Very fundamental and strong question, I will try to write my thoughts, I do not claim that I can answer such a big question. First of all lets examine what is the meaning of a dimensionless constant: We (humans) have defined some physical quantities by relating each other in such a way that, sometimes, a specific function of them can be dimensionless. What does it mean? That our local universe should also posse such a dimensionality and in advance a constant in its evolution one? I doubt a lot about. So, lets go to the second class of dimensionful constants. What exactly are those constants? Roughly speaking they are just the appropriate coefficient of analogy at our model involved for a specific phenomenon. We make our experiments and see that output Y is a , say, quadratic function of input X. We do our regression and find the beta coefficients b0, b1,b2 of the relation Y=b0+b1X+b2X2. Now the coefficients beta should be of an appropriate dimension and "constant". But who really guarantees us that our simple regression (given the data of , how much, three centuries?) has constant coefficients? Nobody. We should had data in the cosmological time range in order to argue that all those constants do not change.

Of course this is a fundamental question we normally hardly bother. Going by the reality, all these constants are experimentally observed  like charge of an electron, mass, c, h cross, G etc. These may change - there is no theoretical value for these constants. A constancy of these means we assume that interactions always remain unaltered over a period of time - do they - we do not know. Yes they do not change in some generations that is our experience.  If these change with time that would mean a dynamism of the universe and self generating or decaying. If they change it would be interesting to observe the rate of change of these as then we can predict future singularities.

However there are other constants which are mathematical notations like pi and natural base e. These can not change.

Development of unified field theories (e.g., the Standard Model or theories of Quantum Gravity aimed at unifying General Relativity and quantum mechanics) brought up the issue of temporal and spatial variations of fundamental constants. Astrophysical data suggesting that such changes may have already occurred on the cosmological timescale have also brought attention to the search. The analysis relies on small spectral shifts in the absorbtion lines of metal ions in interstellar clouds that are being observed in the light emitted by distant quasars. This matter is of utmost importance for quantum metrology, as there is a program to link basic units of the International System of Units (SI) with fundamental constants. Rapid progress development of atomic frequency standards has recently made it possible to perform laboratory searches for present-day variation of fundamental constants. The availability of highly accurate, reliable, and robust measurement technologies is very important to research and development, manufacturing quality control and industry. It is estimated that novel quantum sensors and high-precision clocks may reveal a drift of the fine structure constant and allow achieving geodesy precisions in the millimeter range, while testing concepts of Einsteinian gravity.

The fundamental laws of physics, as we presently understand them, depend on about 25 parameters, such as Planck's constant h, the gravitational constant G, and the mass and charge of the electron. It is natural to ask whether these parameters are really constants, or whether they vary in space or time.

Interest in this question was spurred by Dirac's large number hypothesis. The "large number" in question is the ratio of the electric and the gravitational force between two electrons, which is about 10^40; there is no obvious explanation of why such a huge number should appear in physics. Dirac pointed out that this number is nearly the same as the age of the Universe in atomic units, and suggested in 1937 that this coincidence could be understood if fundamental constants — in particular, G — varied as the Universe aged. The ratio of electromagnetic and gravitational interactions would then be large simply because the Universe is old. Such a variation lies outside ordinary general relativity, but can be incorporated by a fairly simple modification of the theory. Other models, including the Brans-Dicke theory of gravity and some versions of superstring theory, also predict physical "constants" that vary.

Over the past few decades, there have been extensive searches for evidence of variation of fundamental "constants." Among the methods used have been astrophysical observations of the spectra of distant stars, searches for variations of planetary radii and moments of inertia, investigations of orbital evolution, searches for anomalous luminosities of faint stars, studies of abundance ratios of radioactive nuclides, and (for current variations) direct laboratory measurements.

One powerful approach has been to study the "Oklo Phenomenon," a uranium deposit in Gabon that became a natural nuclear reactor about 1.8 billion years ago; the isotopic composition of fission products has permitted a detailed investigation of possible changes in nuclear interactions. Another has been to examine ratios of spectral lines of distant quasars coming from different types of atomic transitions (resonant, fine structure, and hyperfine). The resulting frequencies have different dependences on the electron charge and mass, the speed of light, and Planck's constant, and can be used to compare these parameters to their present values on Earth. Solar eclipses provide another sensitive test of variations of the gravitational constant. If G had varied, the eclipse track would have been different from the one we calculate today, so the mere fact that a total eclipse occurred at a particular location provides a powerful constraint, even if the date is poorly known.

So far, these investigations have found no evidence of variation of fundamental "constants." The current observational limits for most constants are on the order of one part in 10^10 to one part in 10^11 per year. So to the best of our current ability to observe, the fundamental constants really are constant.

