Protective measuremet ad de Broglie-Bohm theory Sha Gao Uit for History ad Philosophy of Sciece & Cetre for Time, SOPHI, Uiversity of Sydey Email: sgao7319@ui.sydey.edu.au We ivestigate the implicatios of protective measuremet for de Broglie-Bohm theory, maily focusig o the iterpretatio of the wave fuctio. It has bee argued that de Broglie-Bohm theory gives the same predictios as quatum mechaics by meas of quatum equilibrium hypothesis. However, this equivalece is based o the premise that the wave fuctio, regarded as a Ψ-field, has o mass ad charge desity distributios. But this premise turs out to be wrog accordig to protective measuremet; a charged quatum system has effective mass ad charge desity distributig i space, proportioal to the square of the absolute value of its wave fuctio. The i de Broglie-Bohm theory both Ψ-field ad Bohmia particle will have charge desity distributio for a charged quatum system. This will result i the existece of a electrostatic self-iteractio of the field ad a electromagetic iteractio betwee the field ad Bohmia particle, which ot oly violates the superpositio priciple of quatum mechaics but also cotradicts experimetal observatios. Therefore, de Broglie-Bohm theory as a realistic iterpretatio of quatum mechaics is problematic accordig to protective measuremet. Lastly, we briefly discuss the possibility that the wave fuctio is ot a physical field but a descriptio of some sort of ergodic motio (e.g. radom discotiuous motio) of particles. Key words: wave fuctio; de Broglie-Bohm theory; protective measuremet; Ψ-field; mass ad charge desity; ergodic motio of particles 1. Itroductio De Broglie-Bohm theory is a otological iterpretatio of quatum mechaics iitially proposed by de Broglie ad later developed by Bohm (de Broglie 198; Bohm 195) 1. Accordig to the theory, a complete realistic descriptio of a quatum system is provided by the cofiguratio defied by the positios of its particles together with its wave fuctio. Although the de Broglie-Bohm theory is mathematically equivalet to stadard quatum theory, there is o clear cosesus with regard to its physical iterpretatio. I particular, the iterpretatio of the wave fuctio i this theory is still i hot debate eve today. The wave fuctio is geerally take as a objective physical field called Ψ-field. As stressed by Bell (1981): No oe ca uderstad this theory util he is willig to thik of Ψ as a real objective field rather tha just a probability amplitude. However, there are various views o exactly what field the wave fuctio is. It has bee regarded as a field similar to electromagetic field (Bohm 195), a active iformatio field 1 Amog other differeces, de Broglie s dyamics is first order while Bohm s dyamics is secod order. It should be poited out that the wave fuctio is also regarded by some authors as omological, e.g. a compoet of physical law rather tha of the reality described by the law (Dürr, Goldstei ad Zaghì 1997; Goldstei ad Teufel 001). We will ot discuss this view i this paper. But it might be worth otig that this o-field view may have serious drawbacks whe cosiderig the cotigecy of the wave fuctio (see, e.g. Valetii 009), ad the results obtaied i this paper seemigly disfavor this view too.
