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the notion of a projective quantum measurement was introduced by von neumann@xcite in his treatise on the mathematical foundations of quantum mechanics . the idea is the following : ( i ) every quantum measurement yielding the data `` yes '' or `` no '' to an experimental question is described by the projection operators @xmath0 ( for `` yes '' ) or @xmath1 ( for `` no '' ) , @xmath2 ( ii ) if the initial state of a quantum object _ before _ a measurement is @xmath3 , and if the measurement yields the experimental answer `` yes '' to a question @xmath4 , then the state of the quantum object _ after _ the measurement @xmath5 is constructed according to the _ collapse _ of the quantum state rule @xmath6 eq.(2 ) is part of a calculation scheme which is _ not _ a unitary development of the quantum state _ during a measurement_. the unitary behavior ( implicit in the schrdinger equation for a quantum object ) was thought by von neumann to hold only _ between measurements_@xcite .    to see how this works , consider a beam of photons which has passed through an `` upwards '' polarizer as in figs.1(a ) and ( b ) . if a second polarizer has an axis at an angle of @xmath7 with respect to the upward direction , then one writes the photon wave function @xmath8 in the basis @xmath9 the probability to pass through the second polarizer is then @xmath10    the cases shown in fig.1 below work as follows : in fig.1(a ) , the second polarizer is set orthogonal to the first polarizer so that no photons will pass ; i.e. @xmath11 this probability can be increased , as in fig.1(b ) below , by placing an intermediate polarizer between the first and last orthogonal polarizers .    in fig.1(b ) , a fraction @xmath12 of the @xmath8 photon beam will pass through the intermediate polarizer and _ collapse into the state @xmath13_. then , a fraction @xmath14 of the @xmath13 photon beam will pass through the final polarizer and again _ collapse into the state @xmath15_. the above two _ collapse of the quantum state processes _ dictate that one multiply probabilities . the final probability that a photon in the upward polarized beam passes through both the intermediate and final polarizers is then given by @xmath16 that an intermediate measurement can _ increase _ the probability of passing through a polarizer stack , say @xmath17 , is an example of what is thought to be a quantum zeno effect @xcite    a textbook quantum zeno effect problem in quantum mechanics@xcite is shown in fig.2 . a beam of photons in state @xmath8 is incident on a stack of @xmath18 polarizers each of which has a polarization axis rotated by an angle of @xmath19 relative to the previous polarizer . according to the von neumann projection postulate@xcite , the quantum state of each photon will _ collapse _ @xmath18 times , yielding a probability @xmath20 of passing through the total stack as pictured in fig.2 . note , in the formal limit of an _ infinite number of quantum state collapse events _ , @xmath21 theoretically _ every photon passes through the stack_. in real experiments , there is always some attenuation of a photon beam passing through many polarizers . although the large @xmath18 quantum zeno effect does appear to be verified in the laboratory@xcite , we have _ a very uneasy feeling _ about attributing such observations to the rule of _ collapsing quantum states_. after all , long before von neumann considered projective measurements , maxwell had developed a reliable electromagnetic radiation theory which perfectly well determined how an electromagnetic wave will ( or will not ) pass through stacks of polarizers . _ and not even once _ did maxwell feel obliged to discuss the collapse of the electromagnetic radiation field functions . for that matter , neither bohr nor einstein@xcite had _ ever _ felt obliged to discuss the so - called wave function collapse . our purpose is to discuss why the projection postulate ( collapse of the quantum state ) , _ certainly may and probably should _ be eliminated from ones thinking in quantum mechanics . the application of this view towards photon polarization measurements is typical of a more general state of affairs . if one returns to the original copenhagen interpretation of the quantum mechanical formalism , then the quantum state describes an _ ensemble of experiments_. 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 mechanical amplitudes to include the small diffraction effects .    in sec.ii , it will be shown how the schrdinger equation for photons in the vacuum is equivalent to the vacuum maxwell field equations . in sec.iii , the schrdinger equation for photons propagating in continuous media ( e.g. polarizers ) will be explored . the photon wave function follows from the maxwell electromagnetic theory in continuous media . however , the electromagnetic wave attenuates if the continuous media exhibits dissipation in the form of finite conductivity . the complete quantum system ( polarizers plus photons ) obeys _ a big monster _ schrdinger equation with a huge number of degrees of freedom and _ does not at all require the projective measurement _ formalism of eq.