Quantum
Optics as a Relativistic Theory of Light *
* We dedicate this essay
to the memory of Professor David Joseph Bohm (19171992), only with whose
brilliant exposition of the meaning of relativity can we here convey the
idea.
In analogy with Bohm's
elucidation of special relativity, in this paper we criticize the Copenhagen
interpretation and seek the way to reinterpret quantum theory in the light of
new evidence of quantum optics. In doing this, we are led to regard the
real photon as only expressing an elemental relationship
established between a quantized light field and a detector in the actual
detection processes. With such a kind of relational approach to quantum
physics, new concepts then are needed to describe physical phenomena, and the
mathematical structure of quantum theory of radiation is viewed as a conceptual
map, in the same way as the Minkowski diagram in Einstein's special relativity,
which already has the perspective of the observer implicit in it.
Keywords:
relational approach to physics; basic physical laws as invariant relationships;
inherent ambiguity in the meanings of physical objects, according to the
Copenhagen interpretation; mathematical structure of quantum optics as a
conceptual map; intrinsic unfalsifiability of EinsteinPodolskyRosen paradox.
I.
Introduction
Modern
physics owes much to Galileo, who was the first to advocate experimentation and
mathematical description of natural processes. In his opinion, any reasonable
theory in physics should be based on measured facts of observation, for it might
extricate, to some extent, description and deduction from ambiguity and
groundless speculation. Along Galileo's line, Newton discovered classical
mechanics that well explains the motion of mansized objects, and, at the same
time, he also granted more precise definitions to those absolute concepts of
space, time, and physical objects.
A
great transition in modern physics, however, was rendered by Einstein, who
realized that those observable properties (immediate facts) are only
relational properties. In other words, by measurement we establish
physical relationships between an "object" and the measuring apparatus. This
is sharply in contrast to Newtonian physics, in which an observation or
measurement is that of the absolute property of the "object" itself. In this
way, the objectivity of nature then will manifest itself as the possibility of
finding what is invariant underlying those relational properties that are
observable, and we shall consider those "objects", like space,
time, and physical objects, not as absolute, but as relative
invariants, which are abstracted from those relationships within certain
observation domains. As the domain under investigation is broadened, we may
expect to come to new invariant relationships, containing the older ones as
approximations and limiting cases [1]. Thus, this bilateral trait of physical
observation has permanently cast the basic pattern of cognition by which man
learns of the world. It is just as the principle of relativity goes: "The
general laws of nature are to be expressed by relationships which are generally
invariant, and which are in principle observable."
Early
this century, Einstein proposed that space and time coordinates are only
relative properties consisting of relationships of objects and events to the
measuring instruments, and usually fail to be independent beings in broader
domains. Instead they fuse into one, as a fourdimensional continuum in which
the existent exists. Thus Einstein radically reformed two concepts out of the
three absolute notions granted by Newton.
The
concept of physical objects, among these three basic notions, is at the heart of
our metaphysics and physics thinking. Objects with figure and size, motion under
the law of causality, are indeed more concrete and direct than space and time.
Newton wrote as follows: [2]
'It seems probable to
me, that God in the beginning form'd matter in solid, massy, hard, impenetrable,
moveable particles, of such sizes and figures, and with such other properties,
and in such proportion in space, as most conduced to the end for which he form'd
them; and that these primitive particles being solids, are incomparably harder
than any porous bodies compounded of them; even so hard, as never to wear or
break in pieces; no ordinary power being able to divide what God himself made
one in the first Creation.'
With
this particle model of matter, not only did Newton succeed in describing the
behavior of mansized objects, but also could man in later centuries overstep
the bounds to exploit the atomistic structure that qualitatively accounted for a
variety of natural phenomena ranging from electricity and magnetism to
chemistry. Thus when this model came out at the end of the nineteenth century,
it had already seemed as if an "inevitable" truth, and many people then believed
that the foundation of physics had been soundly established based on the
mechanistic concept of nature.
As
atomsized phenomena were further investigated in the early days of this
century, classical physics incurred several grave problems such as "orbital
collapse" of the Rutherford model. To save this planetary structure of atom,
Bohr had no choice but to introduce quantum rules. Those phenomenological rules,
indeed, could explain the spectral data of simple atoms like hydrogen, yet they
seemingly could not go on to quantitatively predict more complex systems, since
formulas of this old quantum theory contained physical quantities (electronic
orbits of definite dimensions and periods) lacking definition of measurement
[3]. However, what concerned the physicists of this time more was how to
construct a theory. In positivism's eyes, the theory based on quantities without
measuring meaning was a kind of metaphysics. Influenced by the idea of employing
only observable quantities in Einstein's relativity theory, Heisenberg made a
great step forward in 1925 [4]. He successfully bypassed those unobservables by
using other existing quantitative relationships of empirical facts to found
quantum formalism effectively. For over seventy years since its establishment,
this formalism has been crowned with amazing success in nearly all branches of
physics, and few people now defy its reigning position over the formal
foundation of modern science.
The
theory of special relativity exerted a great impact on Heisenberg's ideas for
the development of a quantum formalism, as has been discussed in many books on
the history of quantum mechanics [5, 6]. Indeed, in both theories the physicists
emphasized that only measurable quantities, that is, observables, belong in a
theory. In quantum mechanics this has been referred to as "a guiding
philosophical principle," and in Einstein's theory it is also regarded as "one
of the keys to special relativity." [5] However, it seems that Heisenberg did
not fully realize the implications of Einstein's relational approach to physics,
which was only many years later amplified exhaustively by Bohm in his book on
this subject [1]. When in 1927 Heisenberg tried to interpret the new formal
structure that he had established [7], rather than question the older ideas of
physical objects in Newtonian physics, he was still, in a way, trying to retain
them in the new mechanics. Indeed, such a habitual tendency to regard older
modes of thought as inevitable might be quite natural, as was analyzed in detail
by Bohm in the same book from a perception background, but it ultimately would
also lead to ambiguity and confusion.
Thus,
in this paper, in analogy with Bohm's elucidation of special relativity we
intend to interpret quantum physics as another relativistic theory. To this end,
the outline will be as follows. We begin in Section II with a preliminary
analysis of some of the main facts underlying our use of particle and wave
pictures in quantum physics (parallel to that in Chaps. XI and XIII of Ref.
[1]), which are to be revealed as only consisting of relationships established
in the processes of interaction. This analysis will help us not only appreciate
Heisenberg's success in formulating a quantum formalism, but also realize no
inevitability of the older notions in interpreting the formalism.
There
then follows with a detailed discussion of Heisenberg's interpretation that
proceeds again in contrast with Bohm's of Lorentz theory (Chaps. IVX).
Considerable emphasis is placed on its efforts to try and retain Newtonian
concepts, the consequence of which is not that the interpretation disagrees with
experiment, but that those concepts entering into the Heisenberg interpretation
are inherently ambiguous.