(Bohm ad Hiley 1993), a field carryig eergy ad mometum (Hollad 1993), ad a causal aget more abstract tha ordiary fields (Valetii 1997) etc 3. I this paper, we will examie the validity of the field iterpretatio of the wave fuctio i de Broglie-Bohm theory i terms of protective measuremet (Aharoov, Aada ad Vaidma 1993; Aharoov ad Vaidma 1993; Aharoov, Aada ad Vaidma 1996). It has bee argued that the time averages of Bohmia particle s positios typically differ markedly from the esemble averages, ad this result based o weak measuremet ad protective measuremet raises some objectios to the reality of Bohmia particles (Eglert, Scully, Süssma ad Walther 199; Aharoov ad Vaidma 1996; Aharoov, Eglert ad Scully 1999; Aharoov, Erez ad Scully 004). O the other had, it seems that these objectios ca be aswered by oticig that protective measuremet is i fact a way of measurig the effect of the Ψ-field rather tha that of the Bohmia particle (see, e.g. Drezet 006). However, our ew aalysis will show that this aswer caot really save the de Broglie-Bohm theory from the attack of protective measuremet; o the cotrary, protective measuremet will pose a more serious threat to the reality of the Ψ-field i the theory. The pla of this paper is as follows. First, we will argue that there are good reasos to thik, ad i particular, protective measuremet already implies that a quatum system with mass m ad charge Q, which is described by the wave fuctio ψ, has effective mass ad charge desity distributios mψ ad Qψ i space respectively. The we ivestigate the implicatios of this result for de Broglie-Bohm theory. To begi with, takig the wave fuctio as a Ψ-field will lead to the existece of a electrostatic self-iteractio of the field for a charged quatum system. This ot oly violates the superpositio priciple of quatum mechaics but also cotradicts experimetal observatios. Secodly, there will also exist a electromagetic iteractio betwee the field ad the Bohmia particle, as they all have charge desity distributio i space for a charged quatum system. This cotradicts the predictios of quatum mechaics ad experimetal observatios too. These results idicate that the field iterpretatio of the wave fuctio caot be right, ad thus the de Broglie-Bohm theory, which takes the wave fuctio as a Ψ-field, is problematic. Lastly, we briefly discuss the possibility that the wave fuctio is ot a physical field but a descriptio of some sort of ergodic motio (e.g. radom discotiuous motio) of particles. 3 Note that there is a commo objectio to the field iterpretatio of the wave fuctio, which claims that the wave fuctio ca hardly be cosidered as a real physical field because it is a fuctio o cofiguratio space, ot o physical space (see, e.g. Moto 00, 006). However, this commo objectio is ot coclusive, ad oe ca still isist o the reality of the wave fuctio livig o cofiguratio space by some metaphysical argumets (see, e.g. Albert 1996; Lewis 004; Wallace ad Timpso 009). Differet from the commo objectio, I will i this paper propose a more serious objectio to the field iterpretatio, accordig to which eve for a sigle quatum system the wave fuctio livig i real space caot be take as a physical field either. Moreover, the reaso is ot metaphysical but physical, i.e. that the field iterpretatio cotradicts both quatum mechaics ad experimetal observatios.
. How do mass ad charge distribute for a sigle quatum system? The mass ad charge of a charged classical system always localize i a defiite positio i space at each momet. For a charged quatum system described by the wave fuctio ψ, how do its mass ad charge distribute i space the? We ca measure the total mass ad charge of the quatum system ad fid them i some regio of space. Thus the mass ad charge of a quatum system must also exist i space with a certai distributios if assumig a realistic view. Although the mass ad charge distributios of a sigle quatum system seem meaigless accordig to the orthodox probability iterpretatio of the wave fuctio, it should have a physical meaig i a realistic iterpretatio of the wave fuctio such as de Broglie-Bohm theory 4. As we thik, the Schrödiger equatio of a charged quatum system uder a exteral electromagetic potetial already provides a importat clue. The equatio is ψ h iq ih = A + Qϕ + V ψ (1) t m hc where m ad Q is respectively the mass ad charge of the system, ϕ ad A are the electromagetic potetial, V is a exteral potetial, h is Plack s costat divided by π, c is the speed of light. The electrostatic iteractio term Qϕψ i the equatio seems to idicate that the charge of the quatum system distributes throughout the whole regio where its wave fuctio ψ is ot zero. If the charge does ot distribute i some regios where the wave fuctio is ozero, the there will ot exist ay electrostatic iteractio ad the electrostatic iteractio term will also disappear there. But the term Qϕψ exists i all regios where the wave fuctio is ozero. Thus it seems that the charge of a charged quatum system should distribute throughout the whole regio where its wave fuctio is ot zero. + Furthermore, sice the itegral Qψ dx is the total charge of the system, the charge desity distributio i space will be Qψ. Similarly, the mass desity ca be obtaied from the Schrödiger equatio of a quatum system with mass m uder a exteral gravitatioal potetial V G : 4 Ufortuately it seems that the orthodox probability iterpretatio of the wave fuctio still iflueces people s mid eve if they already accept a realistic iterpretatio of the wave fuctio. Oe obvious example is that few people admit that the realistic wave fuctio has eergy desity (Hollad (1993) is a otable exceptio). If the wave fuctio has o eergy, the it seems very difficult to regard it as physically real. Eve if Bohm ad Hiley (1993) iterpreted the Ψ-field as active iformatio, they also admitted that the field has eergy, though very little. Oce oe admits that the wave fuctio has eergy desity, the it seems atural to edow it with mass ad charge desity, which are two commo sources of eergy desity.
