arXiv:quant-ph/0612005v1 1 Dec 2006 owti Physics.[1] tried within restric he to when even was “complementarity,” it term whatever the just under convey on readers completely nor agree students, papers, his his of None successful marginally philosophy. only was promulgating total he that a show of to hi seems foundation for History the became precept, it universal until even, a science of boundaries customary nPyis B the Physics, of in popularization were the they Following insi ontology. holistic antinomies, scale re- preternatural, atomic as albeit to deep, recognized unable as first redubbed I been just at having were that them. what however, legitimize move case, and the features arguably weird is these exactly capture esrmn cee xii lentl ihrteproper particles. the interacti or either the waves of of alternately scale exhibit the scheme, on en measurement depending that entities, effect the all to of theory bles quantum of feature that complement to thi crystallized: refers of has consensus Out a identity. compl time particle-wave over of erature of notion issue the the on focused mentarity have literature secondary other t rgntr B becaus originator, is its this Mostly, (QM). 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S e mo h htnprdg feeti hre particle charged electric of paradigm photon the of sm hsinto ghts ELLERI i writer. his mcmlmnaiyoc h htncneti not is concept photon the once complementarity um cs a ices, n n escnrvd h nlcnlso sthat is conclusion final The contrived. less and ent nor on ticles sem- term lit- s arity only aiyaeraaye.I sage ht o each, for that, argued is It reanalyzed. are tarity een ties ted me .” he he [3] of v- b- to m d; e- n, at d e 1 - - t tcnnvgt,adwe si oinr hma h intervent the as them ignore that experimenter? to so an it objects of is with when and communicate navigate, to can supposed it it is e the When it know object ters? material wave particular i any any of used can purpose in those How difference except measurements. boundaries p material with a all connection that to perhaps react conception it, can contradictory wave abandoned internally have the to of seem because notion this of Proponents hm fti ae,wihcnen h te uctgr,t cent subcategory, waves. the other electromagnetic the not of concerns behavior is particle-like which it paper, this independently; of pursue t theme to successful, invited is is effort reader this Whether rationa to phenomena.[4] ‘Stocha seeks as quantum which known theory theory a a (SED), of Electrodynamics’ basis tic the as mo radiat hypothesized their is background that of electromagnetic random virtue the by through particles electromag tion to classical are attached waves waves pilot background such that effect the to oE to uly oee,E however, tually, aeeuto wt ore) sdrvdfo M from derived as sources), (with inhomogen equation the with wave associate to difficult are ab combina solutions In because such instantaneous. othe shockwave-like, seemingly rapid, be On be to can sorption point-like. thought be are they to occasions seems absorption lateral because small with pack section, radiation” such “needle elsewhere as envisioned hand, present are one thereby ets the and on vacuum, universe the the So of throughout vague. ‘mode’ is a conception considered commo the times today, the even inco captured Still, the subdivided, that imagination. somehow idea, is the that itself later radiation was it ‘packets,’ in occurs cteigo lcrmgei aito rmeetos i electrons, from radiation electromagnetic of scattering equations. ihu rnighmefa goau.Ti,i pt fthe of spite M photo in quantum of This, non stream of ignoramus. applicability a spectacular an is himself light branding fundamentally, without that, commentator modern note par- No to waves, paradigm. fails for ‘photon’ model the a as uni nowadays virtually as become packages, has waves, of electromagnetic eff ticularly notion photoelectric the the of case, application any particular a just ner, hs rvnb h osqecso h omro h other the on former the of consequences antennas, the in sources by as driven acting those those variants: two exper interaction fo are ‘photocurrents,’ of there the to reach restricted of the is which beyond mechanism science, completely mental the is fact particles In charged s of as waves, detected. electromagnetic never that write is are this cause so; the is that this suggest why to just on speculate to interesting is C OMPTON atcelk eairo ae snwdy rdtdmostl credited nowadays is waves of behavior Particle-like idea wave pilot the of extention an proposed has writer This diinlspotfrpcae aito a on nthe in found was radiation packaged for support Additional INSTEIN fet hsefc s nacraneeetr man- elementary certain a in is, effect This Effect. ncneto ihtePoolcrcEfc.Ac- Effect. Photoelectric the with connection in INSTEIN rpsdol hteeg transfer energy that only proposed AXWELL AXWELL hoy It theory. which r ncoun- c.In ect. e,the .e., likes r cross- versal ming netic eous tion, uch, me- and lize ilot ion ion ral he he ns s- ’s i- n y n - - r - 2 end in detectors.2 In spite of the vividness of the images from attributable to prevalent or incoming noise so that of