After
bringing out the difficulty as a result of the retention in the Heisenberg
interpretation, we go on, in Section III, to the adoption of relational approach
to quantum optics (similar to that of Einstein as in Chaps. XIII and XIV), in
which the photon is regarded as expressing a relationship established
between a quantized light field and a photodetector in the detection. On the
basis of the observed fact of the wavy modulation in the probability amplitude
of the actually detected counting signals along the propagation direction, one
sees that the observer cannot assign a particle trajectory for light. Thus, it
is clear that in the new domain of quantum investigation we need new notions for
describing physical phenomena, which admit the older ones, such as the particle
concept, as limiting cases.
In
the discussions following, we stress the role of the field and
interaction as basic in the relativistic theory of light, instead of that
of the object and its motion, which are basic in Newtonian theory
(whereas the event and process are basic in Einstein's special
relativity, in Chaps. XXVI XXX). This leads us on to an "interactive" pattern
provided by the framework of quantum optics, with its invariant structure that
unite particle and wave pictures as two sets of relative invariant features of
the same field in different frames of detection. On the basis of this
unification, it is made clear that the quantum failure of observers with
different detection frames to acquire the same physical picture in no way falls
into "subjectivism," since the framework of quantum optics already has the
perspective of the observer implicit in it.
Finally,
we end this section by borrowing Bohm's discussion of the relationship between a
conceptual map and reality itself (Chap. XXXI), to further remove the confusion
in the waveparticle duality.
The
main text is then concluded in Section IV, and followed by an appendix on
quantum paradoxes. Again, with Bohm's discussion on the falsification of
theories (Chaps. XXV), we show that EinsteinPodolskyRosen (EPR) paradox will
become unfalsifiable within the Copenhagen interpretation. Since, according to
the WignerArakiYanase theorem, in the theory a great many of the physical
quantities are in fact unmeasurable. In the same appendix, we also summarize our
resolution of the Schrodinger cat paradox.
In
the whole paper, if we have seemed to belabor excessively an analogy of Bohm's
exposition in his book, this has been deliberate. However, if the reader could
be persuaded to believe that quantum optics is also a relativistic theory of
light, we know that we had achieved our aim.
II.
Observation foundation in quantum physics
In
physics the concept of a noninteracting object does not exist because its
presence could not be established. For the same reason, only by means of
interaction can one discover the objective world. Consequently, relationships
established as a result of interaction exhaust all the physical facts. To
shed light on what this means to quantum theory, we begin with an analysis of
some of the main facts behind our use of particle and wave models.
Two
experiments that led to notions of particle and wave in atomic physics were
Wilson photographs and Davission and Germer's diffraction of matter waves. When
high energy rays pass through a cloud chamber, they cut line tracks across the
vapor. From this experiment, as Heisenberg described in his book "Die
Physikalischen Prinzipien der Quantentheorie" [8], we are likely to regard the
rays as consisting of minute "particles" at high speeds, with the tracks of
condensed droplets indicating their trajectories. However, as Heisenberg also
noticed, the formation of tracks is due to ionization when flying "particles"
collide with the vapor atoms in their way, i.e., is as a result of interaction,
by which the emerging ions then turn into original kernels causing the
condensation of supersaturated vapor around them, thereupon droplets arrange
themselves along the flying paths to shape tracks that are directly observed by
us. Then, one sees that Wilson's chamber registered only the occurrence of
interactions.
From
the description above, one now may see that, in the similar way as all the facts
underlying space and time notions as Bohm analyzed in his book [1], the physical
facts here consist also only of sets of relationships as a result of
interaction involved in the registration (exchange of energy and momentum), in
which no absolute particle is ever to be seen.
If
the particle concept is only as a relative invariant extracted from those
physical facts with certain experimental arrangement (the particle frame of
detection), what then is the origin of the Newtonian idea of an absolute
particle, supposed to be like a solid, massy, hard and impenetrable substance,
essentially independent of all relationships? "Evidently it does not come
primarily from experiment and observation," as Bohm suggested [1], but rather
from the continuation in modified form of our "commonsense" view of physical
objects. In this view, matter is formed from discrete particles, each of which
has a certain place, size, and form. Thus particle is in effect "substantialized" and taken as an absolute.
Similarly,
Davission and Germer's detectors also recorded only the exchange processes of
energy and momentum happening in the detection, wave as a notion merely
represents a relative, rather than absolute, invariance of the relationships of
observed facts in the corresponding circumstances (the wave frame of
detection).
Through
this parallel analysis to Bohm's once the nature of physical facts and our
concepts as relative invariant features are clarified, the implications are
farreaching. First, if one recalls the success of Heisenberg in 1925, we should
have come to realize that it hinged essentially on considerations involving the
relational properties associated with two Bohr states (spectral lines that
characterize the relative energy changes), rather than any absolute property of
an electron itself, tied to single Bohr orbit: two instead of one,
as Dirac briefly commented [9]. Second, if one looks into Heisenberg's
interpretative attempt in 1927, we shall find that his validating classical
concepts to interpret the quantum formalism was, fatally, an effort to retain
our ordinary notions beyond their proper domain, where the theoretical frame
excludes the possibility of complete description of the particle concept
(simultaneous momentum and position) [6]. At this point, in order to exhibit
more clearly the nature of the problems to which the older concepts gave rise in
quantum mechanics, we still need to go in some detail into the Heisenberg
interpretation.
In 1900 Planck's study of the
properties of radiation undoubtedly opened a new page for twentieth century
physics, for it constituted the first evidence that sharply denied the basic
assumption of continuity, which is essential to classical physics. It ultimately
would trigger a whole revolution in our concept of physical objects. Yet it must
not be expected that this should be completed in one move. Indeed, as was only
natural, radical changes only occur after a long series of alternative
interpretations are tried and fail, with the object of saving our "commonsense"
notion of particle that is behind Newton's laws of motion. In this respect, even
Heisenberg's interpretation was without exception, no matter how radical he was
when he established the quantum formalism.
Heisenberg
began by accepting the assumption that classical notions remain valid in quantum
mechanics, who wrote: "All concepts which can be used in classical theory for
the description of a mechanical system can also be defined exactly for atomic
processes in analogy to the classical concepts." [7]. However, his basic new
step was to study the dependence of the measurement of position and momentum on
the relationship between the physicality of apparatus and its irreducible
participation in the measurement. To do so, he constructed the famous
gedanken microscope experiment to measure very accurately the position of
an electron [8]. Heisenberg showed that, when the indivisible quanta of action
must be taken into account in the measurement process, the uncontrollable
disturbance to the electron eventually made it impossible to assign
simultaneously precise values of position and momentum, as regulated by an
uncertainty relation. Thus, in the way of considering that the apparatus was
part of this physical world and must undertake the same irreducible interaction
to observe, which in effect disturbed what is to be measured, Heisenberg's
interpretation preserved the particle notion within the new quantum framework,
i.e., lead to a reconciliation. (At this point let us compare this with
Lorentz's way of trying to reconcile the ether hypothesis with the result of the
MichelsonMorley experiment, as discussed in Bohm's book [1]. When considering
that the arms of the interferometer were composed of atoms and should undergo
the same shift now called the Lorentz contraction, Lorentz actually did
proved that no fringe shift could ever be detected by the apparatus of Michelson
and Morley.) Nor is this all, he could even develop a whole set of uncertainty
relations to imply that in quantum mechanics because of the irreducible
disturbance, all the complete descriptions of classical notions will be
impossible.
