This site dedicated to education about the Planck quantities and their role as a set of natural units. It's divided up so as to cover Planck quantities from several perspectives, those of a general reader, a student in a college physics course, and a reader interested in human-scale power-of-ten variants of the natural units as a postmetric system.
One of the best recent expositions of Planck units occurred in the context of the series of three articles "Scaling Mount Planck" by Frank Wilczek. The first two were published in June and November issues of Physics Today. The third was in the August 2002 issue of the same journal and was quoted on-line by J. Gallas:
"Scaling Mount Planck I: A View from the Bottom,"
Physics Today,
June 2001, page 12
"Scaling Mount Planck II: Base Camp,"
Physics Today,
November 2001, page 12
"Scaling Mount Planck III: Is That All there Is?,"http://www.if.ufrgs.br/~jgallas/wilczek.html
Quote from paper III of the series: [[Frank Wilczek, the author of this text (in Physics Today, August 2002), is the Herman Feshbach Professor of Physics at the Massachusetts Institute of Technology in Cambridge, Massachusetts.
Scaling Mount Planck III: Is that all there is?
Let's quickly recollect the main points of the two earlier columns in this series. Gravity appears extravagantly feeble on atomic and laboratory scales, ultimately because the proton's mass mp is much smaller than the Planck mass MPlanck =(h-bar,c/GN)1/2, where h-bar is Planck's quantum of action, c is the speed of light, and GN is Newton's gravitational constant. Numerically, mp /MPlanck » 10-18. If we aspire, in line with Planck's original vision and with modern ambitions for the unification of physics, to use the natural (Planck) system of units constructed from c, h-bar, and GN (see "Scaling Mount Planck I: A View from the Bottom," Physics Today, June 2001, page 12), and if we agree that the proton is a natural object, then the very small ratio appears at first blush to pose a very big embarrassment. It mocks the central tenet of dimensional analysis, which is that natural quantities expressed in natural units should have numerical values close to unity.
Fortunately, we have a deep dynamical understanding of the origin of the proton's mass, thanks to quantum chromodynamics. The value of the proton's mass is determined by the scale LQCD, at which the interaction between quarks --parameterized by the energy-dependent "running" QCD coupling constant gs(E)--starts to dominate their quantum-mechanical resistance to localization (see "Scaling Mount Planck II: Base Camp," Physics Today, November 2001, page 12). More precisely, the criterion for the quark-binding interaction to dominate is that the QCD analog of the fine structure constant as(E) º gs(E)2/(4ph-barc) becomes of order unity: as(LQCD) » 1. Because the energy dependence of as(E) is very mild, a long run in E is required to change its value significantly. Indeed, we find in this way that our QCD-based estimate of the proton mass, using as(mpc2) » 1, corresponds to gs(MPlanckc2) » 1/2! So the extravagantly small value of mp/MPlanck does not contradict the idea that MPlanck is the "natural" fundamental unit of mass, after all. Whereas naive analysis founders on the value of mp, deeper understanding aims instead at gs(MPlanckc2)--the basic coupling at the basic energy--as the primary quantity, from which mp is derived. And gs(MPlanckc2) is of order unity!
A conceptually independent line of evidence likewise points to MPlanckc2 as a fundamental energy scale. By postulating the existence of an encompassing symmetry at that scale, and weaving the separate gauge symmetries SU(3) ´ SU(2) ´ U(1) of the standard model into a larger whole, we can elucidate a few basic features of the standard model that would otherwise remain cryptic. The scattered multiplets of fermions and their peculiar hypercharge assignments click together like pieces of a disassembled watch. And, most impressively, the disparate coupling strengths we observe at low energy are derived quantitatively from a single coupling--none other than our friend gs(MPlanckc2)--at the basic scale.
In all those previous considerations, gravity itself has figured only passively, as a numerical backdrop. It has supplied us with the numerical value of GN, but that's all. Now, in this concluding column, I examine how (and to what extent) gravity, as a dynamical theory, fits within this circle of ideas.
A lot of portentous drivel has been written about the quantum theory of gravity, so I'd like to begin by making a fundamental observation about it that tends to be obfuscated. There is a perfectly well-defined quantum theory of gravity that agrees accurately with all available experimental data. (I have heard two grand masters of theoretical physics, Richard Feynman and J. D. Bjorken, emphasize this point on public occasions.)
