Is it correct to say that when we use ladder operators $\hat{a}$ and $\hat{a}^\dagger$, we are describing the states in the system using these as variables instead of the usual $\mathbf{r}$ and $\mathbf{p}$ ?

I.e. while I know the expressions for the ladder operators, I am curious on what kind of property or symmetry it exploits such that it can describe states very nicely in terms of using these operators to move up or down the states in the quantum system?

Is using ladder operators simialr to perform some kind of coordinate transformation, which is why it makes the maths nicer?

Still I am no very familar with the expression $-i\hbar\frac{1}{V}$ although I have seen simailr things like $-i\hbar\nabla$, $i\epsilon\hat{L}_k$, so I felt I like I am oversimplfying something in my above picture

NB (off topic): Because I am such a visual learner in steroids, I have word versions of the pictures I am drawing. For example the above statement is actually the same as an illustration if one use it to draw a picture

This is how I tried to gain some physical intuition by essentially reducing eveyr problem into a geometry problem, along with doing more problems in textbooks an…

@ACuriousMind @Slereah
I learnt from today's class that the vacuum state is defined as $\hat{a}(\mathbf{k}|0\rangle)=0, \forall k$ whic has zero momentum and total energy. The professors said that it is an important state when things start interacting

And also in the complex field example of the harmonic oscillator in QFT, you have two different lowering ladder operators that can anihilate the ground state into the vacuum state

Is this vacuum state important in virtual particle related phenomenon such as casimir effect?

@ACuriousMind
Probably influenced by popsci magazines often saying how virtual particle pairs pop in and out of existence thus the vacuum is not actually empty, and that gives a pressure. And the casmir effect if I recall, the region between the plates exclude some of the modes thus the pressure outside the region pushses the plates together

Given now I finalyl have some beginning understanding of QFT, I think I should update myself on what is the rigoruos version of the popsci's "virtual particles pop in and out of existence in the vacuum"

$\hat{H}_{ren}=\hat{H}-\langle 0| \hat{H} | 0\rangle = \int d^3\mathbf{k} \omega_{\mathbf{k}\hat{a}^\dagger(\mathbf{k})\hat{a}(\mathbf{k})+\frac{1}{2}\delta^3(0)-+\frac{1}{2}\delta^3(0)}=\int d^3\mathbf{k} \omega_{\mathbf{k}\hat{a}^\dagger(\mathbf{k})\hat{a}(\mathbf{k})}$

We are also being introduced the concept of renormalisation, which one professor said it can be interpreted as after summing up all the oscillators, (thus the ZPE of them will add up to give a diverging term), and then pull this diverging ZPE like term down so it form the baseline of measuring the energy and we then set t…

The professor then elaborated on that the reason why the total energy is not globally defined on the spacetime manifold is because e.g. at one point and the neighbourhood on the manifold, the total energy may be E_1, but at another point, it becomes E_2

He then say that QFT still works at small enough regions on the manifold where the spacetime is pretty much flat, hence the gravity is weak and thus there isn't too much interaction between gravity and the energy density of fields present there