Nevertheless,
the Heisenberg interpretation of the microscope experiment is formulated in
terms of position and momentum of an electron, measured by apparatus that is
supposed to have an irreducible disturbance to the electron. Therefore, the
measured values ought to be corrected, to take into account the effect of
the participation before we can know what they really mean. But if the
Heisenberg interpretation is right, there can be no way thus to give exactly the
simultaneous values of position and momentum. The simultaneous position
and momentum that define a particle in classical dynamics are therefore
inherently ambiguous, because they drop out of all observable relationships that
can be found in actual measurement and experiments.
Therefore,
the Heisenberg interpretation has also brought about "a novel kind" [1] of
problem, which "goes to the root of basic notions that are at the foundation of
physics." Just as the Lorentz theory on space and time [1], the difficulty of
this mainstream of the Copenhagen interpretation [10] is not its disagreement
with experiment. On the contrary, it is in accord with all that has been
observed since then. The problem essentially is rather that the fundamental
concepts entering into the interpretation, e.g., the notion of particle, are
in fact completely ambiguous. For, as we have seen, it was deduced on the
basis of Heisenberg's uncertainty relation itself that no means at all could
ever be found to give precisely to a particle simultaneous values of position
and momentum. Indeed, since the complete description of classical notions of a
particle cancel out of all observable results, it makes no difference whether we
need such a classical concept of particle in quantum mechanics or not.
From
the above discussion we have seen the remarkable similarity rooted in both the
Heisenberg interpretation and Lorentz theory as detailed by Bohm [1]. Both
theories were developed during a time of crisis in physics when new evidence
showed certain straightforward contradiction to some basic hypotheses of
classical physics (Sir Kelvin's two clouds). To retain the older notions in the
new formalism frames established by the new evidence, both theories need to
refer to a mechanism of the action of apparatus in the measurement, which in
effect distorts or cancels our exact knowledge of these notions. However, as a
direct result those basic notions have become intrinsically ambiguous.
According
to Einstein's relational approach to physics [1], however, the resolution of
this fundamental ambiguity involves a radical change in thinking by basing
ourselves as far as possible on the facts and on hypotheses that are in
principle testable. What are these facts? At the beginning of this section we
have already analyzed one aspect of the relevant facts, viz., that all our
actual knowledge of physical objects is based on observable relationships
established by interaction. To avoid ambiguity in our fundamental notions of
physical objects, it is therefore necessary to express the whole content of
physical law in terms of such relationships, and not in terms of a particle with
intrinsically untestable properties (e.g., simultaneous values of position and
momentum) that are inherently ambiguous.
In
the next section we shall show that the quantum theory of radiation, or its
development since 1960s into quantum optics, provides a clear notion for the
description of detection processes, which is decisive for the study of physical
content in terms of those relationships.
III.
Relational approach to the quantum theory of radiation
Since
the time of Faraday and Maxwell, physics has been developing a field theoretical
description of nature. Thus, our knowledge nowadays of fundamental processes is
viewed through various fields and their interactions. To develop a relational
approach to quantum physics, however, it is not necessary to go too far in this
direction, but to concentrate our discussion on the quantum theory of light, for
the reason that "in quantum optics it is often possible to address such
questions from essentially first principles and to carry out accurate tests of
the theory in the laboratory." [11]
According
to Maxwell's electromagnetic theory, light is a transverse field. In vacuum it
is described by
where
its total energy and momentum are
It is
appropriate to say that the rise of two of the most important principles of
physics in this century, relativity and quantum mechanics, was connected to the
studies of field theories. Indeed, Einstein's special relativity was created out
of the investigation of the electrodynamics of moving bodies. In contrast,
quantum mechanics brought about new interpretations of the "meaning" of field
theories. This began with Schrodinger who introduced a wave equation. Based on a
particle notion, Born interpreted the wave function as a probability amplitude,
the square of which is the probability of finding the particle at a particular
point in space. When this is applied to light, the particle is called a
photon. This interpretation indeed is very fascinating for it can account
for all phenomena that have been observed. But, such a "success" is also at the
cost of the key notions in the interpretation being inherently ambiguous, as we
have discussed in the previous section.
According
to this statistical interpretation, one, quantum mechanically, cannot think of a
classical particle as being defined by its position and momentum, but must
instead introduce the probability of finding the particle. In other
words, the interpretation, on one hand, emphasizes explaining quantum phenomena
in terms of the particle concept, but, on the other hand, it is also inferred
from this interpretation itself that the completeness of description of a
particle trajectory is impossible. Or, that is to say, the particle
interpretation is essentially ambiguous. This leads to much confusion. To the
lay mind it seems like moving without passing through intervening space, and to
the expert it likes a "fuzzy ball." [12] Indeed, if one cannot tell how a
particle moves from one spot to the other in the space, then this particle
notion would be "just purely conceptual inventions, like dotted lines that we
sometimes draw in our imaginations, when we apply geometrical theorems," [1] in
order to draw conclusions concerning the real observations.
Such
a problem is not just a purely theoretical one, which arises only as a result of
the analysis of the Copenhagen interpretation. It is also a factual problem.
For, although in nonrelativistic quantum mechanics it is still possible to give
a statistical interpretation over the position of a particle in the
configuration space, such an interpretation can no longer exist in the
relativistic frame. All the considerations from (1) finite velocity of light
[13]; (2) impossibility of constructing a position operator [14]; and (3) the
gauge invariance [13, 15] indicate that the probabilistic definition of position
is formally possible only in the limiting case of negligible de Broglie
wavelength [16].
The
appearance of the quantum theory of radiation a few years later resulted in the
second kind of the interpretation based on field quantization, that is,
socalled second quantization. This scheme regards a field dynamically as
a set of harmonic oscillators. Therefore, the quantization becomes a procedure
to replace the pairs of normal variables into pairs of operators, which have the
following relations:
Then,
one can describe all the physical properties of a field in terms of these pairs.
For example, the "global" properties
and
the "local"
where
e_{i}
is a unit vector of polarization, while
the quantum state of the field is represented by a vector y in
Fock space {n_{i}}>.
