of a direct link between the quantum and the classical description ... QED like any other quantum theory, in the words of Bohm ...... from his book [21].
-~~-
Acta Physica Austriaca, Supp!. XVIII, 111-151 (1977) © by Springer-Verlag 1977
~ ~
r
~l
CLASSICAL
LIMIT OF QUANTUM
ELECTRODYNAMICS+
by
I. BIALYNICKI-BIRULA Inst. of Theoretical Warsaw,
Physics
Poland
and Department Univ.
of Physics
of Pittsburgh,
Pittsburgh,PA.
CONTENTS
1. Introduction 2. Classical
limit of nonrelativistic
2.1. Hydrodynamic
mechanics
formulation
2.2. Weyl-Wigner-Moyal 3. Classical
quantum
transforms
limit of relativistic
quantum
mechanics
4. Classical limit of the quantum theory of the EM field 4.1. Classical limit in terms of photon number states
.-.....-I~ ~"~_, 'j
n
4.2. Classical
limit in terms of coherent
4.3. Classical
limit and the low-frequency
+ Lecture
given at XVI.lnternationale Kernphysik,Schladming,Austria,February
states limit
Universitatswochen fUr 24-March 5, 1977.
,...
wu
.._.
112
\~
113
1. INTRODUCTION The relationship electrodynamics
between
is a complex
not all of which
what
to compare.
as a perturbation
Quantum
series
tion scheme
strong
which
indications
from these
,
there is no systematic
now
of energy
approximare-
po-
physical
in-
and indubitable
that after
counterpart, shape,
causality,
A very good historical
review
of the funda-
problems
of classical
electrodynamics
can be found
article
by Rohrlich
is the correct
link between
description
only emphasizes
approach
is easily
[2], "presupposes
statements
we should
charged
particles.
dynamics,
the relationship
classical extreme derstood
theory
an external
with
external
simplified
mechanics
models
versions
the quantum
to uncover.
of charged
There
this level". rather
Accepting than abandon
level of QED, especially
in
of this theory.
of quanta
and relativity
that
of QED and it is precisely
the com-
I believe
of quantum
effects
(controlled
constant
n) and relativistic
velocity
of light c) that makes
effects
by the Planck
(controlled
by the
and the QED so difficult
can be fully un-
moving
in
is the
the classical
limit of
I
•
II !
c,
!
fine structure framework
since
II
1\
it implies
formula
of the quantum classical +
constant
the simple a
theory,
theory
->-
00.
0, the laws of quantum
"n
in powers 2 a = e /nc.
->-
0" rule
Such statements
as:
tr tend to zero in any
one obtains
formula"
calculational
an expansion
that if one makes
ponding when tr
\.
- means
In this perturbative
"It is well known
that these two greatly
theory
The only available
of the dimensionless
does not work
field interacting
some aspects
to reach.
tool - perturbation
One is the
particles
do help us to understand
bination of electro-
are two
field and the other
of the electromagnetic sources.
level and the correctness
as complete
and mathematically.
electromagnetic
theory
truncated
the relationship
both physically
of Bohm
field and interacting
between
is easier
cases in which
nonrelativistic
Maxwell
In various
that
electrodynamics
understood
of the electromagnetic
in the words
intensify
for the classical
led to the emergence theories
by observing
[1].
apply to quantum
electrodynamics
theory,
describing
It was the fusion and classical
and the classical
however,
a classical
concepts
and that lack
this fact. Such a pragmatic
refuted,
view of all the successes These
the very nature
theory
the quantum
QED like any other quantum
the search in a recent
possible
the need for this understanding
of a direct
this proposition, mental
and it is quite
all it is QED, not its classical
which
of classical terpretation.
quantum that there
that can be learned
far from understanding
One may question arguing
have at the same time
invariance,
two simple models
that we are still
that
to give finite
the fullfledged
It seems, however,
of this relationship.
we are
exists
between
theories.
is much more to that relationship
,
Clearly,
is not in a much better
of relativistic
definitness
understood.
electrodynamics
sults in each step and which would
sitive
aspects,
since we do not even know
that would be guaranteed
the properties
with many
is at best only asymptotic.