ψ h ih = + mvg + V ψ () t m The gravitatioal iteractio term mv G ψ i the equatio also idicates that the (passive gravitatioal) mass of the quatum system distributes throughout the whole regio where its wave fuctio ψ is ot zero, ad the mass desity distributio i space is mψ. The above result ca be more readily uderstood whe the wave fuctio is a complete realistic descriptio of a sigle quatum system as i dyamical collapse theories. If the mass ad charge of a quatum system does ot distribute as above i terms of its wave fuctio ψ, the other supplemet quatities will be eeded to describe the mass ad charge distributios of the system i space, while this obviously cotradicts the premise that the wave fuctio is a complete descriptio. I fact, the dyamical collapse theories such as GRW theory already admit the existece of mass desity (Ghirardi, Grassi ad Beatti 1995). I additio, eve i de Broglie-Bohm theory, which takes the wave fuctio as a icomplete descriptio ad admits supplemet hidde variables (i.e. the trajectories of Bohmia particles accompayig the wave fuctio), there are also some argumets for the above mass ad charge desity explaatio (Hollad 1993; Brow, Dewdey ad Horto 1995). It was argued that sice the Ψ-field depeds o the parameters such as mass ad charge, it may be said to be massive ad charged (Hollad 1993, p.79). I additio, Brow, Dewdey ad Horto (1995), by examiig a series of effects i eutro iterferometry, argued that properties sometimes attributed to the particle aspect of a eutro, e.g., mass ad magetic momet, caot straightforwardly be regarded as localized at the hypothetical positio of the particle i Bohm s theory. They also argued that it is hard to uderstad how the Aharoov-Bohm effect is possible if that the charge of the electro which couples with the electromagetic vector-potetial is ot co-preset i the regios o all sides of the cofied magetic field accessible to the electro (Brow, Dewdey ad Horto 1995, p.33). Oe may object that de Broglie-Bohm theory seemigly ever admits the above mass desity explaatio, ad o existig iterpretatio of quatum mechaics icludig dyamical collapse theories edows charge desity to the wave fuctio either. As we thik, however, protective measuremet provides a more covicig argumet for the existece of mass ad charge desity distributios 5. The wave fuctio of a sigle quatum system, especially its mass ad charge desity, ca be directly measured by protective measuremet. Therefore, a realistic iterpretatio of quatum mechaics should admit the existece of mass ad charge desity i some way; if it caot, the it will be at least problematic cocerig its iterpretatio of the wave fuctio. 3. Protective measuremet ad its aswer I this sectio, we will give a brief itroductio of protective measuremet ad its 5 It is very strage for the author that most supporters of a realistic iterpretatio of quatum mechaics igore protective measuremet ad its implicatios. Admittedly there have bee some cotroversies about the meaig of protective measuremet, but the debate maily ceters o the reality of the wave fuctio. If oe isists o a realistic iterpretatio of quatum mechaics such as de Broglie-Bohm theory, the the debate will be mostly irrelevat ad protective measuremet will have strict restrictios o the realistic iterpretatio.