Since
applying a_{i} and
a_{i}^{+} to
Fock states causes the states to shift, we call them photon annihilation and
creation operators, respectively. It therefore suggests one may interpret the
field described by n_{1}n_{2}...n_{i}...>
as an ensemble of n_{1}
particles with energy hw_{1} and
momentum hk_{1} ...
and n_{i}
particles
with energy hw_{i}
and
hk_{i}
[13,
17]. In this way, once again one obtains a particle notion but avoiding the
aforementioned formal difficulties. Yet it seems this time that the attempt to
adjust the particle notion to observed facts has led us to a situation of more
confusion in which it is no longer clear what is meant by our photon notion as a
particle or what can be done with it.
In
view of this deep ambiguity and confusion that has developed from the
application of the intuitive notion of the particle beyond its proper domain,
our approach must be, as we remarked earlier, to begin our inquiry afresh by
basing on the facts in our actual processes of light detection. Such a notion
has been developed by Glauber in 1963 [18] that clearly describes light
detection processes based on photoionization. He showed that, for an ideal
photodetector being put at point r in a radiation field, the
probability of observing a photoionization, the counting signal, in the detector
between t and t + dt is proportional to
WI(rt)
dt with
where
E(+) and
E()
are
the positive and negative frequency components of electric field in (5a), and
the state y>
specifies the properties of a field at all times (the Heisenberg picture). It
formulates the exact mathematical expression of what we have discussed on
physical facts at the beginning of Sec. II.
For
simplicity, we shall consider the onedimensional propagation of a onephoton
state (y>=Sc_{i}a_{i} 0>,
c_{i}s are
probability amplitudes), constructed by CohenTannoudji et al. [17] It is
easy to show that the detection probability propagating along the x
direction is
This
probability propagation of observing photoionization within detectors
also has reproduced the probabilistic wave of quantum phenomena that propagates
without deformation with the light speed c. However, it is more essential
in physical content, since the expression itself automatically takes into
account the role of apparatus in the detection processes.
We are now ready to adopt the relational
approach to quantum physics: We shall regard the photon as a kind of
elemental "record" expressing a relationship of a light field to an actual
detection process in which this record is registered. That is, it is only a
portion of energy and momentum which is transferred from a light field to a
detector and by which the record is realized; And our point of departure will be
that in terms of actually measurable "records" of this kind, the interaction
between a light field and a detector in the detection processes is described by
a probabilistic law as expressed by Eq. (7).
We do
not regard the above result as a deduction from the Copenhagen interpretation,
but as a basic hypothesis which is evidently subject to experimental
tests and which has in fact already been confirmed in all the experiments up
till now.
To
see more clearly what this hypothesis implies with regard to the meaning of the
notion of particle trajectory in quantum physics, let us reconsider the
Wilsontype experiment. To indicate a particle trajectory for light (or other
fields), we need arrange an array of photodetectors along the x direction
to record this trajectory. However, The fact is that the detection
processes now must follow a probabilistic law, in which the probability
amplitude of counting signals in detectors along the x direction is
wavily modulated as expressed by Eq. (7), since as we have seen, experiments
show this to be the case. Therefore, one can no longer draw a particle
trajectory for light. because at some points the trajectory may actually
discontinue.
This
is a major break with Newtonian ideas, because one cannot use the notion of
particle trajectory to describe quantum phenomena of light. It must be
emphasized, however, that for light the establishment of notion of particle
trajectory is based only on an indirect deduction, the result of an
organization, which put together counting signals in the detectors along the
x direction. Particle trajectory is therefore no longer an immediate
fact corresponding to light ray in our everyday experience. For it is now
seen to depend, to a large extent, on a purely conventional procedure of
assembling detection signals in the propagation direction. This convention seems
natural and inevitable to our "common sense", but it leads to unambiguous
results, a trajectory can be assigned only under conditions in which geometrical
optics is a good approximation [19]. When the characteristic de Broglie
wavelength can no longer be regarded as effectively infinitely small, then the
experimental facts of physics make it clear that the absolute notion of particle
trajectory should be abandoned.
It
cannot be emphasized too strongly that in this relational approach one does not
deduce Eq. (7) as a consequence of the disturbances of observing instruments as
indivisible quanta are needed for measurement, and from this, infer that a
causal description is impossible in quantum physics. Rather, one begins with the
experimentally wellconfirmed hypothesis of the probability of interaction
described by Eq. (7), as actually measured. This needs no explanation
(e.g., in terms of disturbance of instruments due to indivisible quanta), but is
just our basic starting point in further work. With this starting point, one may
expect to discover new concepts from the quantum formalism, taking the notion of
particle trajectory as a
limiting case.
Thus,
the new notions emerging from the framework of quantum optics are in terms of
quantum fields (such as light field and electron field) and
interactions (such as the detection of light by photodetector). Light, as
a whole, is described by a field in a quantum state y>,
whose "global" properties are characterized by only those conservative
quantities, such as energy and momentum, in the corresponding operator form of
Eq. (4), acting in Fock space, while the "local" properties, such as
propagation, are described by Eq. (5). However, to get any information
concerning the field, an observer needs a number of photodetectors. That is, by
the interaction between light and detectors one gets the immediate facts on the
field. To demonstrate how the physical phenomena now are described in terms of
the field and interaction, in what follows we shall especially focus our
discussion on the propagation properties of light.
Generally,
the regularity and order in the propagation properties of light can be summed up
in the notion of frames of detection. This is essentially the placement of an
array of detectors in a particular way, set up to make possible the expression
of the results of different detections in a common language, and thus to
facilitate the establishment of relationships among these detections. For
example, in a particle frame of detection, one puts a series of
photodetectors in the propagation direction of light. Here, the important fact
is that there exists a set of invariant parameters among different
detection processes, for example, the velocity of light signal propagation
(emission and then absorption) c, which enables us not only to
characterize a "trace" but also relate the "trace" to a portion of energy and
momentum (a photon) transferred from light to a detector in the interaction, to
form a particle picture (p = E/c).
There
also exists a wave frame of detection. In this frame the light is split
into two paths so as to interfere with each other. To see the effect, one also
need to put an array of detectors on the interfering plane, from which one can
infer another set of invariant parameters, such as the frequency, wavelength,
and also phase velocity from the interference fringes formed. Thus one
constructs a wave picture. Indeed, as far as Newtonian mechanics is
concerned, such a wave frame of detection seems not necessary, and it makes
sense to ascribe a particle notion as the only invariant feature to all the
cases in the domain.
Of
course, all this experience depends on the circumstance that the de Broglie
wavelength is so small that on the ordinary scale of distance and time, the wave
modulation in this kind of counting signal detection can be neglected. This is
equivalent to assuming an infinitely small de Broglie wavelength of matter. When
the finite de Broglie wavelength of matter is taken into account, as it was the
case in DavissonGermer experiment, and as light itself also behaves like a
wave, new problems of "waveparticle duality" do in fact arise, which ran
through the famous BohrEinstein dialog and which is still a key issue in recent
interpreting of quantum mechanics [20, 21].
