Can cosmological redshift only decrease photons' energy by increments (that correspond to Planck's constant)?

I have read this question:

The equations that govern quantum mechanics predict that the angular momentum (that is, spinning or orbiting) in a system can't take on any value, but instead come in lumps. The "reduced Planck constant" ℏ=h/2π is the size of a lump of angular momentum. An electron orbiting a nucleus can do so with zero angular momentum, with angular momentum ℏ, with angular momentum 2ℏ, and so on, but for a hypothetical value like 1.37ℏ there are no solutions to the electron's equations of motion. Light also carries angular momentum: any process that emits or absorbs a photon must involve an angular momentum change of ℏ (or a larger integer multiple of ℏ).

What exactly is Planck's constant? how did they calculate it?

Now as far as I understand it, spacetime is continuous. When cosmological redshift affects light's wavelength, that is, the wavelength is stretched, the photons building it up will lose energy. But since spacetime is continuous, this energy loss can come in arbitrarily small amounts, and yes, smaller then Planck's constant.

So cosmological redshift can transform theoretically a photon to certain energy levels that cannot be expressed by the multiples of Planck's constant.

So basically, if spacetime expansion is continuous (not-quantized), then it (through cosmological redshift) can transform photons into energy levels that cannot be expressed as multiples of Planck's constant, but if spacetime expansion is quantized (so spacetime can only expand in increments, such as units of Planck length), then the wavelength can only be stretched in increments so that the energy of the photons always is expressible as multiples of Planck's constant.

Question:

1. Can cosmological redshift only decrease photons' energy by increments (that correspond to Planck's constant)?
• Planck's constant has units of action, not units of energy, so it is unclear what you mean by decreasing energy by "increments of Planck's constant". Commented Aug 9, 2021 at 21:58
• Maybe I misunderstand the question, but doesn't cosmological redshift affect the frequency of the photons? And while $\hbar$ is of course a constant, I would not have thought that the $\omega$ in $E=\hbar\omega$ is discrete. Commented Aug 9, 2021 at 22:34
• @ACuriousMind thank you I edited. Commented Aug 10, 2021 at 1:14
• I don't see how your edit addresses the problem - what does "correspond" now mean? The only way a number an "correspond" to another is via an equation, so write down the equation you're thinking of! Commented Aug 10, 2021 at 7:51

3 Answers

As mentioned in a comment, Planck's constant governs the quantization of action (or angular momentum, which has the same units), and not energy. Photon energies can come in any amount, and indeed are observed to come in any amount. Where quantization comes in is in the relationship between energy and frequency: photons of a particular energy must have a particular frequency, and vice-versa. So as they are redshifted and decrease in frequency, they continuously decrease in energy so as to maintain the relation $$h = \frac{E}{f}$$

• Thank you, are you saying that because of redshift, the energy of the photon can decrease by amounts that correspond to changes smaller then Planck's constant? Commented Aug 10, 2021 at 1:23
• Yes. Planck's constant governs only the product of frequency and energy, not either one on its own. Commented Aug 10, 2021 at 11:37

While not a particularly conventional argument, your question highlights the fact that General Relativity (GR) and Quantum Mechanics (QM) are not fully compatible.

If (when) a quantum theory of GR is realized, it may well suggest that the "stretching" of spacetime may indeed be quantized. However, in its current form, GR is more closely aligned with classical theories and the redshift is allowed to to occur on a continuous basis.

When cosmological redshift affects light's wavelength, that is, the wavelength is stretched, the photons building it up will lose energy. But since spacetime is continuous, this energy loss can come in arbitrarily small amounts, and yes, smaller then Planck's constant.

So cosmological redshift can transform theoretically a photon to certain energy levels that cannot be expressed by the multiples of Planck's constant.

The spin of a photon can only be in one of two possible states and it has no effect on the energy level of the photon.

The redshift has no effect on spin.