implicatio for the existece of mass ad charge desity distributios. Differet from the covetioal measuremet, protective measuremet aims at measurig the wave fuctio of a sigle quatum system by repeated measuremets that do ot destroy its state. The geeral method is to let the measured system be i a o-degeerate eigestate of the whole Hamiltoia usig a suitable iteractio, ad the make the measuremet adiabatically so that the wave fuctio of the system either chages or becomes etagled with the measurig device appreciably. The suitable iteractio is called the protectio. As a typical example of protective measuremet (Aharoov, Aada ad Vaidma 1993; Aharoov, Aada ad Vaidma 1996), cosider a quatum system i a discrete odegeerate eergy eigestate ψ (x). The protectio is atural for this situatio, ad o additioal protective iteractio is eeded. The iteractio Hamiltoia for measurig the value of a observable A i the state is: H I = g( PA (3) where P deotes the mometum of the poiter of the measurig device, which iitial state is take to be a Gaussia wave packet cetered aroud zero. The time-depedet couplig g ( is T ormalized to g( dt = 1, where T is the total measurig time. I covetioal vo 0 Neuma measuremets, the iteractio H I is of short duratio ad so strog that it domiates the rest of the Hamiltoia (i.e. the effect of the free Hamiltoias of the measurig device ad the system ca be eglected). As a result, the time evolutio exp( ipa/ h) will lead to a etagled state: eigestates of A with eigevalues a i are etagled with measurig device states i which the poiter is shifted by these values a i. Due to the collapse of the wave fuctio, the measuremet result ca oly be oe of the eigevalues of observable A, say a i, with a certai probability p i. The expectatio value of A is the obtaied as the statistical average of eigevalues for a esemble of idetical systems, amely < A >= i p i a i. By cotrast, protective measuremets are extremely slow measuremets. We let g ( = 1/ T for most of the time T ad assume that g ( goes to zero gradually before ad after the period T. I the limit T, we ca obtai a adiabatic process i which the system caot make a trasitio from oe eergy eigestate to aother, ad the iteractio Hamiltoia does ot chage the eergy eigestate. As a result, the correspodig time evolutio exp( ip < A > / h) shifts the poiter
by the expectatio value < A >. This result strogly cotrasts with the covetioal measuremet i which the poiter shifts by oe of the eigevalues of A. It should be stressed that T is oly a ideal situatio 6, ad a protective measuremet ca ever be performed o a sigle quatum system with absolute certaity because of the tiy uavoidable etaglemet (see also Dass ad Qureshi 1999) 7. For example, for ay give values of P ad T, the eergy shift of the above eigestate, give by first-order perturbatio theory, is < A > P δ E =< H I >= (4) T Correspodigly, we ca oly obtai the exact expectatio value < A > with a probability very close to oe, ad the measuremet result ca also be the expectatio value < A >, with a probability proportioal to 1/ T, where refers to the ormalized state i the subspace ormal to the iitial state ψ (x) as picked out by first-order perturbatio theory (Dass ad Qureshi 1999). Therefore, a esemble, which may be cosiderably small, is still eeded for protective measuremets. Although a protective measuremet ca ever be performed o a sigle quatum system with absolute certaity, the measuremet is distict from the stadard oe: i o stage of the measuremet we obtai the eigevalues of the measured variable. Each system i the small esemble cotributes the shift of the poiter proportioal ot to oe of the eigevalues, but to the expectatio value. This essetial ovel poit has bee repeatedly stressed by the ivetors of protective measuremet (see, e.g. Aharoov, Aada ad Vaidma 1996). As we kow, i the orthodox iterpretatio of quatum mechaics, the expectatio values of variables are ot cosidered as physical properties of a sigle system, as oly oe of the eigevalues is observed i the outcome of the stadard measurig procedure ad the expectatio value ca oly be defied as a statistical average of the eigevalues. However, for protective measuremets, we obtai the expectatio value directly for a sigle system ad ot as a statistical average of eigevalues for a esemble. Sice the expectatio value of a variable ca be directly measured for a sigle system, it must be a physical characteristic of a sigle system, ot of a esemble (e.g. as a statistical average of eigevalues). This is a defiite coclusio we ca reach by the aalysis of protective measuremet. I the followig we will show that the mass ad charge desity ca be measured by protective measuremet as expectatio values of certai variable for a sigle quatum system (Aharoov ad Vaidma 1993). Cosider agai a quatum system i a discrete odegeerate eergy eigestate ψ (x). The iteractio Hamiltoia for measurig the value of a observable A i the state assumes the same form as Eq. (3): H = g( (5) I PA 6 Note that the spreadig of the wave packet of the poiter also puts a limit o the time of the iteractio (Dass ad Qureshi 1999). 7 It ca be argued that oly observables that commute with the system s Hamiltoia ca be protectively measured with absolute certaity for a sigle system (see e.g. Rovelli 1994; Uffik 1999).
