Canonical vs. path integral quantization

[earth2]

Hey folks,

i have a question concerning canonical and path integral quantization.
From what I have understood so far, these two techniques are different and independent but equivalent.

My problem is that I don't really see where the quantum character enters in the path intregral formulation.

I mean, sure, it is based on bra and kets and all that stuff but don't the appearing fields in a path integral have to fullfil commutation relations, too?

So, before i can build a path integral, i have to make sure that the appearing fields fullfil certain (anti-)commutators, right? But doesn't this mean that the path integral formulation needs a canical formulation first?

Thanks for your answers!
Cheers,
earth2

 

[xepma]

No, everything that is put into the path integral is classical. That's why the formalism is so appealing: you don't have those nasty non-commuting operators. The path integral formalism does make use of the Hilbert space of the theory, but you do not need to introduce commutation relations between the fields.

The quantum nature arises in the computation of amplitudes: you sum over all "possible" contributions to an amplitude, and weighs these with a complex Boltzmann factor e^{i S}. The fact that this factor is complex is what causes the quantum mechanical nature of the amplitudes.

 

[earth2]

Thank you so much! Why isn't it written this clearly in textbooks? :)

 

[Haelfix]

It is, for instance try Feynman and Hibbs