The third section of the Como paper deals with how complementarity resolves
the apparent paradox of the uncertainty relationship. Since these relations
express the crux of the non-classical aspect of quantum theory, understanding
them through complementarity will improve our appreciation of Bohr's point
of view.
Bohr's analysis was essentially coeval with Heisenberg's discovery,
but he never intended complementarity as merely an explanation of
the uncertainty principle. Einstein's criticisms forced Bohr repeatedly
into an analysis of experimental situations designed to overcome the limits
of the principle. Thus, in the years following the Como paper, Bohr's original
approach from an analysis of the application of the classical modes of
description was largely lost from sight.
The belief that Bohr designed complementarity simply as his analysis
of the uncertainty principle is historically unfounded, but it is a natural
mistake, for he was anxious to show how complementarity explains the uncertainty
principle. For physicists raised in the framework of classical physics,
the discovery of the uncertainty principle was a stunning surprise. Thus
it is hardly remarkable that in speaking to a group of physicists at Como,
Italy, Bohr was very eager to show how his new framework of complementarity
was able to interpret this principle.
As Bohr understood it, both his complementarity and Heisenberg's uncertainty
principle were the consequences of the quantum postulate. Complementarity
was the consequence for a conceptual framework in which phenomena are described,
while Heisenberg's discovery was its formal mathematical consequence. In
quantum mechanical formalism, these parameters are not independent of each
other, but are linked reciprocally by the measure of discontinuity in change
of state symbolized by Planck's constant.
It follows mathematically from uncertainty relations that the classical
formalism can only be defined within limits fixed by the element of discontinuity
involved in the change of state of a quantum system. Since the initial
conditions cannot be defined beyond this point, theoretical predictions
of the future state of the system are "indeterministic" or probabilistic
in nature. However, Bohr believed that even though the uncertainty principle
was logically entailed by the quantum postulate, the formalism itself could
not tell us whether this limitation was merely a limit on knowledge or
if it reflects a deeper ontological "indeterminacy".
Nevertheless, classical formalism must be regarded as a special case
of statistical determinism in which the statistical spread of determinism
is negligible relative to the precision of the measurement. Since these
are in fact the sort of interactions which are described classically (in
line with Bohr's correspondence principle) the predictions quantum mechanics
make will approach those of classical mechanics as the exchange of energy
or momentum involved in the exchange increases in size relative to that
expressed by Planck's constant. In other words, as we deal with larger
and larger objects, statistical determinacy will converge to strict determinism.
Statistical predictions in quantum mechanics might be interpreted as
predictions describing behavior between large collections of atomic systems,
each of which can be described by as yet undiscovered deterministic laws.
In fact, the statistical spread of the formalism can only be confirmed
by measurements on many atomic systems. Bohr categorically rejected this
interpretation, however. He writes:
One might perhaps believe that the properties of the elements do
not inform us directly of the behavior of individual atoms but, rather,
that we are always concerned only with statistical regularities holding
for the average conditions of a large number of atoms. ...The elements
have, however, other properties which permit of more direct conclusions
being drawn with respect to the states of motion of the atomic constituents.
Above all, we must assume that the quality of the light which the elements
in certain circumstances emit and which is characteristic of each element
is essentially determined by what occurs in a single atom. Niels Bohr,
The Atomic Theory
Bohr proposed the quantum postulate to develop a mechanics which would
represent atomic systems as mechanically stable. This property of stability
must be attributed to each individual atom, thus the quantum postulate
expresses a fact about the behavior of individual atomic systems. Since
the postulate entails the statistical nature of the predictions of the
formalism, the loss of determinism must be a consequence of the behavior
of individual atoms. Thus Bohr declares as "vain the...