Here it is. Take classical general relativity as it stands: the Einstein-Hilbert action for gravity, with minimal coupling to the standard model of matter. Expand the metric field in small fluctuations around flat space, and pass from the classical to the quantum theory following the canonical procedure. This is just what we do for any other field. It is, for example, how we produce quantum chromodynamics from classical gauge theory. Applied to general relativity, this approach gives you a theory of gravitons interacting with matter.
More specifically, this procedure generates a set of rules for Feynman graphs, which you can use to compute physical processes. All the classic consequences of general relativity, including the derivation of Newton's law as a first approximation, the advance of Mercury's perihelion, the decay of binary pulsar orbits due to gravitational radiation, and so forth, follow from straightforward application of these rules within a framework in which the principles of quantum mechanics are fully respected.
To define the rules algorithmically, we need to specify how to deal with ill-defined integrals that arise in higher orders of perturbation theory. The same problem already arises in the standard model, even before gravity is included. There we deal with ill-defined integrals using renormalization theory. We can do the same here. In renormalization theory, we specify by hand the values of some physical parameters, and thereby fix the otherwise ill-defined integrals. A salient difference between how renormalization theory functions in the standard model and how it extends to include gravity is that, whereas in the standard model by itself we need only specify a finite number of parameters to fix all the integrals, after we include gravity we need an infinite number. But that's all right. By setting all but a very few of those parameters equal to zero, we arrive at an adequate--indeed, a spectacularly successful--theory. It is just this theory that practicing physicists always use, tacitly, when they do cosmology and astrophysics. (For the experts: The prescription is to put the coefficients of all nonminimal coupling terms to zero at some reference energy scale, call it e, well below the Planck scale. The necessity to choose an e introduces an ambiguity in the theory, but the consequences of that ambiguity are both far below the limits of observation and well beyond our practical ability to calculate corrections expected from mundane, nongravitational interactions.)
Of course the theory just described, despite its practical success, has serious shortcomings. Any theory of gravity that fails to explain why our richly structured vacuum, full of symmetry-breaking condensates and virtual particles, does not weigh much more than it does is a profoundly incomplete theory. This stricture applies equally to the most erudite developments in string and M theory and to the humble bottom-up approach used here. This gaping hole in our understanding of Nature is the notorious problem of the cosmological term. Perhaps less pressing, but still annoying, is that the above-mentioned ambiguity in the theory of gravity at ultralarge energy-momentum makes it difficult to address questions about what happens in ultraextreme conditions, including such interesting situations as the earliest moments of the Big Bang and the endpoints of gravitational collapse.
Nevertheless it makes good sense to take our working theory of gravity at face value and to see whether it fits into the attractive picture of unification we have built for the strong, weak, and electromagnetic interactions. Again, a crucial question is the apparent disparity between the coupling strengths. For the standard model interactions, logarithmic running of couplings with energy was a subtle quantum-mechanical phenomenon, caused by the screening or antiscreening effect of virtual particles. With gravity, the main effect is much simpler--and much bigger. Gravity, in general relativity, responds directly to energy-momentum. So the effective strength of the gravitational interaction, when measured by probes carrying larger energy-momentum, appears larger. That is a classical effect, and it goes as a power, not a logarithm, of the energy.
Now on laboratory scales, gravity is much weaker than the other interactions--roughly a factor 10-40. But we've seen that unification of the standard model couplings occurs at a very large energy scale, precisely because their running is logarithmic. And at this energy scale, we find that gravity, which runs faster, has almost caught up to the other interactions! Since the mathematical form of the interactions is not precisely the same, we cannot make a completely rigorous comparison, but simple comparisons of forces or scattering amplitudes give numbers like 10-2. Gravity is still weaker, but not absurdly so. Given the enormity of the original disparity, and the audacity of our extrapolations, this relatively slight discrepancy qualifies, if not quite as full success in achieving, at least as further encouragement toward trusting, the ideal of unification.
Let me summarize. Planck observed in 1900 that one could construct a system of units based on c, h-bar, GN. Subsequent developments displayed those quantities as conversion factors in profound physical theories. Now we find that Planck's units, although preposterous for everyday laboratory work, are very suitable for expressing the deep structure of what I consider our best working model of Nature, as sketched in this three-part series of columns. Planck proposed, implicitly, that the mountain of theoretical physics would be built to purely conceptual specifications, using just those units. Now we've taken the measure of Mount Planck from several different vantage points: from QCD, from unified gauge theories, from gravity itself--and found a consistent altitude. It therefore comes to seem that Planck's magic mountain, born in fantasy and numerology, may well correspond to physical reality. If so, then reductionist physics begins to face the awesome question, compounded of fulfillment and yearning, that heads this column.]]
