Photon Polarization Measurements Without the Quantum Zeno Effect

Home , Photon polarization

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Each photon certainly does not carry its own wave function which might then undergo a collapse whenever a photon disappears (e.g. whenever a photon is absorbed in a polarizer). The wave function (in reality) describes an ensemble of many photons. The collapse of the quantum state whenever a photon is absorbed closely resembles the collapse of a probability table (at the automobile insurance company) every time a car crashes. Actually, the probability table at the insurance company stays intact during a car crash. It is the automobile and not the probability that collapses. This constitutes an altogether different pile of scrap metal, even if the cross section for two automobiles to scatter off one another were computed with a quantum mechan- FIG. 2. A vertical polarized photon beam enters the stack ical amplitudes to include the small diffraction effects. of N=8 polarizers, and a horizontally polarized photon beam In Sec.II, it will be shown how the Schrödinger equa- leaves the stack. tion for photons in the vacuum is equivalent to the vacuum Maxwell field equations. In Sec.III, the Schrödinger equation for photons propagating in continuous media A beam of photons in state |↑> is incident on a stack (e.g. polarizers) will be explored. The photon wave func- of N polarizers each of which has a polarization axis ro- tion follows from the Maxwell electromagnetic theory in tated by an angle of ∆θN = (π/2N) relative to the previ- continuous media. However, the electromagnetic wave ous polarizer. According to the von Neumann projection attenuates if the continuous media exhibits dissipation postulate [15], the quantum state of each photon will col- in the form of finite conductivity. The complete quan- lapse N times, yielding a probability tum system (polarizers plus photons) obeys a big monster Schrödinger equation with a huge number of degrees N 2 N 2 π PN = cos (∆θN ) = cos (7) of freedom and does not at all require the projective mea- 2N surement formalism of Eq.(2). This feature of quantum n o of passing through the total stack as pictured in Fig.2. mechanics is valid for many systems as will be discussed Note, in the formal limit of an infinite number of quantum in Sec.IV. state collapse events, In Sec.V we compute, for a realistic model of polarizers, the Maxwell equation formulation of the Zeno effect. The lim PN =1, (Quantum Zeno Effect). (8) classical Maxwell wave computation (without projection N→∞ operators) yields the same result as the quantum photon Theoretically every photon passes through the stack. In Schrödinger amplitude (without projection operators). real experiments, there is always some attenuation of a In an appropriate regime, the projection postulate does photon beam passing through many polarizers. indeed give an accurate answer, but this may only be Although the large N quantum Zeno effect does appear proved by solving the full Maxwell theory without the to be verified in the laboratory [16], we have a very uneasy projection postulate. One might be led to ask why should feeling about attributing such observations to the rule one ever add to the quantum theory a projection the- of collapsing quantum states. After all, long before von ory of measurements. A hint as to why the projection Neumann considered projective measurements, Maxwell postulate is so popular might be connected to the fact had developed a reliable electromagnetic radiation theory that the Maxwell theory computation (to be presented) which perfectly well determined how an electromagnetic required a fast computer. More than mere speed, we wave will (or will not) pass through stacks of polarizers. required algorithms sufficient to maintain ∼ 103 signifi- And not even once did Maxwell feel obliged to discuss cant figure accuracy during the entire calculation of the the collapse of the electromagnetic radiation field func- photon transmission probability. Without recently devel- tions. For that matter, neither Bohr nor Einstein [17] oped precise numerical algorithms, one might be coerced

2 into employing the projection postulate. The photon Schrödinger Eq.(12) then appears equivalent In the concluding Sec.VI, we question the utility to the vacuum Maxwell theory; i.e. of replacing the wisdom of experimentalists and engi- ∂E(r,t) neers who design and build laboratory equipment with = c curl B(r,t), (17) a merely philosophical view of projective measurements. ∂t It is somewhat unreasonable to represent real laboratory and equipment with mathematical projection operators. ∂B(r,t) = −c curl E(r,t), (18) ∂t II. PHOTON WAVE FUNCTIONS with the vacuum Gauss law Eq.(15) constraints For a spin S particle [18] one requires three Hermitian divB(r,t)=0, divE(r,t)=0. (19) matrices S = (S1,S2,S3) obeying 2 2 2 It is of interest to discuss the relativistic Lorentz sym- [Si,Sj]= iǫijkSk, S + S + S = S(S + 1). (9) 1 2 3 metry of the photon Schrödinger Eq.(12). The proof The photon is a spin one (S = 1) particle so we may of Lorentz symmetry is actually well known since we choose the spin matrices have already proved equivalence with the Maxwell theory. However, the discrete Lorentz symmetries are still worthy 00 0 0 0 i of further discussion [22–28]. If we consider the classical S1 = 0 0 −i , S2 = 0 00 , (10) Lorentz force on a charge equation f = e(E+(v × B)/c), 0 i 0 ! −i 0 0 ! and then allow the charge to change sign e → −e, then the force f will remain the same if we also allow E →−E and and B →−B. Thus, the electromagnetic field Eq.(16) is 0 −i 0 odd under charge conjugation, S3 = i 0 0 . (11) CˆF = −F. (20) 0 0 0 !

