Propagator for the Dirac Equation

The relativistic propagator, $S_F(x^\prime,x;A)$ is defined to satisfy a Green's function equation

$$\left[ \gamma_\mu \left( i\frac{\partial}{\partial x_\mu^\pr... ...}}(x^\prime,x;A) = \delta_{\alpha\beta} \delta^4(x^\prime-x).$$ (6.64)

The propagator is a $4\times 4$ matrix corresponding to the dimensionality of $\gamma$ matrices. Note that the definition of the relativistic propagator differs from the nonrelativistic counterpart: the differential operator $i\partial_{t^\prime} - \hat{H}(x^\prime)$ occurring in the nonrelativistic equation has been multiplied by $\gamma^0$ in the relativistic equation in order to form the covariant operator $i\not{\nabla}^\prime -ie\not{A}^\prime -m$.

Suppressing the indices we have

$$(\hat{\not{\;\!\!\!p}}^\prime - e\not{\!\!A}- m)S_F(x^\prime,x;A) = \delta^4(x^\prime-x).$$ (6.65)

We can compute the free-particle propagator, $S_F(x^\prime,x)$, using

$$(\hat{\not{\;\!\!\!p}} - m) S_F(x^\prime;x) = \delta^4(x^\prime-x)$$ (6.66)

by Fourier transforming to momentum space. $S_F(x^\prime;x)$ depends only on the interval $(x^\prime-x)$. This property is a manifestation of the homogeneity of space and time, and is general would not be valid for the interacting propagator $S_f(x^\prime,x;A)$. We have

$$S_F(x^\prime;x) = S_F(x^\prime-x) = \int \frac{d^4p}{(2\pi)^4} e^{-ip\cdot(x^\prime-x)} S_F(p) .$$ (6.67)

This gives

$$(\not{\;\!\!\!p}- m)_{\alpha\lambda} S_{F_{\lambda\beta}}(p) = \delta_{\alpha\beta} .$$ (6.68)

Solving for the Fourier amplitude $S_F(p)$ and reverting to the matrix shorthand, we find

$$S_F(p) = \frac{1}{\not{\;\!\!\!p}- m} \equiv \frac{\not{\;\!\!\!p}+ m}{p^2 - m^2} \quad\textrm{for}\ p^2\ne m^2.$$ (6.69)

A prescription for how to handle the singularity at $p^2=m^2$ or ( $p_o=\pm\sqrt{\vec{p}^2+m^2} = \pm E$) is needed. This comes from the boundary conditions put on $S_F(x^\prime-x)$ in the integration.

The interpretation given to the Green's function $S_F(x^\prime-x)$ is that it represents the wave produced at the point $x^\prime$ by a unit source located at the point $x$. The Fourier components of such a localized point source contain many momenta larger than $m$, the reciprocal of the electron Compton wavelength, and we expect that positrons as well as electrons may be created at $x$ by the source. However, a necessary physical requirement of the hole theory is that the wave propagating from $x$ into the future consist only of positive-energy electron and positron components. Since positive-energy positrons and electrons are represented by wave functions with positive frequency time behaviour $S_F(x^\prime-x)$ can contain in the future $x_0^\prime>x_0$, only positive-frequency components.

We perform the $dp_0$ integration along the contour in the complex $p_0$-plane. For $t^\prime > t$, the contour is closed in the lower half-plane and includes the positive-frequency pole at $p_0=+\sqrt{\vec{p}^2+m^2} = +E$ only. This gives

$$S_F(x^\prime-x) = \int \frac{d^3p}{(2\pi)^3} e^{i\vec{p}\cdot(\vec{x}^\prime-\vec{x... ...int \frac{dp_0}{2\pi} \frac{e^{-ip_0(t^\prime-t)}}{p^2-m^2} (\not{\;\!\!\!p}+m)$$

