** MOTION OF AN ELECTRON IN A STATIC UNIFORM ELECTROMAGNETIC FIELD**

**3.2 The field-free electron propagator**

The electron propagator is the Green's function of the Dirac equation for the electron. In the field-free vacuum, Dirac's equation may be written in the form

(?$ - m) ip(x) = 0 , (3.2-1)

where := i$ (the Schrödinger representation) and i^(x) is the four-

component wave function. In (3.2-1), and in other equations in this thesis, the bispinor indices are suppressed. The electron propagator

G(x-x') is given by the solution of the equation

(?$ - m) G(x,x') = **6^4^(x** **-x') ** (3.2-2)

The solution of (3.2-2) for the causal Feynman electron

propagator may be written in the form of a vacuum expectation value [see e.g. Berestetskii

*et at.*

1971, §76],
G (x-x') = **-i (o| **T 4<(x) $(x') 1**0 **> , **(3.2-3)**
**r**

where T is the chronological order operator

^ ^ [ $(x) $(xf) for t > t ’

T ip(x) $(x') :=

*)*

*[*

-?(x') *Hx)*

for t < t ’ , (3.2-4)
/ \ ~

and ^(x) and ^(x) are the operators in the interaction picture

corresponding to the wave function ^(x) and its Dirac conjugate *Kx)
(:= ip (x)Y , where t denotes the Hermitian conjugate) respectively.
For a free particle of definite momentum and energy (a plane wave),
the operators ^(x) and ^(xT) are related to the bispinors u a (p) and
v a (j3) (of normalization u a (p) u a ,(p) = 2m 6a a ,_{ = -va (p) va'(P)) by}

$ (x) = 2

* a* J
,3

d

*z*

1
Sa(P) V E ? e" 1PX + ^ a (P> va (E>

\ ipx (2n) 3

_{(2z)h}

e
>
and
A
'l'(x') = 2
a J
J 3
d _{(2z)h}

*z*

1
4 (p )
V. eiPX + ea (£>
P_ipx _{(3.2-5)}(27T) 3

_{(2e) 2}h

**0**

_{*}with p° = £ := (m2 +

_{I}

pI2 )^2 In (3.2-5) aa (p) and t>a (p) are the

annihilation operators for particles and antiparticles respectively of

/s, **j* ** **/\ j***

momentum p and spin state Ot, and aa (p) and ba (p) are the corresponding creation operators. The normalization in (3.2-5) corresponds to one

3 3

particle in the range d p / (2^) at p in the volume V = 1.

The Feynman electron propagator (3.2-3) may be calculated by using (3.2-5) and noting that the only relevant non-zero vacuum expectation values are

**<°l **sa (j.) ( p ’) **|0> = (0| ba (p) **b + , ( p ' ) |0> = **(2tt) **3 «a a .« ( 3 ) ( p - p ' ) •
(3.2-6)
T his g ives
G (x - x') = - i
F
*A* 3
d p

r: 2 I« (p) u (p) **e ( t - t ' ) ** exp [-ip (x - x') ]

**(2**

**tt**

**) 3 2e **

**'**** a**

- v a (p) v a (p) 6(t' - t) e x p [ i p ( x - x 1)]} , (3.2-7) w i t h p° = £. The s u m o v e r the spin sta t e s in (3.2-7) m a y be p e r f o r m e d u s i n g the c o m p l e t e n e s s relat i o n s 2 u a (p) ua (p) = * 5+m and 2 v (p) v (p) =

**i**

- m. (3.2-8a,b)
**i**

**a **

**~ **

**~ **

**a **

**~ **

**~**

T h i s g ives
G (x - x ' )
F
w i t h p° = £ .
**d £ 1 r**

7— 0 ( t - t f) exp [-ip (x - x ’ ) ]
( 2 **TT )**

*^*

+

**(~i>**

+ m) 6 ( t T - t) exp [ip (x - x ' ) ]} ,
(3.2-9)
**(~i>**

The F e y n m a n e l e c t r o n p r o p a g a t o r (3.2-9) m a y be d e r i v e d d i r e c t l y f r o m (3.2-2) by F o u r i e r t r a n s f o r m i n g to m o m e n t u m space, w h i c h

c o r r e s p o n d s to e x p a n d i n g in plane w a v e s . S o l v i n g the r e s u l t i n g a l g e b r a i c e q u a t i o n for the e l e c t r o n p r o p a g a t o r gives

