"All is changed,
changed utterly,
a terrible beauty is born."

The poet William Butler Yeats wrote in the aftermath of the
1916 Irish Rebellion. In the same year Albert Einstein finished
the second relativistic theory.  

Download a concise or full-length copy in PDF; some QM papers
difficult to find on decay in microcavity and quantum phase.

Book "Relational Approach to Quantum Physics" at Amazon










Quantum Optics as a Relativistic Theory of Light *


* We dedicate this essay to the memory of Professor David Joseph Bohm (1917-1992), 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 re-interpret 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 Einstein-Podolsky-Rosen 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 man-sized 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 four-dimensional 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 man-sized 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 atom-sized 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. IV-X). 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 wave-particle 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 Einstein-Podolsky-Rosen (EPR) paradox will become unfalsifiable within the Copenhagen interpretation. Since, according to the Wigner-Araki-Yanase 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 "common-sense" 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 far-reaching. 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 "common-sense" 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 Michelson-Morley 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 straight-forward 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, so-called 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 ei  is a unit vector of polarization, while the quantum state of the field is represented by a vector y in Fock space |{ni}>.

Since applying ai and ai+  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 |n1n2...ni...> as an ensemble of n1 particles with energy hw1 and momentum hk1 ... and ni particles with energy hwi and hki [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 one-dimensional propagation of a one-photon state (|y>=Sciai |0>, cis are probability amplitudes), constructed by Cohen-Tannoudji 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 Wilson-type 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 well-confirmed 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 Davisson-Germer experiment, and as light itself also behaves like a wave, new problems of "wave-particle duality" do in fact arise, which ran through the famous Bohr-Einstein 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 "observer-participancy" [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 ai (and creation operator ai+ ) as the amplitude with a modulating phase factor ei(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 ai and creation operator ai+ , 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 ei(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 particle-wave rather than "particle and wave," the hyphen emphasizing the new kind of unification.

It should be noted that in spite of the above-described 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 ai and ai+  are operators but the phase factors ei(ki.r - w it) (e-i(ki.r - w it)) are c-numbers. 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 ai and ai+ .

The implication of the framework of quantum optics can be made clearer, by which much of our confusion in the wave-particle 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 co-existence 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 ai and ai+ . Therefore, the exact information of the phase factor ei(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 ai and ai+ ) 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 wave-particle 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 ever-improving 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 (second-order) domain, "Heisenberg's microscope experiment breaks down." [10] The Newtonian notion of particle can no longer explain long-distant correlation phenomena without violating the special relativity. Because there must be a non-local 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 second-order 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 dPwf = WII(rt, r't')dtdt', and by definition the probability of no joint counts dPpf = 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 second-order 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 second-order 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 particle-wave map with a correct structure, "and in this way to develop an ever-growing 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 fact-oriented 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 wave-particle 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 EPR-type 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 man-sized 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 fact-oriented 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 Winger-Araki-Yanase (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 self-adjoint 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 self-adjoint 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 self-adjoint 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 atom-sized 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 jump-like 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 "all-or-nothing" 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 time-dependent 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 double-track description, we shall discuss an example of a two-level 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 |ya> and |yb> are the eigenfunctions corresponding to the eigenvalues of upper energy Ea and lower energy Eb. In the Weisskopf-Wigner approximation, the probability cb(t) that the system can potentially take the eigenvalue Eb 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 ca(t) (or cb(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., no-go, 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 multi-level 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 micro-maser, 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 multi-mode {0} expresses the vacuum state of radiation, and {1r} the state of one photon in the rth mode and none in the other, ca,{0} and cb,{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 so-called 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].




1. D. Bohm, The Special Theory of Relativity (Benjamin, New York, 1965).

2. I. Newton, Optiks (Dover Publication, New York, 1952), p.400.

3. M. Born, My Life and My Views (Scribner's, New York, 1968). p. 171.

4. W. Heisenberg, Z. Phys. 33, 879 (1925).

5. J. Mehra and H. Rechenberg, The Historical Development of Quantum Theory Vol.2 (Springer-Verlag, New York, 1982), p.261.

6. M. Jammer, The Philosophy of Quantum Mechanics (Wiley, New York, 1974).

7. W. Heisenberg, Z. Phys. 43, 172 (1927). For an English translation, see Quantum Theory and Measurement ed. J. A. Wheeler and W. H. Zurek (Princeton Univ. Press, New Jersey, 1983), pp. 62-84.