In
the dialog, the point in dispute was the problem of physical reality, for "the
dependence of what is observed upon the choice of experimental arrangement"
seems so "bizarre and counterintuitive" to our common experience. However, this
"observerparticipancy" [20] is not peculiar only to the quantum world. As a
matter of fact, it was shown in Bohm's book with substantial scientific evidence
that it is a common character to our actual mode of perception of the
world, the implication of which is best understood from a relativistic point of
view. (It would seem that this participating nature looks strange to us, mainly
because of our limited and inadequate understanding of the domain of common
experience, as Bohm suggested and discussed in detail in the appendix of
Ref. [1].) Here then is our task for the following discussion.
In
the procedure described above, we see that the analysis of light into
constituent objects (photon particles) has been replaced by its analysis in
terms of quantized fields and interactions (while in Einstein's special
relativity, the analysis was replaced in terms of events and
processes [1]), organized, ordered, and structured so as to correspond to
the characteristics of the light system that is being studied [22]. It follows
that the particle picture and the wave picture taken jointly constitute
the means by which the characteristics of physical phenomena are to be treated.
In this sense, particle and wave pictures together are playing a role similar to
that played by the particle picture alone in Newtonian mechanics. That is to
say, the nature of light is being described in terms of a kind of "interactive"
pattern between a field and the detection of the observer, as exhibited in the
framework of quantum optics.
In an
interactive pattern, for example, of any interactive kit developed in recent
multimedia culture, there is a thoroughgoing unification of its different flows
of knowledge or entertainment whose courses the user can affect, based on the
fact that each of the flows can be related to the others by means of some kind
of directory. The question then naturally arises as to whether, in the
"interactive" pattern of particle and wave pictures taken together, there is not
a similar unification structure of particle and wave pictures, so that "these
two aspects can be causally related with each other," as Einstein firmly
believed [6]. (Recall that in Newtonian mechanics, a wave is derivatively
considered as a periodic motion of particles, so that the particle concept is
more basic and no such equal unification occurs there.)
To
see that there is in fact such a kind of unification of particle and wave
pictures in the framework of quantum optics it is necessary only to refer to Eq.
(5), in which de Broglie's idea [23] is now expressed by the operator
E(rt) (= E()(rt)
+ E(+)(rt)),
in terms of annihilation operator a_{i} (and
creation operator a_{i}^{+} ) as
the amplitude with a modulating phase factor
e^{i}^{(ki.r

w
it)} (and
its conjugate e^{i(ki.r

w
it)}).
This expression evidently contains both information of propagation properties of
light in the two different frames of detection. The propagation of annihilation
operator a_{i} and
creation operator a_{i}^{+} ,
which physically describe events of absorption and emission of light,
determines that in the particle frame of detection light signal travels
at the speed c; and the phase factor e^{i}^{(ki.r

w
it)} in
Eq. (5),
due
to its modulation effect into the probability expression of Eq. (6),
reflects that in the wave frame of detection interference occurs of
counting signals of detection. Thus Eq. (5) implies both what one can observe in
different frames of detection.
It
seems clear then that in the framework of quantum optics, two pictures of
particle and wave are united as two sets of features of the same field in
two different frames of detection, in which they can be related to each other in
such a way that Eq. (5) is invariant. This unification can be characterized by a
term called particlewave rather than "particle and wave," the hyphen
emphasizing the new kind of unification.
It
should be noted that in spite of the abovedescribed unification of particle and
wave brought about in the framework of quantum optics, there remains a rather
important and peculiar distinction between them, resulting from the fact that
a_{i} and
a_{i}^{+} are
operators but the phase factors
e^{i}^{(ki.r

w
it)} (e^{i(ki.r

w
it)}) are
cnumbers. On the basis of this distinction, it is also made clear that
the modulation wave in the probability amplitude of counting signals as if
"moved" at a velocity (phase velocity) greater than c in de Broglie
matter systems in no way confuses us on the maximum speed of propagation of
signals, provided that a signal propagation is described by the annihilation and
creation operators a_{i} and
a_{i}^{+} .
The
implication of the framework of quantum optics can be made clearer, by which
much of our confusion in the waveparticle duality can be avoided, if one still
follows Bohm's discussion of the Minkowski diagram to explain it as a kind of
conceptual map. As we know in special relativity, the diagram of Minkowski also
serves as an invariant structure by which one can relate the measurements of
space and time coordinates in different frames of reference to the same
event [1].
Because
of the relativistic unification of particle and wave pictures into a single
expression Eq. (5), there appears an illusion of coexistence of wave and
particle pictures. However, a little reflection shows that this view of the
framework of quantum optics must be very far from the truth indeed. Consider,
for example, that an observer wants to measure the speed of a light signal, he
must construct a particle frame of detection that registers both where and when
a light signal is emitted and then absorbed. (The propagation of a light signal
is in fact a subject of special relativity). Such an observer cannot survey the
whole of Eq. (5). On the contrary, he can only know of the propagation of
annihilation and creation operators a_{i} and
a_{i}^{+} .
Therefore, the exact information of the phase factor
e^{i}^{(ki.r

w
it)} is
unknown to him; that needs an interference experiment.
The
real situation, as experienced by an observer at one of the frames of detection,
is indeed strikingly different from what is shown in Eq. (5). An observer's
knowledge is restricted to the part of Eq. (5) (for example, the amplitude part
a_{i} and
a_{i}^{+} )
that is in the particle frame, and he never sees what is found of the other part
(the phase factor) in the wave frame, as it is represented in Eq. (5).
For in any frame of detection we are experiencing only what is actually present
in that frame. What we see in the wave frame no longer actually exists in the
particle frame. What is left of the wave experiment done before is only a record
of detection. This record may be in our memories, or in a photographic plate.
From these records we reconstruct a wave picture in our thoughts, as well
as with the aid of pictures and models.
Of
course, as Bohm has also conclusively illustrated with the example of the
Minkowski diagram [1], our notions of physical phenomena are in fact all based
on a reconstruction, "in accord with appropriate geometrical, dynamical,
structural principles that have been abstracted from a wide range of past
experiences." In this sense, the framework of quantum optics will be also a kind
of conceptual map, having a structure that is similar to that of real sets of
light fields and interactions that can actually be observed. "Any map of this
kind is what the world is not." That is, the framework of quantum
optics consists of operators and states of an operator calculus in Hilbert
space, while the experiments in the real world contain laser sources, beam
splitters, photodetectors, and so on. But as happens with the framework, it
implies a structure similar to the structure of what is.