...repeated, expressed hopes of avoiding the essentially statistical
character of quantum mechanical description by the assumption of some causal
mechanism underlying the atomic phenomena and hitherto inaccessible to
observation...Above all such hopes would seem to rest upon an underestimate
of the fundamental differences between the laws with which we are concerned
in atomic physics and the everyday experiences which are comprehended so
completely by the ideas of classical physics...the peculiar stability properties
of atomic structures which are in obvious contrast with the properties
of any mechanical model, but which are so intrinsically connected with
the existence of the quantum of action, form the very condition for the
existence of the objects and measuring instruments, with the behavior of
which classical physics is concerned. Letter From Bohr to John Slater
Different Interpretations of the Uncertainty Principle
The difficulty many philosophers and physicists have with indeterminism
makes it easy to misinterpret what Bohr was trying to say. To him, indeterminism
was a theoretical reflection of a fundamental fact about the behavior of
individual atoms. Heisenberg's discovery of the uncertainty principle in
1927 was no surprise for Bohr, as it was for most other physicists of the
time. His whole orientation had prepared him for just such a revolution.
Before going into Bohr's analysis of the uncertainty principle, two
points should be made clear. First, the mathematical formalism expressing
relationships encompassed by the "uncertainty principle" are straight forward
deductive consequences of the quantum postulate. All too often, discussions
of the principle begin with a series of thought experiments intended to
demonstrate that observations which determine the value of one observable
within a certain range require physical conditions which preclude determining
its canonically conjugate observable within a range that would contradict
the uncertainty principle.
It is then easy to get the mistaken impression that the principle expresses
an empirical generalization derived from analyzing physical situations
and that such a presumed empirical discovery is then injected as a postulate
into the theoretical formalism. However, this is not the case. Heisenberg
first developed his formalism for theoretical representation of the atomic
system processes, and then showed that the consequence of his formalism
was that the system could not be characterized by a state which was defined
in terms of precise values for both canonically conjugate parameters. Since
this analysis of complementarity doesn't require studying the mathematical
formalism, there is no need to consider the derivation of the principle
itself. The concern here is to give Bohr's theoretical conclusions some
physical significance.
Heisenberg's Interpretation
Heisenberg was led to his discovery by proposing that "nature allowed
only experimental situations to occur which could be described within the
framework of the formalism of quantum mechanics". In considering various
thought experiments, he, and later Bohr, tried to show it was impossible
to determine with arbitrary precision the values of both of the canonically
conjugate observables. The reason for this is notoriously ambiguous in
Heisenberg's own presentations, however.
In some cases, he seems to imply that uncertainty arises because the
observation "disturbs" the experiment. In this case, observation would
require interaction, and the uncertainty arises there. However, interactions
do not arise continually in atomic systems, therefore this interpretation
must accept the classical presumption that all physical systems
exist in well-defined states theoretically represented by the concept of
the classical state. Here it is meaningful to talk about the classical
state of the quantum system, even though we cannot precisely know the parameters
that define any particular state.
The Verificationist Viewpoint
However, in other cases, Heisenberg seems to imply a more "verificationist"
view that since it's impossible to verify empirical statements for atomic
systems, assertions making use of this concept are meaningless. He comments:
Any
use of the words "position" and "velocity" with an accuracy exceeding that
given by [the uncertainty principle] is just as meaningless as the use
of words whose sense is not defined...one should remember that the human
language permits the construction of sentences which do not involve any
consequences and which therefore have no content at all - in spite of the
fact that these sentences produce some kind of picture in our imagination...
one should be especially careful in using the words "reality", "actuality",
etc., since these words very often lead to statements of the type just
mentioned. From Heisenberg interviews
The verificationist view is associated with the logical positivist view,
which we have already seen was pervasive during that time. It is worth
noting a rather obvious inconsistency in the same work from which the above
passage was quoted, Heisenberg seems to make a claim about what the electron
"really" is. Looking at how Bohr approached this same problem, it's easy
to see how different his thinking was.
