Sunday 10 December 2017

quantum field theory - Propagator and probability amplitude that a particle propagates


My QFT knowledge has very much rusted and i got confused by these few lines from Peskin and Schroeder:



p.27: " [..] the amplitude for a particle to propagate from $y$ to $x$ is $\langle 0| \phi(x) \phi(y) |0\rangle $. We will call this quantity $D(x — y)$."




(The relation with the commutator is explicitly calculated at (2.53) p.28, + bottom of p.29: $$ [\phi(x) , \phi(y)] = \cdots = D(x-y)- D(y-x) = \langle 0|[\phi(x) , \phi(y)]|0\rangle$$ the r.h.s. are implicitly understood as being proportional to the $\mathbb{1}$ operator)


Finally the expressions of the retarded and Feynman propagators are given (2.55) p.30



$$D_R := \theta (x^0 -y^0) \langle 0|[\phi(x) , \phi(y)]|0\rangle $$



and (2.60) p.31 (without commutators)



$$D_F := \theta (x^0 -y^0) \langle 0|\phi(x) \cdot \phi(y)|0\rangle + \theta (y^0 -x^0) \langle 0|\phi(y) \cdot \phi(x)|0\rangle $$



which by definition of "propagator" or "Green's function" satisfy $(\square +m^2) G(x,y)= -i\delta^4(x-y)$.





Now my confusion comes from the fact that I remember that propagators had the interpretation of the amplitude of propagation, cf. e.g. wikipedia or Peskin last 2 lines p.82, but this is wrong? (three different functions obviously cannot have the exact same interpretation!)


Remark: I am aware that from the defining relation of a Green's function, one can express the "evolution" of a solution of the (Klein-Gordon) equation, so that in some sense propagators expresses an idea of propagation




The first question is too easy so here is a second: Are propagators always combinations of the $D(x-y)$?



  • for interacting fields

  • for more general PDE?


Remark: I'm not trying to relate interacting theories to the free one, so $D(x-y)$ stands for the amplitude of propagation in each theory not the free one. The underlying idea is that $D(x-y)$ has a clear interpretation while the propagators would not have an easy interpretation on their own unless they are just simple functions of these $D(x-y)$.





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