the whole paradigms of what transpires in between, either as electromag- population, a certain portion at any given instant has noise en- netic waves or photon streams, all effective experience for mor- ergy bringing them up to nearly the escape level. Then when a tals is actually limited to these currents, information on which coherent stimulation signal arrives, those electrons at this other- is then used mentally to infer just how these two currents have wise temporary ‘almost’ escaped energy level are boosted over interacted in detail across space and through time. This un- the threshold very quickly, virtually instantly. In fact, LAMB derstanding makes photo detection the crux of the matter, and and SCULLY long ago have shown with such ideas that photo- arguably, the essential reason for the utility of the “photon” con- electric phenomena do not entail quantum theory; semiclassical cept. ideas suffice.[8] Here we come to the point of this paper. The digitization, The vital point made here is: the digitization of the detector or ‘quantization’ if one prefers, of charges, is not a result of reaction to incoming radiation need not necessarily be attributed quantum theory, but an independent empirical fact that serves to particle-like packaging of the incoming stimulus, i.e., pho- as an hypothetical input into both classical and quantum theo- tons. All the behavior involved can be explained fully also in ries of physical phenomena; thus, the “quantum” behavior of terms of continuous, wave-like radiation of the classical sort electromagnetic waves as ensembles of particles, may not be at known to radio, radar and communication engineers, where the all “quantum,” but simply a consequence of the discreteness of discrete aspects of low intensity measurements are due to dis- the charges making up the material of detectors. If we take this creteness of detector charges. In other words, because charges observation as a scientific proposition to be tested for internal are the ineluctable intermediaries between mortals and electro- (theoretical) consistency and external (empirical) verification, magnetic waves, their contribution to the nature of an observa- then it cannot be regarded as anything other than just another tion cannot be evaded. task for work-a-day science. In this spirit, it shall be addressed herein.
III. COMPLEMENTARITY IN OPTICAL EXPERIMENTS II. PROTOTYPICAL WAVE DETECTORS It is the purpose here to parse the logic of some modern Electromagnetic wave detectors come in a variety of con- optics experiments plumbing the mysteries of complementar- structions, each intended for a particular frequency range. In ity of electromagnetic interaction. These are, first, two ‘quan- those lower portions of the electromagnetic spectrum where tum eraser experiments, and then ‘AFSHAR’s’ experiment; all current technology enables following time development of elec- three are intended to focus specifically on the issue of delimit- tromagnetic fields, there is little of special interest for funda- ing complementarity in light. mental quantum physics analysis. It is only in that range in which this is no longer possible that quantum phenomena seem to arise.[6] Detectors in this range do not follow the undulatory time development of incoming radiation. Instead, they absorb A. Quantum Eraser an undetermined number of cycles, and respond then with some bulk transition. There are several versions of experiments demonstrating this Photographic plates are a good example. Incoming radiation conception proposed first by SCULLY and DRUHL¨ in 1982.[9] at some point triggers a chemical reaction in molecules embed- The basic idea is to elaborate on a standard two-slit experiment ded in an emulsion that, as differentiated compounds, then serve so as to be able to mark the signal passing through at least one during development as seeds for some macroscopically visible of the slits, thereby permitting the determination of which slit effect. Likewise, for the photoelectric effect. Incoming radia- the purported ‘photon’ passes through. An extra feature in this tion eventually results in the transition of a valence electron to proposal was the notion that, with clever design, it could be the conduction band where it is then drawn off into circuitry possible to “erase information” so as to flip the case between to be amplified and registered as the ‘detection event.’ The wave and particle, most spectacularly after-the-fact even. As is historical photoelectric effect considered by EINSTEIN has an well known, textbook orthodoxy nowadays would have it that if additional feature that is widely taken as symptomatic of an es- the slit of passage can be determined (a particle-like property), sential quantum character. It is that the stimulated photocurrent then no interference pattern (a wave-like property) will result. appears virtually instantaneously after the start of illumination. The argument that this is to be explained only by assuming that The difficult trick in conceiving of and performing such an the incoming energy is bundled in compact packages, nowadays experiment is to find a method to mark a signal on passage called ‘photons,’ assumes that the electrons, before illumina- through a slit. The original proposal was to employ multilevel tion, were all resting at an energy so low that it would require atoms in place of slits. The atoms were chosen to have, at least a measurable duration of exposure to the incoming radiation in principle, the same emitted decay radiation but different fi- before they could be energized sufficiently to enable them to nal states readable by some other measurement. This scheme overcome binding forces.