$$= -i\int \frac{d^3p}{(2\pi)^3} e^{i\vec{p}\cdot(\vec{x}^\prime-\vec... ...E\gamma_0 - \vec{p}\cdot\vec{\gamma} + m}{2E} \quad\textrm{for}\quad t^\prime>t$$ (6.70)

so that the wave at $(\vec{x}^\prime,t^\prime)$ contains positive-frequency components only. For $t^\prime< t$, the contour can be closed above, including the pole at $p_0=-\sqrt{\vec{p}^2+m^2}=-E$. This gives

$$S_F(x^\prime-x) = -i\int \frac{d^3p}{(2\pi)^3} e^{i\vec{p}\c... ...}\cdot\vec{\gamma} + m}{2E} \quad\textrm{for}\quad t^\prime<t$$ (6.71)

showing the propagator to consist of negative-frequency waves for $t^\prime<t$. Any other choice of contour leads to negative-energy waves propagating into the future or positive-energy waves propagating into the past. Moreover, the negative-energy waves propagating into the past that we have just found are welcome; they are the positive-energy positrons. The origin of the negative-energy waves is the pole at $p_0=-\sqrt{\vec{p}^2+m^2}$, which was not present in the nonrelativistic theory.

The choice of the contour is summarised by adding a small positive imaginary part to the denominator, or simply taking $m^2\rightarrow m^2 - i\epsilon$, where the limit $\epsilon\rightarrow 0^+$ is understood:

$$S_F(x^\prime-x) = \int \frac{d^4p}{(2\pi)^4} \frac{e^{-ip\cdot(x^\prime-x)}}{p^2 - m^2 + i\epsilon} (\not{\;\!\!\!p}+ m) .$$ (6.72)

The two integrations can be combined by introducing projection operators and changing $\vec{p}$ to $-\vec{p}$ in the negative-frequency part:

$$S_F(x^\prime-x) = -i \int \frac{d^3p}{(2\pi)^3} \frac{m}{E} ... ...bda_-(p) e^{ip\cdot(x^\prime-x)} \theta(t-t^\prime) \right] ,$$ (6.73)

with $p_0=E>0$.

Equivalently, writing for normalized plane-wave solutions, we find

$$S_F(x^\prime-x) = -i\theta(t^\prime-t) \int d^3p \sum_{r=1}... ... d^3p \sum_{r=3}^4 \psi_P^r(x^\prime) \overline{\psi}_P^r(x) .$$ (6.74)

We see that $S_F(x^\prime-x)$ carries the positive-energy solutions $\psi^{(+)}$ forward in time and the negative-energy ones $\psi^{(-)}$ backward in time

$$\theta(t^\prime-t) \psi^{(+)}(x^\prime) = i\int d^3x S_F(x^\prime-x) \gamma_0 \psi^{(+)}(x),$$ (6.75)

$$\theta(t-t^\prime) \psi^{(-)}(x^\prime) = -i\int d^3x S_F(x^\prime-x) \gamma_0 \psi^{(-)}(x).$$ (6.76)

The minus sign in the second equation results from the difference of the direction of propagation in time between (6.75) and (6.76).

$S_F(x^\prime-x)$ is known as the Feynman propagator. This spin-1/2 propagator is related to the Klein-Gordon propogator by

$$S_F(x^\prime-x) = (i\not{\partial}-m)\Delta_F(x^\prime-x) .$$ (6.77)

