Download Classical limit of quantum electrodynamics

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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-
-*
<v>
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
<v> 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<v>
are remarkably
-*
<v>
dt<r>
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
<r> 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><ql
o
2TIn8
these relations
or its equivalents
defined
and completeness
t
(q,p) (1/I 1w(q,P)1/I2)
(1/I3Iw(q,P)1/I4)=(1/I111/l4)
(1/1311/12).
(12b)
1
In proving
the uncertainty
of t and ~, but the Weyl-
In the simplest
the help
f(t,~,t).
the circumstances.
degree
orthogonality
with
(r,t). It describes
mechanics
to the distribution
of the following
simpler
+
p
the possibility
function
classical
are often made
[16].
density
distribution
function
[14] and Wigner
distribution
the probability
the velocity.
in quantum
of the distribution
The classical
Moyal
and introduce
goes back to Weyl
was fully developed
ralizes
in imitating
W(q,p)
(q,p) Tr{W(q,p) (j }e-~(qpl-pql)
1
(10)
= f~2TIfi
(14a)
Ip><pl
!qp
<qlp>
e
rr
(14b)
where
e
W(q,p)
rri
(qP-pQ)
(11)
With
the help of Eq. (12b) we can invert
and obtain
a representation
for
0
the formula
in terms
(10)
of its WWM
transform
dr(q,p)
momentum
=
(2TIl'l:)-1
dq dp,and
operators.
to dynamical
The definition
operators
Calculations
Q and P are the position
i
(10) may be applied
and to the density
involving
and
the family
operators.
(j = f ar
(q,p) f
-fi (qp -pq
I
ar
(q I , p') W (q ,p) e
I )
0 (q
I
,
p') .
(15)
of unitary
The WWM transforms
of dynamical
operators
(also known
as
122
123
Xi
Weyl representations
the expressions
cal variables,
ducts
of these operators)
for the corresponding
provided
we always
Wigner
functions)
distribution
classical
quantum
of density
functions.
'l,
and
dynami-
i
e
W(r,k)
(known as
(19)
In particular,
for a pure
state we obtain
of classical
In order to justify
this claim
+
we will express
-+- -+- -+- -+-
ii(r·K-k.R)
-+- -+-
pro-
operators.
matrices
are the counterparts
~
with
fully symmetrize
of Q and P when we construct
The WWM transforms
coincide
in terms of WWM transforms
+
p(r,k)=
the average
+
+
1
7
J d 3 ~1jJ(r+i)1jJ
+ ~
* (r-i)exp{-fix·[K+cJdctA(r+i-ctR.)]}.(20)
+ ~
i7
7 e
7" + X
+
o
value
<A> of the dynamical
state described
operator
by the density
A evaluated
operator
in the
P,
In the derivation
zation
<A>
j dr (q,p)A(q,p)p (q,p)
Tr {Ap}
formula
is obtained
from
(15) and
the same form as in the classical
general
validity
a genuine
function:
it is nonnegative
EM field, we will
The
it is real
of the WWM transform
K
rather
[17] which
particles
invariant
employs
than the canonical
=
mV
=
the formula
version
functions
the kinetic
momentum
P,
equation
P - ~c A .
"'"
p(r',k')=
i (-+- -'"k' "'"k -+-')
-J dr(r,k)Tr{W(r,k)p}e
-+- "'"
-+- -+n r. -·r
the simultaneous
function
dctr'A(R-ct~)]}
.
(21)
n
density
in it. However,
operator
Liouville
Cl t
(18)
"
Tr{ Wp
transformations
potential.
of the presence
taken on
of the wave
as we shall show now, the
for p (~,k,t) does have a simple
O. To this end we calculate
-+-
the invariance
limit can not be directly
(20), because
of motion
gauge
and of the vector
of p (~,k,t), assuming
(17)
In this case
-+-
1
J
in Eq. (20) guarantees
The classical
in an
when
K
"'"
line integral
of p (~,k) under
only for a very re-
a gauge
+
the
of the wave
introduce
i
p (q,p) is not
even though
of charged
generali-
(13) was used
o
class of states.
To study the motion
momentum
Despite
+
fir.p ihr.k
.
W(~,k)=e
e
exp{-R[k'R+~
(12a). It has
theory.