With p = −i¯h∇, the vacuum Schr¨odinger equation for Under time reversal, E → E and B →−B so that free photons reads TˆF = F∗. (21) ∂ψ(r,t) i¯h = c S · p ψ(r,t). (12) Finally, under parity E →−E and B → B so that ∂t ˆ ∗ Writing P F = −F . (22)

F1(r,t) Note ψ(r,t)= F2(r,t) , (13) ˆ ˆ ˆ F3(r,t) ! T CP =1, (23) allows one to cast the photon Schr¨odinger Eq.(12) into as would be expected from relativistic wave equations. the (perhaps) more simple vector form

∂F(r,t) III. PHOTON WAVE FUNCTIONS IN i = c curl F(r,t). (14) ∂t CONTINUOUS MEDIA The vector field F(r,t) should be constrained to obey The Maxwell theory for an electromagnetic distur- bance moving through continuous media [29–34] may be divF(r,t)=0. (15) formulated starting from Eq.(15) is equivalent to only allowing two independent divE(r,t)=4πρ(r,t), divB(r,t)=0, (24) spin states (say superpositions of only two transverse po- larizations), thereby eliminating a third (longitudinal po- and larized) “zero energy” mode. Finally, Eqs.(14) and (15) take on a clearly recognizable form if one defines the real ∂E(r,t) = c curl B(r,t) − 4πJ(r,t), (25) and imaginary parts of F(r,t) to be, respectively, the ∂t electric field E(r,t) and the magnetic field B(r,t) [19] [20] [21]; i.e. ∂B(r,t) = −c curl E(r,t). (26) F(r,t)= E(r,t)+ iB(r,t). (16) ∂t

3 ∞ It is sufficiently general to supplement Eqs.(24), (25) and Σ(r, r′,s)F(r′,t − s)d3r′ds. (36) (26) by employing a linear causal relationship between Z0 Z the current and the electric field, i.e. where the photon self-energy part is defined as ∞ J(r,t)= K(r, r′,s)·E(r′,t − s)d3r′ds. (27) Σ(r, r′,s)= −2πiK(r, r′,s)(1 + Tˆ). (37) Z0 Z Using Eqs.(13) and (36), one may write the photon If, for a complex frequency ζ where ℑmζ> 0, Schrödinger equation in the form −iζt E(r,t)= ℜe E0(r, ζ)e , (28) ∂ψ(r,t) i¯h = c S · p ψ(r,t)+ ∂t −iζt J(r,t)= ℜe J0(r, ζ)e , (29) ∞ ¯h Σ(r, r′,s)ψ(r′,t − s)d3r′ds. (38) Eqs.(27), (28) and (29) serve to define the non-local con- 0 ′ Z Z ductivity σ(r, r , ζ), i.e. In principle the calculation of the self energy part Σ(r, r′,s) is equivalent to the calculation of the con- ′ ′ 3 ′ J0(r, ζ)= σ(r, r , ζ)·E0(r , ζ)d r , (30) ductivity σ(r, r′, ζ) as is evident from Eqs.(21), (31), Z and (37). In a practical calculations involving the pho- where ton Schrödinger equation in a polarizer, one should use ∞ the conductivity and dielectric response functions as σ(r, r′, ζ)= eiζsK(r, r′,s)ds. (31) those of the optical engineers who designed the device Z0 [35–39]. Finally, the non-local form of Eq.(38) is shared Some workers prefer to complete the Maxwell equations by many other quantum mechanical systems as we shall in continuous media by employing a constitutive equa- now briefly explore. tion relating the Maxwell displacement field D(r,t) to E r the electric field ( ,t). In the frequency domain, the ¨ dielectric response ǫ(r, r′, ζ) of the continuous media is IV. NON-LOCAL SCHRODINGER EQUATIONS defined as Although we do not adhere to the non-unitary collapse ′ ′ 3 ′ D0(r, ζ)= ǫ(r, r , ζ)·E0(r , ζ)d r . (32) of the quantum state Eq.(2) as postulated by von Neu- Z mann, we do find the notion of projection operators (also Just so long as the various response functions are main- introduced into quantum mechanics by von Neumann) tained as non-local (in space and time), which of the to be more than just convenient. Eq.(1) will thereby many equivalent response functions are actually used in be invoked in what follows. For a macroscopic labo- a calculation remains a matter of convenience. For ex- ratory quantum system we here insist that a monster ample, Eqs.