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Let me first present what is my understanding of a propagator. What we measure in the experiment is a probability of scattering. We try to construct a theory predicting these measurements. What we are able to calculate is the $S$ matrix encoding an amplitude of probability.

In the case of a scalar field $\phi(x)$ the first order of the $S_{x\rightarrow y}$ matrix is given by a Green function $<\Omega\mid T\{\phi(x)\phi(y)\}\mid\Omega>$ which is the propagator from $x$ to $y$. I.e. a propagator describes the amplitude of probability of propagation from $x$ to $y$.

Now, I have no idea what is the meaning of a propagator in the case of a Dirac field. As it is 4x4 matrix equation, the propagator is also 4x4 entity given by $<\Omega\mid T\{\Psi(x)\bar{\Psi}(y)\}\mid\Omega>$. As it is a matrix it can not anyhow describe an amplitude of probability, we have to get somehow a number from it.

My idea is the following. In the case of the Dirac equation, we have two extra degrees of freedom describing a particle (spins). Although Dirac spinors consist of 4 elements only 2 of them are independent (guaranteed by the Dirac equation). So my idea was that we have to contract the propagator matrix with spin states depending on the process of which amplitude of probability we want to get.

Example 1: Firstly let's say that I am interested in finding the probability of propagation between 1/2 spin electron in $x$ to 1/2 spin electron in $y$. Does it mean that I have to bracket the propagator by a normalised superposition of two states representing 1/2 spin from both sides (I say 2 states of 1/2 spin, as we have 4 components, but only 2 are independent we got that we have 2 spinors describing spin 1/2)?

Example 2: I am interested in just a propagation between $x$ and $y$. Do I have to bracket the propagator by a complete set of 4 spinors from both sides?

Or in other words: A propagator is a matrix of amplitudes of probabilities, then I have to ask myself propagation of which spin states I want to calculate and then contact the matrix with respective states.

Is that idea anyhow correct? If not, what is the meaning of the propagator being a matrix and how to calculate the amplitude of probability of propagations?

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  • $\begingroup$ Yes you're basically right, see (e.g.) Section 3.1.3 of staff.uni-mainz.de/jkopp/qft2-2017-material/qft.pdf $\endgroup$
    – Andrew
    Mar 19, 2022 at 16:03
  • $\begingroup$ @Andrew I don't know if I got your point, exactly such a sum as in the chapter you mention appears in the propagator of the Dirac field. But even when we have one particular spin instead of this sum we have $u^{s}(p)\bar{u}^{s}(p)$ which still is a matrix. How do we get probability amplitude from it? $\endgroup$
    – Paweł Korzeb
    Mar 19, 2022 at 17:52
  • $\begingroup$ You normally end up taking traces over the spinor indices to average over unknown spin polarizations. $\endgroup$
    – Andrew
    Mar 19, 2022 at 18:02

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