(16), the function
distribution
and normalized
stricted
of
the following
formula
(16)
i+
This
of this formula
of the Baker-Hausdorff
the time derivative
that the time evolution
is determined
limit,
of the
by the von Neumann-
equation,
(t)}
For simplicity,
-1
(il'i) Tr{ (WH - HW) p (t)} .
we have assumed
(22)
that the EM field is
124
125
time independent.
additional
Otherwise
term in Eq.
we would
have to include
(22): the time derivative
an
of W.
+
The calculation
simple with
of the r.h.s.
of
the help of the following
+
-±+
v-±+
[at + V'V' + e(~(r)+-xe(r»'V'
rev
(22) is made
++
1 +
fi
n
1 +
V'k - '2 r) W
(- I
11
1+
e
[-V' + -k + i r
2
c
f
0
=
+
WR
k.
mentum
(23a)
(23b)
are small
+ + n
1+
+
daarxB(-~V' + -r - ar)]W
~ k
2
=
+
WK
(24a)
n
1+ +
7V' + -r-ar)]W
~ k
2
KW , (24b)
can be derived
with
that under the integral
the help of
in formula
(18) we can make
the
Since
- I
,
1;!
~
11
r
,
,
.
From Eqs.
(18) and
A
+ +
CL(r,v,t)
Similar
of the
equations
can be
variables:
from Eq.
•
(28)
(22) only in the sign of
we can immediately
limit of Eq.
write
down
(28),
+
+ +
+ +
[v'V'r + F(r) 'V'v]ACL(r,v,t)
(29)
(26a)
+
where
F(r)
rator
in the square bracket
is the Lorentz
bracket
force. The differential
is the Liouville
with
ope-
operator
the Hamiltonian:
(26b)
at ACL
0
and they disappear
-(i11)-l Tr {(WH - HW)A(t)}
(27) the classical
i.e. the Poisson
+
corrections
(25b)
+
11
k + 7 V'
~ r'
n
particles.
this differs
+
+
1'1:
r + - I V'k'
fields
These
(25a)
at
+
+ k'
n.
by
the substitutions
the time evolution
for all dynamical
the time derivative,
from
+
Ilk + r'
can be obtained
mo-
(21). We also notice
replacements
11
of
of kinetic
fields.
(22) determines
at Tr {WAtt)}
which
into powers
of charged
written
+
corrections
for homogeneous
Eq.
instead
(23-24) in which
for slowly varying
completely
states
~ r
relations
(25-26) are made,
+
RW
1+
e 1
+ +
-k - - J da(l-a)rxB(2
c
o
[ ~V'
All quantum
expanding
=
(27)
identities:
where we have used the velocity
(- I V'k + '2 r) W
o ,
]p (r,v,t)
(22-26) we obtain
in the limit, when
(30)
{ACL' H} .
,
Since the Hamiltonian
is time independent
and the Poisson
'l'
bracket
is invariant
particular
under
under
canonical
the time translation,
transformation,
we can write
in
,...
126
Eq.
127
(29) in the form
~
+
Difficulties
+
+ +
+
r(t)
dynamics.
and t(t)
obey the equations
The solution
be written
generalize
this approach
to include
field quantization.
of this equation,
->-
CL r,v,t)
therefore,
3. CLASSICAL
can
moving
LIMIT
->-
ACL(r(t),
v(t),t
0)
( 32)
Creation
the quantum
analogy
of the particle
trajectory
we must specify
A(r,v,t)
assuming
it to be
of the position
operator
QUANTUM
->- ->-
R(t),
MECHANICS
->- ->-
special
R(r',v',t)=Jdr(r,k)Tr{W(r,k)R(t)}e
a collection
the classical
equations
together
with its
obeys in the classical
static
at a time,
ever,
fields
the description
interacting
mutually,
by a potential
the nonrelativistic
understanding
counterparts
provided
external
moving
the interaction
mechanics
of the classical
of classical
and this is usually
framework.
fields
It is only in the
that we may consider
variables
limits, when
since in most
of particles
because
static
fields
one
creation
do not take place.
can never
of the energy
case of a uniform
n
->-
pair creation
We will begin
is des-
of spinless
Thus,
annihilation
in
we have a complete
limit. Quantum
dynamical
they tend to their classical
each describing
At first
occur
conservation.
in
How-
[18], even in a very
electric
field one finds con-
from the vacuum.
in an
field or even
term in the Hamiltonian.
quantum
we must
in terms of WWM
tinuous
to the case of many particles
(also time dependent)
functions,
as was shown by Schwinger
simple
transforms
is the fixed number
one may think that these processes
limit
of motion.
