Physical constants can take many dimensional forms: dimension refers to the constituent structure of all space (cf. volume) and its position in time; a dimensionless quantity is a quantity without an associated physical dimension. No association to time, length and always has a dimension of 1.

A. Dimensionless ratios:

The fine structure constant α, is the coupling constant characterizing the strength of the electromagnetic interaction (the parameter that describes coupling between light and relativistic electrons and is traditionally associated with quantum electrodynamics) that keeps the atom and the whole universe together. The name comes from the fact that it determines the size of the splitting or fine-structure of the hydrogenic spectral lines. The numerical value of α is the same in all systems of units.

Nobody knows where the alpha number for a coupling comes from, that's why it is 'magic': is it related to pi (as the reduced Plancks constant hbar) or perhaps to the base of natural logarithms? Attempts to find a mathematical basis for this dimensionless constant have continued. It's one of the greatest mysteries of physics: a magic number, we don't know what kind of dance to do on the computer to make this number come out; 29 and 137 being the 10th and 33rd prime numbers. The difference between the 2007 CODATA value for α and this theoretical value is about 3 x 10-11, about 6 * S.E. for the measured value. Today the most accurate value is 137,035999084, the three last unsure. In QED, using the quantum Hall effect or the anomalous magnetic moment of the electron (the "Lande g-factor", g), or with "quantum cyclotron" apparatus.

hbar*c = Plancks charge. So it is elemental charge quantized/Plancks charge, or Plancks charge is the ratio of elemental charge and √α? The Planck charge is α^{-1/2} or approx. 11.706 times greater than the elementary charge e carried by an electron.

Stable matter, and therefore life and intelligent beings, could not exist if its value were much different. If alpha were bigger than it really is, we should not be able to distinguish matter from 'ether' (the vacuum, nothingness, grid), (Einstein showed there was no ether). The fact that alpha has just its value 1/137 is certainly no chance but itself a law of nature.

Arnold Sommerfeld introduced the fine-structure constant in 1916.

B. Dimensional ratios: God's units, Unit of measurement, Planck units are physical fundamental units of measurement defined exclusively in terms of five universal physical constants listed below, in such a manner that these five physical constants take the value one when expressed in terms of these units. Planck units elegantly simplify particular algebraic expressions appearing in physical law. Planck units are only one system of natural units among other systems, but are considered unique in that these units are not based on properties of any prototype object, or particle (that would be arbitrarily chosen) but are based only on properties of free space. The constants that Planck units, by definition, normalize to 1 are the:

• Gravitational constant, G;

• Reduced Planck constant, ħ;

• Speed of light in a vacuum, c;

• Coulomb constant, (sometimes ke or k);

• Boltzmann's constant, kB (sometimes k).

Lev Okun in "Trialogue on the number of fundamental constants" Part I "Fundamental constants: parameters and units" 2002:

There are two kinds of fundamental constants of Nature: dimensionless (like α about 1/137) and dimensionful (c — velocity of light, hbar — quantum of action and angular momentum, and G — Newton’s gravitational constant). To clarify the discussion I suggest to refer to the former as fundamental parameters and the latter as fundamental (or basic) units. It is necessary and sufficient to have three basic units in order to reproduce in an experimentally meaningful way the dimensions of all physical quantities. Theoretical equations describing the physical world deal with dimensionless quantities and their solutions depend on dimensionless fundamental parameters. But experiments, from which these theories are extracted and by which they could be tested, involve measurements, i.e. comparisons with standard dimensionful scales. Without standard dimensionful units and hence without certain conventions physics is unthinkable.The third fundamental unit is the fundamental particles.

It is clear that the number of constants or units depends on the theoretical model or framework and hence depends on personal preferences and it changes of course with the evolution of physics. At each stage of this evolution it includes those constants which cannot be expressed in terms of more fundamental, hitherto unknown, ones. At present this number is a few dozens, if one includes neutrino mixing angles. It blows up with the inclusion of hypothetical new particles.

Usually combined with QED, perturbation, but also with gauge and Yukawa couplings in the quantum elemental particle framework with extensions.

Universal fundamental units of Nature.