electrodynamics
since to my knowledge,
well
and classical
of the two theories
theory, with
the perturbation Classical
subject,
problem,
is the true content
supposed
quantum
are at present
it can not be a simple
of the relationship and classical
the corres-
[3] or "In the limit,
mechanics
must reduce
114
115
to those of classical applicable
mechanics"
to QED, because
QED in its present large values
it is very unlikely
form can be extended
with higher
spin is also a source
the classical
limit. Contrary
originated
with Pauli
not necessarilly problem
with the classical
disappear
clarified
by de Broglie
affect
precisely
in wave mechanics
in
analysis
problems
the cross
of a particle
there are strong For example,
indications
backward
Historically,
of the relativ-
the ratio of the differential
classical
that the spin
himself
limit, but
cross
MECHANICS
the first example
of the classical
theory was, of course,
the transition
law for the black body radiation formula.
[8] "for a vanishingly fi, the general
In the words
small value
to the
of Planck
of the quantum
formula
section E)..
2 c n 1 -5- ---cn ).. exp(k)..T)-l
(2 )
into Raleigh's
formula"
( 1)
in the differential predicted
is clearly
a systematic
is
a 2 1 - (- sin -) c 2
1/2 particles,
after
LIMIT OF NONRELATIVISTIC
Raleigh-Jeans
of action,
to that of a spinless
v
direction,
in
that this indeed happens.
do(O) in the Born approximation
The decrease
approximation
is
degenerates do (1/2) do (0)
of the
limit in the nonrelativistic
QUANTUM
is studied
in the. classical
do (1/2) for a spin 1/2 particle particle
of validity
to these problems
2. CLASSICAL
in the
in all scattering
we can not prove
section
and the classical
limit,
[7], who have
from the Planck
influences
the regions
of the classical
limit of a quantum
istic scattering
in the classical
[6] and later
the trajectory
of exact solutions
from here that the
theory.
problems. In the absence
conclusion
However,
do not overlap.
We will return
opinion,
on nuclei.
limit. This
and Keller
with
general
in a.
(of the order of l/fi). This
what we are dealing
electrons
the scattering
Born approximation
of spin do
in the classical
limit if the motion
over large distances
terms
to the widespread
by Rubinov
shown that the spin may classical
spin influences
limit of
of complications
[5], the effects
was first discussed
was further
order
of relativistic
we can not draw a definite
to arbitrarily
because
in connection
The electron
which
scattering
that
of a.
Not all the problems QED arise
[4] are just not
cross
section
by this formula
seen in experiments
ckT
E)..
o
This
first example
( 3)
in the for spin on the
..
~
feature
exhibits
of the classical
comparing
very clearly
limit. Namely,
the same physical
quantity
one important we should be
(the spectral
energy
117
116
density)
calculated
according
to two different
It is only in this restricted term classical between
limit,
always
the same concept
The second correspondence
this principle the intensities equal
of quantum
of spectral
to the intensity
the same situation characteristic theories
mechanics.
lines become
Fourier
as before
According
computed
to
numbers
derived
riables:
the probability
defined
through
charged
particle
dynamic
the study of the classical
va-
p and the velocity
moving
current,
in an external
relation
between
S/fi of the wave
j
=
v
+
pv. For a
EM field we ob-
the real amplitude
function
and the hydro-
variables
R2
p
(4 )
from the two v
introduction
density
the probability
and the phase
+
this historical
wave
we have
are compared.
After
the complex
by the set of four real hydrodynamic-like
R
that the same physical
(light intensity)
was dis-
function
tain the following
classically
Again
[11], we replace
which
+
asymptotically
amplitude".
by Madelung
formulation,
+
a crucial
of high quantum
of radiation
from the corresponding
covered
limit is the
played
formulation
In the hydrodynamic
by the two theories.
of Bohr, which
[9] "in the region
2.1. HydrodynamiC
use the
in mind a connection
of the classical
principle
role in the discovery
having
described
example
theories.
sense that we will
m
-1
(V S
e c
-
+ A)
(5)
let us turn to
limit of quantum
+
The Schrodinger
mechanics.
equation
rewritten
in terms of p and v
reads Several
approaches
gate this limit, which
differ
sical characteristics the quantum bility
jectories
evaluated problem
of the phy-
theory.
average
in this theory with classical
was investigated
mechanics.
equation
he derived
what
called
theorem.