One can generalize
des-
. (33)
(31), this function
(velocity)
of wave
by external
mechanical
this new situation
of the state vector
and annihilation
time derivative
characteristic
case of static
particle
to Eq.
the quantum
caused
theory.
->-->-
-fi(mr·v'-k.r')
->- ->- ->-
of a single particle
of particles
To deal with
one component
mechanics
done in the field-theoretic
i
->-
whose main
introduce
->-
the WWM transform
cribed
and
field is not a selfconsistent
force us to abandon
of particles.
->- ->-
quantum
and annihilation
cription,
In order to obtain
arbitrary
OF RELATIVISTIC
in an external
fields
According
theory
in the form
(->-->-
->-
to a relativistic
of classical
Relativistic
A
when we try to
+
a tAC L (r,v ,t)=[ v (t) ."r (t) +F (r (t)) ."v (t)]ACL (r ,v ,t), (31)
provided
are encountered,
only those
our discussion
particles.
Assuming
of particles
solutions
with
the simpler
that the creation
case
and
do not occur we can consider
of the Klein-Gordon
equation
mechanical
exist
and
[(ina
o.
'i
~
-
§:A ) (Hia u
c ~
e ~
2 2
-A ) - m c ]~(x)
c
o ,
( 34)
1
.~
~
129
128
t
"t
which describe particles. The time dependence of such
·
.
.
f requenc i es (so 1utlons
con t alns
on 1y POSl.t lve
e iw t ,
w
>
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<jl
H
only one particle
at a time. Following
described
What
~A)
2
c
+ (-:-'V
time dependent
The equations
c
rz:»
[c/m c
the hydrodynamic
no need to exclude
situation
root" of the Klein-Gordon
v~ and scale the density
[19], the
u
= -
m
-1
Cd
u
S+~A)
c
(37b)
u
in the relativistic
and
theory
and itceases
of the external
to be identical
field.
with the argument
In other words,
the center
~
of mass
T
can not be identified
with
the center
of charge.
why the WWM does not have an unambiguous
pretation,
theory
even though
based on Eq.
problems
it can be formally
and deriving
physical
inter-
applied
to the
again the hydrodynamic
from it the relativistic
analog
Eqs.
theory
based on the Klein-Gordon
in the same manner
except
formulation
of the Ehren-
v~ a
that we introduce
the velocity
a~
(lv
u v
~np + mp v v
(36) can also be written
u
p
u
v -,vA
is obtained
as in the nonrelativistic
4m
propagation
of the relativistic
equation
n;
p
(37c)
as the equations
Of P and v~ along the stream
for the
lines,
formulation
fest theorem.
The hydrodynamic
= -
That is
(35). We can shed some light on these
by considering
2
~V
-p
a u v~
(38a)
tr2
e fA v p +d (p-l/2C] pl/2)
mc
p
2m2 A
(38b)
case,
in the form of
\ )
~
The quantization
condition
also a relativistic
for the velocity
form [20]:
takes on now
4
l
130
131
fE do ~v [m(8 ~ vv
a v v)~
-
+
e
-f
]
c ~v
=
2rrttn
(39)
.
.}
In the classical
(38) become
moving
limit the equations
identical
in an external
condition
with those
of motion
for a charged
and apply
the Klein-Gordon
relativistic
compose
into the phase
dust
maining
spinor
field and the quantization
n
con-
(36) we can obtain
analog
by integration
of the Ehrenfest
a
the
u
(u
+
0
r P v
Pv
+
PV
v
0
f d 3r
Ic
not appear explicitely
+
1+ +
appearance
of factors
case, the Planck
constant
does
in these equations.
However,
the
coincide
with the center
become
of mass
of charge
between
jections
for the Dirac
the center
the Dirac
(44)
the same as obtained
raised
o
by Pauli,
[6,7] are valid
in this
is basically
limit. We may obtain
+
the obhere.
about
that of uniqueness
different
equation
is confined,
the result
fixed while
of Pauli.
l/n, then small corrections
(41)
magnetic
of fi, which
moment,
making
For example,
domain,
taking
of the
behaviour,
variables.
if we hold the size of the spatial
particle
limits,
the asymptotic
0, of various physical
in the classical
to which
the
the limit, then
If this size increases
to particle
trajectories
are due to the effects
get multiplied
as
of
of the
by l/tt and contribute
limit.
form
I believe
[.(~na
by de Broglie
assumptions
the order
- ~A ) + mc]1jJ
from the
u
classical
when IT
of charge
of Zitterbewegung.