"In
all maps (conceptual or otherwise) there arises the need for the user to locate
and orient himself by seeing which point on the map represents his
position and which line represents the direction in which he is looking."
In doing this, one recognizes that every act of "actualization" [21] as in the
discussion of waveparticle duality yields a unique perspective on the world.
But with the aid of the framework of quantum optics, one can relate what is seen
from one perspective (the particle frame) to what is seen from another (the wave
frame), in this way by abstracting out what is invariant under change of
perspective, and leading to an everimproving knowledge and understanding of the
actual character of the radiation under investigation. Thus, when an observer,
doing experiments with different frames of detection, is to understand what he
sees, he need not puzzle, regarding to which view is "right" and which view is
"wrong" (wave or particle). Rather, he consults to the map  Eq. (5), and try
to come to a common understanding of why in each way detecting the same light
field has a different perspective and comes therefore to his one view, related
in a certain way to that of the other (for example, the de Broglie relation
p = h/l).
In
this way, we unite two pictures of wave and particle as two sets of invariant
features of the same light field in different frames of detection. The notion of
particle in Newtonian mechanics now is as an approximation under the
circumstance that the effective de Broglie wavelength is infinitely small,
whereas in another limiting case when the average photon number is large enough
so that discrete phenomena of quantization are washed out, we recover the
concept of electromagnetic wave of Maxwell's theory.
Of
course, the story of quantum relativity does not stop here. In the "signal
counting signal" [17] domain of detection discussed above, the notion of
particle can still be contained as an approximation, which enables Heisenberg to
account for quantum phenomena by means of a disturbance. But, in the double
counting signal (secondorder) domain, "Heisenberg's microscope experiment
breaks down." [10] The Newtonian notion of particle can no longer explain
longdistant correlation phenomena without violating the special relativity.
Because there must be a nonlocal informing mechanics between two separated
particles. (This was also realized by Bohm [24, 25].) These phenomena, however,
can be explained as the correlation of local interactions of a global
quantized field in a state y>.
Thus, when we come to this new domain of experience, it is not surprising that
new concepts are needed, leading to understanding of the new phenomena under
investigation [25].
Milonni
has showed that this kind of phenomena can be unitedly described in terms of
secondorder correlation function WII(rt,r't')
in the detection theory [25, 26]. The probability of double counting signals
that a photoionization occurs at r between t and t +
dt and another one at r' between t' and t' +
dt' is proportional to WII(rt,
r't')dtdt' where
with
m, n summing over polarization axes x, y, and
z.
This
joint counting probability of Eq. (8), by its nature, suggests an invariant
relationship in this domain. If we take the coincidence of two photoionization
events (the complete correlation) as the manifestation of the wave
feature, whereas the particle feature means no joint counts ever
occur, then, for a light field in state y>,
we have the following identity
where
the differential probability dP_{wf} =
WII(rt,
r't')dtdt', and by definition the probability of no
joint counts dP_{pf} = 1 
WII(rt,r't')dtdt'.
Therefore, in this way we also unify the wave feature and the particle feature
into a continuum within this secondorder domain. Thus, we see that not only the
way the quantized field is detected but also how it is generated (in different
quantum state y>)
plays an important role in the physical laws in the quantum theory of
radiation.
Finally,
we shall follow Bohm [1] to reach the following summary. In Newtonian mechanics
the role of the observer was very much underemphasized. Since physicists may
have thought that the perspective of the observer need not appear in the
fundamental laws of physics, though they may have always learned that each
observer does have a perspective. Rather, they described physical phenomena in
terms of motion of "absolute" particles that are independent of the way in which
they are measured and observed, so that no part is played by the observer at all
in these laws. On the other hand, according to the relational approach to
physics, it is clear that the framework of quantum optics is a map corresponding
to what will be observed in a frame of detection arranged in a certain way.
Therefore, this map has already taken into account some of the observer's
perspective. Moreover, as we have seen, not only the way of detecting a field
but also the way of preparing it (in a state y>)
has a different perspective to the field in the secondorder detection domain.
Thus, whether we consider what is seen by different observers or by the same
observer in different frames, it is always necessary to relate the results of
these detections, by referring to a particlewave map with a correct structure,
"and in this way to develop an evergrowing knowledge and understanding of what
is invariant and therefore not dependent on the special perspective of each
observer."
IV.
Further discussion and conclusion
The development of modern physics has
shown its striking tendency of more and more getting away from "absolute"
notions. Newtonian mechanics, as the first main theory of physics of this kind,
had already incorporated a number of relativistic ideas that underlie our use of
the Galilean transformation. But in the theory the basic three, the notions of
space, time, and physical objects, were still treated as
absolute.
Radical
revolution in our concepts of space and time initiated by Einstein, in some way,
depends on how to understand a new transformation (the Lorentz transformation)
discovered in electrodynamics [5, 27, 28]. In terms of the old notions of
Newtonian physics, i.e., the "real" (or "true") time that Lorentz called, he
thought that the time which entered into the transformation relations was the
"apparent" time. But, the "very famous point of Einstein" was essentially to
base on facts and hypotheses that are in principle testable. "There is not one
'apparent' and another 'real' time; there is just one 'real' time, and that is
what Lorentz called 'apparent' time."[27] In terms of this real time that
consists only of a relationship between the observed phenomenon and apparatus,
new concepts concerning space and time then are necessary.
The
fundamental changes in our notion of physical objects, however, reside in the
quantum formalism, especially, in the factoriented framework of quantum optics.
Such a new formalism might also be expected to lead us to new concepts, which
would contain the older ones as approximations and limiting cases. With
Einstein's relational approach to physics, in this paper, it is proposed that
the real photon, entering the new framework of quantum optics, also
expresses only an elemental relationship between a light field and a
photodetector that we can really observe in detection, and it is hoped that this
treatment will bring about a "turning around of the physical picture."
In
Einstein's special relativity, the role of the event (such as emission or
absorption of signal) and process (such as the transmission of a signal)
was introduced to replace that of object and its motion, which are
basic in Newtonian theory, where the Minkowski diagram serves as a conceptual
map that already has the observer's perspective implicit in it. In the quantum
relativity discussed here, however, an analysis in terms of the quantized
field and interaction is further suggested to account for those
events of emission and absorption in the domain of quantum phenomena, in which
the framework of quantum optics, the same, becomes a map that can tell us what
are to be observed in different frames of detection.
It
can be said that, in Einstein's relational approach, the whole task of physics
is assumed to find out what is relatively invariant in the study of
relationships between various aspects of this universe. Such a guiding
epistemology should also be of utmost importance to quantum physics. Since it
seems that conceptual difficulties arise whenever we refer particles and waves
as more or less permanent objects, rather than regard them as relative
invariants which have been abstracted from a variety of relationships of
observation; and it seems that once we can decompose the problems into
fundamental processes in terms of the interaction between a light field and
detectors, one would in principle access the key to the problems.