(30) and (32) are equivalent if we choose Schrödinger equation with a huge number of degrees of freedom still holds true, 4πiσ(r, r′, ζ) ǫ(r, r′, ζ)= 1δ(r − r′)+ . (33) ∂ ζ i¯h |Ψ(t) >= Htot|Ψ(t) > . (39) ∂t The dielectric, magnetic, or conductivity response We do not contemplate the ultimate monster wave func- functions conventionally used to describe continuous me- tion of the whole universe, not only because we are not dia now make an appearance into the Schrödinger equa- smart enough to find it, but because we do not know how tion for a photon moving through the continuous media to build an experimental ensemble of universes to test a via the photon “self-energy” insertion. To see what is hypothetical quantum state of our universe. We leave involved, we may write the Maxwell Eqs.(24) and (25) as such considerations to people more wise than are we. We are happy to say something meaningful about a divF(r,t)=4πρ(r,t), (34) very small physical piece of a laboratory quantum state; and e.g. a “projected piece” ∂F(r,t) ψ(t) = P Ψ(t) . (40) i = c curl F(r,t) − 4πiJ(r,t). (35) ∂t We leave the rest of the monster wave function, where Eq.(16) has been invoked. Employing Eqs.(21), (27) and (35), we find our central result for the photon φ(t) = Q Ψ(t) , (41) Schrödinger equation in continuous media to the engineers who design the experiment. With the

∂F(r,t) little piece of the wave function Eq.(40) and the big piece i = c curl F(r,t)+ ∂t of the wave function Eq.(41), we rewrite Eq.(39) as 4 ∂ |ψ(t) > H V |ψ(t) > −iEt/h¯ i¯h = , (42) ψ(t) = ψE e . (48) ∂t |φ(t) > V † H′ |φ(t) > Eqs.(46), (47) and (48) imply † where H = PHtotP, V = PHtotQ, V = QHtotP and ′ i¯h H = QHtotQ. H + ∆(E) − Γ(E) ψE = E ψE , (49) We can only hope to find |ψ(t) > which obeys 2 n o where ∂ i¯h ψ(t) = H ψ(t) + V φ(t) , (43) i¯h 1 ∂t ∆(E) − Γ(E)= V V †. (50) 2 E − H′ + i0+ so we solve for the monster piece That the effective energy dependent Hamiltonian on the ∂ i¯h φ(t) = H′ φ(t) + V † ψ(t) , (44) left hand side of Eq.(49) ∂t i¯h H (E)= H + ∆(E) − Γ(E) (51) subject to the causal boundary condition eff 2 ∞ i ′ is not Hermitian, is due to the “Fermi platinum rule” φ(t) = − e−iH s/h¯ V † ψ(t − s) ds. (45) ¯h transition rates to leave the small subspace and wander Z0 into the remaining monster subspaces. The Fermi plat- From Eqs.(43) and (45) we find the general non-local inum rule Schrödinger equation for the piece of the wave function 2π that is of interest; It is Γ(E)= Vδ(E − H′)V † (52) ¯h ∂ ∞ i¯h ψ(t) = H ψ(t) +¯h Σ(s) ψ(t − s) ds, is just a little bit more valuable than the more usual per- ∂t Z0 turbative Fermi golden rule, since Eq.(52) is rigorously (46) true to all orders of perturbation theory. The fixed energy E case corresponds to having a fixed photon frequency where ω = (E/¯h) for the electromagnetic problem at hand. Let us return to this problem. i ′ Σ(s)= − Ve−iH s/h¯ V †. (47) ¯h2 V. MAXWELL THEORY AND POLARIZERS The full monster Schrödinger Eq.(39) for complete wave function |Ψ(t) > reduces to the non-local in time Schrödinger Eq.(46) for the small subspace piece of the The Maxwell theory of a transverse wave moving wave function |ψ(t) >. A particular example of Eq.(46) is through a polarizer may be formulated following the dis- the photon Schrödinger Eq.