Time ordering for Dirac spinors

[kelly0303]

Hello! The time ordered product for Dirac spinors is defined as: $$<0|\psi(x)\bar{\psi}(y)|0>-<0|\bar{\psi}(y)\psi(x)|0>$$ Can someone explain to me how should I think of the dimensionality of this. For a Dirac spinor, $\psi(x)$ is a 4 dimensional column vector, so the first term in that expression is a 4x4 matrix while the second one is a number. Is a multiplication by the 4x4 identity matrix implied for the second term? Thank you!

 

[Gaussian97]

Well, not sure where did you get this result, but the time-ordered product of two Dirac spinors is usually written as $$T\left\{\psi(x)\bar{\psi}(y)\right\}=\begin{cases}\psi(x)\bar{\psi}(y)&\text{if} &x^0>y^0\\-\bar{\psi}(y)\psi(x)&\text{if} &y^0>x^0\end{cases}$$ Or, equivalently,$$T\left\{\psi(x)\bar{\psi}(y)\right\}=\theta(x^0-y^0)\psi(x)\bar{\psi}(y)-\theta(y^0-x^0)\bar{\psi}(y)\psi(x)$$

But this, as you point, has no much sense. The time-ordered product of two spinors is defined by its components, so really when you see the expressions above you must understand
$$T\left\{\psi_a(x)\bar{\psi}_b(y)\right\}=\theta(x^0-y^0)\psi_a(x)\bar{\psi}_b(y)-\theta(y^0-x^0)\bar{\psi}_b(y)\psi_a(x)$$

 

[kelly0303]

Well, not sure where did you get this result, but the time-ordered product of two Dirac spinors is usually written as $$T\left\{\psi(x)\bar{\psi}(y)\right\}=\begin{cases}\psi(x)\bar{\psi}(y)&\text{if} &x^0>y^0\\-\bar{\psi}(y)\psi(x)&\text{if} &y^0>x^0\end{cases}$$ Or, equivalently,$$T\left\{\psi(x)\bar{\psi}(y)\right\}=\theta(x^0-y^0)\psi(x)\bar{\psi}(y)-\theta(y^0-x^0)\bar{\psi}(y)\psi(x)$$

But this, as you point, has no much sense. The time-ordered product of two spinors is defined by its components, so really when you see the expressions above you must understand
$$T\left\{\psi_a(x)\bar{\psi}_b(y)\right\}=\theta(x^0-y^0)\psi_a(x)\bar{\psi}_b(y)-\theta(y^0-x^0)\bar{\psi}_b(y)\psi_a(x)$$

Oh I see! Thank you so much (also sorry for the typo, I missed the step function...). However, I am a bit confused now. The definition, mathematically, makes sense now, in terms of components of spinors (i.e. not row or columns vectors) but I am not sure I know how to think of it physically. For a scalar field, I was thinking of the propagator as something that creates a particle at $x$ and propagates it to $y$ i.e. it gets annihilated at $y$. But for spinors, as the propagator is defined for each element of the Dirac spinor, it seems like the propagator creates a component of a spinor ($a$ ) at $x$ and annihilates another component of the spinor ($b$ ) at $y$. But I am not sure how can one think of just a component of a spinor alone, as the spinor is an object whose components should transform together in a way given by some representation of the Lorentz group. It looks like the propagator doesn't propagate the whole spinor, but just one component at a time i.e. one can compute 16 propagators for a given spinor. So what is the physical meaning of the propagator in this case? Thank you!

 

[vanhees71]

It's pretty difficult to understand the meaning of the propagator in the way you try to explain it. The reason is that due to the demand of microcausality (locality) the quantum fields are constructed such as to transform as the classical analogues "locally under Poincare transformations" and for this you need the expansion of the free-field operators to contain both positive and negative frequencies. To have positive energies you have to write an annihilation operator in front of the negative-frequency modes, leading to the prediction that for each particle you should also have an antiparticle. So a field operator $\psi_a$ annihilates a particle and/or creates an anti-particle and vice versa for $\bar{\psi}_b$.

The spinor indices label in a specific way the spin states, i.e., each one-particle mode is characterized by a momentum eigenvalue $\vec{p}$ and a spin state $\sigma_z =\pm 1/2$ (with the usual meaning of spin in the restframe of the particle). That's why the spinor field has four components: two spin states for a particle and two spin states for an antiparticle.

In the formalism of QFT concerning calculations of observable quantities in perturbation theory (in "vacuum QFT" these are usually the S-matrix elements) a propagator is depicted as an internal line connecting two vertices, which describe interaction processes. In QED the most simple diagram, where an electron-positron propagator occurs is Compton scattering, i.e., the elastic scattering of an electron (or positron) with a photon. The propagator provides the transition-matrix element from the in state (characterized by an asymptotic free electron with a given momentum and spin/polarization parameter and an asymptotic free photon with a given momentum and helicity/polarization parameter) to an outstate (characterized again by the described properties of the asymptotic free electron (positron) and photon).

Note that there are two diagrams, and the electron lines carry an arrow indicating the flow of electric charge. The external legs denote the asymptotic free particle states with defined on-shell momenta and polarizations. An external line with an arrow pointing into a vertex can now mean either an incoming particle (electron) in the initial or and outgoing particle (positron) in the final state, depending on the direction of the momentum relative to the direction of the arrow indicating the electric-charge flow. That's also why in QED each electron/positron-photon vertex must have an incoming and an outgoing electron/positron line, i.e., the incoming charge must equal the outgoing charge due to charge conservation.

The propagator contains both electron and positron annihilation and creation operators, and thus it automatically describes all kinds of possible charge flows through a diagram, i.e., depending on what's scattered in a specific diagram it carries momentum and spin of either an electron, a positron or both with their corresponding electric charge. In this way the maths of Feynman diagrams take care of all the symmetries and conservation laws, including also the necessary relative signs under exchange of external fermion lines, which ensure that at any order of perturbation theory you take into account the correct antisymmetrization of the states you sum/integrate over for a given process.