where A is a ormalized projectio operator o small regios V havig volume v, which ca be writte as follows: A 1, = v 0, x V x V (6) The a protective measuremet of A will yield the followig result: 1 ψ = ψ A = ( x) dv v (7) v It is the average of the desity ψ ( x) over the small regio V. Whe v 0 ad after performig measuremets i sufficietly may regios V we ca fid the whole desity distributio ψ ( x). For a charged system with charge Q the desity ψ ( x) times the charge yields the effective charge desity Qψ (x). I particular, a appropriate adiabatic measuremet of the Gauss flux of the electric field comig out of a certai regio will yield the value of the total charge iside this regio, amely the itegral of the effective charge desity Q ψ ( x) over this regio (Aharoov ad Vaidma 1993; Aharoov, Aada ad Vaidma 1996). Similarly, we ca measure the effective mass desity of the system i priciple by a appropriate adiabatic measuremet of the flux of its gravitatioal field. It ca be show that protective measuremets ca ot oly measure the odegeerate eergy eigestates of a sigle quatum system, but also measure its time-depedet quatum states via Zeo effect by frequet covetioal measuremets i priciple (Aharoov ad Vaidma 1993). Thus the above results hold true for ay give wave fuctio. This provides a strog argumet for associatig physical reality with the wave fuctio of a sigle quatum system. Although oe may still object to this associatio, the objectio will be irrelevat i a realistic iterpretatio of the wave fuctio such as de Broglie-Bohm theory. Therefore, we ca always test the realistic iterpretatios of quatum mechaics by the above results of protective measuremet, which show that the mass ad charge of a sigle quatum system described by the realistic wave fuctio ψ (x) is distributed throughout space with effective mass desity m ψ ( x) ad effective charge desity Q ψ ( x) respectively. 4. Implicatios for de Broglie-Bohm theory Now we will ivestigate the implicatios of the existece of mass ad charge desity for de Broglie-Bohm theory. For the sake of simplicity, we will restrict our discussios to the wave fuctio of a sigle quatum system. The coclusio ca be readily exteded to may-body
systems. It has bee argued that de Broglie-Bohm theory gives the precisely same predictios as quatum mechaics by meas of quatum equilibrium hypothesis. Cocretely speakig, the quatum equilibrium hypothesis provides the iitial coditios for the guidace equatio which make de Broglie-Bohm theory obey Bor s rule i terms of positio distributios. Moreover, sice all measuremets ca be fially expressed i terms of positio, e.g. poiter positios, this amouts to full accordace with all predictios of quatum mechaics. However, this equivalece is based o the premise that the wave fuctio, regarded as a Ψ-field, has o mass ad charge desity distributios. If the wave fuctio has mass ad charge desity distributios as protective measuremet implies, the, as we will argue below, takig it as a Ψ-field will lead to some predictios (e.g. the existece of electrostatic self-iteractio) that cotradict both quatum mechaics ad experimetal observatios. If the wave fuctio is a physical field such as a Ψ-field, the its mass ad charge desity will be simultaeously distributed i space. This has two disaster results at least. Oe is that charge will ot be quatized; the total charge iside a very small regio ca be much smaller tha a elemetary charge for a sigle quatum system. This obviously cotradicts the commo expectatio that charge should be quatized. But maybe our expectatio eeds to be revised. So this result is ot fatal for the field iterpretatio of the wave fuctio. The other is that the wave fuctio will ot satisfy the superpositio priciple of quatum mechaics. For example, for the wave fuctio of a sigle electro, differet spatial parts of the wave fuctio will have gravitatioal ad electrostatic iteractios, as these parts have mass ad charge simultaeously. Let s aalyze the secod result i more detail. Iterestigly, the so-called Schrödiger-Newto equatio, which was proposed for other purposes (Diosi 1984; Perose 1998), just describes the gravitatioal self-iteractio of the wave fuctio. The equatio for a sigle quatum system ca be writte as ψ h ψ ψ ( x, 3 ih = Gm d x ψ Vψ t m x + (8) x x where m is the mass of the quatum system, V is a exteral potetial, ad