  The offsets between the radial velocities of the rotational transitions of
carbon monoxide and the fine structure transitions of neutral and singly
ionized carbon are used to test the hypothetical variation of the fine
structure constant, alpha. From the analysis of the [CI] and [CII] fine
structure lines and low J rotational lines of 12CO and 13CO, emitted by the
dark cloud L1599B in the Milky Way disk, we find no evidence for fractional
changes in alpha at the level of |$\Delta \alpha/\alpha$| < 3*10^-7. For the
neighbour galaxy M33 a stringent limit on Delta alpha/alpha is set from
observations of three HII zones in [CII] and CO emission lines: |$\Delta
\alpha/\alpha$| < 4*10^-7. Five systems over the redshift interval z = 5.7-6.4,
showing CO J=6-5, J=7-6 and [CII] emission, yield a limit on |$\Delta
\alpha/\alpha$| < 1.3*10^-5. Thus, a combination of the [CI], [CII], and CO
emission lines turns out to be a powerful tool for probing the stability of the
fundamental physical constants over a wide range of redshifts not accessible to
optical spectral measurements.
