8 1. CONCEPTUAL APPROACHES TO SPACETIME
so that d D u D vt. e coordinates assigned to the traveler in a co-moving frame evolve
as
x
0
D
ct
0
; x
0
D .c; 0/
and so the elapsed time registered on the traveler’s clock is D t=. is result is consistent
with the usual presentation of the twin scenario.
1.3 THE “PROPER TIME” FORMALISM IN QED
Although this book focuses on relativistic classical mechanics, we make a brief digression into
the application of spacetime parameterization methods by Schwinger and Feynman in devel-
oping quantum electrodynamics. In his 1951 calculation of vacuum polarization in an external
electromagnetic field, Schwinger [10] represented the Green’s function for the Dirac field as
a parametric integral and formally transformed the Dirac problem into a dynamical theory in
which the integration variable acts as an independent time. Applying his method to the Klein–
Gordon equation, we express the Green’s function as
G D
1
.p eA=c/
2
C m
2
i
so that writing
G.x; x
0
/ D hxjGjx
0
i D i
Z
1
0
dse
i.m
2
i/s
hxje
i.peA=c/
2
s
jx
0
i (1.3)
the function
G.x; x
0
Is/ D hx.s/jx
0
.0/ihxje
i.peA=c/
2
s
jx
0
i
satisfies
i
@
@s
hx.s/jx
0
.0/i D
p
e
c
A
2
hx.s/jx
0
.0/i (1.4)
with the boundary condition
lim
s!0
hx.s/jx
0
.0/i D ı
4
.x x
0
/:
Schwinger regarded x
.s/ and
.s/ D p
.s/
e
c
A
.s/ as operators in a Heisenberg picture
that satisfy canonical relations
Œx
;
D i
Œ
;
D
ie
c
F
(1.5)
iŒx
; K D
dx
ds
iŒ
; K D
d
ds
; (1.6)
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