It is true that we can rewritte
limit resulting
different
we obtain
in the "squared"
(43b)
since the classical
The problem
is even more pronounced
case owing to the presence
c ~
.
does not
of mass.
more complicated
The difference
and the center
f
vP
AP
is essentially
vOlc on the l.h.s. is a clear in-
of the fact that the center
Things
(43a)
decomposition
(40b)
c
dication
u
equations:
(40a)
ep (E + -vXB)
As in the nonrelativistic
[Y~(ina
in the limit, when
,
0
= ~mc
vA
1jJ= e(i/n)S
equation.
we may deand the re-
+
f d3r
Ic
=
u v~)
However,
3
exp(iS/n)
for
theorem:
u
at m f d r
developed
In particular,
factor
field u and obtain
0, the classical
+
v~a
f d 3r
equation.
(36) or
the methods
disappears.
From Eqs.
at
to this equation
e
. u e u
2 2 en u v
--A ) (~na --A ) -m c - -0
f
U
~ c ~
c
2c
~v
'1
o
(42)
•
also contribute
how to calculate
that the pair creation
in the classical
contributions
processes
may
limit if we only knew
from pair creation
summed
1I
133
132
over all orders
the external
of perturbation
theory with respect
to
means
field.
large quantum
intensity
numbers
which
in turn imply high
,
of the field.
(1
i'
In the absence
Klein-Gordon
of the exact
and Dirac
pair creation
equations
in time-varying
leave this problem
solutions
of the
that would
exhibit
external
We may give a more
this view by noticing
field, we must
t
unresolved.
LIMIT OF THE QUANTUM
The principal
difference
limit of the quantum
quantum
mechanics
by Sakurai
theory
THEORY
between
was explained
and we will begin by quoting
diation
of the EM field into a Fourier
potential
-+
is achieved
when the number
garded
as a continuous
lopment
mates
variable.
of the classical
the dynamical
-+L
k ,A
In contrast,
mechanics
Newton's
the classical
is the mechanics
equation
that in the very beginning
and the particle
nature
This description
pondence
principle.
After
only the wave nature
viewed
as a quantum
a certain
oscillator
number,
elements
(45)
system
quantized
of the EM field
and this can happen
and creation
as l/In so that the average
as l/n.
the transition
of charged
quantum
to the classical
split
let us consider
particles,
described
limit
an ar~
by non-
mechanics,
interacting
with the
operator
of the system
can be
EM field.
The Hamiltonian
into three parts
of light
H
limit for the EM
(46)
HF + HA + HI '
where
corres-
are the excitations
of photons
characterizing
values
values
of annihilation
go to infinity
were appearent".
in Bohr's
classical
go to infinity
relativistic
it was no coincidence
of the EM field and hence the number
series,
+ h.c.
limit the average
their
numbers
bitrary
of photons.
obeying
all, photons
ik·r
for the EM field systematically
of a single mass point
contained
-+
(k,A)a(k,A)e
To describe
deve-
wave
of the classical
field is also indirectly
photon
of ra-
wave approxi-
of trillions
of the electron
of the
becomes
The space-time
Thus
£
2vw
only if matrix
[21].
limit of Schrodinger's
of motion.
in the expansion
-+-+
-+ -+
In the classic~l
may as well be re-
electromagnetic
behavior
theory
of photons
number
2
must approach
very clearly
from his book
limit of the quantum
so large that the occupation
all creation
vector
operators
"The classical
multiplies
the classical
of the EM field and of the
of particles
Ih
'\!
,.f
confirming
operators
A(r)
OF THE EM FIELD
that
argument
and annihilation
' •.
4. CLASSICAL
formal
can be
the state
of the EM field. Classical
limit
of
'(
.
-+
HF
3
~ Jd r
1
oA 2
(~(at)
c
-+
2
+ (VXA) )
.
,
(47a)
t;~
I
,
i'"
135
134
S
~
~
~
~.