(2 ) . this feature of quantum mechanics is valid for many systems as will be discussed in sec.iv .    in sec.v we compute , _ for a realistic model of polarizers _ , the maxwell equation formulation of the zeno effect . the _ classical maxwell wave _ computation ( without projection operators ) yields the same result as _ the quantum photon schrdinger amplitude _ ( without projection operators ) .    in an appropriate regime , the projection postulate does indeed give an accurate answer , but this may only be proved by solving the full maxwell theory without the projection postulate . one might be led to ask why should one _ ever _ add to the quantum theory a projection theory of measurements . a hint as to why the projection postulate is so popular might be connected to the fact that the maxwell theory computation ( to be presented ) required a fast computer . more than mere speed , we required algorithms sufficient to maintain @xmath22 significant figure accuracy during the entire calculation of the photon transmission probability . without recently developed precise numerical algorithms , one might be coerced into employing the projection postulate .    in the concluding sec.vi , we question the utility of replacing the wisdom of experimentalists and engineers who design and build laboratory equipment with a _ merely _ philosophical view of projective measurements . it is somewhat unreasonable to represent real laboratory equipment with mathematical projection operators . for a spin @xmath23 particle@xcite one requires three hermitian matrices @xmath24 obeying @xmath25=i\epsilon_{ijk}s_k , \ \ s_1 ^ 2+s_2 ^ 2+s_3 ^ 2=s(s+1).\ ] ] the photon is a spin one ( @xmath26 ) particle so we may choose the spin matrices @xmath27 and @xmath28 with @xmath29 , the vacuum schrdinger equation for free photons reads @xmath30 writing @xmath31 allows one to cast the photon schrdinger eq.(12 ) into the ( perhaps ) more simple vector form @xmath32 the vector field @xmath33 should be constrained to obey @xmath34 eq.(15 ) is equivalent to only allowing two independent spin states ( say superpositions of only two transverse polarizations ) , thereby _ eliminating _ a third ( longitudinal polarized ) `` zero energy '' mode . finally , eqs.(14 ) and ( 15 ) take on a clearly recognizable form if one defines the real and imaginary parts of @xmath33 to be , respectively , the electric field @xmath35 and the magnetic field @xmath36@xcite@xcite@xcite ; i.e. @xmath37 the photon schrdinger eq.(12 ) then appears equivalent to the vacuum maxwell theory ; i.e. @xmath38 and @xmath39 with the vacuum gauss law eq.(15 ) constraints @xmath40    it is of interest to discuss the relativistic lorentz symmetry of the photon schrdinger eq.(12 ) . the proof of lorentz symmetry is actually well known since we have already proved equivalence with the maxwell theory . however , the discrete lorentz symmetries are still worthy of further discussion@xcite . if we consider the classical lorentz force on a charge equation @xmath41 , and then allow the charge to change sign @xmath42 , then the force @xmath43 will remain the same if we also allow @xmath44 and @xmath45 . thus , the electromagnetic field eq.(16 ) is odd under charge conjugation , @xmath46 under time reversal , @xmath47 and @xmath45 so that @xmath48 finally , under parity @xmath44 and @xmath49 so that @xmath50 note @xmath51 as would be expected from relativistic wave equations . the maxwell theory for an electromagnetic disturbance moving through continuous media@xcite may be formulated starting from @xmath52 and @xmath53 @xmath54 it is sufficiently general to supplement eqs.(24 ) , ( 25 ) and ( 26 ) by employing a linear causal relationship between the current and the electric field , i.e. @xmath55 if , for a complex frequency @xmath56 where @xmath57 , @xmath58 @xmath59 eqs.(27 ) , ( 28 ) and ( 29 ) serve to define the non - local conductivity @xmath60 , i.e. @xmath61 where @xmath62 some workers prefer to complete the maxwell equations in continuous media by employing a constitutive equation relating the maxwell displacement field @xmath63 to the electric field @xmath35 . in the frequency domain , the dielectric response @xmath64 of the continuous media is defined as @xmath65 just so long as the various response functions are maintained as non - local ( in space and time ) , which of the many equivalent response functions are actually used in a calculation remains a matter of convenience . for example , eqs.(30 ) and ( 32 ) are equivalent if we choose @xmath66    the dielectric , magnetic , or conductivity response functions conventionally used to describe continuous media now make an appearance into the schrdinger equation for a photon moving through the continuous media via the photon `` self - energy '' insertion . to see what is involved , we may write the maxwell eqs.(24 ) and ( 25 ) as @xmath67 and @xmath68 where eq.(16 ) has been invoked . employing eqs.