The Wheeler–Feynman absorber theory (also
called the Wheeler–Feynman time-symmetric
theory), named after its originators, the
physicists Richard Feynman and John Archibald
Wheeler, is an interpretation of electrodynamics
derived from the assumption that the solutions
of the electromagnetic field equations must
be invariant under time-reversal transformation,
as are the field equations themselves. Indeed,
there is no apparent reason for the time-reversal
symmetry breaking, which singles out a preferential
time direction and thus makes a distinction
between past and future. A time-reversal invariant
theory is more logical and elegant. Another
key principle, resulting from this interpretation
and reminiscent of Mach's principle due to
Tetrode, is that elementary particles are
not self-interacting. This immediately removes
the problem of self-energies.
== T-symmetry and causality ==
The requirement of time-reversal symmetry,
in general, is difficult to conjugate with
the principle of causality. Maxwell's equations
and the equations for electromagnetic waves
have, in general, two possible solutions:
a retarded (delayed) solution and an advanced
one. Accordingly, any charged particle generates
waves, say at time
t
0
=
0
{\displaystyle t_{0}=0}
and point
x
0
=
0
{\displaystyle x_{0}=0}
, which will arrive at point
x
1
{\displaystyle x_{1}}
at the instant
t
1
=
x
1
/
c
{\displaystyle t_{1}=x_{1}/c}
(here
c
{\displaystyle c}
is the speed of light), after the emission
(retarded solution), and other waves, which
will arrive at the same place at the instant
t
2
=
−
x
1
/
c
{\displaystyle t_{2}=-x_{1}/c}
, before the emission (advanced solution).
The latter, however, violates the causality
principle: advanced waves could be detected
before their emission. Thus the advanced solutions
are usually discarded in the interpretation
of electromagnetic waves. In the absorber
theory, instead charged particles are considered
as both emitters and absorbers, and the emission
process is connected with the absorption process
as follows: Both the retarded waves from emitter
to absorber and the advanced waves from absorber
to emitter are considered. The sum of the
two, however, results in causal waves, although
the anti-causal (advanced) solutions are not
discarded a priori.
Feynman and Wheeler obtained this result in
a very simple and elegant way. They considered
all the charged particles (emitters) present
in our universe and assumed all of them to
generate time-reversal symmetric waves. The
resulting field is
E
tot
(
x
,
t
)
=
∑
n
E
n
ret
(
x
,
t
)
+
E
n
adv
(
x
,
t
)
2
.
{\displaystyle E_{\text{tot}}(\mathbf {x}
,t)=\sum _{n}{\frac {E_{n}^{\text{ret}}(\mathbf
{x} ,t)+E_{n}^{\text{adv}}(\mathbf {x} ,t)}{2}}.}
Then they observed that if the relation
E
free
(
x
,
t
)
=
∑
n
E
n
ret
(
x
,
t
)
−
E
n
adv
(
x
,
t
)
2
=
0
{\displaystyle E_{\text{free}}(\mathbf {x}
,t)=\sum _{n}{\frac {E_{n}^{\text{ret}}(\mathbf
{x} ,t)-E_{n}^{\text{adv}}(\mathbf {x} ,t)}{2}}=0}
holds, then
E
free
{\displaystyle E_{\text{free}}}
, being a solution of the homogeneous Maxwell
equation, can be used to obtain the total
field
E
tot
(
x
,
t
)
=
∑
n
E
n
ret
(
x
,
t
)
+
E
n
adv
(
x
,
t
)
2
+
∑
n
E
n
ret
(
x
,
t
)
−
E
n
adv
(
x
,
t
)
2
=
∑
n
E
n
ret
(
x
,
t
)
.
{\displaystyle E_{\text{tot}}(\mathbf {x}
,t)=\sum _{n}{\frac {E_{n}^{\text{ret}}(\mathbf
{x} ,t)+E_{n}^{\text{adv}}(\mathbf {x} ,t)}{2}}+\sum
_{n}{\frac {E_{n}^{\text{ret}}(\mathbf {x}
,t)-E_{n}^{\text{adv}}(\mathbf {x} ,t)}{2}}=\sum
_{n}E_{n}^{\text{ret}}(\mathbf {x} ,t).}
The total field is retarded, and causality
is not violated.
