# Is there a mixed field-particle representation of QED?

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?