newtons gravitational laws (top down) or from the simple equation CT = R
The mass of any volume of vacuum space is given by M = Rcubed / (GxT squared) according to FH Theory….