The assumption that the free field is identically
zero is the core of the absorber idea. It
means that the radiation emitted by each particle
is completely absorbed by all other particles
present in the universe. To better understand
this point, it may be useful to consider how
the absorption mechanism works in common materials.
At the microscopic scale, it results from
the sum of the incoming electromagnetic wave
and the waves generated from the electrons
of the material, which react to the external
perturbation. If the incoming wave is absorbed,
the result is a zero outgoing field. In the
absorber theory the same concept is used,
however, in presence of both retarded and
advanced waves.
The resulting wave appears to have a preferred
time direction, because it respects causality.
However, this is only an illusion. Indeed,
it is always possible to reverse the time
direction by simply exchanging the labels
emitter and absorber. Thus, the apparently
preferred time direction results from the
arbitrary labelling.
== T-symmetry and self-interaction ==
One of the major results of the absorber theory
is the elegant and clear interpretation of
the electromagnetic radiation process. A charged
particle that experiences acceleration is
known to emit electromagnetic waves, i.e.,
to lose energy. Thus, the Newtonian equation
for the particle (
F
=
m
a
{\displaystyle F=ma}
) must contain a dissipative force (damping
term), which takes into account this energy
loss. In the causal interpretation of electromagnetism,
Lorentz and Abraham proposed that such a force,
later called Abraham–Lorentz force, is due
to the retarded self-interaction of the particle
with its own field. This first interpretation,
however, is not completely satisfactory, as
it leads to divergences in the theory and
needs some assumptions on the structure of
charge distribution of the particle. Dirac
generalized the formula to make it relativistically
invariant. While doing so, he also suggested
a different interpretation. He showed that
the damping term can be expressed in terms
of a free field acting on the particle at
its own position:
E
damping
(
x
j
,
t
)
=
E
j
ret
(
x
j
,
t
)
−
E
j
adv
(
x
j
,
t
)
2
.
{\displaystyle E^{\text{damping}}(\mathbf
{x} _{j},t)={\frac {E_{j}^{\text{ret}}(\mathbf
{x} _{j},t)-E_{j}^{\text{adv}}(\mathbf {x}
_{j},t)}{2}}.}
However, Dirac did not propose any physical
explanation of this interpretation.
A clear and simple explanation can instead
be obtained in the framework of absorber theory,
starting from the simple idea that each particle
does not interact with itself. This is actually
the opposite of the first Abraham–Lorentz
proposal. The field acting on the particle
j
{\displaystyle j}
at its own position (the point
x
j
{\displaystyle x_{j}}
) is then
E
tot
(
x
j
,
t
)
=
∑
n
≠
j
E
n
ret
(
x
j
,
t
)
+
E
n
adv
(
x
j
,
t
)
2
.
{\displaystyle E^{\text{tot}}(\mathbf {x}
_{j},t)=\sum _{n\neq j}{\frac {E_{n}^{\text{ret}}(\mathbf
{x} _{j},t)+E_{n}^{\text{adv}}(\mathbf {x}
_{j},t)}{2}}.}
If we sum the free-field term of this expression,
we obtain
E
tot
(
x
j
,
t
)
=
∑
n
≠
j
E
n
ret
(
x
j
,
t
)
+
E
n
adv
(
x
j
,
t
)
2
+
∑
n
E
n
ret
(
x
j
,
t
)
−
E
n
adv
(
x
j
,
t
)
2
{\displaystyle E^{\text{tot}}(\mathbf {x}
_{j},t)=\sum _{n\neq j}{\frac {E_{n}^{\text{ret}}(\mathbf
{x} _{j},t)+E_{n}^{\text{adv}}(\mathbf {x}
_{j},t)}{2}}+\sum _{n}{\frac {E_{n}^{\text{ret}}(\mathbf
{x} _{j},t)-E_{n}^{\text{adv}}(\mathbf {x}
_{j},t)}{2}}}
and, thanks to Dirac's result,
E
tot
(
x
j
,
t
)
=
∑
n
≠
j
E
n
ret
(
x
j
,
t
)
+
E
damping
(
x
j
,
t
)
.