[7] Such an assumption is not, how- should allow the experimenter to do, even well after-the-fact, ever, beyond reproach. It could just as well be assumed that or not do the second measurement; as a consequence, the in- the electrons in the detector mass had a distribution of energies terference pattern should appear or not appear. (The sense in which the pattern appears or not results from the alternate data reduction calculations feasible with the existence of different elements in the data sets, after-the-fact. It should not be un- 2 Actually, every charge vis-`a-vis all others, plays both roles; but, to date derstood that some visual image of an object is made to come there is no widely accepted closed form of electromechanics taking this into and go.) We gather that exactly this setup was never realized, account.[5] however. 3
140
120
100
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40 Coincidence counts in 800 s 20
0 -4 -3 -2 -1 0 1 2 3 4 FIG. 1 A setup for optically testing the ‘quantum eraser’ effect. The Detector D position (mm) basic trick is to try to use the idler emission to mark the portion of s the signal passing through each of the slits in a double-slit diffraction experiment. FIG. 3 The antifringe pattern seen when the polarizer in the idler beam is horizontal.[11]
140 behind the slits with points seen from the idler beam are ana- 120 lyzed, then the interference effect can be made to reappear—in
100 the coincidencecounts. That is, if the polarizerin the idler beam (let us assume from a Type-I PDC) is set to horizontal, then the 80 signal in the vertical channel behind the slits will yield corre- lated “hits,” (Fig. 2) by effectively filtering out those hits in the 60 horizontal channel for lack of a mate in the idler channel. If the idler polarizer is changed to vertical, then the correlated hits are 40 in the horizontal channel behind the slits, as now the vertical component is filtered out (Fig. 3). If the polarizer is removed Coincidence counts in 800 s 20 from the idler beam, then no correlations with the slits can be made; the only hit pattern available shows no interference, and 0 -4 -3 -2 -1 0 1 2 3 4 it is said that “which-way information has been erased” (Fig. 4). Detector Ds position (mm) This behavior is said to illustrate quantum complementary. FIG. 2 The fringe pattern seen when the polarizer in the idler beam is But does it? First note that the various signals used are just dif- vertical.[11] ferent states of polarization. This is very significant because po- larization is a phenomenon fully explained by STOKES circa 50 years before the need for quantum theory was known.[12] It is 1. Eraser and double-slit diffraction a fully classical phenomenon. How then, can this phenomenon be used to plumb a principle of quantum mechanics? The only A particularly clean and purely optical ‘quantum eraser’ ex- even remotely quantum aspect to this story is the implicit as- periment involving two slits, inspired by a conceptually simi- sumption that the signals are made up of ‘quantum’objects, i.e., lar proposal of SCULLY et al. [10], was carried out by WAL- photons, so that it is presumed their behavior must be regulated BORN et al. and reported in 2002.[11] The essential feature by quantum mechanics. But, as was argued above, the possi- of this setup is that the beam sent towards the slits is the sig- bility of fully non quantum character has not been rigorously nal output of a parametric down conversion crystal (PDC)3; the excluded. idler is directed separately through another polarizer and then Moreover, since there are two slits and two different inter- to a detector. See Fig 1. Here the marking in the slits is con- ference patterns, with arguments that are brief and loose (i.e., sidered achieved by placing polarizers before each slit. If the “it is clear that ...”) in the customary interpretation, an associ- two polarizers before the slits are set to orthogonal positions, ation is made to ‘which-way’ information. This is done in spite then the signals impinging on the registration screen are inde- of the fact that each interference pattern is indisputably a two pendent and do not form an interference pattern. This feature slit pattern. Thereafter, even while observing that the sum of is thought to conform with the complementarity principle, i.e., these two patterns is what is observed when the idler polarizer that which-way information provided by a polarization tag put is removed, is is said that this results from erasing some infor- on the signals as they pass through the slits, is said to destroy mation. One ought be excused for finding