We will discuss
ep
~
represents
tensor
some aspects
Tik
- p 4m ViVk £np + mp vivk
There
are more
fi2
is now
(7)
.
solutions
to Eqs.
(6) than to the Schrodin-
the Ehren-
a convenient
of the classical
Tik is
This
since we have replaced
two real functions
forsolutions
to Eqs.
(6) correspond
tool solutions
to explain
(6b)
'
Starting
here with the help of the hydrodynamic
which by itself
B)k - Vi Tik
the stress
by four. Only those mulation,
x
where
ger equation, fest theorem
(E +
nature
for the first time by Ehrenfest
[10] in the early days of quantum
the Ehrenfest
+
mdt(pvk)
trajectories
trajectories.
from the Schrodinger
(6a)
-V (p~)
possi-
the tra-
In view of the statistical
we must compare
d tP
between
One natural
as those characteristics
of particles. mechanics,
in the choice
to investi-
that are to be compared
and the classical
is to choose
of quantum
have been developed
of the original
Schrodinger
equation
limit. satisfy
\'
\
the following
quantization
condition
which
to
118
f d"t
119
(mY'
-* v
x
+
e-*
c
2rrnn
B)
(8)
There
is no direct
contribution
to the Ehrenfest
equation
E
(9b) from the stress for every
surface
construct
E. If this does not hold we can not
~ from
the condition satisfied
-*
and v. It is sufficient
p
(8) on initial
values
of this tensor
to impose
equations
-*
of v to have it
where
at all times [12] .
enters
unlike
the wave
equation, ponding
functions
limit comes with
the recognition
n
-* 0, the r.h.s.
of
the separation
smaller
with
there
theory.
between
the decrease
is no restriction The Ehrenfest
way of taking
over the whole results
Eq.
as the classical
this
rent fashion
con-
average luated
to anything,
the allowed
values
of n. In the classical
equations
d
-*
e«E
-*
1-
-*
c
+ --
E«r»
-* v
+ -
c
x
the brackets
can be obtained
from Eqs. (6)
-*
(6a) by r and integrating Eq.
inequality
the probability
it
becomes
density
but this can not happen
(6b). The
averages
the spreading
when
-*
B»
(i.e. integrals
to the force eva-
and average
velocity
-*-*
B«r»
n
(9b)
with p over the whole
space).
/
..
\'
tensor.
following
for all times due to is caused
term i.e. the n-depen-
In the classical
disappears
of classical
in an external
its classical
limit,
and the hydrodynamic
the form of the equations
ensemble
when
0 function,
This spreading
pressure
-* 0, the spreading
(6) assume
to the Dirac
uniformly
of wave packets.
(charged dust) moving
quantum-mechanical
less and less pronounced
shrinks
by the so called quantum
for a statistical
the standard
the
limit
in the form
denote
because
v.
particle where
x
position
time in a diffe-
counterparts
not equal
position
-*-*
The average
vary with
than their classical
< (E + - x B» c
(9a)
-*
modifications
do not have the same content
of motion.
velocity
at the average
equations m dt
are remarkably
-*
dt
equations
We may note
[13].
force is, in general,
-* v
-*
gets
-*
-*
in the hydrodynamic
under nonlinear
equations
dent term in the stress d
appears
equation
and the average
(8) is equi-
quantization
space and next integrating
can be written
look; the only place
that the Ehrenfest
The Ehrenfest
(8)
In the limit, when
on the velocity
theorem
classical
they do not change
The above by first multiplying
(6b). This is why the Ehrenfest
constant
of the Schrodinger
condition
(8) can be made equal
because
stable;
It is worthwhile
that Eq.
to one of the Bohr-Sommerfeld of the old quantum
(6),
only the divergence
the corres-
concepts.
limit. The correct
ditions
with
of the quantization
in the classical
valent
Eqs.
the Schrodinger
compared
classical
what becomes
satisfying
obeying
can be directly
statistical
noticing
variables
because
is in the first term of this tensor.
here in passing The hydrodynamic
Eq.
have a purely
the Planck
equation
tensor
charged
EM field,
trajectory.