How Bohr Saw Heisenberg's Discovery as Complementarity
Typically the two parameters involved in the uncertainty principle are
of position and momentum, or of energy and time. Bohr saw Heisenberg's
discovery from the point of view of the complementarity of the two classical
modes of description. Bohr saw the determination of position or time as
the goal of the mode of space-time coordination, while seeing the determination
of momentum or energy as a goal of applying the "claim of causality" through
the conservation principles. Just as the uncertainty principle allows one
to determine position and time or momentum and energy with an arbitrary
degree of precision, so the two classical modes could be used together,
but only in a complementary way.
From Bohr's point of view, the uncertainty principle does not
express any consequence of "disturbing" the system by observation. But
at the same time, his analysis does depend on recognizing that observation
involves an interaction that is "uncontrollable" once the quantum postulate
is accepted. Bohr tried to show that the classical state of the system
is only attainable by assuming it is possible to apply both spatial-temporal
and causal modes of description to the system at the same time. Since the
quantum postulate expressly says this cannot be done, the classical concept
of the atomic system is no longer defined. For this reason, the disturbance
principle is at odds with the analysis of the uncertainty principle from
the view of complementarity.
In fact, Bohr already arrived at the need for a complementary point
of view before he knew of the uncertainty principle, but quite naturally
saw Heisenberg's discovery as confirmation of his own analysis. He even
considered using the word "reciprocal" (derived from the mathematical way
of expressing the uncertainty principle between the canonically conjugate
parameters) instead of "complementary" to refer to the two modes of description.
The Disturbance Principle
Though it forms no part of complementarity, the disturbance principle
was frequently defended as part of the Copenhagen Interpretation and often
identified with Bohr's view in the years following Heisenberg's discovery.
From the perspective of Heisenberg, it appeared that the basis of the disturbance
principle lay in the fact that the instruments doing the observing "disturbed"
the observed system such that its state after observation is no longer
what was determined in the measurement. This interpretation compares observation
of atomic systems to measuring, for example, the inner workings of a wrist
watch using a yardstick.
However, the disturbance interpretation plays havoc with the facts behind
the genesis of the uncertainty principle and its status within the mathematical
formalism of quantum mechanics. The principle is a straight forward deductive
consequence of the quantum theoretical formalism which provides a highly
confirmed means of predicting the outcome of interaction between radiation
and matter. There is no mention of disturbance in the derivation of the
principle itself, nor of how to go about determining the relevant parameters.
The design of experiments is relevant only to interpreting the physical
significance of the principle. The assumption that the classical system
really exists in a classical mechanical state supposes the question of
whether an experiment could be designed which will yield greater knowledge
about the state of the atomic system than the uncertainty principle allows.
If this could be done, the theory would be properly judged incomplete.
The disturbance interpretation mistake becomes apparent when we realize,
that according to it, we could only approach classical ideals of strict
determinism if our measuring instruments were the size of atoms. However,
it is only an immense difference between the dimensions of ordinary
human experience and those involved in atomic processes that made strict
determinism a nearly obtainable goal. If our instruments were the same
size as atoms, then the role of the quantum in an interaction would be
ever increasing rather than decreasing, as the disturbance interpretation
suggest.
In classical mechanics, the observation also "disturbs" the observed,
but the disturbance is either negligible or "controllable" and so can be
accounted for in defining the state of an isolated system after the observation
interaction. In quantum theory, ordinarily the effect of the interaction
cannot be considered negligible nor "controllable". Since the disturbance
interpretation makes it appear that the uncertainty principle is a empirical
generalization, it's unable to explain why this alleged disturbance cannot
be determined in the quantum framework, and allows a return to classical
deterministic formalism.
The Epistemic Interpretation
A second possible interpretation, the "epistemic interpretation", holds
that the uncertainty principle is a consequence that it was developed to
describe not the properties of atomic systems, but rather what we
can know about nature as we experience it. If we accept that the
adequacy of a theory rests on its success in predicting observable phenomena,
then the uncertainty principle gives no reason to hold that quantum theory
is inadequate. The theory is in no way incomplete because it allows only
either a measurement on position and time, or momentum and energy, because
in fact no phenomena allows us to determine such information. The uncertainty
principle merely expresses this fact and warrants accepting it as a mathematical
instrument for predicting relevant phenomena.