this apologia at least the wave-information revealed by the interference pattern. lexicographically pathological. Now, however, if coincident counts between data points taken Thus, those who find such imagery based on quantum me- chanics at all obscure and contrived may be well disposed to consider the following classical rendition. Let us take it that the signal directed towards the slits is a vertically polarized classical 3 PDC crystals are two types: Type I in which the polarization of idler and electromagnetic beam. If there are no polarizer filters in front of signal are anticorrelated; and, Type-II in which which they are correlated. the slits, after passing through both slits the signal shows inter- 4 ference on the registration screen. Next let us place polarizers 120 with axes set at π/4 to the vertical beforeoneslit, and at +π/4 − before the other slit. Now, each polarizer will split the incom- 100 ing vertically polarized electric field into two components one vertical and one horizontal, each with amplitude 1/√2 1/√2 80 times the ‘mother’ signal as determined by MALUS’s Law,× once as a projection onto the polarizer axis and then again to find the 60 component in either the vertical or horizontal direction. In turn, each orthogonalcomponentor ‘daughter’signal will result in an interference pattern on the registration screen. But, one pattern 40 will be fringes (Fig. 2), while the other is antifringes (Fig. 3). Because the horizontal diffraction pattern is the sum of oppos- Coincidence counts in 400 s 20 ing components, a phase term of π is inserted and the pattern comprises antifringes; in contrast the vertical components are 0 -4 -3 -2 -1 0 1 2 3 4 codirectional and in phase. These two patterns together add up Detector D position (mm) to a total pattern without interference (Fig 4). That is, if the s daughter horizontal interference pattern can be written FIG. 4 The sum of the fringe and antifringe patterns seen when there ax2 2 is no polarizer in the idler beam.[11] Ih(x)= ke− cos bx (1) and the daughter vertical pattern4
ax2 2 Iv(x)= ke− sin bx, (2) then the sum
ax2 2 2 ax2 Ih(x)+ Iv(x)= ke− (cos bx + sin bx)= ke− , (3) shows no interference. In the terminology of this explanation, it can not be said that information has been erased, although it has been concealed, using, as it were, a kind of secretwriting. It can be refound(ren- dered legible) as follows. As the signal was one output from a parametric down conversion crystal (PDC), the conjugate idler then, depending on whether the crystal was of type I or II, will be horizontal or vertical polarization respectively. If type I, then the vertical contribution to the total pattern behind the slits will have nearly perfect correlated partner pulses in the horizontal FIG. 5 A schematic showing the setup of the delayed choice quantum- mode in the idler branch. By counting only those hits in the sig- eraser experiment. Salient features include the long optical path length nal pattern behind the slits for which there is a companion hit of signal in relation to the idler branch, and, the use of passive beam in the horizontal idler mode, effectively the horizontal contri- splitters (BS) that, in effect, radomly compile ensembles that do or do bution to the signal pattern is filtered out, revealing the vertical not correlate with the idler branch. interference pattern. This procedure can be called ‘coincidence filtering.’ When such coincidences are the result of polarization, such filtering is a purely non quantum procedure. The salient feature in this experiment, and distinguishing it On the basis of this analysis it can be said that such a “quan- from the one just considered, is the use of beam splitters. At tum eraser” is neither quantum nor an eraser; as such, it has lit- the so-called single photon level they are known to be virtually tle to contribute to a debate regarding the significance of quan- absolutist in splitting a beam by whole photons; either a whole tum complementarity. Furthermore, the only feature resembling photon goes through,or a whole photon is reflected, not both by quantum theory is the fact that the intensity of the signals used dividing incoming energy. In this experimenttheir role is to take is so low that detailed structure, i.e., its composition as a stream the decision regarding which correlation shall be registered out of point charges, that is electrons, is visible as individual “hits.” of the hands of a mortal, i.e., the experimenter, and put it in the hands of ZEUS. Then, after-the-fact, each subensemble marked by which detector combination fired, is separately analyzed for 2. Delayed choice eraser intensity as a function of lateral detector displacement in the idler branch. Ensembles that correspond, again, to those made Another particularly interesting ‘quantum eraser’ experiment by coincidences triggered by just one of the components in one put the decision to measure wave- or particle-like characteristics or the other dimension, corresponding to Eqs. (1) or (2), show to the internals of the setup, thereby taking it away from the interference. experimenter. The experimental setup is depicted in Fig 5. Beam splitters operating on a beam of the intensity yielding single hits in the detectors, show strict anticorrelated hits, which is interpreted nowadays as reflecting the essential, ineluctable, distinct identity of photons. According to current understand- 4 The phase difference of π/2 in these expressions for intensity equals π in ing, the random choice of transmission or reflection is a reflec- E-field amplitudes. tion of the fundamental uncertain, statistical, i.e., “quantum” 5
