# 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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