The epistemic interpretation may have been motivated by talk that properties
of a transphenomenal nature is meaningless and irrelevant to science. Classically,
the notion of an independent reality "existing" apart from its phenomenal
manifestation could be regarded as that concept of an isolated system.
Since classical formalism describes the motions of particles and waves,
the reality we experience should be consistent regarding such particles
and waves. However, what quantum theory tells us is that if its description
of its atomic systems is considered complete, since it cannot define a
classical mechanical state for these systems, they cannot be considered
particles or waves. It follows that the notion of an independent
reality cannot be described as previously thought from the classical viewpoint.
While the concept of an independently existing reality can no longer
be assumed to be describable, it doesn't necessarily follow that this concept
of an independent reality bears no relation to the world as experienced
or as described in scientific theory. However, if the epistemic interpretation
cannot establish that the notion of an independently existing reality is
meaningless, how does it justify its claim that our knowledge is limited
in the way expressed by the uncertainty principle?
On one hand, if the meaningfulness of any claims concerning the character
of an independent reality is denied, then it cannot be said the reason
we cannot know, for example, the simultaneous position and momentum of
a particle is because in reality there are no properties to be known.
But if there is no such reason for the conclusions of the uncertainty principle,
why not instead conclude that an altogether different theoretical formalism
will one day be invented to escape these limitations and return to strict
determinism?
On the other hand, if the possibility of describing an independently
existing reality is allowed, then the theory is complete in providing
everything that can be known about such objects. The fact that there are
limitations on how we precisely determine the parameters is a consequence
of a deeper limitation on what it is that we can actually know. In this
interpretation, the fact that nature has thus far hidden from us any means
of simultaneously determining position and momentum of a particle may be
revealed as only a limitation on our imagination, with deeper mysteries
lurking underneath that.
Either way, the epistemic interpretation cannot be made consistent with
Bohr's claim that quantum theory is complete without admitting that the
limitation on knowledge expressed by the uncertainty principle results
from a character of a reality behind the phenomena. If it is accepted,
as is the classical viewpoint, that parameters do correspond to properties
of an independent reality, since quantum theory uses such parameters implies
that nature possesses the corresponding properties. The proper conclusion
would then be that quantum theory is indeed incomplete and that one day
we will find a complete one.
Einstein and Bohr
Einstein seemed to think Bohr was defending the epistemic interpretation
with his framework of complementarity. By this time, Einstein had totally
repudiated the view that science is only concerned about developing a formal
means for predicting phenomena. This is the reason he repeatedly attempted
to devise thought experiments which would reveal more about the system
than the quantum formalism would permit, thereby dis-proving Bohr's belief
it was complete. In fact, if Bohr had defended the epistemic interpretation
of the uncertainty principle, as Einstein believed, Einstein's attitude
would have expressed open-minded characteristics of scientific progress
while Bohr would be seen as dogmatic.
Bohr's Interpretation of the Uncertainty Principle
Finally, let's turn to Bohr's interpretation of the uncertainty principle.
He writes According to the quantum theory, just the impossibility of
neglecting the interaction with the agency of measurement means that every
observation introduces a new uncontrollable element. Indeed, it follows
from the above considerations that the measurement of the positional co-ordinates
of a particle is accompanied not only by a finite change in the dynamic
variables, but also the fixation of its position means a complete rupture
in the causal description of its dynamical behavior, while the determination
of its momentum always implies a gap in the knowledge of its spatial propagation.
Just this situation brings out most strikingly the complementary character
of the description of atomic phenomena which appears as an inevitable consequence
of the contrast between the quantum postulate and the distinction between
object and agency of measurement. Como Lecture
In his debates with Heisenberg, Bohr would insist that in order for
the theoretical representation of the physical system to have any empirical
content at all, it must be possible to derive from it a description of
the observed phenomena, treated as a causal result of the behavior of the
atomic system. To give classical terms the empirical reference they must
have requires interacting with the system, however. But the quantum postulate
states that the interacting whole has an individuality which prohibits
unambiguously defining the state of the system while it is in observation.
