Lecture Notes for MATH4017 Quantum Field Theory — Feynman propagator for the Dirac field

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5.4 Feynman propagator for the Dirac field

To perform perturbative QFT calculations in the spirit of Chapter 4 that involve also interacting Dirac quantum fields, we have to understand the Feynman propagator for the Dirac field. From our explicit expressions for the Heisenberg picture field operators (5.81), we immediately observe that the $2$-point correlation functions

\begin{flalign} \expect {0}{\Psi _\alpha (x)\,\Psi _\beta (y)}{0} = 0\quad ,\qquad \expect {0}{\overline {\Psi }_\alpha (x)\,\overline {\Psi }_\beta (y)}{0} = 0 \end{flalign} for two Dirac fields or two adjoint Dirac fields are zero. (The indices $\alpha ,\beta $ denote the components of spinors.) The mixed $2$-point correlation functions $\expect {0}{\Psi _\alpha (x)\,\overline {\Psi }_\beta (y)}{0}$ and $\expect {0}{\overline {\Psi }_\alpha (x)\,\Psi _\beta (y)}{0}$ between Dirac and adjoint Dirac fields are, as we will see, nontrivial. To avoid notational clutter due to using further indices $\alpha ,\beta $, it makes sense to introduce the matrix-valued correlation functions

\begin{flalign} \expect {0}{\Psi (x)\,\overline {\Psi }(y)}{0}\,\in \, \mathrm {Mat}^{}_{N\times N}(\bbC )\quad ,\qquad \expect {0}{\overline {\Psi }(x)\,\Psi (y)}{0}\,\in \, \mathrm {Mat}^{}_{N\times N}(\bbC )\quad , \end{flalign} whose components are defined by

\begin{flalign} \expect {0}{\Psi (x)\,\overline {\Psi }(y)}{0}_{\alpha \beta } \,:=\, \expect {0}{\Psi _\alpha (x)\,\overline {\Psi }_\beta (y)}{0}\quad ,\qquad \expect {0}{\overline {\Psi }(x)\,\Psi (y)}{0}_{\alpha \beta } \,:=\, \expect {0}{\overline {\Psi }_\beta (x)\,\Psi _\alpha (y)}{0}\quad . \end{flalign} (Note the flip of indices in the second definition, which is a convention that will be useful later.) Let us compute the first one in detail. Using (5.81) and that the vacuum state is annihilated by all annihilation operators, we compute

\begin{flalign} \nn \expect {0}{\Psi (x)\,\overline {\Psi }(y)}{0} &= \sum _{s,r}\int _{\bbR ^{2(d-1)}} \frac {e^{\ii \,k\,x - \ii \,q\,y}}{\sqrt {2\omega _{\mathbf {k}}}\,\sqrt {2\omega _{\mathbf {q}}}}~\expect {0}{a_s(\mathbf {k})\,a^\dagger _r(\mathbf {q})}{0}~ u^s(\mathbf {k})\,\overline {u^r(\mathbf {q})} \,\frac {\dd \mathbf {k}}{(2\pi )^{d-1}}\,\frac {\dd \mathbf {q}}{(2\pi )^{d-1}}\\ \nn &= \int _{\bbR ^{d-1}} \frac {e^{\ii \,k\,(x-y)}}{2\omega _{\mathbf {k}}}~\sum _{s} u^s(\mathbf {k})\,\overline {u^s(\mathbf {k})} \,\frac {\dd \mathbf {k}}{(2\pi )^{d-1}}\\ \nn &= \int _{\bbR ^{d-1}} \frac {e^{\ii \,k\,(x-y)}}{2\omega _{\mathbf {k}}}~\big ({-}\ii \,\slashed {k} + m\big ) \,\frac {\dd \mathbf {k}}{(2\pi )^{d-1}}\\ &=\big ({-}\slashed {\partial }_x + m\big )\,\int _{\bbR ^{d-1}} \frac {e^{\ii \,k\,(x-y)}}{2\omega _{\mathbf {k}}} \,\frac {\dd \mathbf {k}}{(2\pi )^{d-1}} = \big ({-}\slashed {\partial }_x + m\big )\,W_2(x,y)\quad , \end{flalign} where $\slashed {\partial }_x = \gamma ^\mu \,\tfrac {\partial }{\partial x^\mu } $ denotes $x$-derivatives. In the second step we used the anticommutation relations (5.82), in third step the spin sum identity (5.58) and in the last step we have recognized our old friend, the $2$-point Wightman function (3.49b) of the Klein-Gordon field. In words, the correlation function $\expect {0}{\Psi (x)\,\overline {\Psi }(y)}{0}$ for the Dirac field is obtained by applying the “wrong sign Dirac operator” $(-\slashed {\partial }_x + m)$ on the correlation function $W_2(x,y)$ of the Klein-Gordon field. That’s very nice. By a similar calculation, one finds that

