Is electric field a property of a charge or is it a spatial distribution of electrostatic force?

Question

I am a bit confused about what gives rise to an electric field. I can look at it in two different ways as follows.

1. When two charges are separated by a distance, the electric field gives the distribution of magnitudes of electrostatic force at different sets of two different positions of the two charges within the field suggesting that the electric field exists only if the electrostatic force exists. And will not exist if there is only one charge present.

Or

2. As quantum theory of fields suggests, the electrostatic field is a property of a charge and thus exists even when there is only one charge present suggesting electric field to be independent of electrostatic force.

Which of the above two statements is correct? Is electric field linked to the charge or the electrostatic force?

Answer 1

The classical definition of an electric field at a position $x$ is the force that a small test charge would feel if it was placed at $x$ (divided by the test charge).

More formally,

\begin{equation} \vec{E}(\vec{x}) = \lim_{q\rightarrow 0} \frac{\vec{F}({\rm on\ }q{\rm\ at\ } \vec{x}\ {\rm\ due\ to\ other\ charges})}{q} \end{equation}

In particular, there is no problem defining the electric field due to a single point charge.

In quantum field theory, the electric field is an operator that is typically defined in terms of the electromagnetic tensor $F_{\mu\nu}$ \begin{equation} E_i \equiv F_{0i} = \partial_0 A_i - \partial_i A_0 \end{equation} where $A_\mu$ is the vector potential. There need not be a source of the electric field in quantum field theory; vacuum fluctuations have physical effects such as the Casimir effect, or renormalization of the electron mass.