From the free propagator $S_F(x^\prime-x)$ we may formally construct the complete Green's function and the $S$-matrix elements, that is, the amplitudes for various scattering processes of electrons and positrons in the presence of force fields. The exact Feynman propagator $S_F(x^\prime,x;A)$ satisfies (6.65) and can be expressed in terms of a superposition of free Feynman propagators

$$(\hat{\not{\;\!\!\!p}}^\prime - e\not{\!\!A}^\prime - m )S_F(x^\prime,x;A) = \delta^4(x^\prime-x) ,$$

$$(\hat{\not{\;\!\!\!p}}^\prime-m)S_F(x^\prime,x;A) = \delta^4(x^\prime-x) + e\not{\!\!A}^\prime S_F(x^\prime,x;A) ,$$ (6.78)

$$= \int d^4y \delta^4(x^\prime-y) [ \delta^4(y-x) + e\not{\!\!A}(y) S_F(y,x;A)] ,$$

$$= \int d^4y (\hat{\not{\;\!\!\!p}}^\prime -m) S_F(x^\prime;y) [ \delta^4(y-x) + e\not{\!\!A}(y) S_F(y,x;A)] ,$$

$$S_F(x^\prime,x;A) = \int d^4y S_F(x^\prime-y) [ \delta^4(y -x) +e\not{\!\!A}(y) S_F(y,x;A) ],$$

$$= S_F(x^\prime-x) + e\int d^4y S_F(x^\prime-y) \not{\!\!A}(y) S_F(y,x;A) .$$ (6.79)

This is the relativistic counterpart of the Lippmann-Schwinger equation. Another notation for $S_F(x^\prime,x;A)$ is $S_F(x^\prime;x;A)$. This integral equation determines the complete propagator $S_F(x^\prime;x;A)$ in terms of the free-particle propagator $S_F(x^\prime;x)$.

Proceeding in analogy to the nonrelativistic treatment, the iteration of the integral equation yields the following multiple scattering expansion

$$S_F(x^\prime,x;A) = S_F(x^\prime-x)$$ (6.80)

$$+ e\int d^4x_1 S_F(x^\prime-x_1) \not{\!\!A}(x_1) S_F(x_1-x)$$

$$+ e^2 \int d^4x_1d^4x_2 S_F(x^\prime-x_1) \not{\!\!A}(x_1) S_F(x_1-x_2) \not{\!\!A}(x_2) S_F(x_2-x)$$

$$+ \ldots .$$

Equation 6.78 can be viewed as an inhomogeneous Dirac equation of the form $(\hat{\not{\;\!\!\!p}}-m)\Psi(x) = \rho(x)$ which is solved by the Green's function techniques as follows

$$\Psi(x) = \psi(x) + \int d^4y S_F(x-y)\rho(y) .$$ (6.81)

The exact solution of the Dirac equation with the Feynman boundary conditions, is

$$\Psi(x) = \lim_{t\rightarrow-\infty} \int d^4x^\prime S_F(x,x^\prime;A) \psi(x^\prime) ,$$

$$= \lim_{t\rightarrow-\infty} \int d^4x^\prime [ S_F(x,x^\prime) + e\int d^4y S_F(x,y) \not{A}(y) S_F(y,x^\prime) ] \psi(x^\prime) ,$$

$$= \psi(x) + e\lim_{t\rightarrow-\infty} \int d^4y S_F(x,y) \not{A}(y) \int d^4x^\prime S_F(y,x^\prime) \psi(x^\prime),$$

$$\fbox{$ \displaystyle \Psi(x) = \psi(x) + e\int d^4y S_F(x-y) \not{\!\!A}(y) \Psi(y) $}\ ,$$ (6.82)

where $\psi(x)$ is the free-particle solution. The scattering wave contains only positive frequencies in the future and negative frequencies in the past

$$\Psi(x) - \psi(x) \rightarrow \int d^3p \sum_{r=1}^2 \psi_P^r(x) [-ie \int d^4y \overline{\psi}_P^r(y) \not{\!\!A}(y) \Psi(y) ] \quad\textrm{as}\quad t \rightarrow +\infty ,$$ (6.83)

$$\Psi(x) - \psi(x) \rightarrow \int d^3p \sum_{r=3}^4 \psi_P^r(x) [+ie \int d^4y \overline{\psi}_P^r(y) \not{\!\!A}(y) \Psi(y) ] \quad\textrm{as}\quad t \rightarrow -\infty .$$ (6.84)