L
= a,.
HA
1 +2
1
2m. Pi + 2
4.1. Classical
eiej
L
r-
2
ei
e.
= - L. -m.~ c
I
~
+
--z-2 : +2
A (r.)
~
+ + +
p. ·A(r.) + L
~
~
i 2m.c
~
We will
:
,
(47c)
~
first calculate
operator
a large number
the method
described
calculation
will
the colons
annihilation
denote
ordering
of creation
the sum
and
study the time evolution
in the interaction
picture,
by the Hamiltonian
in which
HO
=
harmonic
the time evolution
e
.
-:!:Ht
- e tt
0
(48 )
V(t)
radiation
k
Ik> +
k
limit. Generalization
number
1jJ
point
state vector
of the phase
c
t
states
In+m> by
¢
( 52)
An arbitrary
L
follow
[22]. The
for a single mode
of the phase
to
1jJ
(49)
We will
,
¢
where
. t
_ T
-fi
J0 dT H I (T)
exp
V (t) _
photon
n is a fixed reference
function
states
i.e. only one term in
where
00.
number
papers
the classical
functions
im
In + m> + e
out
i
-RHt
(¢ )
and m varies
from -n
can be represented
as a
eim¢
(53)
¢
L
m=-n
c
n+m
and
and the scalar product
_
i
-H 0 t
n
HI(T) -e
H
.
~ t
--H
110
I
(50)
e
(<I>
I '1')
J21l
d¢
21l
o
The EM field evolves
to the free Maxwell
in the interaction
picture
Annihilation
-+
-+
=
L
k, A
+ -+
+
E(k,A)a(k,A)e
~ 2
++
-iwt+ik'r
+ h.c.
q,
takes on the form
* (¢ ) 'I' (¢)
(54)
according
equations:
functions
A(r,t)
,
fi
of the
is straightforward.
wave
Hp + HA is separated
of photons.
first be performed
Let us represent
of this system
elements
photon
in our recent
(51) survives
to many modes
operators.
We will
governed
normal
matrix
V(t) between
containing
of qlectromagnetic
where
states
,_It I
evolution
H
number
(47b)
+ I
i;;tjI +ri-r
j
~
limit in terms of photon
(51)
a
vw
o
+
e-i¢
and creation
in the following
(n +
!
i
1-)1/2
3¢
operators
act on these wave
way,
( 55a)
1
.'<:,
{~
136
at
137
*
(n +
-+
_8_)1/2 ei~ .
1
(55b)
8~
l
i
We also obtain
1 8
If the reference
selves
to values
ximate
formulas
pOint
is large and if we restrict
of m much
smaller
T exp
VCL(t~~)
I~J
I ~ .
ata -+ n +
(56)
where
our-
e
i
HCL(t)=-l: mc
Iii
than n, we can appro-
t
CL
dT HI (T)}
{-R ~
(59)
t~
2
-+
-+ -+
ei 72 -+
Pi (t) 'ACL(ri,t~~)+~~ACL(ri,t~~).
12m.c
(60)
1
(55) by expanding
l/In and keeping
them into powers
of
only two lowest terms
Formulas
a -+ e
-i~
_1__ 8 )
owing
1 _8 )
at -+ In (1 + 2ni 8 ~ ei~
(59) and
(60) for the evolution
same as in the case of a classical
(57a)
In (1 + 2ni 8~
to the operatorial
obtain
from them the transition
number
states.
term in the expressions
formula
the classical
I
we may keep only the first
(57). Since
for the vector
< n +m V ( t)
I n>
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~<n+mlv(t)
VCL(t~~)
In> .
(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
L<n+miV(t)in>PA<nivt
m
(64)
(t)in+m>
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
<5
(.p
-.p')
(65)
between
small and the effective
at
system
emission
21T
PA (t)
=
f
0
d
~
t
VCL(t;.p)PA VCL(t;.p)
a single
Therefore,
the density
operator
the limit of large n is related
in the presence
averaging
quantum
of an external
operation.
classical
approach
of the atomic
EM field through
is carried
one uses the so-called
in the description
but it is introduced
system
to the density
Such averaging
optics whenever
in
operator
the phase
fluence
states
(for harmonic
averaging
are equivalent).
oscillations
system
of this section
constant
between
making
the
the
likely.
