## 1 Introduction

The fundamental laws of particle physics, in our current understanding, depend on 28 constants including the gravitational constant, *G*, the mass, *m*_{e}, and charge, *e*, of the electron, the masses of six quarks, *m*_{u}, *m*_{d}, *m*_{c}, *m*_{s}, *m*_{t}, and *m*_{b}, the Planck constant, ℏ, the Sommerfeld constant α, the coupling constants of the weak, *g*_{w}, and strong, *g*_{s}, interactions, etc. The numerical values of these constants are not calculated within the Standard Model and remain, as Feynman wrote about the fine structure constant α in 1985, “one of the greatest mysteries of physics” [1]. However, it is natural to ask whether these constants are really constants, or whether they vary with the age of the universe, or over astronomical distances.

The idea that the fundamental constants may vary on the cosmological time scale has been discussing in different forms since 1937, when Milne and Dirac argued about possible variations of the Newton constant *G* during the lifetime of the universe [2, 3]. Over the past few decades, there have been extensive searches for persuasive evidences of the variation of physical constants. So far, there was found no one of them. The current limits for dimensionless constants such as the fine structure constant, , and the electron to proton mass ratio, , obtained in laboratory experiments and from the Oklo natural reactor are on the order of one part in 10^{15} − 10^{17} [4-6] and one part in 10^{14} − 10^{16} [7-9] per year, respectively. The detailed discussion of ideas behind laboratory experiments can be found in a review [10].

Assuming that the constants are linearly dependent on the cosmic time, the same order of magnitude constraints on the fractional changes in and in are stemming from astronomical observations of extragalactic objects at redshifts [11-15]. Less stringent constraints at a percent level have been obtained from the cosmic microwave background (CMB) at [16-18] and big bang nucleosynthesis (BBN) at [19, 20]. We note that space and/or time dependence of α based on optical spectra of quasars and discussed in the literature [[21], and references therein] is still controversial and probably caused by systematic effects since independent radio-astronomical observations, which are more sensitive, show only null results for both and [22, 23].

Surprisingly, it looks as if the Einstein heuristic principle of local position invariance (LPI) — *the outcome of any local non-gravitational experiment is independent of where and when in the universe it is performed* — is valid all over the universe, i.e., at the level of neither α no μ deviate from their terrestrial values for the passed 10^{10} yr. In the Milky Way, it was also found no statistically significant deviations of from zero at even more deeper level of [24-26].

However, the violation of the LPI was predicted in some theoretical models such as, for example, the theory of superstrings which considers time variations of α, *g*_{w}, and the QCD scale Λ_{QCD} (i.e., μ since ) and thereby opening a new window on physics beyond the Standard Model [[27], and references therein]. If the fundamental constants are found to be changing in space and time, then they are not absolute but dynamical quantities which follow some deeper physical laws that have to be understood. Already present upper limits on the variation of the fundamental constants put very strong constraints on the theories beyond the Standard Model [[28], and references therein]. This motivates the need for more precise laboratory and astronomical tests of the LPI. Of course, there are also other attempts to look for the new physics. For example the electric dipole moments (EDMs) of the elementary particles are very sensitive to the different extensions of the Standard Model. Present limit on the EDM of the electron significantly constrains supersymmetrical models and other theories [29, 30].

In this review we will consider tests of LPI which are based on the analysis of microwave and submillimeter1 astronomical spectra and which are essentially more sensitive to small variations in α and μ than the test based on optical spectral observations of quasars.