G is Newto s gravitatioal costat. Much work has bee doe to study the mathematical properties of this iterestig equatio (see, e.g. Harriso, Moroz ad Tod 003; Moroz ad Tod 1999; Salzma 005). Some experimetal schemes have bee also proposed to test its physical validity (Salzma ad Carlip 006). As we will see, although such gravitatioal self-iteractios caot yet be excluded by experimets 8, the existece of electrostatic self-iteractio already cotradicts experimetal observatios. If there is also a electrostatic self-iteractio, the the equatio for a free quatum system 8 It has bee argued that the existece of a self-iteractio term i the Schrödiger-Newto equatio does ot have a cosistet Bor rule iterpretatio (Adler 007). The reaso is that the probability of simultaeously fidig a particle i differet positios is zero. However, i a realistic iterpretatio of quatum mechaics where the wave fuctio is regarded as a real physical etity rather tha as a probability amplitude, the existece of gravitatioal self-iteractio term seems quite atural. For example, the field iterpretatio ca be cosistet with covetioal quatum measuremet via a dyamical collapse process. As we thik, oe covicig objectio is that if there is a self-gravitatioal iteractio for the wave fuctio of a charged particle, the there will also exist a electrostatic self-iteractio because the charge desity always accompaies the mass desity, while the existece of electrostatic self-iteractio is already icosistet with experimetal observatios (see below). If this objectio is valid, the the Schrödiger-Newto equatio will be wrog eve as a approximatio, ad moreover, the approach of semiclassical gravity will also be excluded (cf. Salzma ad Carlip 006).
with mass m ad charge Q will be ψ h ψ ( ψ x, 3 ih = + ( kq Gm ) d x ψ (9) t m x x x where k is the Coulomb costat. Note that the gravitatioal self-iteractio is a attractive force, while the electrostatic self-iteractio is a repulsive force. It has bee show that the measure of 4Gm the potetial stregth of a gravitatioal self-iteractio is ε = for a free particle hc with mass m (Salzma 005). This quatity represets the stregth of the ifluece of self-iteractio o the ormal evolutio of the wave fuctio; whe ε 1 the ifluece is sigificat. Similarly, for a free charged particle with charge Q, the measure of the potetial 4kQ stregth of the electrostatic self-iteractio is ε =. As a typical example, for a free hc electro with charge e, the potetial stregth of the electrostatic self-iteractio will be 4 ke 3 ε = 1 10. This idicates that the electrostatic self-iteractio will have hc sigificat ifluece o the evolutio of the wave fuctio of a free electro. If such a iteractio ideed exists, it should have bee detected by precise experimets o charged microscopic particles. As aother example, cosider the electro i the hydroge atom. Sice the potetial of its electrostatic self-iteractio is of the same order as the Coulomb potetial produced by the ucleus, the eergy levels of hydroge atoms will be sigificatly differet from those predicted by quatum mechaics ad cofirmed by experimetal observatios. Therefore, takig the wave fuctio as a Ψ-field will lead to the existece of electrostatic self-iteractio that cotradicts both quatum mechaics ad experimetal observatios 9. Moreover, de Broglie-Bohm theory makes the situatio worse by addig the Bohmia particles. Iasmuch as the wave fuctio has charge desity distributio i space for a charged quatum system, there will exist a electromagetic iteractio betwee it ad the Bohmia particles. This is icosistet with the predictios of quatum mechaics ad experimetal observatios either. Certaily, oe ca elimiate the electromagetic iteractio betwee the Ψ-field ad Bohmia particles by deprivig the Bohmia particles of mass ad charge. But they will be ot real particles ay more. The i what sese the de Broglie-Bohm theory provides a realistic iterpretatio of quatum mechaics? Oe may also wat to deprive the Ψ-field of mass ad charge desity to elimiate the electrostatic self-iteractio. But, o the oe had, the theory will break its physical coectio with quatum mechaics, as the wave fuctio i quatum 9 Oe may object to the argumet here with the example of classical electromagetic field. Electromagetic field is a field, but it has o self-iteractio. Thus a field does ot require the existece of