------------------------------------------------------------------------

http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/constants.html

 However, more recent observations of distant galaxies suggest that billions of years ago the strength of the electromagnetic interaction was different at opposite ends of universe. That's the surprising conclusion of a group of physicists in Australia, who have studied light from ancient quasars. The researchers found that the fine-structure constant has changed in both space and time since the Big Bang. the fine-structure constant, known as the alpha constant, actually varies throughout the universe. In one direction, the constant seemed to grow larger the farther astronomers looked; in another direction the constant took on smaller values with greater distance. If confirmed this represent a challenge to astrophysicists and particle physicists for an explanation. However, many researchers remain skeptical.

The problem is that the data under investigation could suffer from fairly large statistical fluctuations that could produce fake trend. There are much too controversy in the scientific community for a solid conclusion.The general view is that unless results are confirmed independently we should be careful about the results. Therefore,the result that billions of years ago alpha was about one part in 100,000 smaller than it is today remains a controversial result that is not accepted by all physicists.

Laser manipulation of atomic and molecular internal and external degrees of freedom (quantum engineering) enables physicists to prepare atoms (and soon molecules) in well defined states and even cool them down to the quantum limit. Quantum engineering of matter produced new breakthroughs in atomic physics and quantum optics. It also opened new horizons in quantum metrology for testing of fundamental physical laws and achieving unparalled accuracies in measurements on space and time. The advent of late quantum technologies has evolved to high precision atomic clocks with applications in geodesy, inertial sensing, satellite navigation (GPS) and laser ranging. Matter-wave interferometry has proven itself to be one of the most promising candidates for next generation quantum sensors which are at the basis of tomorrow’s most advanced technology. Thus state of the art quantum engineering of light fields yielded extremely accurate measurements on space, which open new dimensions towards the realization of gravitational wave detectors. These devices exhibit sensitivies for relative length changes at a level of 10^(-22), where direct measurement of gravitational waves is expected to occur in the near future. There is a constant preoccupation in the world scientific community to convert any performed measurements into frequency measurements as the second is the most precisely realised unit. Recent progress in the techniques of laser cooling and trapping of atoms, realization of lasers with long coherence time (1 s) have truly unleashed the potential of the optical clocks which exhibit superior performances. 

https://www.ptb.de/cms/fileadmin/internet/publikationen/mitteilungen/2009/PTB-Mitteilungen_2009_Heft_2_en.pdf

@Neven

I don't agree with your statement that because particle physicists usually work in units c=h_bar=G_N=1, particle physicists do not care if their was time variations in these constants. If confirmed, time variations in these constants could ultimately  lead to new physics and reshape physicists' understanding of cosmology on all cosmological time scales. It will pose a serious challenge to astronomers, physicists (including particle physicists) for an explanation.

Probably so, but Quantum metrology does care about time variations and extremely high-precision spectroscopy also does, and I refer to the most accurate measurements on the gyromagnetic ratio of the electron, of leptonic magnetic moments, mass spectrometry and quantum physics using particle traps, as one of the most accurate tool to investigate physics under minimum perturbation conditions thus enabling manipulation of individual quantum systems (quantum engineering). And I mean testing the basis of quantum physics and the transition between microscopic and macroscopic.

@Bogdan

Thank you for for bringing us up to date with the extremely accurate measurements of state of the art quantum engineering. This will certainly have a big impact on the accuracy and interpretation of astronomical data, it will remove any systematic biases which can mimic the appearance of varying alpha.

Dear Isaam,

I would agree with you if you defined a dimensionful fundamental constant as a measurable quantity. For Duff's main point is that experiments measure only dimensionless quantities. That is, in a measurement one compares an unknown dimensionful quantity with one or more known dimensionful quantities and, as a result, one obtains just a number.

For example, one may define the speed of light as the number of

 wavelengths of the orange-red emission line of the krypton-86 atom multiplied by 1 650 763.73 that a light ray travels during the time of 1/11.317 halflives of the beta+ decay of magnesium-22 into sodium-23. It is quite possible (e.g., due to some unknown physical law that causes a variation of the electro-weak coupling constant) that such a measurement would give different results at different times, so the speed of light defined in this way may vary with time.

However, if you define the speed of light as a fundamental constant as it was done in Particle Data, it will remain constant forever.

Dear All,

Based on several assumptions to deduce a cosmological model with three fundamental constants including the speed of light in vacuum, the Planck constant, and the gravitational constant, along with the dimensionless electroweak coupling constant turned into functions of the gravitational potential. Initial research of this model has indicated solutions to avoid the singularity in both special relativity and general relativity.

This paper is under open review in RG, any comments will be appreciated.

Best Regards

Article Functions of the Gravitational Potential

Dear Issam

sorry for misspelling your name in my previous post

of elementary particles N in the visible universe is proportional with square cosmic time t . This hypothetical be cause gravitational constant G decreasing inversely with the growth of time.