In
terms of the field and interaction, basic change in our notions of quantum
measurement is to be expected, in which one no longer regards the interaction as
a disturbance factor from an observing apparatus to an object, and, from this,
infer that all the complete descriptions of the object are impossible as
specified by Heisenberg uncertainty relations. Rather, one should utilize
the interaction as a means by which we build up relationships between the
observing apparatus and the observed universe, so as to find out the invariant
structure of the universe.
In
terms of the field and interaction, the probabilistic "wave" of quantum
phenomena in Eq. (7) represent only the probability of interaction
event (emission or absorption
of a portion of energy and momentum) happening in the processes of detection. We
hope that this kind of probabilistic hypothesis would have pleased Einstein, for
he opposed only the ontological probability.
In
terms of the field and interaction, problems such as interference between
independent laser beams will no longer puzzle us by posing questions such as:
How does a photon interfere? With itself [29] or with others [30]? Since Eq. (6)
describes the probability of interaction event, in which a portion of energy and
momentum (a photon) is transferred from one of the light fields to a detector,
the detector certainly does not discriminate from which field it received the
photon, and the phenomenon can be well explained by the interference of
transition amplitudes [17] in the framework of quantum optics.
In
terms of the field and interaction, not only waveparticle duality, discussed
above, but also the other kinds of duality revealed recently [31] can be
understood by taking the framework of quantum optics as a map that has implied
the role of the observer. Thus, the "potentiality under the actualization" [21]
should be understood as a potentiality of interaction in which the
observer can choose freely his detection frame, rather than an
ontological potentiality by which one thereby falls into a kind of
"subjectivism." The measurement of polarization discussed in Ref. [21] is just
such an example, which not only illustrates the principle of superposition of
states in quantum mechanics [29], but also plays a great role in Bohm's EPRtype
experiment [24]. The main facts establishing the superposition law, however, are
still based on interaction. Every light field does have its definite (rather
than "ambiguous") polarization direction. But to measure the polarization one
must employ a polarizer, the function of which is to "project" an unknown but
definite mode onto two known perpendicular directions. That is, by absorbing a
photon of the mode and then emitting one of the two modes with probability, we
can then assign the polarization. As we know, in the experiment the probability
depends on the orientation of the polarizer that we can freely choose. Thus, we
see that quantum theory does emphasize the special role of each observer in a
way that is different from what is done in Newtonian mechanics. "But the
recognition of this unique perspective serves, as it were, to clear the ground
for a more realistic approach to finding out what is actually invariant and not
dependent on the perspective of the observer." [1]
To
arrive at the final conclusion of this paper, let us recall a priori
assumptions of Newton in the introduction. After we have in effect followed
Bohm's relativistic melody to hum a quantum song, it will be realized that it is
our mankind selves who "in the beginning form'd matter in solid, massy, hard,
impenetrable, moveable, ..., as most conduced to" our everyday life in the
mansized domain. Such a notion generally is adequate only in this domain of
validity, so that as we go beyond this domain, one may expect to come to the
development of new concepts. In the progress of this process, twin ideals of
Galileo's scientific methods, experimentation and mathematization, along with
Einstein's relational approach to physics, will forever have won.
Appendix:
Quantum paradoxes  "EPR paradox" and "the case of Schrodinger's cat"
After
we have adopted the factoriented framework of quantum optics to bring out new
notions for describing quantum phenomena, in this appendix let us return to the
land of quantum mechanics where fundamental problems originally arose.
One
of the problems of the Copenhagen interpretation that concerned Wigner very much
is that of "unmeasurable quantities." [32] Early in 1952 [33] Wigner showed that
quantities which do not commute with all additive conserved quantities can not
be precisely measured. Araki and Yanase [34] later further proved the theorem
mathematically for the general case, and a detailed discussion of the physical
implications has been provided in Wigner's Princeton Lectures [32].
The
WingerArakiYanase (WAY) theorem has in fact posed the severest problems to the
"standard interpretation," not only because the results "blur the mathematical
elegance of von Neumann's original postulate that all selfadjoint operators are
measurable," but also
because, if those quantities, in a strict way, are unmeasurable, it makes no
difference whether we assume that there are such quantities or not,
according to Bohm's opinion, which was intensively discussed in his book
[1].
The
"difficulties inherent in the measurement of a great many, if not most,
operators" were shown to be the "internal" problems of the standard
interpretation. Therefore, if those quantities that von Neumann called also
observeables are in fact unobservables, one then should not interpret quantum
mechanics in terms of those notions, since in quantum theory the "very famous
point of Einstein" was also emphasized that only measurable quantities belong in
a theory [27], so as to avoid making unnecessary and unprovable assumptions
concerning those quantities that are unmeasurable.
Henceforth
position by itself, and other quantities of this kind by themselves that do not
commute with conserved quantities, "are doomed to fade away into mere shadows"
[28] (as bare operators in quantum formalism), and only those conservative
quantities will preserve the physical reality.
Thus
position, whose physical reality must be unequivocally determined by the
structure of quantum mechanics, is "first deposed from its high seat, " and new
concepts are needed as discussed in the main text.
In
view of the "very famous point of Einstein," the WAY theorem that only
quantities which commute with all additive conserved quantities are precisely
measurable can also offer us new sights into EPR paradox [35]. In 1935,
Einstein, Podolsky, and Rosen has studied a system in which two "particles"
interact with each other at first and then the interaction ceases (it is so
arranged that the measurement of one of the two "particles" can be performed
without in any way disturbing it). According to the "standard interpretation"
that every observable quantity corresponds to a selfadjoint operator, they
reached the conclusion that (I) when the operators corresponding to two physical
quantities do not commute, the two quantities do have simultaneous reality. This
obviously contradicts the property of selfadjoint operators in Hilbert space:
(II) when operators corresponding to physical quantities do not commute, the two
quantities can not be simultaneously measured, that is, a paradox. Quantum
mechanics was so successful in explaining phenomena in the atomsized world that
they did not question what later Bohm called "the inherent ambiguity" in the von
Neumann axiomatic system, but instead doubted the completeness of quantum
mechanics.
Now,
in accordance with the WAY theorem, we see that, the two quantities (position
and momentum) which EPR has assigned, the measurability of position whose
corresponding operator does not commute with Hamiltonian, is actually completely
ambiguous. Thus, it is a sure thing that the paradox, due to the intrinsic
vagueness of the measurability of "a great many" quantities, will be forever
"unfalsifiable" within the standard interpretation, a term that Sir
Popper used to describe any such a kind of proposition. (For the detailed,
please see Chapter XXV of Bohm's book "The falsification of theories" [1].)