\begin{flalign} \expect {0}{\overline {\Psi }(x)\,\Psi (y)}{0} = - \big (\slashed {\partial }_x + m\big )\,W_2(x,y) = - \big ({-}\slashed {\partial }_y + m\big )\,W_2(x,y)\quad , \end{flalign} which is again very nice.

The time-ordered product from (3.54) can be extended to Dirac fields $\Psi $ and their Dirac adjoints $\overline {\Psi }$, with the slight, but by now familiar, modification that there will be a minus sign whenever two odd operators are permuted. For instance, we have

\begin{flalign} \nn \TO \big (\Psi _\alpha (x)\overline {\Psi }_\beta (y)\big ) &= \begin{cases} \Psi _\alpha (x)\,\overline {\Psi }_\beta (y)&,~\text {if }x^0\geq y^0\quad ,\\ -\overline {\Psi }_\beta (y)\, \Psi _\alpha (x)&,~\text {if }y^0\geq x^0\quad . \end {cases} \end{flalign} We define the Feynman propagator for the Dirac field as the vacuum expectation value

\begin{flalign} \nn \expect {0}{\TO \big (\Psi (x)\overline {\Psi }(y)\big )}{0} &= \begin{cases} \expect {0}{\Psi (x)\,\overline {\Psi }(y)}{0}&,~\text {if }x^0\geq y^0\quad ,\\ -\expect {0}{\overline {\Psi }(y)\,\Psi (x)}{0}&,~\text {if }y^0\geq x^0\quad , \end {cases}\\[5pt] \nn &= \begin{cases} \big ({-}\slashed {\partial }_x + m\big )\,W_2(x,y)&,~\text {if }x^0\geq y^0\quad ,\\ \big ({-}\slashed {\partial }_x+ m\big )\,W_2(y,x)&,~\text {if }y^0\geq x^0\quad , \end {cases}\\[5pt] &=\big ({-}\slashed {\partial }_x + m\big )\,\Delta _F(x-y)\quad , \end{flalign} where $\Delta _F(x-y)$ is the Feynman propagator for the Klein-Gordon field. Using our Fourier integral formula (3.57) for the latter, we obtain the following Fourier integral formula

\begin{equation} \label {eqn:FeynmanDirac} S_F(x-y) := \expect {0}{\TO \big (\Psi (x)\overline {\Psi }(y)\big )}{0} = \lim _{\epsilon \to 0} \int _{\bbR ^d} \frac {-\ii \,\big ({-}\ii \,\slashed {k} +m\big )}{k^2 + m^2 -\ii \,\epsilon }~e^{\ii \,k\,(x-y)}\,\frac {\dd k}{(2\pi )^d} \end{equation}

for the Feynman propagator for the Dirac field. Note that this Feynman propagator is a Green’s function for the Dirac equation, i.e.

\begin{flalign} \nn \big (\slashed {\partial }_x+m\big ) \, S_F(x-y)\,&=\,\big (\slashed {\partial }_x+m\big )\, \big ({-}\slashed {\partial }_x + m\big )\,\Delta _F(x-y)\\ \,&=\, (-\partial ^2 +m^2)\,\Delta _F(x-y) \,=\, -\ii \,\delta (x-y)\quad . \end{flalign}

Taking again care of the signs associated with permuting odd operators, Wick’s Theorem 4.1 generalizes to the Dirac field, and so does its variant from Corollary 4.3 that is the key for perturbative QFT calculations. For example, using our shorthand notation from Section 4.2, we find that

\begin{flalign} \label {eqn:Dirac4pointfree} \expect {0}{\TO \big (\Psi _1 \,\overline {\Psi }_2 \,\Psi _3\,\overline {\Psi }_4\big )}{0} =(S_F)_{12}\,(S_F)_{34} - (S_F)_{14}\,(S_F)_{32} \quad , \end{flalign} where the minus sign arises from the odd permutation $(1234)\to (1432)$. Comparing this with the case of the Klein-Gordon field from Example 4.4, we note that there is no $(S_F)_{13}\,(S_F)_{24}$ contribution. This follows from our observation above that the correlation functions for the Dirac field are only nontrivial when evaluated among $\Psi $ and $\overline {\Psi }$, i.e. all correlation functions involving only $\Psi $’s or $\overline {\Psi }$’s are zero. From this observation one also deduces that