(21 ) , ( 27 ) and ( 35 ) , we find our central result for the photon schrdinger equation in continuous media @xmath69 @xmath70 where the photon self - energy part is defined as @xmath71 using eqs.(13 ) and ( 36 ) , one may write the photon schrdinger equation in the form @xmath72 @xmath73    in principle the calculation of the self energy part @xmath74 is equivalent to the calculation of the conductivity @xmath60 as is evident from eqs.(21 ) , ( 31 ) , and ( 37 ) . in a practical calculations involving the photon schrdinger equation in a polarizer , one should use the conductivity and dielectric response functions as those of the optical engineers who designed the device@xcite . finally , the non - local form of eq.(38 ) is shared by many other quantum mechanical systems as we shall now briefly explore . although we do not adhere to the non - unitary collapse of the quantum state eq.(2 ) as postulated by von neumann , we do find the notion of projection operators ( also introduced into quantum mechanics by von neumann ) to be more than just convenient . eq.(1 ) will thereby be invoked in what follows . for a macroscopic laboratory quantum system we here insist that a _ monster _ schrdinger equation with a huge number of degrees of freedom still holds true , @xmath75 we do not contemplate the _ ultimate monster wave function _ of the _ whole universe _ , not only because we are not smart enough to find it , but because we do not know how to build an experimental _ ensemble of universes _ to test a hypothetical quantum state of _ our universe_. we leave such considerations to people more wise than are we . we are happy to say something meaningful about a very small physical piece of a laboratory quantum state ; e.g. a `` projected piece '' @xmath76 we leave the rest of the monster wave function , @xmath77 to the engineers who design the experiment . with the little piece of the wave function eq.(40 ) and the big piece of the wave function eq.(41 ) , we rewrite eq.(39 ) as @xmath78 where @xmath79 , @xmath80 , @xmath81 and @xmath82 . we can only hope to find @xmath83 which obeys @xmath84 so we solve for the monster piece @xmath85 subject to the causal boundary condition @xmath86    from eqs.(43 ) and ( 45 ) we find the general non - local schrdinger equation for the piece of the wave function that is of interest ; it is @xmath87 where @xmath88    the full monster schrdinger eq.(39 ) for complete wave function @xmath89 reduces to the non - local in time schrdinger eq.(46 ) for the small subspace piece of the wave function @xmath90 . a particular example of eq.(46 ) is the photon schrdinger eq.(38 ) . the point is that when a photon propagates inside of a continuous media device , say a polarizer , the _ monster wave function _ includes _ both _ the photon _ and _ the polarizer degrees of freedom . in a one photon projected subspace , the quantum state of the polarizer is fixed . in addition to the polarizer , there is but one propagating photon . if the photon is absorbed , then an electronic excitation is produced which heats up the polarizer . the piece of the monster quantum state representing this absorbed photon situation is not contained in the single photon subspace . the single photon subspace wave function is then attenuated due to photon absorption in the polarizer . all of this ( including the attenuation of the maxwell wave in the medium ) is described by the one photon schrdinger equation , _ without _ recourse to the projection postulate . maxwell was so lucky to be able to write down the schrdinger equation for the photon in matter , including attenuation , _ without _ knowing anything about quantum states ( be they monster or otherwise ) . it is of interest to look for a fixed energy solution of the subspace wave function eq.(46 ) of the form @xmath91 eqs.(46 ) , ( 47 ) and ( 48 ) imply @xmath92 where @xmath93 that the effective energy dependent hamiltonian on the left hand side of eq.(49 ) @xmath94 is not hermitian , is due to the `` fermi platinum rule '' transition rates to leave the small subspace and wander into the remaining monster subspaces . the fermi platinum rule @xmath95 is just a little bit more valuable than the more usual perturbative fermi golden rule , since eq.(52 ) is rigorously true to all orders of perturbation theory . the fixed energy @xmath96 case corresponds to having a fixed photon frequency @xmath97 for the electromagnetic problem at hand . let us return to this problem . the maxwell theory of a transverse wave moving through a polarizer may be formulated following the discussion of born and wolf@xcite . ( i ) if a transverse wave propagates in the @xmath98-direction , then the wave may be described by four electromagnetic field components , say @xmath99 if the polarizer is oriented by a rotation angle @xmath7 , as in fig.1(b ) , then the dielectric response tensor will have the form @xmath100 where we have employed the @xmath101 pauli matrices @xmath102 and @xmath103 the complex principal eigenvalues of the dielectric tensor at frequency @xmath104 have the form @xmath105 the real parts of the dielectric eigenvalues @xmath106 determine the velocities of the waves polarized in the principal directions . the conductivities @xmath107 determine the dissipative attenuation of the waves polarized in the principle directions . for a well designed polarizer , a wave with an allowed polarization direction will propagate unattenuated while a wave with a disallowed polarization will be strongly attenuated . _ it is the anisotropic dissipative conductivity that is often modeled by the quantum projective postulate for the 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 invented .    