Here is a quote from the June 2001 article, the first in Wilczek's popularly written series "Scaling Mount Planck".
<< We see that the question it poses is not, "Why is gravity so feeble?" but rather, "Why is the proton's mass so small?" For in natural (Planck) units, the strength of gravity simply is what it is, a primary quantity, while the proton's mass is the tiny number [1/13 quintillion].>>
You probably remember the famous Feynmann quote where he says every physicist should have 1/137 tacked up on the wall as a reminder to think about why that number is what it is. Wilczek is essentially saying the same thing about the number 1/(13 quintillion) except he calls it the square root of N which is confusing because there are so many N's. We had better give Wilczek's number a distinctive name, how about zeta?
Alpha, the fine structure constant (for reasons of laziness called "1/137"), is the value of Coulomb's constant in Planck units.
Zeta, the number Wilczek is talking about (for reasons of laziness we can call it "1/13 quintillion"), is the value of the proton mass in Planck units.
From my perspective two of the most interesting questions in physics are why these numbers are what they are and I have not the clue of a clue.
The URL for Wilczek's article is
http://www.physicstoday.org/pt/vol-54/iss-6/p12.html
and there is a more extensive quote, providing a little more context, further down the page.
The Planck units have been around since Planck proposed them in 1899.
'Über irreversible Strahlungsvorgänge'. Sitzungsberichte der Preußischen Akademie der Wissenschaften, vol. 5, p. 479 (1899).
The values he listed for them then are much the same as those in use today. It is their importance which, in recent decades, has been increasingly recognized by rank and file physicists, and this for reasons which remain essentially the same as when John Archibald Wheeler (one of Feynmann's mentors) called attention to Planck units in the Fifties.
In the excited talk surrounding Planck units today, it is often pointed out that the angular Compton wavelength (Arthur Holly Compton, 1923) of the Planck mass is equal to its Schwarzschild black hole halfradius (1916, the year Karl Schwarzschild died). One can easily get the misconception that the reason Planck units are essential to unifying gravity and the theory of matter is somehow new. The concepts are vintage pre-1925. The awareness in the community is new.
A smaller mass has a correspondingly larger Compton wavelength, representing greater difficulty in confining its location, but it has a smaller radius as a black hole. A mass smaller than Planck, compressed into a black hole as permitted by general relativity, would have a halfradius smaller than its (angular format) Compton wavelength. The black hole's event horizon would have such a small radius that quantum mechanics would not allow the mass to be localized within the confines of that radius. So the prevailing pictures of gravity (general relativity) and matter (quantum mechanics) are incompatible at Planck mass and below, simply because the Compton exceeds the Schwarzschild black hole halfradius. The incompatibility is based on pre-1925 vintage concepts and has been understood at least since the Fifties, by which time a few people had already begun looking for theories of quantum gravity.
An excellent discussion by the physicist John Baez of the Compton and the black hole halfradius and the conflict between theories of gravity of matter can be found on the web. What Baez calls the Schwarzschild radius is actually half the radius, GM/c2, and one needs to remember that the Compton he uses, h-bar/Mc, is the angular wavelength (the extent of a radian of phase and not a cycle). Anyway, if you set GM/c2 = h-bar/Mc and solve for M you get Planck mass.