Besides
the ambiguity of measurability for many important quantities that formed the
Copenhagen interpretation, there are other fundamental weaknesses of the
standard theory. In his Princeton Lectures Wigner also reformulated resolutions
for those problems related to the measurement paradox. To bring out the point
directly, let us begin with the discussion of quantum jumps [36]. In quantum
mechanics, such a process of quantum jumps is described by a system with
coupling (i.e., in interaction with the external surroundings).
From
substantial experimental spectral facts, it is firmly verified that some of
physical elements of atomic systems (for example, the energy) only take discrete
values. Hence, the development of those elements are in a jumplike style, i.e.,
quantum jumps. However, for a system with coupling, the evolution of Schrodinger
wave function with time is continuous. Thus, it is obvious that the
development of physical elements (physical quantities) and the evolution of the
wave function of a quantum system are by no means the same process.
In
classical physics, the continuity is assumed to be a basic feature of physical
systems. That is to say, by interaction energy and momentum are transferred
continuously from one system to another. However, Planck's quantum hypothesis
thoroughly altered this picture to give quantum jumps between discrete
eigenvalues of atomic systems. It is also this "allornothing" nature [25]
of Planck's quanta that gives rise
to intrinsic chance of quantum events of interaction, in which one can only
plead to the objective probability [21], as we have discussed above.
(Probability out of the "or" relation.)
Thus,
it is evident that there needs two different mathematical entities in the
quantum formalism, in order to describe completely quantum phenomena:
eigenvalues, which are related with eigenfunctions determined from
eigenequations of operators in a system, represent physical elements of reality;
the Schrodinger wave function, which is the solution of a timedependent
Schrodinger state equation, give the probability of quantum jumps between those
eigenvalues. (A Schrodinger wave function, in general, is expanded into a series
of eigenfunctions, where the coefficients are explained as the probability
amplitudes of finding the eigenvalue corresponding to that eigenfunction in the
system.)
To appreciate the necessity of the
doubletrack description, we shall discuss an example of a twolevel atomic
system decaying in an electromagnetic background.
If the system is in interaction with a
free space, the effect of the coupling on the two processes are respectively:
(1) For the Schrodinger wave function,
where
y_{a}>
and y_{b}>
are the eigenfunctions corresponding to the eigenvalues of upper energy
E_{a} and
lower energy E_{b}. In
the WeisskopfWigner approximation, the probability c_{b}(t)
that the system can potentially take the eigenvalue E_{b}
is
growing with the time exponentially. It is seen that the Schrodinger wave
function y(t)>
evolves continuously and causally. (2) For the physical elements, the coupling
to the free space provides a possible way out for the developing. Since whenever
a jump occurs at the later times, the quantum emitted to the electromagnetic
field will propagate away in the open space making the transfer irreversible.
Thus, if one follows the behavior of the atom, it will stay all along in the
upper level until a jump occurs, in which the time of jump is distributed with
probability determined by c_{a}(t)
(or c_{b}(t)).
If
the system is put in a closed space (for example, an atom in a microcavity), the
effect of the coupling will lead to: (a) The above evolution of the Schrodinger
wave function is modified and quantum recurrence of Rabi oscillation occurs; (b)
Since the reflection of the electromagnetic field by the cavity walls, the jump
in the development of physical elements is effectively prevented [37], and the
system will always stay in the upper level, i.e., nogo, no matter the
oscillation in the evolution of the Schrodinger wave function. Nevertheless,
when the atom is flying across the microcavity at a speed, because the whole
system of atom and cavity should be taken as a "caviton," which is also in
interaction with the free space. This "caviton" can emit an atom into open space
that will fly away. Therefore, at the moment when the atom leaves the cavity, as
in the first case, the probability of emitting the atom in the upper state is
still determined by the evolution of Schrodinger wave function, that is, by the
interval time of the atom flying across the cavity, as was verified by
experiment [38].
Certainly,
these two dynamics interplay with each other. For instance, in a simple case of
decaying in a multilevel system, once a jump occurs in the developing process
of physical elements, the evolution of the Schrodinger wave function will start
from new initial conditions. And in more complicated systems, there is the
effect of "backaction of measurement," which has now been well studied in
quantum optics to include phenomena such as quantum jumps, dynamics of
micromaser, and continuous photodetection [39]. In these cases, a jump in the
developing process of physical elements will drastically affect the way that
Schrodinger wave function evolves, and as a repay, the evolution determines the
probability of the next jump.
With
the above clarification that eigenvalues of operators describe the physical
elements of reality, whereas the Schrodinger wave function will give the
probability of quantum jumps between these eigenvalues, it seems clear that, if
one attaches the Schrodinger wave function to the physical element, then it "is
consistent with modern quantum mechanics only if the temporal evolution of the
system is such that a coherent superposition of the states does not develop."
[36] Instead, if one pins the physical element on the Schrodinger wave function,
then the physical element will be "blurred," [40] because it can take different
values at the same moment. Thus, it would be "naive" to accept a "blurred model"
for representing reality, according to Schrodinger.
We
have seen that it was the absurdity of a "blurred model" of the Copenhagen
interpretation that upset Schrodinger to propose "the case of cat." However,
there will be no paradox in the case, if one examines his executing device in
detail. The paradox was caused by our ignorance of a very simple fact that
two matter systems never interact directly, but mediated by an
electromagnetic field.
The
derivation of the cat paradox is based on a coherent evolution of the state of
the total system that leads to the entanglement between the state of the
observed system and that of apparatus [6]. However, in all the cases of this
kind of decaying problems, as we have analyzed above, the coherent evolution is
rather between an atom and electromagnetic background, i.e. [41],
than
between the atom and a detector, the cat, where multimode {0} expresses the
vacuum state of radiation, and {1r} the
state of one photon in the rth mode and none in the other,
c_{a,{0}} and
c_{b,{1}r
}
are
probability amplitudes. Thus, it is clear that there is neither a waver between
the "cat alive" and the "cat dead", nor even the states that are assigned to
them in the Copenhagen interpretation. The "alive" or "dead" of an innocent cat
is only an indication of whether a quantum jump occurs in the developing process
of physical elements of the atom, as discussed above, to which the coherent
evolution of the state of an atom and the electromagnetic background in
Eq. (A2) corresponds. Or in other words, long before the box is opened, it has
already been determined that the cat is alive or dead. Therefore, there is
nothing paradoxical in the state of affairs [21].
Although
the Schrodinger cat might no longer bewilder us any more, the study of socalled
Schrodinger cat states in quantum optics out of the "cat affair" is important to
understanding of the quantum statistical properties of radiation. It will be
useful in the state engineering of radiation fields and may find its
applications, for example, in coherent chemistry [42].
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Of course, in the quantum theory of radiation before, the physicists have
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phenomena. Rather, they assumed more or less that the physical objects are
described in terms of an ensemble of photon particles.
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Kis, Phys. Rev. A 50, 1777 (1994).
And his partner.
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