of motion particles each
120
121
2.2. Weyl - Wigner
- Moyal
transforms operators
One can go one step further statistical
mechanics
the counterpart This method
by Moyal
the statistical
principle
In quantum
precludes
distribution
transform
mation
function
mechanics
lations Tr{W(q,p)Wt
[15] and
f(t,~,t)
of both the position
(WWM transform)
gene-
and
of freedom,
(j is
rator
(q' ,p')}
fdr
of having
fdr
(q',p')
re-
(l2a)
(q-q')8 (p_p')
Baker-Hausdorff
Wigner
i
in ~. p e""l\l-
irPQ
efiqP -
W(q ,p)
(13)
as we can get under
the (generalized)
case of a system with only one
the WWM transform
uses the
formula
i
-
is as close an approxi-
function
one repetitiously
authentic
0
(q,p) of a ope-
we adopt
and the completeness eigenfunctions
the following
relations
for
of q and p, for which
normalization
conventions
by the formula fdqlq>
0). It turns out that we can easily
method
of WWM transforms
by taking
the "square
generalize
to the relativistic
a four-vector c
the
of 2. To obtain
equation
riables
+ -
il'idtlP (x)
h 1
and hence by considering particle
previously
the classical
fields
as compared
analog
position
operator
V
T~v
in
va-
e C;p f
"
(36a)
(36b)
vA
where
=
p
,
2R2
(37a)
case
of basic dynamical
and Wigner
must be modified
for the hydrodynamic
of Eq. (27) .
to the nonrelativistic
As was shown by Newton
EM fields.
as that
v variables.
there is
o ,
(o v~)
u
case even for the
interpretation
formulation
read
u
d
or one anti-
fields we can obtain
in the relativistic
of motion
p by a factor
(35)
the same procedure
for static
is the clear physical
(x)
d
limit the relativistic
is lacking
static
+ e
of
field, but
ACL'
amplitudes
we can still
Assuming
again
between
photon
that only one mode of the
we obtain
in the classical
limit
e -im~V CL (t~~) J2Tr d~ 2Tr
potential
operator
(61)
_'t'
o
the representation
(57) can be used for every mode, we obtain
following
external
are the
limit, when ~ -+ 0 and n -+ 00 in such a
way that nn is kept constant
(52) and
character
operator
(57b) EM field is important,
In the classical
the or inversly
in
limit l: eim~ .
(62)
m
A CL (~t.~)= "'t'
-e-
\ ) nn(k,A)c 2v k,A w
-+L
2 -+-+ -+ -+(k A) -i(wt-k'r+~ (k,A)) h £, e + • c. ,
Thus,
(58)
by Fourier
external obtain
where
~ on the l.h.s.
phases.
Owing
for the collection
to its dependence
still an operator, as the corresponding this operator
stands
even though classical
into the formula
on the phases,
Upon
photon Having
is
amplitudes,
it has the same appearance object.
analysing
field with respect transition
the evolution to the phase
amplitudes
opeLator
in an
of the field we
in the limit when n-+oo•
of all ACL
obtained
\'
itially mode
operator
the photon
only).
the classical
we may proceed
of the density
introducing
for V(t) we obtain
I
limit of the transition
to calculate
in this limit,
the time evolution
assuming
field was in the n photon
In the interaction
picture
that in-
state
(one
we obtain the formula
t
"'"
138
139
v ( t) i n> P A < n i V t
P (t)
under
(63)
(t)
the assumption
absorbed where
PA is the initial
system. equal
The atomic
density
density
operator
matrix
PACt)
to the trace of P (t) with respect
photons
of the atomic
photons
~.
to the photon
LPA
which
the depletion
system
are dealing P A (t)
(61) for the matrix
elements
to the presence of the vector
atomic
pOint In
beam by the
systems
at a time
the ability
but scattered
Such difference
to emit
particles
systems may ra-
in behavior
is due
1/1; in the expansion
of the factor
potential.
of
the interaction.
This is the case for bound
levels,
or
This is true only when we
do not possess
photons.
diate many photons.
and using the identity
than the number
of the photon
with not too many
with discrete substituting
during
is negligible.
in addition
copiously of V and vt
smaller
did not change
other words, atomic
states
is much
in the beam n, so that the reference
practically
at time t is
that the number m of emitted
For bound-bound
(45)
transitions,
w can not be very small due to energy
conservation,but
m
for the transition
states w can be
we arrive
atomic
L
eim(.p-.p ')
=
21T
p(z)p(z)