for the classical maxwell calculation , the wave at the beginning @xmath108 of the polarizer is related to the wave at the end @xmath109 of the polarizer by a @xmath110 transfer matrix @xmath111 @xmath112 is defined in eq.(53 ) , and in partitioned matrix form @xmath113 @xmath114 for a stack of @xmath18 polarizers , each of width @xmath115 and oriented at angles @xmath116 , one may repeat the above process going from the beginning @xmath108 of the stack to the end @xmath117 of the stack @xmath118 one multiplies the transfer matrices for each polarizer in the stack @xmath119    once the total @xmath110 transfer matrix is computed , one may find the transmission amplitudes @xmath120 and reflection amplitudes @xmath121 for the two polarization directions @xmath122 by solving the linear algebra problem @xmath123 for an incident photon in the allowed @xmath124-polarization direction , the probability that the photon exits a stack of @xmath18 polarizers ( rotated into the allowed @xmath125-polarization direction ) is given by the absolute square of the transmission amplitude , @xmath126 from eqs.(54 ) , ( 59 ) , and ( 61)-(63 ) , with each of the angles in @xmath116 set equal to @xmath127 , one may compare the the projection postulate @xmath128 in eq.(7 ) to the maxwell theory , or equivalently the photon schrdinger eq.(63 ) , without the projection postulate . the matrix multiplications in eq.(61 ) and the linear algebra in eq.(62 ) are best left to a computer . the results of such a computation are shown in fig.3 . the solid curve in fig.3 is the prediction of the projection postulate eq.(7 ) . for the maxwell theory with a reasonable stack of laboratory polarizers , there may be @xmath129 wavelengths of light in each polarizer in the stack . thus we chose the parameter @xmath130 for purpose of illustration . we further choose @xmath131 to describe the polarization direction for which the polarizer would be transparent and the complex @xmath132 to describe the polarization direction for which the polarizer would be strongly attenuated ( if not absolutely opaque ) . the resulting computed points are the filled circles of fig.3 . for this case , the projection postulate yields a @xmath128 which is quite close to the full maxwell theory .    for some regimes ( perhaps more close to the laboratory situation ) , the maxwell theory begins to diverge from from the projection postulate for large @xmath18 . in the laboratory all polarization directions are subject to small ( but certainly finite ) attenuation . to test this situation , we computed the maxwell theory prediction when the allowed polarization has a weak attenuation response @xmath133 while the disallowed polarization has a stronger attenuation response @xmath134 . the results are the filled squares of fig.3 . for @xmath135 , there is a zeno effect in which @xmath128 increases . however for @xmath136 , the is an attenuation induced decrease in @xmath128 , contrary to the `` zeno effect expectation '' . however , one might regard this attenuation of the allowed polarization direction to be merely an indication of poor experimental design . some final comments about the calculations producing fig.3 are worthy of note . the transfer matrix formalism used to calculate the transmission probability @xmath128 was tedious , but straight forward . however , the numerical computations required to produce fig.3 were not entirely trivial . the first three numerical commercial software programs that we employed for the calculation produced complete nonsense@xcite . the physical problem was that the maxwell waves reflect back and forth within the polarizers . hence , there exist _ exponentially attenuated _ terms as well as _ exponentially amplified _ terms in the transfer matrix . high accuracy was needed for the matrix multiplications in eq.(61 ) . only with high accuracy at intermediate stages of the calculation , could eqs.