The blackbody radiation spectrum describes the emission of photons by electric charges that are not at absolute 0. There is a conservation of angular momentum involved, in that the photons are spin-1 objects, and the object emitting it will have to be able to shed rotation (if you'll allow the sloppy language) to emit the photon. The rotational states are quantized, so the emitted photons have quantized energies.

More broadly and more generalized, particles in bound states have quantized "allowable" states. Among those parameters confined by the boundaries of the system is the energy of a particle.

The blackbody spectrum is emitted by accelerating particles in a bound system - an atom for instance, or an ion in a plasma. The "allowed" energy of those particles in the bound system are quantized, and thus the photons they emit when changing energy states are discrete in frequency, corresponding to the difference in energy between the final and initial state of the particle's transition.


In the Heat Death universe, there are no particles left to be bound to any system. Thus the only boundary left is mere existence. The photons exist "in" the universe. So if the universe has a finite size, then that size quantizes the minimum frequency that can resonate, but it's not quantized unless the size of the universe is quantized. I.e. so long as the boundaries of the universe expand "smoothly" and not like a step function, then there is nothing left to quantize the bound states.

I suspect you'll dither over the propagation of photons in the EM fields being observation by the EM fields, and without that, there are no photons at all... but if those photons propagate through EM fields, then the expansion of spacetime will inevitably stretch their wavelengths to longer and longer lengths, corresponding to lower frequency and thus energy.

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There's an open question in physics about the Conservation of Energy and how it relates to the changing energy of photons as they propagate through spacetime. It is readily observable that photons decrease in energy as they travel through expanding spacetime, but it is not remotely clear where that energy goes.