structure that is used is the BORN association of the probabil- ity of presence (here of a ‘photon’) being proportional to the square of the field. In this case the field is actually the electro- magnetic field, so that the BORN hypothesis completely over- laps the theory of photocurrent generation, i.e., that it is pro- portional to the square of the electric field, a fact that might be seen even as the underpinning for BORN’s interpretation. Moreover, AFSHAR’s analysis calls on the notion that proper- ties of a light beam are considered continously variable from wave to particle, a concept that wreaks havoc with the concept FIG. 6 The basic setup of AFSHAR’s experiment. First the location of ‘particle.’ Once again, there is little to nothing here address- of the nodes of the diffraction pattern on the interferance screen are ing any feature of the complimentary principle from BOHR. On determined as the location for opaque objects (wire grid, seen here this basis it can be taken that this experiment actually addresses on end). Then this screen is removed while leaving the grid in place. the issue of the tenability of the ‘photon’ paradigm. It renders Finally the image of the slits is observed on the image screen. The the ray-optical viewpoint of photon propagation problematical presence of the grid does not affect the quality of the slit image. while endorsing wave propagation. In other words, it is con- ceptually a variation of the extensive experimental program of ROYCHOUDHURI[16], who identified a number of internal in- character of the universe at an atomic scale. This feature might consistencies in photon imagery. seem, therefore, to vest this experiment with an essential quan- tum element. For this experiment, however, it is immaterial how the choice is made; the phenomenaunder study does not depend Conclusions on the mechanism (or lack thereof) of choice. The only relevant feature is that the separate subensembles be compiled; because When is a phenomenon a quantum phenomenon? This ques- only within each does the phenomenon of interest takes place. tion can be deeper than it appears at first sight. A complication The quantum character of beam splitters, although logical arises in that some non quantum structure from classical physics on its surface, is actually not inevitable. For it to be neces- fits perfectly well within the vocabulary, notation, algorithms sary, would require that alternative theories are logically and and interpretation of quantum theory without, however, actu- rigorously excluded. Insofar as non quantum alternatives ex- ally depending on any hypothesis unique to quantum mechan- ist, e.g.:[14], the viability of the photon paradigm is just provi- ics. Vectors of absolutely any sort, for example, can be symbol- sional. ized by ‘bras’ and ‘kets.’ Insofar as the bra-ket notation encom- passes abstract vector space structure, it can be self-consistently used whenever this structure obtains. To this writer’s mind, B.AFSHAR'S experiment this leaves only one option open for defining the category of quantum phenomena, namely: a quantum phenomenon is one This experiment also aims to study complementarity.[15] The that absolutely cannot have its patterns encoded mathemati- basic idea is to set up a double-slit diffractionpatternat a middle cally without calling on a unique fundamental hypothetical in- stage of an optics train which is continued through a lens to end put of Quantum Mechanics. In the final analysis, this means in a sharp image of the slits; see: Fig. 6. Then, its conceptual that the noncommutivity of phase (or quadrature) space must novelty consists in showing that thin opaque obstacles placed be essential. Therefore, any phenomenon that can be explained in the beam exactly at the locations of destructive interference without reference to this noncommutivity, no matter how unap- in the fringe pattern midway through the train have no effect pealing the non quantum argument, is not a genuine “quantum on the intensity of the slit’s image at the end of the train. The phenomenon,”even when it can also be explained with quantum image of the slits can be sharp only if those photons, which mechanics. it would seem have to have been intercepted by the obstacles, Just here there arises a premier example of non quantum are in fact not removed from the beam. Indeed, if one slit is structure embedded in the quantum formalism. The noncom- blocked, the obstacles do substantially degrade the image of the mutivity found in ‘qubit’ space of polarization space or of spin open slit. Thus, one is led to question how to reconcile these space is widely thought to be of a quantum