Path integral in coherent states

[topper]

Hey,

there is something I don't really understand about the path integral (functional integral) formalism in QFT:

Why do you need to introduce a coherent-state representation of the Dirac fields in order to evaluate their path integral?
Where is the crucial point why it doesn't work like in the scalar field case, where you have the operators in the correlation function on the one side and just scalar functions under the integral on the other side?

I hope someone gets my point :)

Thanks,
topper

 

[azntoon]

_42The main difference between a scalar field and a Dirac field is that the former is composed of complex numbers, while the latter is composed of spinor wavefunctions. This means that the path integral for a Dirac field needs to be evaluated in terms of wavefunctions, rather than just scalar functions. To do this, we introduce a coherent-state representation of the Dirac fields, which allows us to express the path integral as an integration over wavefunction amplitudes. This is in contrast to the scalar field, where we can evaluate the path integral directly in terms of scalar functions without introducing a coherent-state representation.