http://math.ucr.edu/home/baez/planck/node2.html
You see me trying to understand why the Planck units are so interesting. I realized over 20 years ago they were interesting but I could not say exactly why. Now the web is full of explanations of why they are interesting. They are the rage with physicists and cosmologists. Which of these explanations is right? I can't say. Here is a more extensive quote from Franck Wilczek's June 2001 article. Again the URL is
http://www.physicstoday.org/pt/vol-54/iss-6/p12.html
<<Soon after Max Planck introduced his constant h-bar in the course of a phenomenological fit to the blackbody radiation spectrum, he pointed out the possibility of building a system of units based on the three fundamental constants h-bar, c, and GN. Indeed, from these three we can define a unit of mass (h-bar c/GN)1/2, a unit of length (h-bar GN/c3)1/2, and a unit of time (h-bar GN/c5)1/2—what we now call the Planck mass, length, and time, respectively. Planck's proposal for a system of units based on fundamental physical constants was, when it was made, formally correct but rather thinly rooted in fundamental physics. Over the course of the 20th century, however, his proposal became compelling. Now there are profound reasons to regard c as the fundamental unit of velocity and h-bar as the fundamental unit of action. In the special theory of relativity, there are symmetries relating space and time—and c serves as a conversion factor between the units in which space intervals and time intervals are measured. In quantum theory, the energy of a state is proportional to the frequency of its oscillations—and h-bar is the conversion factor. Thus c and h-bar appear directly as primary units of measurement in the basic laws of these two great theories. Finally, in general relativity theory, spacetime curvature is proportional to the density of energy—and GN (actually c4/GN ) is the conversion factor. If we accept that GN is a primary quantity, together with h-bar and c, then the enigma of [a number which is the square of 1/13 quintillion's] smallness looks quite different. We see that the question it poses is not, "Why is gravity so feeble?" but rather, "Why is the proton's mass so small?" For in natural (Planck) units, the strength of gravity simply is what it is, a primary quantity, while the proton's mass is the tiny number [1/13 quintillion].>>
It appears that a slow shift (beginning with active minorities such as cosmologists and string theorists) is underway in physics which will gradually cause the metric system to be abandoned in favor of practical versions of the Planck units. These will be conveniently scaled by powers of ten the way scientific units always are, and some will be ascribed conventional values to provide exact metric convertibility. This kind of thing (a change in the dominant system of units) does happen now and then. Judging from the 1899 paper in which he proposed them, Planck actually seems to have (been courageous or stubborn enough to have) had something like this in mind. But it won't happen quickly, I suspect. The inertia in systems of units is formidable.
Several of the essays at this site are addressed to two needs, the need for appropriatetly sized named power-of-ten-scaled versions (in the metric system this is generally addressed with the help of prefixes like giga and nano, and a few named auxilliaries like the liter) and the need for precise counterparts to the Planck quantities having truncated uncertainties and exact metric equivalents. Unless these two needs are addressed, the extreme sizes of the natural units together with the uncertainty in measuring G will retard acceptance.
The essays, some of which which include sample physics problems and examples in several Planckian systems, compiled in an attempt to "test drive" several versions of human-scale Planck units, are listed on the main planck.com index page.
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Thanks to those who have written to us with their reactions and ideas! Special thanks to Andrew Usher both for moral support and for catching several glaring lapses of algebra where I had failed to proof-read. We hope to hear from more readers interested in the Planck units--my address is leonard@endspamplanck.com. Particular acknowledgment and thanks are due to LBL physicist David Goldberg and UCB astronomer Alex Filippenko, both of whose comments and encouragement have been especially helpful. The astronomer John D. Barrows provided information on George Johnstone Stoney, whose 1874 paper "On the physical units of nature" anticipated Planck in identifying certain of the natural units. I also want to thank my editor Peter Saint-Andre because his interest in the Planckian Fables (he actually suggested the name to me) sparked a noticeable fraction of the writing now at this site. He demonstrates that one can be interested in both the Planck quantities and in verse translation of classical poetry.
A Human-scale Model of the Planck Quantities Analogs of the Planck units at human scale. The interrelationships between quantities are the same as those between the natural Planck units but their sizes are adjusted by powers of 105 to allow them to be visualized and experienced. Used thoughout in the earlier set of Planckian Fables.
Planckian Fables is on-line here. An earlier version is also posted at Monadnock Review. Also at this site are several sample chapters of Threepenny Planet, Nature's Proportions in Planck Terms, a book written in 1996.
1. The Planck Units
2. The Earth as a
Threepenny Planet--Gravitational Length
3.Measuring and Visualizing Planck
Quantities
4. Naming the Humanly Scaled Set,
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Two top metrologists have suggested abandoning the platinum-iridium prototype kilogram and replacing it with a definition based on a specified frequency. This is in line with resolution 7 of the 21st CGPM (1999 General Conference on Weights and Measures) which essentially called for a redefinition of the kilogram of this type. Writing in Metrologia, volume 36, NIST physicists Mohr and Taylor, the chair and previous chair of CODATA, offered this redefinition:
The kilogram is the mass of a body at rest whose equivalent energy equals the energy of a collection of photons whose frequencies sum to 135 639 274×1042 [cycles per second].
Mohr and Taylor suggested that the larger Planck's constant be made exactly equal to 2997924582/135 639 274 × 10-42 joule second. Long overdue. (In the Center's ton-pace-minute system we make h-bar exactly 10-40 tonpace minute.) For more on-line about their proposals, see their August, 2000 article in the Electronic Journal of Differential Equations: Quantum electrodynamics and the fundamental constants, by Peter J. Mohr and Barry N. Taylor.
http://ejde.math.swt.edu/conf-proc/04/m1/mohr.pdf
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