4.8. SPEEDS OF LIGHT AND THE MAXWELL LIMIT 93
4.8 SPEEDS OF LIGHT AND THE MAXWELL LIMIT
As discussed in Section 3.7, concatenation—integration of the pre-Maxwell field equations over
the evolution parameter —extracts from the microscopic event interactions the massless modes
in Maxwell electrodynamics, expressing a certain equilibrium limit when mass exchange settles
to zero. In this picture, the microscopic dynamics approach an equilibrium state because the
boundary conditions hold pointwise in x as ! 1, asymptotically eliminating interactions that
cannot be described in Maxwell theory. e Maxwell-type description recovered by concatenat-
ing the microscopic dynamics may thus be understood as a self-consistent summary constructed
a posteriori from the complete worldlines.
We have assumed that 0 c
5
< c and we must check that SHP theory remains finite
as c
5
! 0. First we notice that c
5
appears explicitly three times in the pre-Maxwell equations
(3.20)
@
f
1
c
5
@
f
5
D
e
c
j
'
@
f
5
D
e
c
j
5
'
D
c
5
c
e
'
@
f
C @
f
C @
f
D 0 @
f
5
@
f
5
C
1
c
5
@
f
D 0
twice in the form
1
c
5
@
and once multiplying the event density
'
. e derivative term poses
no problem in the homogeneous pre-Maxwell equation, which is satisfied identically for fields
derived from potentials. Specifically, the fields f
5
contain terms of the type @
5
a
D
1
c
5
@
a
that
cancel the explicit -derivative of f
, evaluated before passing to the limit c
5
! 0. However,
the homogeneous equation does impose a new condition through
c
5
@
f
5
@
f
5
C @
f
D 0 !
c
5
!0
@
f
D 0
requiring that the field strength f
become -independent in this limit. For the fields derived
in Section 4.2 this condition is violated by the multiplicative factor '.
R
/ unless we simul-
taneously require c
5
! 0 ) 1=c
5
! 1, in which case '.x; / ! 1=2 D 1, using (3.12)
for . is requirement effectively spreads the event current j
˛
'
uniformly along the particle
worldline, recovering the -independent particle current
j
'
.
x;
/
D
Z
ds '
.
s
/
j
.
x; s
/
!
Z
ds 1 j
.
x; s
/
D J
.x/
j
5
'
.
x;
/
D
Z
ds '
.
s
/
j
5
.
x; s
/
!
Z
ds j
5
.
x; s
/
@
j
'
.
x;
/
C
1
c
5
@
j
5
'
.
x;
/
! @
J
.
x
/
D 0
associated with Maxwell theory. Generally, because the -dependence of the potentials and fields
is contained in ', the condition ! 1 eliminates all the terms in the pre-Maxwell equations
containing @
. Similarly, the photon mass m
„=c
2
must vanish.
94 4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS
We saw that f
5
is generally proportional to c
5
for fields of the Liénard–Wiechert type.
erefore, we can write the inhomogeneous pre-Maxwell equations in the finite form
@
f
D
e
c
j
'
@
1
c
5
f
5
D
e
c
'
;
where we see that f
5
decouples from the field f
that now satisfies Maxwell’s equations.
To find the limiting form of the electromagnetic interactions, we consider an arbitrary
event X
.
/
, which induces the current
j
˛
'
.
x;
/
D c
Z
ds '
.
s
/
P
X
˛
.
s
/
ı
4
Œ
x X
.
s
/
:
From the field strengths found in Section 4.2 the Lorentz force on a test event moving in the
field induced by this current can be written
M Rx
D
e
c
f
.x; / Px
C f
5
.x; / Px
5
D
e
2
4c
e
j
R
j
=
F
.x; / Px
C c
2
5
F
5
.x; /
1 C
.
c
5
=c
/
2
;
where
F
.x; / D
e
4R
.
z
ˇ
z
ˇ
/
ˇ
2
R
2
"
.
R
/
c
z
ˇ
z
ˇ
R
z
P
ˇ
z
P
ˇ
R C
.
z
ˇ
z
ˇ
/
P
ˇ z
R
2
9
=
;
F
5
.x; / D
e
4cR
z
ˇ
2
C ˇ
R
R
2
"
.
R
/
c
z
C ˇ
Rc
2
=c
2
5
R
C
z
P
ˇ z
cR
2
9
=
;
:
In the limit ! 1 and c
5
! 0, we see that c
2
5
F
5
.x; / ! 0, and so the Lorentz force inter-
action reduces to the -independent expression
M Rx
D
e
2
4c
F
.x/ Px
recovering the Lorentz force in the standard Maxwell form. e parameter c
5
=c thus provides
a continuous scaling of Maxwell’s equations and the Lorentz force to the standard forms in
Maxwell theory. e combined limit ! 1 and c
5
! 0 restricts the possible dynamics in SHP
to those of Maxwell theory, as a system in -equilibrium [9].
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