(62 ) and ( 63 ) be reliably computed . for realistic laboratory polarizers , the required accuracy turns out to be very high indeed . fortunately , aberth@xcite has developed precise numerical algorithms together with publicly available programming code which allows for @xmath137 significant figure accurate computations . much of the required code for the standard functions and standard linear algebra was thus available . we used this code . however , we required @xmath22 significant figure computations to produce the reliable plots shown in fig.3 . in discussions of the foundations of quantum mechanics there has been a tendency to regard all interference of waves as explicitly due to quantum mechanics . this is only sometimes true , since `` waves '' as well as `` particles '' also have classical limits . in many ( perhaps most ) of the electromagnetic experiments and applications , the maxwell wave itself is classical . the _ distinction _ between a classical wave , a classical particle , and a quantum particle has been discussed simply , _ yet clearly _ , by feynman@xcite . \(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 detected . _ one _ photon can never make _ two _ different counters each give you _ one half _ of a click . that is much too smooth for quantum mechanics . in quantum mechanics , each photon counter gives one click or no clicks . the photon is _ certainly not _ a classical wave . the photon is _ certainly not _ a classical particle . the photon is _ certainly not _ a little bit classical wave and a little bit classical particle . ( iii ) the photon is _ neither _ a classical wave _ nor _ a classical particle _ nor _ a little bit of both .    electromagnetic waves , as described by maxwell , are classical waves . if a radio station broadcasts a signal , then the signal is ( for all practical purposes ) smooth . your radio gets some of the signal , and our radio gets some of the signal . there is enough smoothly distributed classical wave to go around to all of the radios . the classical maxwell wave is smooth . one detector can not get a fraction of a photon . remember , photons come in lumps . one photon can get destroyed in at most one radio receiver . one detector can get any fraction of a classical maxwell wave . remember , classical waves are smooth . yet the schrdinger equation for photon is the same ( in a mathematical formalism ) as the maxwell equations for classical electromagnetic waves . this is because the photon has no mass and because the photon is a boson . when very many photons are bose condensed into a single quantum state , the result is a maxwell wave which is classical for all practical purposes . the above picture was not all that clear to einstein ( by his own admission ) , and we do not claim to know more about photons than did einstein . according to einstein , a maxwell wave hits a half silvered mirror and some of the wave goes to one detector and the rest of the wave goes to a second detector . if the wave represents one photon , then only one counter goes click . the other counter gets whacked with a maxwell wave . yet this counter counts 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 out not have a photon . some counters go click and some counters remain mute . you never know . and photons are lumpy . it is a very strange shell game according to einstein@xcite . however , the zeno effect formally attributed to the quantum mechanical projection postulate has here been argued to be merely a classical wave effect . maxwell would not have had to increase his knowledge of electromagnetic theory by even one iota in order to totally understand the polarization version of the zeno effect . no quantum mechanics is required , not to even speak of the projection postulate . 99 j. von neumann , _ mathematical foundations of quantum mechanics _ , princeton university press , princeton ( 1996 ) . r.l . stratonovich and v.p . belavkin , _ international journal of theoretical physics _ , _ 35 _ , 2215 - 28 ( 1996 ) . g.s . agarwal and s.p . lett _ , * 185 * , 139 - 142 ( 1994 ) m.a.b . whitaker , _ interaction - free measurement and the quantum zeno effect _ , _ physics lett a _ , * 244 * , 502 - 6 , ( 1998 ) . o. alter and y. yamamoto , _ phys . rev . a _ * 55 * , r2499 - 2502 , ( 1997 ) . a. beige and g.c . hegerfeldt , _ j. of phys . * 30 * , 1323 - 34 , ( 1997 ) . a. beige , g.c . hegerfeldt , and d.g . sondermann , _ foundations of physics _ , * 27 * , 1671 - 88 ( 1997 ) . s. pascazio , h. nakazato , and m. mamiki , _ j of the phys . soc of japan _ , * 65 * , 285 - 8 , ( 1996 ) . j. pan , y.zhang , and q. dong , _ phys lett a _ , * 238 * , 405 - 7 ( 1998 ) . l.s . schulman , _ phys rev a _ , * 57 * , 1509 - 15 ( 1998 ) . s. pascazio , _ foundations of physics _ , * 27 * , 1655 - 70 ( 1997 ) . b. nagels , l.j.f . hermans , and p.l . chapovsky , _ phys rev lett _ , * 79 * , 3097 - 100 ( 1997 ) . h. nakazato , m. namiki , s. pascazio , and h. rauch , _ phys lett a _ , * 199 * , 27 - 32 ( 1995 ) . j. s. townsend , _ a modern approach to quantum mechanics _ , p.62 , mcgraw - hill , new york ( 1992 ) . stancil and g.e . copeland , _ j. of phys . b _ , * 27 * , 2801 - 7 ( 1994 ) . p. kwiat , h. weinfurter , and a. zeilinger , _ quantum seeing in the dark , scientific american _ 72 , nov . 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we consider a photon beam incident on a stack of polarizers as an example of a von neumann projective measurement , theoretically leading to the quantum zeno effect . 
 the maxwell theory ( which is equivalent to the single photon schrdinger equation ) describes measured polarization phenomena , but without recourse to the notion of a projective measurement .