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Say I have a theory with only one (energy) scale, e.g. one given by the fundamental constants

$$\epsilon=\sqrt{\dfrac{\hbar c^5}{G}}.$$

In this case, where I can't compare to something else, is there a way to argue that

$$\epsilon<\epsilon^2<\epsilon^3<\dots\ ?$$

By that reasoning, can there be a (field?) theory, where values are obtained from some expansion like in a path integral (which needs a hierarchy of that sort)?

If you really only need/have a theory with $\hbar, c, G$, how can energies like particle masses be deduced from the theory (instead of being experimental input)? And then if, at best, the theory predicts some mass of a particle $\phi$ to be $m_\phi=a_\phi\dfrac{\epsilon\ }{\ c^2}$, then the number $a_\phi$ must have some geometrical meaning, right?

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I just want to note that, for the record, the Planck mass of $\epsilon / c^2$ is equal to $10^{22}$ times the mass of the electron. Obviously this is problematic since your proposal now requires $a_{\phi} \ll 1$. I feel better when my geometry factors are between 1 and 10. –  AlanSE Jul 5 '12 at 19:21
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4 Answers 4

You can't do it for real in quantum field theory, there are always adjustible parameters. The reason is that quantum field theory doesn't have a fundamental length, it is defined on the continuum, so it can always be rescaled. But if you have a quantum gravity theory that reduces to quantum field theory at energies less than some large energy, you can get exponentially small masses out without putting them in using coupling constant running. If you say that the coupling of a confining theory is something small (but not absurdly small) at the Planck scale, the mass of the confined particles is extremely small, and the confinement radius is enormous. This is how the proton mass is determined.

In principle, you could have one confining theory generate condensates that then break other symmetries, and so on, and give masses to all particles using only such a mechanism. This is called "technicolor", and it would predict the masses of the low-lying particles from dimensionless coupling constants. But the scaling that produces a continuum means that the full relations between all constants can't be predicted without a theory of quantum gravity, which breaks the continuum limit in field theory.

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This is a very good question. I think there is no quantum field theory which predicts all particle masses.

Masses (measured in Planck unit) are real numbers. The real numbers are NOT predictable, just like the radius of the orbit of Earth moving around Sun (measured in Planck unit) is not predictable. So the real fundamental constants are NOT predictable, and have to be inputted into the theory by hand.

However, those unpredictable real quantities have a predictable property: they are time dependent (but may change very very slowly). This is also just like the average radius of the orbit of Earth moving around Sun. So masses (measured in Planck unit) are not predictable, but may be time dependent.

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Theories don't predict units unless you put units in. A theory which predicts the masses of the fundamental particles would actually only predict the mass ratios $a_\phi$. Presumably they would emerge as eigenvalues of some operator, or perhaps as the zeros of some complicated function.

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Here is my take on it from "history" point of view:

  • With $c$ we understood that there is "no difference" between space and time and we started to use same units to measure them. Same thing for energy and momentum, magnetic and electric fields, e.t.c.

  • Then $\hbar$ appeared and we realized that we could measure energies, momenta, distances and time intervals with same units (like $GeV$s). That the inly fundamental unit we have, right? All other units are just introduced for convenience and can be reduced to $GeV$s.

  • Now we want to include $G$ into the picture. But that would mean that we will leave the last unit we had -- we will measure all the energies in Planck units. Meaning that we will work just with "pure" dimensionless numbers.

So your theory with only $c,\hbar,G$ parameters should be essentially a pure mathematical construction, that gives you dimensionless numbers "in Planck units".

These numbers can:

  1. have some geometric nature (as you said),

  2. be solutions or eigenvalues of some functions or operators (as David Zaslavsky said),

  3. be just "accidental" numbers like Earth radius and even drift with time (as Xiao-Gang Wen said)

Or some combination (or none) of the above.

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