Vacuum energy is zero, not infinite?

https://en.wikipedia.org/wiki/Vacuum_energy

Each point in space could act as an oscillator for an electromagnetic wave. In quantum mechanics, the lowest energy state of a harmonic oscillator has kinetic energy 1/2 hf, where f is the resonance frequency.

Does this imply that each point in space contains substantial energy, and consequently, energy density in space is infinite?

Another way to look at the problem is to imagine a cubic vessel whose walls are perfect mirrors. Photons can form standing waves there at an infinite number of frequencies f. If each frequency has some minimum energy, then the energy content is infinite.

A crystal of atomic matter can contain sound waves. Each atom can be treated like a little harmonic oscillator. The minimum kinetic energy of these atoms is significant. If space is analogous to the crystal, then the energy of the "atoms" in space is infinite.

There might be an error in these arguments. A quantum mechanical oscillator contains a massive particle, often an electron. Even if the oscillator would have zero kinetic energy, it would have the mass-energy of the particle.

In quantum mechanics, we must describe the particle as a wave. A wave packet which is confined in a small space must contain high frequencies. These frequencies mean a high momentum and a lot of kinetic energy.

An electron can only have p = 0 if its wave function is spread over an infinite volume of space.

If we take the electron out of the oscillator, then the oscillator can have zero kinetic energy.

What about photons and space as an oscillator? If there is no photon, then the oscillator is empty, and its total energy can be zero. Thus, the energy content of empty space is zero.

The electromagnetic wave function of empty space is everywhere zero. This is a valid solution of the classical electromagnetic wave equation.

On the other hand, if we have one electron confined into a small space, then the classical Dirac wave function must contain p != 0, that is, it must have kinetic energy.

Working with classical fields, we see that a typical harmonic oscillator really must have p != 0. But a field is allowed to be identically zero if it does not contain mass or energy.

What about a harmonic oscillator frame which only gains mass-energy when it is excited? The lowest energy state of such an oscillator is zero. Empty space can be seen as an oscillator frame which only oscillates if mass-energy is put into it.

In this blog we have suggested the following thought experiment: we have a tense string of zero mass. When we feed a wave into it, the string gains mass-energy. The gained mass-energy is the thing which the string is oscillating when the string moves.

Could this be analogous to empty space and electromagnetic wave energy which is fed into it? No. The mass-energy of the string is larger when we grow the amplitude of waves. Then those waves would move slower. The same is not true for electromagnetic waves.

Dark energy in cosmology might be true vacuum energy. A Higgs type field which is non-zero everywhere might contain energy.