{\displaystyle E^{\text{tot}}(\mathbf {x}
_{j},t)=\sum _{n\neq j}E_{n}^{\text{ret}}(\mathbf
{x} _{j},t)+E^{\text{damping}}(\mathbf {x}
_{j},t).}
Thus, the damping force is obtained without
the need for self-interaction, which is known
to lead to divergences, and also giving a
physical justification to the expression derived
by Dirac.
== Criticism ==
The Abraham–Lorentz force is, however, not
free of problems. Written in the non-relativistic
limit, it gives
E
damping
(
x
j
,
t
)
=
e
6
π
c
3
d
3
d
t
3
x
.
{\displaystyle E^{\text{damping}}(\mathbf
{x} _{j},t)={\frac {e}{6\pi c^{3}}}{\frac
{\mathrm {d} ^{3}}{\mathrm {d} t^{3}}}x.}
Since the third derivative with respect to
the time (also called the "jerk" or "jolt")
enters in the equation of motion, to derive
a solution one needs not only the initial
position and velocity of the particle, but
also its initial acceleration. This apparent
problem, however, can be solved in the absorber
theory by observing that the equation of motion
for the particle has to be solved together
with the Maxwell equations for the field.
In this case, instead of the initial acceleration,
one only needs to specify the initial field
and the boundary condition. This interpretation
restores the coherence of the physical interpretation
of the theory.
Other difficulties may arise trying to solve
the equation of motion for a charged particle
in the presence of this damping force. It
is commonly stated that the Maxwell equations
are classical and cannot correctly account
for microscopic phenomena, such as the behavior
of a point-like particle, where quantum-mechanical
effects should appear. Nevertheless, with
absorber theory, Wheeler and Feynman were
able to create a coherent classical approach
to the problem (see also the "paradoxes" section
in the Abraham–Lorentz force).
Also, the time-symmetric interpretation of
the electromagnetic waves appears to be in
contrast with the experimental evidence that
time flows in a given direction and, thus,
that the T-symmetry is broken in our world.
It is commonly believed, however, that this
symmetry breaking appears only in the thermodynamical
limit (see, for example, the arrow of time).
Wheeler himself accepted that the expansion
of the universe is not time-symmetric in the
thermodynamic limit. This, however, does not
imply that the T-symmetry must be broken also
at the microscopic level.
Finally, the main drawback of the theory turned
out to be the result that particles are not
self-interacting. Indeed, as demonstrated
by Hans Bethe, the Lamb shift necessitated
a self-energy term to be explained. Feynman
and Bethe had an intense discussion over that
issue, and eventually Feynman himself stated
that self-interaction is needed to correctly
account for this effect.
== Developments since original formulation
==
=== Gravity theory ===
Inspired by the Machian nature of the Wheeler–Feynman
absorber theory for electrodynamics, Fred
Hoyle and Jayant Narlikar proposed their own
theory of gravity in the context of general
relativity. This model still exists in spite
of recent astronomical observations that have
challenged the theory. Stephen Hawking had
criticized the original Hoyle-Narlikar theory
believing that the advanced waves going off
to infinity would lead to a divergence, as
indeed they would, if the universe were only
expanding. However, as emphasized in the revised
version of the Hoyle-Narlikar theory devoid
of the "Creation Field" (generating matter
out of empty space) known as the Gravitational
absorber theory, the universe is also accelerating
in that expansion. The acceleration leads
to a horizon type cutoff and hence no divergence.
Gravitational absorber theory has been used
to explain the mass fluctuations in the Woodward
effect (see section on Woodward effect below).
=== Transactional interpretation of quantum
mechanics ===
Again inspired by the Wheeler–Feynman absorber
theory, the transactional interpretation of
quantum mechanics (TIQM) first proposed in
1986 by John G. Cramer, describes quantum
interactions in terms of a standing wave formed
by retarded (forward-in-time) and advanced
(backward-in-time) waves. Cramer claims it
avoids the philosophical problems with the
Copenhagen interpretation and the role of
the observer, and resolves various quantum
paradoxes, such as quantum nonlocality, quantum
entanglement and retrocausality.