(38). The point is that when cussion of Born and Wolf [40]. (i) If a transverse wave a photon propagates inside of a continuous media device, propagates in the z-direction, then the wave may be de- say a polarizer, the monster wave function includes both scribed by four electromagnetic field components, say and the photon the polarizer degrees of freedom. In a E1(z) one photon projected subspace, the quantum state of the E2(z) χ(z)= . (53) polarizer is fixed. In addition to the polarizer, there is  B1(z)  but one propagating photon. If the photon is absorbed,  B2(z)  then an electronic excitation is produced which heats up   the polarizer. The piece of the monster quantum state If the polarizer is oriented by a rotation angle θ, as in representing this absorbed photon situation is not con- Fig.1(b), then the dielectric response tensor will have the tained in the single photon subspace. The single photon form subspace wave function is then attenuated due to pho- ǫ˜1 +˜ǫ2 ǫ˜1 − ˜ǫ2 −iτ2θ iτ2θ ton absorption in the polarizer. All of this (including ǫ(θ)= + e τ3e , (54) 2 2 the attenuation of the Maxwell wave in the medium) is with- described by the one photon Schrödinger equation, where we have employed the 2 × 2 Pauli matrices out recourse to the projection postulate. Maxwell was so lucky to be able to write down the Schrödinger equation 0 1 0 −i τ1 = , τ2 = , (55) for the photon in matter, including attenuation, without 1 0 i 0 knowing anything about quantum states (be they monster or otherwise). and It is of interest to look for a fixed energy solution of 1 0 the subspace wave function Eq.(46) of the form τ3 = . (56) 0 −1

5 The complex principal eigenvalues of the dielectric tensor polarizers (rotated into the allowed 2-polarization direc- at frequency ω have the form tion) is given by the absolute square of the transmission amplitude, 4πiσj ˜ǫj = εj + , j =1, 2 . (57) 2 ω P = |T2(N)| . (63) N The real parts of the dielectric eigenvalues ε1,2 determine From Eqs.(54), (59), and (61)-(63), with each of the an- the velocities of the waves polarized in the principal di- gles in (θ1, ..., θN ) set equal to (π/2N), one may compare rections. The conductivities σ1,2 determine the dissipa- the the projection postulate PN in Eq.(7) to the Maxwell tive attenuation of the waves polarized in the principle theory, or equivalently the photon Schr¨odinger Eq.(63), directions. For a well designed polarizer, a wave with an without the projection postulate. The matrix multiplica- allowed polarization direction will propagate unattenu- tions in Eq.(61) and the linear algebra in Eq.(62) are best ated while a wave with a disallowed polarization will be left to a computer. The results of such a computation are strongly attenuated. shown in Fig.3. It is the anisotropic dissipative conductivity that is of- ten modeled by the quantum projective postulate for the 1 case of photon polarization. On the other hand, conductivity is evidently classical since both Ohm and Maxwell were dead and gone before quantum mechanics had been 0.8 invented. For the classical Maxwell calculation, the wave at the beginning (z = 0) of the polarizer is related to the wave 0.6 at the end (z = a) of the polarizer by a 4 × 4 transfer N matrix P ωa 0.4 χ(a)= M(θ, ξ)χ(0), ξ = , (58) c χ(z) is deﬁned in Eq.(53), and in partitioned matrix form 0.2

M(θ, ξ)= 0 0 5 10 15 20 cos(ξ ǫ(θ)) − sin(ξ ǫ(θ))/ ǫ(θ) τ2 . N τ2 ǫ(θ) sin(ξ ǫ(θ)) τ2 cos(ξ ǫ(θ))τ2 p p p FIG. 3. The Zeno effect probability PN for a photon to p p p (59) pass through N polarizers: (i)The solid curve (evaluated for integer N) is the projection postulate in Eq.(7). (ii) The filled For a stack of N polarizers, each of width a and oriented circles represent the Maxwell theory with zero attenuation at angles (θ1, ..., θN ), one may repeat the above process for the allowed polarization direction. (iii) The filled squares going from the beginning (z = 0) of the stack to the end represent the Maxwell theory with a small attenuation even (z = Na) of the stack for the allowed polarization direction. ωa χ(Na)= MN χ(0), ξ = . (60) c The solid curve in Fig.3 is the prediction of the pro- One Multiplies the transfer matrices for each polarizer in jection postulate Eq.