The field-free electron propagator

In document Vacuum polarization in a static uniform electromagnetic field with applications to pulsars (Page 80-85)

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') 10 > , (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

z

1 4 (p ) V. eiPX + ea (£> P_ipx (3.2-5) (27T) 3 (2e) 2h 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)

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)

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.

3.3

D i r a c ’s equation in the presence of a

In document Vacuum polarization in a static uniform electromagnetic field with applications to pulsars (Page 80-85)