self-iteractio. However, this is a commo misuderstadig. The crux of the matter is that the o-existece of electromagetic self-iteractio results from the fact that electromagetic field itself has o charge. If the electromagetic field had charge, the there would also exist electromagetic self-iteractio due to the ature of field, amely the simultaeous existece of its properties i space. I fact, although electromagetic field has o electromagetic self-iteractio, it does have gravitatioal self-iteractio; the simultaeous existece of eergy desities i differet spatial locatios for a electromagetic field must geerate a gravitatioal iteractio, though the iteractio is too weak to be detected by curret techology.
mechaics has mass ad charge desity accordig to our aalysis, ad o the other had, sice protective measuremet ca measure the mass ad charge desity for a sigle quatum system, the theory will be uable to explai the measuremet results either 10. Although de Broglie-Bohm theory ca still exist i this way as a mathematical tool for experimetal predictios (somewhat like the orthodox iterpretatio it tries to replace), it obviously departs from the iitial expectatios of de Broglie ad Bohm, ad as we thik, it already fails as a physical theory because of losig its explaatio ability. To sum up, de Broglie-Bohm theory caot accommodate the result that the wave fuctio has mass ad charge desity distributios, which is implied by protective measuremet. If the wave fuctio, regarded as a Ψ-field, has charge desity distributio i space for a charged quatum system, the there will exist a electrostatic self-iteractio of the Ψ-field ad a electromagetic iteractio betwee the field ad Bohmia particle. This ot oly violates the superpositio priciple of quatum mechaics but also cotradicts experimetal observatios. Therefore, de Broglie-Bohm theory as a realistic iterpretatio of quatum mechaics is problematic accordig to protective measuremet. 5. Further discussios If the wave fuctio is ot a descriptio of physical field as de Broglie-Bohm theory assumes, the exactly what does the wave fuctio describe? There is already a importat clue. It is that the superpositio priciple i quatum mechaics permits o existece of the self-iteractio of the wave fuctio i real space for a sigle quatum system. This idicates that the mass ad charge desity do ot exist i differet regios simultaeously. How is this possible? It aturally leads us to the secod view that takes the wave fuctio as a descriptio of some kid of ergodic motio of a particle. O this view, the effective mass ad charge desity are formed by time average of the motio of a charged particle, ad they distribute i differet locatios at differet momets. I other words, the mass ad charge desity exists i a time divisio way (by cotraries a field exists throughout space simultaeously). At ay istat, there is oly a localized particle with mass ad charge. Thus there will ot exist ay self-iteractio for the wave fuctio. There are ideed some realistic iterpretatios of quatum mechaics that attempt to explai the wave fuctio i terms of some sort of ergodic motio of particles. A well-kow example is the stochastic iterpretatio of quatum mechaics (e.g. Nelso 1966). Nelso (1966) derived the Schrödiger equatio from Newtoia mechaics via the hypothesis that every particle of mass m is subject to a Browia motio with diffusio coefficiet h / m ad o frictio. I more techical terms, the quatum mechaical process is claimed to be equivalet to a classical Markovia diffusio process. O this iterpretatio, particles have cotiuous trajectories but o velocities, ad the wave fuctio is a statistical average descriptio of their motio. However, it has bee poited out that the classical stochastic iterpretatios are icosistet with quatum mechaics (Glabert, Häggi ad Talker 1979; Wallstrom 1994). Glabert, Häggi ad Talker (1979) argued that the Schrödiger equatio is ot equivalet to a Markovia process, ad the various correlatio fuctios used i quatum mechaics do ot have the properties of the correlatios of a classical stochastic process. Wallstrom (1994) further showed that oe must add 10 Oe caot simply regard the results of protective measuremet of mass ad charge desity as meaigless. These results are proportioal to the module square of the wave fuctio of a sigle quatum system at every locatio of space.
