How Do Matrix Valued Propagators Interpret Fermion Amplitudes?

[Bobhawke]

For scalar fields the propagator is just a number that represents the amplitude for a particle to go from one space time point to another.

For fermions, the propagator is matrix valued. What then is the amplitude for a fermion to go from one point to another? How are the elements of the matrix to be interpreted in terms of probability ampltudes?

 

[malawi_glenn]

row-vector times matrix times column vector = number.

 

[Bobhawke]

The fermion propagator involves the dot product of a 4 vector (momentum) and the gamma matrices. I am pretty sure the result is a matrix.

 

[malawi_glenn]

No, since the states are vectors... (4-spinors)

Have you never worked with relativistic quantum identities? What part of my first post did you not understand?

 

[humanino]

What part of my first post did you not understand?

The signature

 

[malawi_glenn]

The signature

shall I change it?

But seriously, what is wrong with my explanation?

 

[blechman]

The fermion propogator IS a matrix, Bobhawke is correct. It is an outer product of spinors, not an inner product.

I think the answer he is looking for is $S_{\alpha\beta}\equiv\langle T\psi_\alpha\bar{\psi}_\beta\rangle$ is the amplitude for a fermion of polarization $\alpha$ to propagate to a fermion of polarization $\beta$.

Up to signs and whatnot!

Malawi_glenn: don't change your signature for anyone!

 

[humanino]

$S_{\alpha\beta}\equiv\langle T\psi_\alpha\bar{\psi}_\beta\rangle$

I think we agree, there is not much more to this equation than what malawi_glenn was writing in "row-vector times matrix times column vector". It is also written in "the states are vectors... (4-spinors)".

 

[malawi_glenn]

yes, the propagator is matrix, but the amplitude is a number. And that was the question, if the propagator is a matrix, what will happen to the amplitude.

 

[blechman]

For fermions, the propagator is matrix valued. What then is the amplitude for a fermion to go from one point to another? How are the elements of the matrix to be interpreted in terms of probability ampltudes?

I think that is the question I answered. The matrix elements of the propagator are the amplitudes for the polarization states to propagate.

 

[Bobhawke]

Thanks for the replies everyone!

 

[Hans de Vries]

For scalar fields the propagator is just a number that represents the amplitude for a particle to go from one space time point to another.

Actually, the propagator is generally the propagation from a source.

So in QED the photon is propagated from the transition current (the interference
pattern caused by an electron in two momentum states) and the electron is
propagated from the interference term $e\gamma^\mu\,A_\mu\psi$

For fermions, the propagator is matrix valued. What then is the amplitude for a fermion to go from one point to another? How are the elements of the matrix to be interpreted in terms of probability ampltudes?

It is indeed the amplitude per polarization state but there are some interesting
details about the interplay between SU(2) and U(1). For instance in a magnetic
field B the energy will be different per polarization state:


$$\binom{~~\exp(-i[E+\Delta E]t)~~}{~~\exp(-i[E-\Delta E]t)~~} ~~=~~ \binom{~~\exp(-i\Delta Et)~~}{~~\exp(+i\Delta Et)~~}~\exp(-iEt})$$

At the RHS the energy is the same for both states but the spinor represents
a precessing spinor around the direction of the magnetic field. Note that:

$$x\uparrow=\binom{1}{1}, ~~y\uparrow=\binom{1}{i},~~x\downarrow=\binom{1}{-1}, ~~y\downarrow=\binom{1}{-i}$$

(up to an overal factor of $1/\sqrt{2}$)


Regards, Hans.