Importance of Densities in QFT

[physwiz222]

TL;DR Summary

I wonder if there is a way to find charge and particle number densities of Quantum Fields like in regular QM

I have been looking at QFT out of curiosity so I am not an expert. In standard Quantum Mechanics you have something called the Charge or Electron density where you essentially describe a many particle wavefunction in terms of 3 coordinates by integrating the coordinates of all the electrons except for one and repeat for every electron giving you the Total Electron Density as a function of 3 coordinates. In fact many methods in Quantum Chemistry like Density Functional Theory seek to calculate this.

Is a similar thing possible in QFT as the wavefunction in QFT is a wavefunctional an infinite dimensional functional of field configurations but we live in 3 dimensions so is it possible to describe the Quantum Fields in terms of charge and particle densities at least via the Operator Formalism like is there a Particle/Charge Density Operator in QFT which you can take the expectation value of to get the charge or particle density of say the vacuum or the 1 particle state and visualize that.

Also it seems no one ever calculates charge and particle densities in QFT if there is such an operator at least for free fields its probably horribly intractable for interacting fields (from what I know we dont even have a way to describe finite time dynamics of interacting fields at least with the S Matrix and Feynman Diagrams so computing densities for this is out of the question).

With that in mind why is it never calculated for free fields isnt the charge density or Particle Density of Quantum Fields an Important thing to study as it provides a 3 dimensional tangible picture of for example the Vacuum State. I know QFT courses have to focus on Feynman Diagrams and the S Matrix but why cant we also find charge and particle densities of Free Fields at least to visualize the Vacuum and particle states in a more concrete spatial way to see how particles emerge from Fields.

Keep in mind I am not asking for a classical realist picture of electron trajectories. I want a 3 dimensional tangible non classical picture of charge and energy density. I know Quantum Fields arent like classical fields I just want to know if it possible to describe them using charge or particle number density.

 

[vanhees71]

The charge-current and the energy-momentum tensor (containing the energy and momentum density) are described by local field operators. Eg., for the free Dirac field the charge four-current is given by
$$\hat{j}^{\mu}(x)=q :\hat{\bar{\psi}}(x) \gamma^{\mu} \hat{\psi}(x):.$$