(7). For the Maxwell theory with a the stack reasonable stack of laboratory polarizers, there may be ∼ 20 wavelengths of light in each polarizer in the stack. Mtot,N = M(θN , ξ)...M(θ2, ξ)M(θ1, ξ). (61) Thus we chose the parameter ξ = (ωa/c) = 100 for purpose of illustration. We further chooseǫ ˜1 = 1 to describe Once the total 4 × 4 transfer matrix is computed, one the polarization direction for which the polarizer would may find the transmission amplitudes Tj (N) and reflec- be transparent and the complexǫ ˜1 =1+2i to describe tion amplitudes Rj (N) for the two polarization directions the polarization direction for which the polarizer would (j =1, 2) by solving the linear algebra problem be strongly attenuated (if not absolutely opaque). The resulting computed points are the filled circles of Fig.3. 1+ R1(N) T1(N) For this case, the projection postulate yields a PN which R2(N) T2(N) is quite close to the full Maxwell theory. Mtot,N   =   . (62) R2(N) −T2(N) For some regimes (perhaps more close to the laboratory  1 − R1(N)   T1(N)  situation), the Maxwell theory begins to diverge from     For an incident photon in the allowed 1-polarization di- from the projection postulate for large N. In the lab- rection, the probability that the photon exits a stack of N oratory all polarization directions are subject to small

6 (but certainly finite) attenuation. To test this situa- tected. One photon can never make two different coun- tion, we computed the Maxwell theory prediction when ters each give you one half of a click. That is much too the allowed polarization has a weak attenuation response smooth for quantum mechanics. In quantum mechanics, ǫ˜1 =1.1+0.001i while the disallowed polarization has a each photon counter gives one click or no clicks. The stronger attenuation responseǫ ˜2 = 1.1+0.05i. The re- photon is certainly not a classical wave. The photon is sults are the filled squares of Fig.3. For N ≤ 3, there is certainly not a classical particle. The photon is certainly a Zeno effect in which PN increases. However for N ≥ 6, not a little bit classical wave and a little bit classical par- the is an attenuation induced decrease in PN , contrary ticle. (iii) The photon is neither a classical wave nor a to the “Zeno effect expectation”. However, one might classical particle nor a little bit of both. regard this attenuation of the allowed polarization di- Electromagnetic waves, as described by Maxwell, are rection to be merely an indication of poor experimental classical waves. If a radio station broadcasts a signal, design. then the signal is (for all practical purposes) smooth. Some final comments about the calculations producing Your radio gets some of the signal, and our radio gets Fig.3 are worthy of note. The transfer matrix formal- some of the signal. There is enough smoothly distributed ism used to calculate the transmission probability PN classical wave to go around to all of the radios. The clas- was tedious, but straight forward. However, the numer- sical Maxwell wave is smooth. One detector cannot get a ical computations required to produce Fig.3 were not fraction of a photon. Remember, photons come in lumps. entirely trivial. The first three numerical commercial One photon can get destroyed in at most one radio re- software programs that we employed for the calculation ceiver. One detector can get any fraction of a classical produced complete nonsense [41]. The physical problem Maxwell wave. Remember, classical waves are smooth. was that the Maxwell waves reflect back and forth within Yet the Schrödinger equation for photon is the same (in the polarizers. Hence, there exist exponentially attenu- a mathematical formalism) as the Maxwell equations for ated terms as well as exponentially amplified terms in the classical electromagnetic waves. This is because the pho- transfer matrix. High accuracy was needed for the ma- ton has no mass and because the photon is a Boson. trix multiplications in Eq.