\begin{flalign} \expect {0}{\TO \bigg (\prod _{i=1}^n\Psi _i \prod _{j=1}^m\overline {\Psi }_j\bigg )}{0} = 0 \qquad \text {(for $n\neq m$)}\quad , \end{flalign} so time-ordered $n+m$-point functions are trivial unless there is an equal number of Dirac $\Psi $ and adjoint Dirac $\overline {\Psi }$ fields.

As a final but important comment, I would like to add that the two points $x,y\in \bbR ^d$ in the Feynman propagator for the Dirac field (5.96) play very different roles as $x$ is the position of the $\Psi $ operator and $y$ the position of the $\overline {\Psi }$ operator. This in particular means that we have to be more careful when drawing Feynman diagrams involving Dirac fields in order to distinguish between the $\Psi $’s and the $\overline {\Psi }$’s. The standard way to deal with this issue is to draw a directed line

$(5.100) \{begin}{flalign} S_F(x-y) ~=~ \expect {0}{\TO \big (\Psi (x)\overline {\Psi }(y)\big )}{0} ~= \parbox {1.5cm}{ \begin{tikzpicture}[scale=1] \draw [thick,-latex] (0.65,0) -- (-0.1,0); \draw [thick] (0,0) -- (-0.65,0); \draw (0.75,0) node{\footnotesize {$~~y$}}; \draw (-0.75,0) node{\footnotesize {$x~~$}}; \end {tikzpicture}} \{end}{flalign}$

with arrow pointing from $\overline {\Psi }$ to $\Psi $. The direction of this arrow is chosen to coincide with the forward-in-time flow of $a$-type particles; indeed, from (5.81) we see that $\overline {\Psi }$ creates $a$-type particles and that $\Psi $ annihilates $a$-type particles. (Alternatively, the arrow depicts the backward-in-time flow of $b$-type particles.) With these conventions, the time-ordered $4$-point function in (5.98) admits the following graphical representation by Feynman diagrams

$(5.101) \{begin}{flalign} \expect {0}{\TO \big (\Psi (x_1) \,\overline {\Psi }(x_2) \,\Psi (x_3)\,\overline {\Psi }(x_4)\big )}{0}~=~ \parbox {1.75cm}{ \begin{tikzpicture}[scale=1] \draw [thick,-latex] (-0.4,-0.4) -- (-0.4,0.1); \draw [thick] (-0.4,0) -- (-0.4,0.4); \draw [thick,-latex] (0.4,-0.4) -- (0.4,0.1); \draw [thick] (0.4,0) -- (0.4,0.4); \draw (-0.5,0.5) node{\footnotesize {$x_1~~$}}; \draw (-0.5,-0.5) node{\footnotesize {$x_2~~$}}; \draw (0.5,0.5) node{\footnotesize {$~~x_3$}}; \draw (0.5,-0.5) node{\footnotesize {$~~x_4$}}; \end {tikzpicture}} ~+~ \parbox {1.75cm}{ \begin{tikzpicture}[scale=1] \draw [thick,-latex] (0.4,-0.4) -- (0.1,-0.1); \draw [thick] (0.2,-0.2) -- (0.05,-0.05); \draw [thick] (-0.4,0.4) -- (-0.05,0.05); \draw [thick] (0.4,0.4) -- (-0.2,-0.2); \draw [thick,-latex] (-0.4,-0.4) -- (-0.1,-0.1); \draw (-0.5,0.5) node{\footnotesize {$x_1~~$}}; \draw (-0.5,-0.5) node{\footnotesize {$x_2~~$}}; \draw (0.5,0.5) node{\footnotesize {$~~x_3$}}; \draw (0.5,-0.5) node{\footnotesize {$~~x_4$}}; \end {tikzpicture}}\qquad . \{end}{flalign}$

It is important to stress that these arrows are a different concept than the momentum flow arrows that we have used in our momentum space Feynman rules in Section 4.6. Hence, when discussing scattering amplitudes involving Dirac fields one has to keep track of both the flow of $a$-type particles and the momentum flow in Feynman diagrams. We will see later examples of such scattering amplitudes when we study quantum electrodynamics (QED) in Chapter 7.

5.4 Feynman propagator for the Dirac field

Further reading