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In QFT, we can write the amplitude for a field to be in configuration $A_{in}(x)$ at time $0$ and end in configuration $A_{out}(x)$ as $K_t[A_{in},A_{out}]$, for example. Alternatively we could expand this in terms of photons and calculate the amplitudes of photons to start in position $x$ and end in position $y$, $K_t(x,y)$,

With electrons, it really only makes sense to consider them as particles, even though we can formerly write them as Grassmann fields.

I wonder if there is a description of QED which treats the photons as a field but the electrons as particles.

Thus you might have amplitudes $K(A_{in},x_1,x_2,x_3,..;A_{out},y_1,y_2,...)$ for a configuration to start with electromagnetic field $A_{in}$ and electrons in positions $x_1$, $x_2$,...

So you would sum over all electron paths including closed loops and over all fields $A$ consistent with those paths. So treating electrons as particles but keeping the electromagnetic field as a field. The Feynman graphs would still have electron paths but the photons would be treated differently.

I wonder if such a description is known?

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Sure; this amounts to a choice of a basis in the space of states in which your basis states are eigenstates of the electric field operator and have a definite electron number. This can be done for a free theory, or for an interacting theory you can at least do it approximately.

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