The number three corresponds to the three basic entities (notions): space, time and matter. It does not depend on the dimensionality of space, being the same in spaces of any dimension. It does not depend on the number and nature of fundamental interactions. For instance, in a world without gravity it still would be three.

The three basic physical dimensions: L, T, M:

Planck 1899 Stoney 1870 [ ]= dimension

l = hbar/mc, l= e√G/c2 velocity [v] = [L/T]

t= hbar/mc2, t= e√G/c3 angular mom. [J] = [MvL] = [ML2/T]

m= hbar*c/G m = e/√G action[S] = [ET] = [Mv2T] = [ML2/T]

Stoney’s (without hbar) and Planck’s units are numerically close to each other, their ratios being √α . Originally proposed in 1899 by Max Planck, these units are also known as natural units because the origin of their definition comes only from properties of nature and not from any human construct. They are generally believed to be both universal in nature and constant in time. Now this picture is challenged.

Gauge couplings.

A coupling constant (g) is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian of a system can be separated into a kinetic and interaction part. The coupling constant determines the strength of the interaction part with respect to the kinetic part, or between two sectors of the interaction part (similarity - oscillations?). For example, the electric charge of a particle is a coupling constant, or mass.

If g << 1, it is weakly coupled with perturbation, QED, CKM

If g = /> 1, - it is strongly coupled. An example is the hadronic theory of strong interactions, non-perturbative methods , QCD.

This picture change a bit with the asymptotic degrees of freedom.

The reason this can happen, seemingly violating the conservation of energy, at short times, is the uncertainty relation, result -> quantization in the interaction picture/"virtual" particles going off the mass shell. \Delta E\Delta t >/= hbar This is the Planck scale, where particles vanish 'out of sight' into the virtual Dirac Sea. Gravity is usually ignored (non-renormalisability).

(Toms with asympt. degrees of freedom: If we let g denote a generic coupling constant, then the value of g at energy scale E, the running coupling constant g(E), is determined by E ( dg(E)/dE) = β(E, g) And it can be fused with gravity in Einstein.).

Such processes renormalize the coupling and make it dependent on the energy scale, μ at which one observes the coupling. The dependence of a coupling g (μ) on the energy-scale is known as running of the coupling, and the theory is known as the renormalization group. A beta-function β(g) encodes the running of a coupling parameter, g. If the beta-functions of a quantum field theory vanish, then the theory is scale-invariant. The coupling parameters of a quantum field theory can flow even if the corresponding classical field theory is scale-invariant.

If a beta-function is positive, the corresponding coupling increases with increasing energy. An example is quantum electrodynamics (QED). Particularly at low energies, α ≈ 1/137, whereas at the scale of the Z boson, about 90 GeV, one measures α ≈ 1/127. Moreover, the perturbative beta-function tells us that the coupling continues to increase, and QED becomes strongly coupled at high energy. In fact theoretically the coupling apparently becomes infinite at some finite energy. This phenomenon was first noted by Landau, and is called the Landau pole. Furthermore it was seen that the coupling decreases logarithmically, a phenomenon known as asymptotic freedom, see the Nobel Prize in Physics 2004. This theory was first properly suggested 2006 by Robinson and Wilczek. Confirmed this year by Toms.

found

**Fundamental units in old-fashioned quantum string theory**(QST): the quantity hbar is is also fundamental in the “positive” sense: it is the quantum of the angular momentum J and a natural unit of the action S. When J or S are close to hbar, the whole realm of quantum mechanical phenomena appears.

The speed of light c has already been implicitly used in order to talk about the area of a surface embedded in space-time. This fact allows us to replace hbar by a well defined length, λs, which turns out to be fundamental both in an intuitive sense and in the sense of S. Weinberg. Indeed, we should be able, in principle, to compute any observable in terms of c and λs. String theory (OST) only needs two fundamental dimensionful constants c and λs, i.e. one fundamental unit of speed and one of length.

The apparent puzzle is clear: where has our loved hbar disappeared? My answer was

then (and still is):it changed its dress! Having adopted new units of energy (energy being replaced by energy divided by tension, i.e. by length), the units of action (hence of hbar) have also changed.And indeed the most amazing outcome of this reasoning is that the new Planck constant, λs^2 is the UV cutoff.

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