Local gauge transformation in Fock space of charged particles

I’m currently fiddling around with gauge-phase-transformations in Fock space.
Especially, I’m trying to write a local gauge-phase-transformation as
an operator in a basis-independent way.

Here is what I have so far.
Consider a system of indistinguishable particles (each with a charge $q$).

Total charge

Let’s take the total charge operator $hat{Q}$. It can be defined
by its action on the $n$-particle states (using Fock states in the position basis):
$$begin{align}
&hat{Q} |rangle &= & 0 \
&hat{Q} |vec{x}_1rangle &= & q |vec{x}_1rangle \
&hat{Q} |vec{x}_1vec{x}_2rangle &= & 2q |vec{x}_1vec{x}_2rangle \
&…
end{align} tag{1}$$
The operator $hat{Q}$ can be written in a basis-independent way:
$$hat{Q} = qhat{N} = qint d^3x hat{psi}^dagger(vec{x})hat{psi}(vec{x}) tag{2}$$
where $hat{N}$ is the total number operator, and
$hat{psi}^dagger(vec{x})$ and $hat{psi}(vec{x})$ are the canonical
creation and annihilation operators at position $vec{x}$.
It is easy to check that this operator (2) satifies the definition (1).

Global gauge transformation

Now let’s consider a global gauge-phase-transformation $hat{U}(f)$
with a global constant $f$. $hat{U}(f)$ can be defined by its action
on the $n$-particle states:
$$begin{align}
&hat{U}(f) |rangle &= & |rangle \
&hat{U}(f) |vec{x}_1rangle &= & e^{iqf} |vec{x}_1rangle \
&hat{U}(f) |vec{x}_1vec{x}_2rangle &= & e^{2iqf} |vec{x}_1vec{x}_2rangle \
&…
end{align} tag{3}$$
It is easy to guess that $hat{U}(f)$ can be written
in a basis-independent way:
$$hat{U}(f) = e^{ihat{Q}f} tag{4}$$
And indeed, by using $hat{Q}$ from above
it can be verified that (4) satisfies the definition (3).

So far no problem.

Local gauge transformation

And now for the local gauge-phase-transformation $hat{U}(f)$
with a position-dependent function $f(vec{x})$.
Again $hat{U}(f)$ can be defined by its action on the $n$-particle states
(by generalizing the definition (3)):
$$begin{align}
&hat{U}(f) |rangle &= & |rangle \
&hat{U}(f) |vec{x}_1rangle
&= & e^{iqf(vec{x}_1)} |vec{x}_1rangle \
&hat{U}(f) |vec{x}_1vec{x}_2rangle
&= & e^{iqf(vec{x}_1)} e^{iqf(vec{x}_2)} |vec{x}_1vec{x}_2rangle \
…
end{align} tag{5}$$

I was not able to write $hat{U}(f)$ in a basis-independent way
so that it will satisfy the definition (5).

• $hat{U}(f) = int d^3x e^{ihat{Q}f(vec{x})}$
  is obviously wrong, because $hat{U}$ has the dimension of a volume,
  instead of being dimensionless.
• $hat{U}(f) = e^{iint d^3x hat{Q}f(vec{x})}$
  is also wrong, because the exponent has the dimension of a volume,
  instead of being dimensionless.
• $hat{U}(f) = int d^3x hat{psi}^dagger(vec{x})
  e^{ihat{Q}f(vec{x})} hat{psi}(vec{x})$
  is wrong, because when acting on the vacuum state
  the result is $hat{U}|rangle=0$ instead of $hat{U}|rangle=|rangle$.

Any ideas? Is it even possible?