of photons
emitted
(or absorbed)
is large, it is natural
significantly
(or absorption)
to assume
by
that
does not in-
the state of the emitters.
leads us directly
to the concept
This
of coherent
of radiation.
out in
semi-
4.2. Classical
limit in terms of coherent
states
states
manner
of the EM field,
time and phase
Coherent
when we assume
to a classical,
were obtained
with
infrared
appear
in a natural
that the quantized
their appearance
All the results
more
act of emission
assumption
then as an ad hoc time averaging
procedure
of photons
the atomic
coupling
and the EM field increases
If the number
(66)
scattering
external
current.
in QED
EM field is coupled
These
states made
in QED for the first time in connection
divergencies.
In their
fundamental
paper
4
1
1
140
on this subject,
crucial
mation
Bloch
assumption
interactions
and Nordsieck
with
low energy
photons"
of electron
the EM field. The term coherent
century
operators
[24], who recognized
quantum
photons
states
IJ
states
and Nordsieck
of various
of continuous
by Glauber
for
Dynamics
of the sources
these
sources
are so strong
by the emission
what happens
gencies,
current
of radiation
is justified
of photons.
in the analysis
the effective
the decrease
of infrared
with
this is the physical
(low frequency)
an external
coupling
low frequencies
is strong
reason why the emission
photons
can be described
to
Hepp
matical
rigor
theory
current.
is more
of the classical
to the similarity
between
limit and the infrared
only the appearance
of classical
when w
+
O. One may indeed give a physical
when
+
0 we must
w
obtain
For low frequencies
macroscopic
exactly
features
of the radiating
vant and they can always be described
physics.
Moreover
argument
for low frequencies
+
author
and Classical
the weak
to this principle
variables
in co-
than
values
equations
this result with
unitary
operators
(unbounded)
of particles
the use of coherent
0, the expectation
the classical
using
with
theory.
[27]. It was shown that in
All these works were
mechanics
position
of the classical
Re-
full mathe(cf. formula
and momentum
of the classical
with
of quantum
of motion.
W(q,p)
incomplete
interacting
in terms
ope-
in one respect:
limit of the
a field. That is
limit is still unresolved.
that
these
limit.
are rele-
states
are defective
states
in quantum
the number
fields,
way.
arbitrary
where
[25,27].
coherent
referring
in one important
of coherent
states
limit of quantum
used in quantum
of the harmonic
extent
I'
when
the only exception
by classical
of
the classical
of coherent
to Refs. [25-28],because
only the
sources
states were used
Quantum
later formulated
studied
We will not discuss
than.
in the limit,
the classical
i.e. long wavelengths
coherent
context
the problems
divergencies
currents
In a broader
of the dynamical
were
there was no discussion
of
in the analysis
[26]. According
[28] proved
why the problem
There
n
satisfy
cently,
rators.
of soft
in terms
values
of motion
(11)) rather
and
in the
states have the same form as in the classical
operators
increases
coupled
principle
the limit, when
diver-
stored
limit has been shown
between
[25], who
states by the present
(51). As we obconstant
of publications.
by Klauder
Equations
This is pre-
of w, so that the current
field oscillators
when
that they are practically
as can be seen from formula
before,
herent
states
of the classical
the relation
the expectation
in
of coherent
representations,
to investigate
correspondence
of a given external
not affected
aspects
in a large number
of a
of these states
large if the energy
The usefulness
of
for eigenstates
quarter
must be very
field is to be finite.
This
optics.
terms
served
of coherent
the importance
The description
cisely
as given".
was introduced
after the work of Bloch
the
of electron
in first approxi-
is treated
leads to the appearance
of annihilation
[23] made
that in the description
the motion
assumption
with
~
I
141
respect.
oscillator,
mechanics
readers
mechanics
the choice
is to a large
It is only in the theory
states
appear
With
of
in a very natural
-..
143
142
There is a vast textbook
can find information
states
literature,
about the properties
where
(cf. for example
[29,30,31]).
reader has at least a rudimentary
one
of coherent
We hope that the
knowledge
Disregarding
-,
'1
we can write
1
teraction
discussion
of a single mode of radiation
with
The interaction
Hamiltonian
operator
for this system
in the interaction
in the form
(cf. Refs.
eza
z>
an external
There
a
(t)
* (t)
a e
-iwt
+
a (t) a
t
e
i1jJe~. (na t +n *)a
e
e
and the evolution
[29,30]
picture
that coherent
can
quantum
or [32])
analogs
classical
( 67)
t
e
ina
e
classical
.*
( 68)
is
1
where
1'1
.
