What happens in intergalactic space

I might add some further notes to the actual material things existing in intergalactic space. One might wonder but the notion that there is space is already stating that there is more than nothing.

It implies that there is at least vacuum which is a pretty interesting thing on its own.

Quantum Mechanical harmonic oscillator

Maybe you know that the harmonic oscillator has energy levels

$E_n = \hbar \omega \left( n + \frac{1}{2}\right)$

and an astonishing result is that the lowest energy state is $E_0 = \frac{1}{2}\hbar\omega > 0$.

Quantum electrodynamical oscillator

Coming back to the vacuum, the situation is somewhat comparable. Considering Heisenberg's Principle of Uncertainty in its energy-time form,

$\Delta{t}\cdot\Delta{E} \geq \hbar$

we can see already that a state of a quantum system with definite zero energy for all times cannot exist, even though the expectation value might vanish.
Going more into detail, we see that the operator of the vector potential fullfills the wave equation

$\Delta{A_l} - \frac{1}{c^2}\partial_{tt}A_l = 0$

and a Helmholtz equation if one puts $\partial_{tt}\rightarrow{-\omega^2}$. This equation is usually tackled by separation of variables and after some math we arrive at a Hamilton

$H = \frac{1}{2}\sum_{\lambda}\left({p^2_\lambda+\omega_\lambda^2\lambda{q^2_\lambda}}\right)$

where now $\lambda$ accounts for some mode index. And here comes the magic. This is a description equation for harmonic oscillators! But here we run into a conceptional difficulty. The vacuum energy

$E_{vac} = \frac{1}{2}\sum_\lambda{\hbar\omega_\lambda}$

is infinitely large since there are infinitely many modes of the vacuum. But this is not very physical, so most of the time for calculations you just "leave out" this part.

Implications of a vacuum energy

In the case of different separated domains where you are able to allow a different different number of modes (e.g. via metal plates), this energy will be different for those domains resulting in a force which is the famous Casimir effect.

But vacuum energy has other implications. One hope it that it might some day explain the cosmological constant in terms of a unified field theory.

So, I hope, I could convince you that "empty" might be much more one would expect :)

Sincerely

Robert

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