**c(p) = — t - = - f - 1" ■ **

**(3.2-10)**

p - m p - m

T h e four c o m p o n e n t s of the 4-ve c t o r p ^ in (3.2-10) are i n d e p e n d e n t
**0** **2** **2**

v a r i a b l e s , n o t n e c e s s a r i l y rel a t e d b y (p ) = £ .

The b o u n d a r y c o n d i t i o n s for the e l e c t r o n p r o p a g a t o r are e q u i v a l e n t to p r e s c r i p t i o n s for i n t e g r a t i n g a r o u n d the poles in

coordinate space. The Feynman propagator, which corresponds to a

choice of boundary conditions such that for t > t ’, G(x-x')

corresponds to a positive-energy electron propagating forward in time

from r T to r, while for t < t ’, G(x-x') corresponds to a negative- energy electron propagating backwards in time from r' to r is found by

passing above the pole at p ° = £ and below the pole at p° =-£, as in figure 3-1, when performing the p°-integration.

Figure 3-1: The Feynman contour in the complex p°-plane.

Alternatively, the p°-integration may be performed everywhere along the real axis, but with the mass m of the particle given an

infinitesimal negative imaginary part, that is, with

m *-*■* m - iO . (3.2-11)

With this prescription (Feynman’s rule for avoiding the poles) the poles at p° = ±£ are moved off the real axis (see figure 3-2) and integration along this axis is equivalent to integration along the contour of figure 3-1.

p° = -£ + iO

---**>■**---**>**--- **>**---

p° = £ - iO

Figure 3-2: Feynman’s rule for avoiding the poles in the complex p°-plane.

The Feynman field-free electron propagator is given in momentum space by

g f(p) ^ + m p 2 - m 2 + i 0

### (

**3**

### .

**2**

### -

**12**

### )

and in coordinate space by

**Gf(x** **-x’)**

**d4p_____**

**(2tt) 4 (p2 **- m 2 + iO)
+ m )

exp [-ip (x - x ’) ] . (3.2-13)

For t ^ ’, closing the Feynman contour in the lower half-plane includes the pole at p° = £ and gives

d 3p (YUe - Y.p + m )

— 21--- exp [-ie(t - t ’) + ij3. **(x **- x ' ) ]
(3.2-14a)
while, for t < t T, closing the contour in the upper half-plane yields

r d 3p (-Y°e - Y.p + m)
**g f(x** **-xt) - - i I** * --- -— --- *exp[ie(t

**- t ')**+ ip.

**(x - x') ]**G (x - x ’) = - i I 3

**F**

**J****(2tt) 3**

**(2tt)**(3.2-14b) The results (3.2-14a,b) may also be derived from (3.2-13) by using the Plemelj formula (2.3-2). A change of integration variable in

(3.2-14b) from p to -p allows one to write (3.2-14a,b) in the form (3.2-9). This demonstrates that the propagator defined by (3.2-3) is a solution of (3.2-2) with boundary conditions corresponding to

F e y n m a n ’s rule for avoiding the poles.

An alternative representation of the Feynman field-free electron propagator may be obtained from (3.2-13) by using the integral

representation

**1**

(p2 - m 2 + i 0 )

ds e x p [ i s ( p 2 - m 2 + i O ) ] (3.2-15)

Gp (x - x') ds

**16tt2s2**

exp(~im2s) * { * - * ' ) +* m

x exp

**47 (x- x ')2**

(3.2-16)
with Feynman's rule for avoiding the poles understood. The

integration variable s in (3.2-16) plays the role of the proper-time.