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 * $$\alpha = \frac{e^2}{4\pi\varepsilon_0\hbar^2c}= \frac$$,

Planck units and the invariant scaling of nature
Some theorists (such as Dirac and Milne) have conjectured that physical "constants" might actually change over time and they have devised cosmologies that allow for these changes (e.g. Dirac's Large Numbers Hypothesis). Such cosmologies have not gained mainstream acceptance and yet there is still considerable, scientific interest in the possibility that physical 'constants' might change. It is a possibility that challenges even the most brilliant minds and it raises questions that seem to admit different answers. One such question is this: if a physical constant such as the speed of light did change, would we be able to notice it? George Gamow argued in his book Mr. Tompkins in Wonderland that any change in a physical constant, such as the speed of light in a vacuum, would result in obvious changes. However, as shown in Table 2, Planck units are derived from ratios of physical constants. Planck units therefore cannot be used to measure changes in those constants since the units themselves would change. If for example the speed of light c, were somehow suddenly cut in half and changed to c/2 then the Planck length would increase by a factor of 2$\sqrt{2}$ and Planck time would increase by a factor of 4$\sqrt{2}$. Measured in Planck units therefore, the new speed of light would be measured as 1 new Plank length per 1 new Planck time - which is no different to the old measurement. This unvarying aspect of the Planck scale, or of any other set of natural units, leads many theorists to conclude that a change in physical constants can only ever show up as a change in so called 'dimensionless constants of nature'. One such dimensionless constant is the Fine structure constant. The fine structure constant (denoted α) relates the size of atoms - approximately the Bohr radius (denoted a0) - to physical constants and therefore to Planck units in this way:


 * $$a_0 = {{4\pi\varepsilon_0\hbar^2}\over{m_e e^2}}= {{m_P}\over{m_e \alpha}} l_P $$,

where me is the mass of the electron. According to this equation, a change in the Bohr radius can be expressed entirely as a change in the fine structure constant. The Planck length therefore can be used to measure a relative change in the the size of atoms even though it can't measure a change in the other physical constants represented in the equation. John Barrow has addressed the issue in these terms:

There are some experimental physicists who think they have in fact measured a change in the fine structure constant and this has intensified the debate about the measurement of physical constants. According to some theorists there are some very special circumstances in which changes in the fine structure constant can be measured as a change in physical constants. Others however reject the possibility of measuring a change in physical constants under any circumstance. The difficulty or even the impossibility of measuring changes in physical constants has led some theorists to debate with each other whether or not a 'physical' constant has any practical significance at all and that in turn leads to questions about which physical constants are meaningful.

When measuring a length with a ruler or tape measure, one is actually counting tick marks on a given standard, i.e., measuring the length relative to that given standard; the result is a dimensionless value. It is no different for physical experiments, as all physical quantities are measured relative to some other like-dimensioned values. If all physical quantities (masses and other properties of particles) were expressed in terms of Planck units, those quantities would be dimensionless numbers (mass divided by the Planck mass, length divided by the Planck length, etc.) and the only quantities we would measure when observing nature or conducting experiments would be dimensionless numbers. See Duff (2004) and section III.5 (by Duff alone) of Duff, Okun, and Veneziano (2002).

We can notice a difference if some dimensionless physical quantity such as α or the proton/electron mass ratio changes; either change would alter atomic structures. But if all dimensionless physical quantities remained constant (this includes all possible ratios of identically dimensioned physical quantities), we could not tell if a dimensionful quantity, such as the speed of light, c, had changed. And, indeed, the Tompkins concept becomes meaningless in our existence if a dimensional quantity such as c has changed, even drastically.

If the speed of light c, were somehow suddenly cut in half and changed to c/2 (but with all dimensionless physical quantities continuing to remain constant), then the Planck length would increase by a factor of 2$\sqrt{2}$ from the point-of-view of some unaffected "god-like" observer on the outside. But then the size of atoms (approximately the Bohr radius) are related to the Planck length by an unchanging dimensionless constant:


 * $$a_0 = {{4\pi\varepsilon_0\hbar^2}\over{m_e e^2}}= {{m_P}\over{m_e \alpha}} l_P $$

Atoms would then be bigger (in one dimension) by 2$\sqrt{2}$, each of us would be taller by 2$\sqrt{2}$, and so would our meter sticks be taller (and wider and thicker) by a factor of 2$\sqrt{2}$ and we would not know the difference. Our perception of distance and lengths relative to the Planck length is logically an unchanging dimensionless constant.

Moreover, our clocks would tick slower by a factor of 4$\sqrt{2}$ (from the point-of-view of this unaffected "god-like" observer) because the Planck time has increased by 4$\sqrt{2}$ but we would not know the difference. (Our perception of durations of time relative to the Planck time is, axiomatically, an unchanging dimensionless constant.) This hypothetical god-like outside observer might observe that light now travels at half the speed that it used to (as well as all other observed velocities) but it would still travel 299&thinsp;792&thinsp;458 of our new meters in the time elapsed by one of our new seconds ($c/2$ $4\sqrt{2}/2\sqrt{2}$ continues to equal 299&thinsp;792&thinsp;458 m/s). We would not notice any difference.

This contradicts what George Gamow wrote in his book Mr. Tompkins in Wonderland; where he suggested that if a dimension-dependent universal constant such as c changed, we would easily notice the difference.