8. W. Heisenberg, Die Physikalischen Prinzipien der Quantentheorie (Hirzel, Leipzig, 1944).

9. P. A. M. Dirac, Directions in Physics (Wiley, New York, 1978), p.4.

10. D. Bohm, The Uncertainty Principle and Foundations of Quantum Mechanics, ed. W. C. Price and S. S. Chissick (Wiley, New York, 1977), pp. 559-563.

11. J. H. Eberly and P. W. Milonni, Encyclopedia of Modern Physics, ed. R. A. Meyers (Academic Press, New York, 1987), pp.519-550;

12. M. Scully and M. Sargent, Physics Today 25(3), 38 (1972).

13. V. B. Berestetski, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics (Pergamon, Oxford, 1982).

14. E. P. Wigner and T. D. Newton, Rev. Mod. Phys. 21, 400 (1949).

15. A. I. Akhiezer and V. B. Berestetski, Quantum Electrodynamics (Wiley, New York, 1965).

16. We shall reconsider the problem from a different angle in the appendix.

17. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms (Wiley, New York, 1989).

18. R. J. Glauber, Quantum Optics and Electronics, eds. C. de Witt, A. Blandin, and C. Cohen-Tannoudji (Gordon and Breach, New York, 1964), pp. 65-185.

19. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, MA, 1980), pp. 484-492.

20. J. A. Wheeler, Quantum Theory and Measurement ed. J. A. Wheeler and W. H. Zurek (Princeton Univ. Press, New Jersey, 1983), pp. 182-213.

21. A. Shimony,  Scientific American 258, 36 (1988); A. Shimony, The New Physics ed. P. Davies (Cambridge Univ. Press, Cambridge, 1989), pp. 373-395.

22. Of course, in the quantum theory of radiation before, the physicists have probably always realized the notion of field. However, they may have felt that such a concept of field need play no part in the interpretation of physical phenomena. Rather, they assumed more or less that the physical objects are described in terms of an ensemble of photon particles.

23. For example, see W. Schommers, Quantum Theory and Pictures of Reality (Springer-Verlag, Berlin, 1989), pp. 17-20.

24. D. Bohm and Y. Aharonov, Phys. Rev. 108, 1070 (1957).

25. Since there were many very good review articles on the topics in quantum optics, we here do not go deeper, but refer to P. W. Milonni, The Wave-Particle Dualism, ed. S. Diner et al. (Reidel, Dochester, 1984), pp. 27-67.

26. P.W. Milonni, Phys. Rep. 25, 1 (1976).

27. W. Heisenberg, Physics and Beyond: Encounters and Conversations (Harper and Row, New York, 1972), p.174.

28. H. Minkowski, The Principle of Relativity, eds. A. Einstein et al. (Dover, New York, 1923), pp. 75-91.

29. P. A. M. Dirac, The Principle of Quantum Mechanics (Clarendon Press, Oxford, 1958).

30. H. Paul, Rev. Mod. Phys. 58, 209 (1986).

31. D. M. Greenberger, M. A. Horne, and A. Zeilinger, Physics Today 46(8), 22 (1993).

32. E. P. Winger, Quantum Theory and Measurement, eds. J. A. Wheeler and W. H. Zurek (Princeton, New Jersey, 1983), pp.260-314.

33. E. P. Wigner, Z. Phys. 133, 101 (1952).

34. H. Araki and M. Yanase, Phys. Rev. 120, 622 (1960).

35. A. Einstein, B. Podolsky, and N. Rosen,  Phys. Rev. 47, 777 (1935).

36. For a modern treatise, for example, see R. J. Cook, Progress in Optics Vol.XXVIII ed. E. Wolf (North-Holland, Amsterdam, 1990), p.363-416.

37. R. J. Cook and P. W. Milonni, Phys. Rev. A35, 5081 (1987).

38. D. Meschede, Phys. Rep. 211, 201 (1992).

39. P. Meystre, Progress in Optics Vol. XXX ed. E. Wolf (North-Holland, Amsterdam, 1992), p.261-351.

40. E. Schrodinger, Naturwissenschaften  23, 807 (1935). For an English translation, see Quantum Theory and Measurement ed. J. A. Wheeler and W. H. Zurek (Princeton Univ. Press, New Jersey, 1983), pp. 152-167.

41. M. Sargent III, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, MA, 1974), p.239.

42.  J. Janszky, T. Kobayashi, and An.V. Vinogradov, Optics Comm.76, 30 (1990); J. Janszky, P. Adam, An. V. Vinogradov, and T. Kobayashi, Chem. Phys. Lett. 213, 368 (1993); J. Janszky, An.V. Vinogradov, T. Kobayashi, and Z. Kis, Phys. Rev. A 50, 1777 (1994).

And his partner.        

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