=== Shu-Yuan Chu's quantum theory in the presence
of gravity ===
In 1993, Chu developed a model of how to do
quantum mechanics in the presence of gravity,
which combines some of the latest ideas in
particle physics, superstrings, and a time-symmetric
Wheeler–Feynman description of gravity and
inertia. In 1998 he extended this work to
derive Einstein's equation for the "adjunct
gravitational field" using concepts from statistics
and maximizing the entropy.
=== Attempted resolution of causality ===
T. C. Scott and R. A. Moore demonstrated that
the apparent acausality suggested by the presence
of advanced Liénard–Wiechert potentials
could be removed by recasting the theory in
terms of retarded potentials only, without
the complications of the absorber idea.
The Lagrangian describing a particle (
p
1
{\displaystyle p_{1}}
) under the influence of the time-symmetric
potential generated by another particle (
p
2
{\displaystyle p_{2}}
) is
L
1
=
T
1
−
1
2
(
(
V
R
)
1
2
+
(
V
A
)
1
2
)
,
{\displaystyle L_{1}=T_{1}-{\frac {1}{2}}\left((V_{R})_{1}^{2}+(V_{A})_{1}^{2}\right),}
where
T
i
{\displaystyle T_{i}}
is the relativistic kinetic energy functional
of particle
p
i
{\displaystyle p_{i}}
, and
(
V
R
)
i
j
{\displaystyle (V_{R})_{i}^{j}}
and
(
V
A
)
i
j
{\displaystyle (V_{A})_{i}^{j}}
are respectively the retarded and advanced
Liénard–Wiechert potentials acting on particle
p
i
{\displaystyle p_{i}}
and generated by particle
p
j
{\displaystyle p_{j}}
. The corresponding Lagrangian for particle
p
2
{\displaystyle p_{2}}
is
L
2
=
T
2
−
1
2
(
(
V
R
)
2
1
+
(
V
A
)
2
1
)
.
{\displaystyle L_{2}=T_{2}-{\frac {1}{2}}\left((V_{R})_{2}^{1}+(V_{A})_{2}^{1}\right).}
It was originally demonstrated with computer
algebra and then proven analytically that
(
V
R
)
j
i
−
(
V
A
)
i
j
{\displaystyle (V_{R})_{j}^{i}-(V_{A})_{i}^{j}}
is a total time derivative, i.e. a divergence
in the calculus of variations, and thus it
gives no contribution to the Euler–Lagrange
equations. Thanks to this result the advanced
potentials can be eliminated; here the total
derivative plays the same role as the free
field. The Lagrangian for the N-body system
is therefore
L
=
∑
i
=
1
N
T
i
−
1
2
∑
i
≠
j
N
(
V
R
)
j
i
.
{\displaystyle L=\sum _{i=1}^{N}T_{i}-{\frac
{1}{2}}\sum _{i\neq j}^{N}(V_{R})_{j}^{i}.}
The resulting Lagrangian is symmetric under
the exchange of
p
i
{\displaystyle p_{i}}
with
p
j
{\displaystyle p_{j}}
. For
N
=
2
{\displaystyle N=2}
this Lagrangian will generate exactly the
same equations of motion of
L
1
{\displaystyle L_{1}}
and
L
2
{\displaystyle L_{2}}
. Therefore, from the point of view of an
outside observer, everything is causal. This
formulation reflects particle-particle symmetry
with the variational principle applied to
the N-particle system as a whole, and thus
Tetrode's Machian principle. Only if we isolate
the forces acting on a particular body do
the advanced potentials make their appearance.
This recasting of the problem comes at a price:
the N-body Lagrangian depends on all the time
derivatives of the curves traced by all particles,
i.e. the Lagrangian is infinite-order. However,
much progress was made in examining the unresolved
issue of quantizing the theory. Also, this
formulation recovers the Darwin Lagrangian,
from which the Breit equation was originally
derived, but without the dissipative terms.
