A What's the idea behind propagators

[Tomishiyo]

I'm studying QFT by David Tong's lecture notes.

When he discusses causility with real scalar fields, he defines the propagator as (p.38)
$$D(x-y)=\left\langle0\right| \phi(x)\phi(y)\left|0\right\rangle=\int\frac{d^3p}{(2\pi)^3}\frac{1}{2E_{\vec{p}}}e^{-ip\cdot(x-y)},$$
then he shows that the commutator of two scalar fields at arbitrary spacetime points $x,y$ is
$$[\phi(x),\phi(y)]=D(x-y)-D(y-x).$$
When he moves on to discuss the propagator of a fermion, however, he defines
$$iS_{\alpha\beta}=\{\psi_{\alpha}(x),\bar{\psi}_{\beta}(y)\}$$
as the propagator, where $\bar{\psi}=\psi^{\dagger}\gamma^{0}$ is the Dirac adjoint.

Could anyone explain me from where this definition comes from? I understand why the commutator turns into an anticommutator (due to the spinor quantization) and why you need a $\psi$ and $\bar{\psi}$ (due to Lorentz invariance). But I find it confusing, since he didn't call the commutator of two scalar fields as the propagator before, he defined it as the vacuum expectation of the field in two different points as $D(x-y)$. Or are the propagators named differently for scalars and spinors? Why does it has to be a matrix instead of a number (meaning, why define it with the adjoint to the right of the anti-commutator instead of the other way around)? Also, why define it with an $i$ factored out?

 

[vanhees71]

You have to specify which propagator you mean. There are many propagators used in QFT. In the vacuum for perturbation theory you usually only need the time-ordered one (which is the same as the Feynman propagator in this case), in the many-body case you need more (Schwinger-Keldysh real-time formalism).

You can derive the propgators of course from the Wightman function (expectation values of fixed-ordered fiel products) or also from the retarded propagator (which is the commutator for bosons and the anti-commutator for fermions times the $\Theta$ function to make it retared) or the spectral function, which just is the imaginary time of the retarded propagator. For an introduction, see Sect. 2.2.4 of

https://th.physik.uni-frankfurt.de/~hees/publ/off-eq-qft.pdf

as an example for the treatment at finite temperature in the real-time formalism.

 

[Tomishiyo]

Thank you for your answer.

The propagators I'm referring are for the free scalar field and the free spinor field. I'm not sure if this answers your question because I understand so little about propagators that I'm not even used to the terminology.

As for the second part of your answer, I'll take a look at the ref.

 

[Demystifier]

Could anyone explain me from where this definition comes from?

Conceptually, one should distinguish the following two objects:
- propagator, which is another name for the Green function associated with a partial differential equation
- 2-point function, which is another name for the vacuum correlator of fields at two different points
Even though these two functions may eventually turn out to be the same (or closely related) functions, their a priori definitions are very different. Sloppy physicists sometimes say "propagator" when they mean "2-point function", and vice versa. For more details see e.g. Ryder, Quantum Field Theory.

 

[Tomishiyo]

Conceptually, one should distinguish the following two objects:
- propagator, which is another name for the Green function associated with a partial differential equation
- 2-point function, which is another name for the vacuum correlator of fields at two different points
Even though these two functions may eventually turn out to be the same (or closely related) functions, their a priori definitions are very different. Sloppy physicists sometimes say "propagator" when they mean "2-point function", and vice versa. For more details see e.g. Ryder, Quantum Field Theory.

This explains a lot. Then the answer to my other questions, I suppose, must be that the Green function to the Dirac operator is a matrix and for some convenient reason Tong factored out the $i$. Thank you for your answer.

 

[vanhees71]

The point is that there is not "the Green's function" for hyperbolic differential equations, i.e., wave equations. For free particles a Green's function obeys the equation
$$(\Box+m^2) G(x,x')=\delta^{(4)}(x-x').$$
For any solution you can add an arbitrary solution of the homogeneous equation,
$$(\Box+m^2) G_0(x,x')=0,$$
and you get again another Green's function.

You need to know which Green's function you need for your specific problem. E.g., in vacuum QFT to evaluate Feynman rules for $n$-point functions and finally $S$-matrix elements you need the time-ordered Green's function, i.e., for the uncharged Klein-Gordon field
$$\mathrm{i} \Delta(x,x')=\langle \Omega|\mathcal{T}_c \hat{\phi}(x) \hat{\phi}(x')|\Omega \rangle.$$
You can show, using the canonical equal-time commutators, that this function indeed obeys the defining equation for the Green's function.

In linear-response theory you need the retarded propagator which by definition is $\propto \Theta(x^0-x^{\prime 0})$. You can also show, using the equal-time commutators, that this is given by
$$\mathrm{i} G_{\text{ret}}(x,x')=\Theta(x^0-x^{\prime 0}) \langle \Omega|[\hat{\phi}(x),\hat{\phi}(x')]|\Omega \rangle.$$
So you have to be careful to use the right propagator for any given problem.