nature. However, contradictory facts. thegroupin play hereis SU(2) which is homeomorphicto O(3). The very precise final image of the slits gives very precise The latter encodes the structure of rotations on a sphere which is which-way, i.e., particle-type information (in the sense of geo- a totally geometric structure having nothing to do with quantum metrical optics) on which slit the light for each slit image has mechanics. By cause of homomorphism, if O(3) is geometric, passed through. At the same time, knowing just exactly where then SU(2) must be also. The noncommutivityuniquely evident to place the opaque objects requires very precise knowledge of in quantum mechanics is in phase space and is a dynamical fea- the wave-optical character of the beam. Since both types of ture, not just geometry. knowledge are in play and are known to the experimenter at the In the experiments considered above, noncommutivity on same time, this knowledge is, arguably, in violent contradiction phase space plays no role in describing the observed phe- to the complementarity principle. nomenon. Thus, it is concluded that ‘quantum’ is not an ap- Once again, however, a central question is just what is ‘quan- propriate adjective. The term ‘eraser’ is also inappropriate. To tum mechanical’ in the involved phenomena? There is noth- erase, etymologically, is derived from Latin words meaning ‘to ing in the device chain for which otherwise quantum theory scrape out.’ Clearly this meaning is derived from times when is required. Indeed, in AFSHAR’s paper analyzing this exper- writing was often carved in stone. Carving out words is essen- iment, one does not find a single occasion for which a factor of tially different than filling in the letters rendering them illegible PLANCK’s constant is required. The only quantum-appearing (with due attention to color and texture). Competent restoration 6 can recover a concealed message; destroyed informations is lost [2] DE BROGLIE, L., ‘Recherches sur la Th´eorie des Quanta,’ Ann. forevermore. None of this has anything to do with the depths of de Phys., 10e s´erie, III (1925).5 a complementarity principle for electromagnetic radiation. [3] SELLERI, F., ‘Quantum Paradoxes and Physical Reality,’ The key or essential feature of what heretoforegoes under the (Kluwer, Dordrecht, 1990), Chapter IV. rubric of (optical) “quantum eraser” is the phase shift between [4] KRACKLAUER,A. F., Found. Phys. Lett. 12 (5), 441 (1999). the vertical and horizontal components. This shift allows the [5] KRACKLAUER, A. F. and KRACKLAUER, P. T, ‘Electrodynam- ics: action-at-no-distance,’ in Has the Last Word Been Said on two components to fully compliment each other as seen in Eq. Classical Electrodynamics?, CHUBYKALO,A.;ESPINOZA, A.; (3). Because this is a simple geometric effect, it must arise in ONOOCHIN,, V and SMIRNOV-RUEDA, R. (eds.) (Rinton Press, a great variety of situations. The euphoric bedazzlement at its Princeton, 2004) pp. 118-132. implications for the nature of ‘time’ in Quantum Mechanics as [6] PEARSALL, T. P., ‘Photonics Essentials,’ (Mc-Graw-Hill, New reported for some observers in [17], from this point of view, is York, 2003). a reaction to the logical jujitsu entailed in trying to understand [7] GREENSTEIN, G. and ZAJONC, A. G., ‘The Quantum Chal- nature on the basis of inappropriate assumptions, in this case, lenge,’ (Jones and Bartlett, Sudbury, 1997). the photon paradigm. [8] LAMB, W. and SCULLY, M. O., ‘The photoelectric effect without This is not to claim that a complementarity principle has photons,’ in Polarization: Matiere et Rayonement, (Presses Univ. nothing to do with quantum mechanics. Although any such Fr., Paris, 1969) 363-369. [9] SCULLY, M. O. and DRUHL¨ , K., Phys. Rev. A, 25 (4), 2208 principle seems to be an expression of bandwidth consider- (1982). ations from classical wave theory, in fact the quantum one [10] SCULLY,M.O.,ENGLERT, B. G. and WALTER, H., Nature is intimately connected to the HEISENBERG uncertainty rela- (London), 351, 111 (1991). tion, which, unlike electromagnetic bandwidth, is scaled by [11] WALBORN, S. P., TERRA CUNHA,M.O.PADUA, S. and PLANCK’s constant, and must, therefore, refer essentially to MONKEN, C. H., Phys. Rev. A, 65, 033818. a quantum phenomenon. The pertinent question then may be: [12] STOKES,G. G., Trans. Camb. Phil. Soc. 9, 339 (1852). how and why for DE BROGLIE waves (sometimes called ‘mat- [13] KIM, YOON-HO; RONG, YU.; KULIK, S. P.; SHIH, YANHUA ter waves’) is bandwidth regulated in a fundamentally distinct and SCULLY, M. O., Phys. Rev. Lett. 84 (1) 1 (2000). manner from that for electromagnetic waves? But, this issue [14] SANTOS, E., Proc. SPIE, 5866, 36 (2005). pertains to entities that are indisputably particulate in a classi- [15] AFSHAR, S., Proc. SPIE, 5866, 229 (2005). cal limit, and not to continuous, undulatory radiation. [16] ROYCHOUDHURI, C., Proc. SPIE, 5866, 26 (2005). [17] AHARONOV, Y. and ZUBAIRY,M. S. Science 307, 875 (2005).
References
[1] BELLER, M., ‘Quantum Dialogue,’ (U. Chicago Press, Chicago, 1999). 5 The writer’s English translation of this ref., and preprints of refs. [4] and [5], can be downloaded from his webpage.