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A photon with a frequency of less than 1 Hz would have an energy below $$ E = h\nu < 6.626×10^{−34} \;\rm J $$ which would be less than the value of Planck's constant. Do photons with such a low energy exist and how could they be detected? Or does Planck's constant give a limit on the amount of energy that is necessary to create a single photon?

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The Planck constant is $h=6.626\times 10^{-34}\;\rm J\,s$ (joule seconds), you cannot compare it to an energy which is measured in Joules - this is the flaw in your argument. To answer the question: Such low energy photons can exist in principle, however the question is how to actually generate them. I'd propose to take a possibly low energy photon and redshift it (check Doppler effect). It will, however, be in very red radio range and therefore hard to detect.

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    $\begingroup$ The OP has multiplied $h$ by the frequency in Hz (which in this case is 1Hz) so will definitely get an energy in $J$, so they can be compared $\endgroup$ – danimal Mar 17 '15 at 11:59
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    $\begingroup$ the energy of a 1Hz photon is $6.6\times 10^{-34}J$, numerically equal to the value for $h$ in SI units, and this is a tiny energy $\endgroup$ – danimal Mar 17 '15 at 12:38
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    $\begingroup$ Sorry, the second half of the answer is silly. There is absolutely no problem in producing photons of less-than-one-hertz frequency. Just oscillate with a charge at a frequency less than one hertz and be sure that lots of electromagnetic waves are produced. The waves are composed of many photons - at such low frequencies, we actually produce a huge number of photons. What is hard is to "count" them or experimentally prove that the number is integer - it is not hard to produce them. $\endgroup$ – Luboš Motl Mar 17 '15 at 12:47
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    $\begingroup$ danimal: What does it mean physically if two quantities are equal numerically which are of different type? SI units are entirely arbitrary, therefore it is only possible to compare quantities in the same units (or dimensionless quantities without any units at all). $\endgroup$ – Photon Mar 17 '15 at 13:39
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    $\begingroup$ The energy is obviously tiny compared to, say, energies within the optical spectrum. But it doesn't make any sense to compare this energy to the Planck constant (exactly for the reason you described) - and that's what the OP did. $\endgroup$ – Photon Mar 17 '15 at 14:07
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The frequency $\nu$ is in seconds$^{-1}$, which is purely human-based unit having a relation to rotation of the Earth. Thus no reason why 1 Hz was a limit. Planck $h$ value is also not massless unit and it's value has relation to SI system.

Existence: while I don't see a principal reason for non-existence of such a photon, neither I see a physical process, that would generate such a radiation. You would need some anthena of length $\sim 10 ^8$ meters and some process that would correlate a charge across such a distance.

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You don't need a long antenna to radiate at 1 Hz. You need a long antenna to radiate efficiently at 1 Hz. The efficiency is proportional to the cube of the length of the antenna in wavelengths (look up {electrically} 'small antennas'). A big 1 Hz current in a short wire will radiate very little power, but 1 photon a second would be 6.6e-34 Watts, so the numbers may be in favour of radiation. 1 Hz ==> 3e8 m wavelength, so 1 m long wire antenna may have efficiency of order 3e-26, which looks like lots of photons per watt into the antenna (most of the watt goes into resistive, dielectric and magnetic losses in the matching circuit or the generator).

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The shortest answer for why such photons "exist" is that whether a given photon qualifies depends on your rest frame. Take your favourite high-frequency photon in the universe, say of frequency $n$ Hertz in your rest frame. With a Lorentz boost $-\beta$, I multiply this by $\gamma (1-\beta)=\cosh\phi-\sinh\phi=e^(-\phi)$ with $\phi$ the rapidity $\phi=\mathrm{artanh}\beta$. Setting $\phi>\ln n$ (equivalently, $\beta>\frac{1-n^{-2}}{1+n^{-2}}$) does the job.

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