This ensures agreement with theory and experiment,
up to but not including the Lamb shift. Numerical
solutions for the classical problem were also
found. Furthermore, Moore showed that a model
by Feynman and Hibbs is amenable to the methods
of higher than first-order Lagrangians and
revealed chaoticlike solutions. Moore and
Scott showed that the radiation reaction can
be alternatively derived using the notion
that, on average, the net dipole moment is
zero for a collection of charged particles,
thereby avoiding the complications of the
absorber theory. An important bonus from their
approach is the formulation of a total preserved
canonical generalized momentum, as presented
in a comprehensive review article in the light
of quantum nonlocality.This apparent acausality
may be viewed as merely apparent, and this
entire problem goes away. An opposing view
was held by Einstein.
=== Alternative Lamb shift calculation ===
As mentioned previously, a serious criticism
against the absorber theory is that its Machian
assumption that point particles do not act
on themselves does not allow (infinite) self-energies
and consequently an explanation for the Lamb
shift according to quantum electrodynamics
(QED). Ed Jaynes proposed an alternate model
where the Lamb-like shift is due instead to
the interaction with other particles very
much along the same notions of the Wheeler–Feynman
absorber theory itself. One simple model is
to calculate the motion of an oscillator coupled
directly with many other oscillators. Jaynes
has shown that it is easy to get both spontaneous
emission and Lamb shift behavior in classical
mechanics. Furthermore, Jayne's alternatives
provides a solution to the process of "addition
and subtraction of infinities" associated
with renormalization.This model leads to essentially
the same type of Bethe logarithm an essential
part of the Lamb shift calculation vindicating
Jaynes' claim that two different physical
models can be mathematically isomorphic to
each other and therefore yield the same results,
a point also apparently made by Scott and
Moore on the issue of causality.
== Woodward effect ==
The Woodward effect is a physical hypothesis
about the possibility for a body to see its
mass change when the energy density varies
in time. Proposed in 1990 by James Woodward,
the effect is based on a formulation of Mach's
principle proposed in 1953 by Dennis Sciama.If
confirmed experimentally (see timeline of
results in the main article), the Woodward
effect would open pathways in astronautics
research, as it could be used to propel a
spacecraft by propellantless propulsion meaning
that it would not have to expel matter to
accelerate. As previously formulated by Sciama,
Woodward suggests that the Wheeler–Feynman
absorber theory would be the correct way to
understand the action of instantaneous inertial
forces in Machian terms.
== Conclusions ==
This universal absorber theory is mentioned
in the chapter titled "Monster Minds" in Feynman's
autobiographical work Surely You're Joking,
Mr. Feynman! and in Vol. II of the Feynman
Lectures on Physics. It led to the formulation
of a framework of quantum mechanics using
a Lagrangian and action as starting points,
rather than a Hamiltonian, namely the formulation
using Feynman path integrals, which proved
useful in Feynman's earliest calculations
in quantum electrodynamics and quantum field
theory in general. Both retarded and advanced
fields appear respectively as retarded and
advanced propagators and also in the Feynman
propagator and the Dyson propagator. In hindsight,
the relationship between retarded and advanced
potentials shown here is not so surprising
in view of the fact that, in field theory,
the advanced propagator can be obtained from
the retarded propagator by exchanging the
roles of field source and test particle (usually
within the kernel of a Green's function formalism).
In field theory, advanced and retarded fields
are simply viewed as mathematical solutions
of Maxwell's equations whose combinations
are decided by the boundary conditions.
== See also ==
Causality
Symmetry in physics and T-symmetry
Transactional interpretation
Abraham–Lorentz force
Retrocausality
Two-state vector formalism
Paradox of a charge in a gravitational field
Ni Guangjiong
== Notes ==
== Sources ==
Wheeler, J. A.; Feynman, R. P. (April 1945).
"Interaction with the Absorber as the Mechanism
of Radiation" (PDF). Reviews of Modern Physics.
17 (2–3): 157–181. Bibcode:1945RvMP...17..157W.
doi:10.1103/RevModPhys.17.157.
Wheeler, J. A.; Feynman, R. P. (July 1949).
"Classical Electrodynamics in Terms of Direct
Interparticle Action" (PDF). Reviews of Modern
Physics. 21 (3): 425–433. Bibcode:1949RvMP...21..425W.
doi:10.1103/RevModPhys.21.425.