(61). Only with high accuracy When very many photons are Bose condensed into a sin- at intermediate stages of the calculation, could Eqs.(62) gle quantum state, the result is a Maxwell wave which is and (63) be reliably computed. For realistic laboratory classical for all practical purposes. polarizers, the required accuracy turns out to be very The above picture was not all that clear to Einstein (by high indeed. Fortunately, Aberth [42] has developed pre- his own admission), and we do not claim to know more cise numerical algorithms together with publicly available about photons than did Einstein. According to Einstein, programming code which allows for ∼ 104 significant fig- a Maxwell wave hits a half silvered mirror and some of the ure accurate computations. Much of the required code wave goes to one detector and the rest of the wave goes for the standard functions and standard linear algebra to a second detector. If the wave represents one photon, was thus available. We used this code. However, we re- then only one counter goes click. The other counter gets quired ∼ 103 significant figure computations to produce whacked with a Maxwell wave. Yet this counter counts the reliable plots shown in Fig.3. nothing. When there is but one photon, only one counter can speak for her. Some Maxwell wave parts turn out to have have a photon, and some Maxwell wave parts turn VI. CONCLUSIONS out not have a photon. Some counters go click and some counters remain mute. You never know. And photons In discussions of the foundations of quantum mechan- are lumpy. It is a very strange shell game according to ics there has been a tendency to regard all interference Einstein [44]. of waves as explicitly due to quantum mechanics. This However, the Zeno effect formally attributed to the is only sometimes true, since “waves” as well as “parti- quantum mechanical projection postulate has here been cles” also have classical limits. In many (perhaps most) argued to be merely a classical wave effect. Maxwell of the electromagnetic experiments and applications, the would not have had to increase his knowledge of elec- Maxwell wave itself is classical. The distinction between tromagnetic theory by even one iota in order to totally a classical wave, a classical particle, and a quantum par- understand the polarization version of the Zeno effect. ticle has been discussed simply, yet clearly, by Feynman No quantum mechanics is required, not to even speak of [43]. the projection postulate. (i) A quantum particle is certainly not a classical particle. Quantum particle dynamics dictates amplitude superposition, whereas classical dynamics dictates probability superposition. (ii) A quantum particle is certainly not a classical wave. A classical wave is “smooth” while a quantum particle comes in “lumps”. Photons are lumpy. Photon counters yield one click whenever a photon is de-