Thus, the physical conditions necessary for observation are complementary
to those necessary for defining the state of the system. Bohr regarded
the uncertainty principle as directly expressing the formal consequence
of this complementarity between the mode of space time coordination and
the mode of causality description. Since the quantum postulate denies the
classical justification for regarding the parameters used to define the
state as pictures of the properties of an independent reality, Bohr repudiated
the classical correlation of the parameters with the properties of the
object regarded as isolated from any observed interaction. The "pictures"
we can form using these concepts of an isolated object refer not to a concrete
reality lying behind the phenomena, but to what Bohr called "abstractions":
On the whole, it would seem scarcely justifiable, in the case of
the interaction problem, to demand a visualization by means of space-time
pictures. In fact all our knowledge concerning the internal properties
of atoms is derived from experiments on their radiation and collision reactions
[i.e., on interactions], such that the interpretation of experimental facts
ultimately depends on the abstractions of radiation in free space and free
material particles [i.e., on systems as isolated]. Hence, our whole space-time
view of physical phenomena, as well as the definition of energy and momentum
depends ultimately on these abstractions. In judging the application of
these auxiliary ideas, we should only demand inner consistency, in which
connection special regard has been paid to the possibilities of definition
and observation. Como Lectures
The use of classical terms to describe atomic systems as "particles"
or "waves" interacting with the measuring apparatus refers not to a concrete
underlying reality, but to an abstraction which is necessary for describing
the phenomena as interactions between the measuring agencies and the physical
systems decried in quantum theory. Since the disturbance interpretation
holds that the objects of the atomic domain really do have properties corresponding
to the parameters that define their classical state, Bohr's interpretation
is that the limit expressed by the uncertainty principle is merely a consequence
of the formalism we must use to describe what it is we are observing.
Common Misunderstandings About Bohr's Viewpoint
Because we must still describe observations in terms of the classical
framework, it is easy to view the uncertainty principle as a limit on
knowledge. However, this misunderstanding is a consequence of presupposing
that the descriptive concepts of "particle" and "wave" picture the object
to which they refer as existing in isolation away from any observation.
In other words, we fail to make the quantum postulate a true postulate
in our description of nature. Consequently, we imagine somehow that the
limitations expressed by the postulate, and the uncertainty principle,
are not inherent in nature. And one day they will be circumvented when
we gain improved empirical information.
Bohr, on the other hand, argued against this classical tendency to interpret
the concepts as also referring unambiguously to an independent and real
object. He concluded that the concepts by which we express what we learn
in an observational interaction refer to properties of phenomenal objects
which can be described only in the interaction of the system with the observing
instruments. By recognizing the complementarity between space-time coordination
and causal description, Bohr avoided the dilemma that gives trouble to
the epistemic interpretation, since each phenomenal appearance is a unique
individual event.
Wave-particle dualism is not a feature of reality, but rather a feature
of our way of representing reality, and of course then the possibility
is open for different descriptions which may remove such a dualism. The
fact that Bohr emphatically believed such a search was futile reveals his
conviction that complementarity was a result of the very way we observe
our reality.
As Bohr would say, "we must relearn the presuppositions governing the
use of our most elementary concepts". Complementarity tells us that since
specific parameters of a physical theory have empirical evidence only
if interpreted as referring to specific properties of observed phenomena,
when those same parameters are used to define the state of the system isolated
from observation, we cannot assume they can be interpreted as referring
to properties possessed by that system apart from observational interactions.
At the same time, the framework of complementarity cannot be coherent
if it ignores the question of how to refer to this "independent reality"
which in interacting with observational instruments produces the phenomena
which are described by the theory. Facing up to the epistemological and
ontological consequences concern the next two parts of this review. This
ends Part 4 of this review. Thanks for reading! Comments
are appreciated!