by had a quatizatio coditio, as i the old quatum theory, i order to recover the Schrödiger equatio, ad thus the Schrödiger equatio ad the Madelug hydrodyamic equatios are ot equivalet. I fact, Nelso (005) also showed that there is a empirical differece betwee the predictios of quatum mechaics ad his stochastic mechaics whe cosiderig quatum etaglemet ad olocality. I additio, it has bee geerally argued that the classical ergodic models that assume cotiuous motio caot be cosistet with quatum mechaics (Aharoov ad Vaidma 1993; Gao 010). Classical ergodic models are plagued by the problems of ifiite velocity ad acceleratig radiatio (Aharoov ad Vaidma 1993). I particular, a particle udergoig cotiuous motio, eve if it has ifiite velocity, caot move throughout two spatially separated regios where the wave fuctio of the particle may spread. Besides, the classical ergodic models etail the existece of a fiite ergodic time, which is also icosistet with the existig quatum theory (Gao 010). Based o these egative results, it has bee suggested that the wave fuctio may describe radom discotiuous motio of particles (Gao 006a, 006b, 010). This ew iterpretatio of the wave fuctio ca avoid the problems of classical ergodic models, ad it also provides a atural realistic alterative to the orthodox view. O this iterpretatio, the square of the absolute value of the wave fuctio ot merely gives the probability of the particle beig foud i certai locatios, but also gives the objective probability of the particle beig there. Moreover, it seems that the theory of radom discotiuous motio ca also provide a promisig solutio to the otorious quatum measuremet problem (Gao 006a, 006b). However, the theory is still at its prelimiary stage, ad much study is still eeded before a defiite coclusio ca be reached about the true meaig of the wave fuctio. Ackowledgmets I am deeply grateful to Atoy Valetii for his isightful criticism ad ecouragemet ad to Dea Rickles ad Huw Price for their helpful advice ad suggestios for may improvemets. This work was supported by the Postgraduate Scholarship i Quatum Foudatios provided by the Uit for History ad Philosophy of Sciece ad Cetre for Time (SOPHI) of the Uiversity of Sydey. Refereces Adler, S. L. (007). Commets o proposed gravitatioal modificatios of Schrödiger dyamics ad their experimetal implicatios. J. Phys. A 40, 755-764. Aharoov, Y., Aada, J. ad Vaidma, L. (1993). Meaig of the wave fuctio, Phys. Rev. A 47, 4616. Aharoov, Y., Aada, J. ad Vaidma, L. (1996). The meaig of protective measuremets, Foud. Phys. 6, 117. Aharoov, Y., Eglert, B. G. ad Scully M. O. (1999). Protective measuremets ad Bohm trajectories, Phys. Lett. A 63, 137. Aharoov, Y., Erez, N. ad Scully M. O. (004). Time ad Esemble Averages i Bohmia
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