(69)
=-nJdTa(T)e~wT
o
11jJ
is a phase
on the vacuum,
It is believed
1
states
factor which
the evolution
is unobservable.
operator
produces
Acting
Third,
the values
an external
state
current
ei1jJe-lnI2/2
v (t) 10>
eH lin> .
einatlo>
(70)
z
1
(t)
in
t
J
state
and
Iz>
intensity
of z. This belief
the average
the product
and antihermitian
its minimal
saturate
value
of
parts
of coherent
the uncertainty
re-
of z(t) for a mode driven
a(t) are identical
case and in the quantum
the
a + at in the coherent
of the hermitian
functions
lation).
a coherent
state, whose
to z + z*. Second,
(gaussian
by given
mode,
facts. First,
of the field operator
a attains
(i.e. the
by the intensity
is the phase
state
dispersions
states
that the coherent
value
z> is equal
field are the
radiation
of the classical
on the following
indicate
and not by statistical
field is characterized
zl2 and the phase
which
are characterized
For a single
of the operator
and e
arguments
of pure classical
states which
is the analog
an a
heuristic
of field variables
is based
1 t
(71)
states of the radiation
distributions).
iwt
H e-lnI2/2
form
- z alo>
the phase.
V(t)
state in the canonical
(70),
*
t
are various
values
HI
in the formula
of the in-
current.
be written
the coherent
factor
of this
subject.
Let us begin with a brief
the phase
by
in the classical
case,
iwT
(72)
dT a (T) e
o
A coherent
another
classical
herent
state is also obtained
coherent
state. Therefore,
currents
states.
it suffices
when V(t) acts on
in the world
to consider
The analogy
of
states
only co,j
between
coherent
can be extended
states
and pure classical
to statistical
mixtures
of such
states.
Statistical
mixtures
of coherent
states
formed
,
j
144
145
according
to the rules of quantum
sity matrix
position
tained
of a mixed
mechanics
state is equal
of the projectors
(the denfunction
to the super-
on the pure states con-
I!
(J
in the mixture)
determined
This
displaces
=
"
d
f
Z
-TI--
1
follows
(73)
z> p(z)<zl
the variable
In the interaction
correspond
pure classical
sity matrices
integration
to classical
states.
plane
(73) is extended
of the variable
the EM field is very similar
matter
mechanics
amenable
to classical
The possibility
distribution
context
function
(68) and
*
+ zoa)
(zoat
exp
state by zo0
we obtain
z>p(z)<zlv
t
(t)
TI
2
The
=Jd/I
over the
2
d
z+in (t»P(z)<z+in
(t)I=J
TI
zl z>P(z-in (t))<zl.
(74)
z.
in the theory
and these two concepts,
closely
are used to formulate
in terms of statistical
of
d
J-zi
V(t)
This
of
formula
i
to the role of WWM trans-
of fact, are mathematically
Both concepts
from formula
(73) of den-
as the P-representation.
The role of the P-representation
forms in quantum
mixtures
The representation
is known
in the formula
complex
statistical
PI (t)
theory.
z of the coherent
picture
2
should
whole
statement
from the fact that the operator
2
P
by the classical
--H
n
t
0
.
t
to the Schrodinger
i
-H
PI (t)
en
picture
reads
t
0
related.
the quantum
distributions
P (t)
as a
e
translated
which
theory
e
are in turn
2
Jd TI zIZ>P(z-in(t»<zl
-~wa a
interpretation.
to interpret
P(z) as a classical
was stressed
by Sudarshan
of the so-called
optical
equivalence
d2
z
J --TI
1
.
. t
z>P(ze ~wt. - ~n(t)e ~w )<z 1
(75)
in the
theorem
in time described
The change
by the formula
(cf. Ref. 29 p. 192).
We are now ready to state the following
correspondence
ternal
currents
property
valid
for systems
(for simplicity,
classical
driven
P t (z)
p(zeiwt
_ in(t)eiwt)
we give it for one mode):
is the same as the change
The time evolution
pressed
p(z),
as the change
coincides
with
of the density
matrix,
in time of the weight
the change
( 76)
by ex-
predicted
when ex-
by the classical
have obtained
function
of the distribution
1) i
IJ
of the distribution
theory.
the same result,
P(z) with P (q,p,t) and solving
oscillator.