7 Acknowledgement [25] M. Ya. Agre and L.P. Rapoport, Optics and Spectroscopy, 75, 622-3n (1993). This work has benefited from the allocation of time at [26] M.W. Evans, Foundations of Phys. Lett,, 7, 67-74 (1994). the Northeastern University High-Performance Comput- [27] S.L. Braunstein and J.A. Smolin, Phys. Rev. A 55, 945- ing Center (NU-HPCC). 50, (1997). [28] D.T. Gillespie, Phys. Rev. A, 49 (1994). [29] G.Y. Guo, H. Ebert, W.M. Temmerman, and P.J. Durham, Phys. Rev. B, 50, 3861-8, (1994). [30] S.P. Vyatchanin and E.A. Zubov, Optics and Spec- troscopy, 74, 544-6, (1993). [31] V. Sochor, Proceedings of the SPIE - The International Society for Optical Engineering, 2799, 185-7 (1996). [1] J. von Neumann, Mathematical Foundations of Quantum [32] G.Z.K. Horvath, R.C. Thompson, and P.L. Knight, Con- Mechanics, Princeton University Press, Princeton (1996). temporary Physics, 38, 25-48 (1997). [2] R.L. Stratonovich and V.P. Belavkin, International Jour- [33] M. Namiki, S. Pascazio and C. Schiller, Phys Lett A, 187, nal of Theoretical Physics, 35, 2215-28 (1996). 17-25, (1994). [3] G.S. Agarwal and S.P. Tewari. Phys. Lett, 185, 139-142 [34] N.L. Manakov, A.V. Meremianin, A.F. Starace, Phys Rev (1994) A, 57, 3233-44 (1998). [4] M.A.B. Whitaker, Interaction-free Measurement and the [35] L.S. Schulman, Foundations of Physics, 27, 1623-36, Quantum Zeno Effect, Physics Lett A, 244, 502-6, (1998). (1997). [5] O. Alter and Y. Yamamoto, Phys. Rev. A 55, R2499- [36] M. Kitano, Optics Communications, 141, 39-42 (1997). 2502, (1997). [37] D. Home and M.A.B. Whitaker, Annals of Phys, 258, [6] A. Beige and G.C. Hegerfeldt, J. of Phys. A 30, 1323-34, 217-85 (1997). (1997). [38] M.J. Gagen, H.M. Wiseman, and G.J. Milburn, Phys Rev [7] A. Beige, G.C. Hegerfeldt, and D.G. Sondermann, Foun- A, 48, 132-42 (1993). dations of Physics, 27, 1671-88 (1997). [39] T.E. Phipps Jr., Physics Essays, 11, 155-63, (1998). [8] S. Pascazio, H. Nakazato, and M. Mamiki, J of the Phys. [40] M. Born and E. Wolf, Priciples of Optics, Chapt.XIV, Soc of Japan, 65, 285-8, (1996). Pergamon Press, Oxford (1980). [9] J. Pan, Y.Zhang, and Q. Dong, Phys Lett A, 238, 405-7 [41] Explicit brand names of commercial software will not be (1998). mentioned to protect the guilty. [10] L.S. Schulman, Phys Rev A, 57, 1509-15 (1998). [42] O. Aberth, Precise Numerical Methods Using C++, Aca- [11] S. Pascazio, Foundations of Physics, 27, 1655-70 (1997). demic Press, San Diego (1998). [12] B. Nagels, L.J.F. Hermans, and P.L. Chapovsky, Phys [43] R. P. Feynman, R.B. Leighton, and M. Sands, The Feyn- Rev Lett, 79, 3097-100 (1997). man Lectures on Physics, III, Chapt.1, Addison Wesley, [13] H. Nakazato, M. Namiki, S. Pascazio, and H. Rauch, Reading (1965). Phys Lett A, 199, 27-32 (1995). [44] A. Fine, The Shaky Game; Einstein Realism and the [14] J. S. Townsend, A Modern Approach to Quantum Me- Quantum Theory, The University of Chicago Press, chanics, p.62, McGraw-Hill, New York (1992). Chicago (1996). [15] P.C. Stancil and G.E. Copeland, J. of Phys. B, 27, 2801- 7 (1994). [16] P. Kwiat, H. Weinfurter, and A. Zeilinger, Quantum See- ing in the Dark, Scientific American 72, Nov. (1996); P. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger and M.A. Kasevich, Phys. Rev. Lett. 74, 4763 (1995). [17] N. Bohr, Discussions with Einstein on Epistemological Problems in Atomic Physics in Albert Einstein Philoso- pher Scientist, Edited by P. A. Schilpp, Open Court, La Salle (1997). [18] M. Ban, Phys Rev A, 49, 5078-85 (1994). [19] S.A. Gurvitz, Phys. Rev. B, 56, 15215-23 (1997). [20] Y. Japha, A.G. Kofman, and G. Kurizki, Acta Physica Polonica A, 93, 135-45 (1998). [21] S. Machida and A. Motoyoshi, Foundations of Physics, 28, 45-57 (1998). [22] M. Bordag, K. Kirsten, and D.V. Vassilevich, J of Phys A, 31, 2381-9 (1998). [23] F. Mihokova, S. Pascazio, and L.S. Schulman, Phys Rev A, 56, 25-32 (1997). [24] N.L. Manakov and V.D. Ovsiannikov, J of Phys B, 30, 2109-17(1997).