All the results
function
For example,
equating
we would
the function
Eq. (27) for the harmonic
obtained
in this subsection
,
146
147
can be generalized
to include many modes
In most instances
inserting
this generalization
sums over all the modes
of the EM fields.
will
of the radiation
and it will not lead to any clarification
important
problem.
use external,
and whether
Under what precise
classical
literature
with the paper by Bloch
the use of classical
Hence,
to describe
a transition
and Nordsieck
currents
we can then obtain
we can
d~
•
low frequencies
=
r0
2
(78)
by either
taking
the limit
of the energy-momentum
p
not
~
+ bk
=
~
determined
current
by the requirements
conservation
and long wavelenghts,
electron,
as if the photons
wave vectors
and the
keeping
limit
With the help of these
one can establish
quantum
results
of Lorentz
low-frequency
also the connection
in several
the Klein-Nishina
amplitudes
formula
simple
equal
is the simultaneous
between
theorems,
classical
For example,
and
from
2
ro
dn
'2
2
--(l-cose)
2
1 + cos e
{l+mc
nUl
2
2
nUl
[H (l-cose) --2]
(Hcose)
[1+ (l-cose) --2
mc
mc
transi-
for finite amounts
over soft-photon
to some extent,
by Brown
the main part of the change
to some unspecified
with
an in-
and even
in the electron
four-
the soft photons.
in four-momentum
nonelectromagnetic
and it was not shown that the change
is introduced
/I
lW
:ii
and Goble
contributions
a change
i:i','
interactions.
They have been able to perform
However,
to photons
9.Ii.
obtained
due to the interaction
actions
n to zero
Such a simultaneous
momentum
attributed
scattering
with
beam to infinity,
in electromagnetic
some results
summation
to include,
finite.
and the
to be difficult
transition
if we are to account
[34] are encouraging.
finite
seems
of the photon
transfered
In this respect
properties.
the Compton
being
as the initial
had the frequencies
to zero. What
their product
of energy
are uniquely
or low-energy
cases.
describing
ampli-
invariance,
analytic
are kept
fixed when we take the limit b ~ 0, then even virtual
after it was shown
and some simple
four-momenta
must have the same momenta
limits of QED transition
0.
Ul ~
a result
at each vertex:
electrons
limit and the low-frequency
that such soft-photon
the limit
is simply
conservation
~
tion is essential
by Low [33]
° or
but also as
of the quantum
studied
~
q . If the electron
to achieve
tudes have been extensively
n
of the two limits
and with the intensity
The low-frequency
formula
2
(1 + cos e)
This equivalence
starting
descriptions.
4.3. Classical
Thompson
1
for very soft
numbers
the classica
2
dO'
limit. From
limit is understood
the equivalence
one obtains
U
[23] we learn that
case of large photon
to very
field
the sources
problems
is justified
if the classical
only as limiting
classical
conditions
on the infrared
I.
of the one very
this can be done in the classical
the existing
photons.
currents
just involve
in a self-consistent
was
inter-
attributed
way.
nUl
2
Complete
}
.]
,
(77)
program.
scattering
II II
calculation
still remains
Even in a very simple
of an electron,
the bremsstrahlung
spectrum
an unfulfilled
case of the Coulomb
the classical
[35],
formula
for
~
~4l
...
149
148
I wish
222
l~ ~
(~)2
3 c
mc 2
I (w)
2
(Amv )
Ze 2w
(~)2 ~n
v
has not been obtained
connected
(79)
fj
have convinced
troduction
from full QED in the classical
to end here my review
with the classical
of various
you that the first sentence
is indeed
topics
limit of QED, hoping
to
of the In-
true.
If
limit.
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of runaway
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appear
in QED. First
acceleration
extended
electron
classical
certain
mation
whose
electron
to quantum
classically
disappear
remains
that pre-
when one assumes
[37]. Next
they summed
in the nonrelativistic
theory,
showing
the electrostatic
L. Thus,
two being the part due to the magnetic
whether
the results
relativistic
ximation
theory,
strongly
the classical
its quantum
extension
ot cau.ality
avoided
an explanation
for classical
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an
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.,'
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to